is equivalent to the absolute derived category of the exact category of
strongly contraflat left ${\mathcal{E}}$-contramodules.

###### Proof.

Part (a) follows from Lemma B.4.7(a) and the dual version of Proposition
A.5.6. For part (b), see Corollary A.6.2 or [51, Section 3.8 and Remark 3.7].
Part (c) can be deduced from part (b) together with the fact that strongly
contraflat left ${\mathcal{E}}$-contramodules have finite projective
dimensions in ${\mathcal{E}}{\operatorname{\mathsf{--contra}}}$ and
Proposition A.5.6. Alternatively, notice that countable products of strongly
contraflat ${\mathcal{E}}$-contramodules have finite projective dimensions by
(the proof of) Corollary B.4.8(b), so Corollary A.6.2 applies. ∎

######  Corollary B.4.10.

In the situation of Corollary B.4.6, assume additionally that the coring
${\mathcal{E}}$ is a projective left $B$-module. Then the data of a dualizing
complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for the corings
${\mathcal{C}}$ and ${\mathcal{E}}$ induces an equivalence of triangulated
categories
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{E}}{\operatorname{\mathsf{--contra}}})$,
which is provided by the derived functors
${\mathbb{R}}\operatorname{Hom}_{\mathcal{C}}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}^{\mathbb{L}}\nobreak{-}$.

###### Proof.

In addition to what has been said in Corollaries B.4.2, B.4.6 and Theorem
B.4.9, we point out that in our present assumptions the assertions of Lemmas
B.4.3(a) and B.4.4(a) apply to any strongly contraflat left
${\mathcal{E}}$-contramodule ${\mathfrak{F}}$. Indeed, in view of Corollary
B.4.8(b) one only has to check that the functor
${\mathcal{N}}\odot_{\mathcal{E}}{-}$ takes short exact sequences of strongly
contraflat left ${\mathcal{E}}$-contramodules to short exact sequences of
abelian groups for any right ${\mathcal{E}}$-comodule ${\mathcal{N}}$. This
follows from part (a) of the same Corollary, as
$\operatorname{Hom}_{\mathbb{Z}}({\mathcal{N}}\odot_{\mathcal{E}}{\mathfrak{F}},\>{\mathbb{Q}}/{\mathbb{Z}})\simeq\operatorname{Hom}^{\mathcal{E}}({\mathfrak{F}},\operatorname{Hom}_{\mathbb{Z}}({\mathcal{N}},{\mathbb{Q}}/{\mathbb{Z}}))$
and the left ${\mathcal{E}}$-contramodule
$\operatorname{Hom}_{\mathbb{Z}}({\mathcal{N}},{\mathbb{Q}}/{\mathbb{Z}})$ is
$B$-cotorsion by Lemma 1.3.3(b). Therefore, one can construct the derived
functor
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{\mathcal{E}}^{\mathbb{L}}{-}$
using strongly contraflat resolutions. ∎

###  B.5. Base ring change

Let ${\mathcal{C}}$ be a coring over an associative ring $A$. Given a right
${\mathcal{C}}$-comodule ${\mathcal{N}}$ and a left ${\mathcal{C}}$-comodule
${\mathcal{M}}$, their _cotensor product_
${\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}$
is an abelian group constructed as the kernel of the natural pair of maps
${\mathcal{N}}\otimes_{A}{\mathcal{M}}\birarrow{\mathcal{N}}\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{M}}$.
Given a left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ and a left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, their group of
_cohomomorphisms_
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{M}},{\mathfrak{P}})$ is
constructed as the cokernel of the natural pair of maps
$\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}{\mathcal{M}},\>{\mathfrak{P}})\birarrow\operatorname{Hom}_{A}({\mathcal{M}},{\mathfrak{P}})$
[50, Sections 1.2.1 and 3.2.1].

Let ${\mathcal{C}}$ be a coring over an associative ring $A$ and
${\mathcal{E}}$ be a coring over an associative ring $B$. A _map of corings
${\mathcal{C}}\longrightarrow{\mathcal{E}}$ compatible with a ring
homomorphism $A\longrightarrow B$_ is an $A$-$A$-bimodule morphism such that
the maps $A\longrightarrow B$, ${\mathcal{C}}\longrightarrow{\mathcal{E}}$,
and the induced map
${\mathcal{C}}\otimes_{A}{\mathcal{C}}\longrightarrow{\mathcal{E}}\otimes_{B}{\mathcal{E}}$
form commutative diagrams with the comultiplication and counit maps in
${\mathcal{C}}$ and ${\mathcal{E}}$ [50, Section 7.1.1].

Let ${\mathcal{C}}\longrightarrow{\mathcal{E}}$ be a map of corings compatible
with a ring map $A\longrightarrow B$. Given a left ${\mathcal{C}}$-comodule
${\mathcal{M}}$, one defines a left ${\mathcal{E}}$-comodule
${}_{B}{\mathcal{M}}$ by the rule
${}_{B}{\mathcal{M}}=B\otimes_{A}{\mathcal{M}}$, the coaction map being
constructed as the composition $B\otimes_{A}{\mathcal{M}}\longrightarrow
B\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{M}}\longrightarrow
B\otimes_{A}{\mathcal{E}}\otimes_{A}{\mathcal{M}}\longrightarrow{\mathcal{E}}\otimes_{A}{\mathcal{M}}\simeq{\mathcal{E}}\otimes_{B}(B\otimes_{A}{\mathcal{M}})$.
Similarly, given a right ${\mathcal{C}}$-comodule ${\mathcal{K}}$, there is a
natural right ${\mathcal{E}}$-comodule structure on the tensor product
${\mathcal{K}}_{B}={\mathcal{K}}\otimes_{A}B$. Given a left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, one defines a left
${\mathcal{E}}$-contramodule ${}^{B}{\mathfrak{P}}$ by the rule
${}^{B}{\mathfrak{P}}=\operatorname{Hom}_{A}(B,{\mathfrak{P}})$, the
contraaction map being constructed as the composition
$\operatorname{Hom}_{B}({\mathcal{E}},\operatorname{Hom}_{A}(B,{\mathfrak{P}}))\simeq\operatorname{Hom}_{A}({\mathcal{E}},{\mathfrak{P}})\longrightarrow\operatorname{Hom}_{A}({\mathcal{E}}\otimes_{A}B,\>{\mathfrak{P}})\longrightarrow\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}B,\>{\mathfrak{P}})\simeq\operatorname{Hom}_{A}(B,\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}}))\longrightarrow\operatorname{Hom}_{A}(B,{\mathfrak{P}})$.

Assuming that ${\mathcal{C}}$ is a flat right $A$-module, the functor
${\mathcal{M}}\longmapsto{}_{B}{\mathcal{M}}\colon{\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow{\mathcal{E}}{\operatorname{\mathsf{--comod}}}$
has a right adjoint functor, which is denoted by
${\mathcal{N}}\longmapsto{}_{\mathcal{C}}{\mathcal{N}}\colon{\mathcal{E}}{\operatorname{\mathsf{--comod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
and constructed by the rule
${}_{\mathcal{C}}{\mathcal{N}}={\mathcal{C}}_{B}\mathbin{\text{\smaller$\square$}}_{\mathcal{E}}{\mathcal{M}}$.
In particular, the functor
${\mathcal{N}}\longmapsto{}_{\mathcal{C}}{\mathcal{N}}$ takes a coinduced left
${\mathcal{E}}$-comodule ${\mathcal{E}}\otimes_{B}U$ into the coinduced left
${\mathcal{C}}$-comodule ${\mathcal{C}}\otimes_{A}U$ (where $U$ is an
arbitrary left $B$-module) [50, Section 7.1.2].

Given a coring ${\mathcal{C}}$ over an associative ring $A$ and an associative
ring homomorphism $A\longrightarrow B$, one can define a coring
${}_{B}{\mathcal{C}}_{B}$ over the ring $B$ by the rule
${}_{B}{\mathcal{C}}_{B}=B\otimes_{A}{\mathcal{C}}\otimes_{A}B$. The counit in
${}_{B}{\mathcal{C}}_{B}$ is constructed as the composition
$B\otimes_{A}{\mathcal{C}}\otimes_{A}B\longrightarrow
B\otimes_{A}A\otimes_{A}B\longrightarrow B$, and the comultiplication is
provided by the composition
$B\otimes_{A}{\mathcal{C}}\otimes_{A}B\longrightarrow
B\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{C}}\otimes_{A}B\simeq
B\otimes_{A}{\mathcal{C}}\otimes_{A}A\otimes_{A}{\mathcal{C}}\otimes_{A}B\longrightarrow
B\otimes_{A}{\mathcal{C}}\otimes_{A}B\otimes_{A}{\mathcal{C}}\otimes_{A}B\simeq(B\otimes_{A}{\mathcal{C}}\otimes_{A}B)\otimes_{B}(B\otimes_{A}{\mathcal{C}}\otimes_{A}B)$.
There is a natural map of corings
${\mathcal{C}}\longrightarrow{}_{B}{\mathcal{C}}_{B}$ compatible with the ring
map $A\longrightarrow B$.

Assuming that ${\mathcal{C}}$ is a flat right $A$-module and $B$ is a
faithfully flat right $A$-module, the functors
${\mathcal{M}}\longrightarrow{}_{B}{\mathcal{M}}$ and
${\mathcal{N}}\longmapsto{}_{\mathcal{C}}{\mathcal{N}}$ are mutually inverse
equivalences between the abelian categories of left ${\mathcal{C}}$-comodules
and left ${}_{B}{\mathcal{C}}_{B}$-comodules. One proves this by applying the
Barr–Beck monadicity theorem to the conservative exact functor
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
B{\operatorname{\mathsf{--mod}}}$ taking a left ${\mathcal{C}}$-comodule
${\mathcal{M}}$ to the left $B$-module $B\otimes_{A}{\mathcal{M}}$ (see [36,
Section 2.1.3] or [50, Section 7.4.1]).

Now let us assume that ${\mathcal{C}}$ is a flat left $A$-module,
${\mathcal{E}}$ is a flat left $B$-module, and $B$ is a flat left $A$-module.
Then the functor ${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}$ takes
$A$-cotorsion left ${\mathcal{C}}$-contramodules to $B$-cotorsion left
${\mathcal{E}}$-contramodules (by Lemma 1.3.3(a)) and induces an exact functor
between the exact categories
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}\longrightarrow{\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$.
The left adjoint functor
${\mathfrak{Q}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}$ to this exact
functor may not be everywhere defined, but one can easily see that it is
defined on the full subcategory of left ${\mathcal{E}}$-contramodules induced
from cotorsion left $B$-modules in the exact category
${\mathcal{E}}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$,
taking an induced contramodule
${\mathfrak{Q}}=\operatorname{Hom}_{B}({\mathcal{E}},V)$ over ${\mathcal{E}}$
to the induced contramodule
${}^{\mathcal{C}}{\mathfrak{Q}}=\operatorname{Hom}_{A}({\mathcal{C}},V)$ over
${\mathcal{C}}$ (where $V$ is an arbitrary cotorsion left $B$-module; see
Lemma 1.3.4(a)).

Finally, we assume that ${\mathcal{C}}$ is flat left $A$-module and $B$ is a
faithfully flat left $A$-module. It is clear from Example 3.2.1 that the
functor
${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}\colon{\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}\allowbreak\longrightarrow{}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
is _not_ an equivalence of exact categories in general (and not even in the
case when ${\mathcal{C}}=A$). The aim of this section is to prove the
following slightly weaker result (cf. Corollaries 4.6.3–4.6.5 and 5.3.3).

######  Theorem B.5.1.

Let ${\mathcal{C}}$ be a coassociative coring over an associative ring $A$ and
$A\longrightarrow B$ be an associative ring homomorphism. Assume that
${\mathcal{C}}$ is a flat left $A$-module and $B$ is a faithfully flat left
$A$-module. Then the triangulated functor
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$
induced by the exact functor ${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}$
is an equivalence of triangulated categories.

######  Corollary B.5.2.

In the assumptions of the previous Theorem, the exact functor
${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}\colon{\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}\longrightarrow{}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
is fully faithful and induces isomorphisms between the Ext groups computed in
the exact categories
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
and
${}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$.
Any short sequence (or, more generally, bounded above complex) in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
which this functor transforms into an exact sequence in
${}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
is exact in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$.

###### Proof.

See [50, Section 4.1]. ∎

The proof of Theorem B.5.1 is based on the following technical result, which
provides adjusted resolutions for the construction of the left derived functor
of the partial left adjoint functor
${\mathfrak{Q}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}$ to the exact functor
${\mathfrak{P}}\longmapsto{}^{B}{\mathfrak{P}}$. For completeness, we
formulate three versions of the (co)induced resolution theorem; it is the
third one that will be used in the argument below.

######  Theorem B.5.3.

(a) Let ${\mathcal{C}}$ be a coring over an associative ring $A$; assume that
${\mathcal{C}}$ is a flat right $A$-module. Then the coderived category
${\mathsf{D}}^{\mathsf{co}}({\mathcal{C}}{\operatorname{\mathsf{--comod}}})$
of the abelian category of left ${\mathcal{C}}$-comodules is equivalent to the
quotient category of the homotopy category of complexes of coinduced
${\mathcal{C}}$-comodules by its minimal triangulated subcategory containing
the total complexes of the short exact sequences of complexes of
${\mathcal{C}}$-comodules termwise coinduced from short exact sequences of
$A$-modules and closed under infinite direct sums.

(b) Let ${\mathcal{C}}$ be a coring over an associative ring $A$; assume that
${\mathcal{C}}$ is a projective left $A$-module. Then the contraderived
category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}})$
of the abelian category of left ${\mathcal{C}}$-contramodules is equivalent to
the quotient category of the homotopy category of complexes of induced
${\mathcal{C}}$-contramodules by its minimal triangulated subcategory
containing the total complexes of the short exact sequences of complexes of
${\mathcal{C}}$-contramodules termwise induced from short exact sequences of
$A$-modules and closed under infinite products.

(c) Let ${\mathcal{C}}$ be a coring over an associative ring $A$; assume that
${\mathcal{C}}$ is a flat left $A$-module. Then the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
of the exact category of $A$-cotorsion left ${\mathcal{C}}$-contramodules is
equivalent to the quotient category of the homotopy category of complexes of
${\mathcal{C}}$-contramodules termwise induced from cotorsion $A$-modules by
its minimal triangulated subcategory containing the total complexes of the
short exact sequences of complexes of ${\mathcal{C}}$-contramodules termwise
induced from short exact sequences of cotorsion $A$-modules and closed under
infinite products.

###### Proof.

This is yet another version of the results of [53, Proposition 1.5 and Remark
1.5], [54, Theorem 4.2.1], the above Propositions A.2.1 and A.3.1(b), etc. The
difference is that the categories of coinduced comodules or induced
contramodules are not even exact. This does not change much, however. Let us
sketch a proof of part (c); the proofs of parts (a-b) are similar (but
simpler).

The contramodule relative bar-resolution, totalized by taking infinite
products along the diagonals, provides a closed morphism with a contraacyclic
cone onto any complex of $A$-cotorsion left ${\mathcal{C}}$-contramodules from
a complex of ${\mathcal{C}}$-contramodules termwise induced from cotorsion
$A$-modules. This proves that the natural functor from the homotopy category
of complexes of left ${\mathcal{C}}$-contramodules termwise induced from
cotorsion $A$-modules to the contraderived category of $A$-cotorsion left
${\mathcal{C}}$-contramodules is a Verdier localization functor [51, Lemma
1.6].

In order to prove that the natural functor from the quotient category of the
homotopy category of termwise induced complexes in the formulation of the
theorem to the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
is fully faithful, one shows that any morphism in
$\mathsf{Hot}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
from a complex of contramodules termwise induced from cotorsion $A$-modules to
a contraacyclic complex over
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
factorizes through an object of the minimal triangulated subcategory
containing the total complexes of the short exact sequences of complexes of
${\mathcal{C}}$-contramodules termwise induced from short exact sequences of
cotorsion $A$-modules and closed under infinite products.

Let us only explain the key step, as the argument is mostly similar to the
ones spelled out in the references above. Let
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be the total
complex of a short exact sequence
${\mathfrak{U}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{V}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{W}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of complexes of $A$-cotorsion ${\mathcal{C}}$-contramodules, and let
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex
of ${\mathcal{C}}$-contramodules termwise induced from cotorsion $A$-modules.
The construction dual to the one explained in [50, Theorem 5.5] provides a
closed morphism onto the complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ from a complex
of ${\mathcal{C}}$-contramodules
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ termwise
induced from flat cotorsion $A$-modules such that this morphism is induced
from an admissible epimorphism of cotorsion $A$-modules at every term of the
complexes (cf. the proof of Theorem B.3.1(b)).

Let ${\mathfrak{R}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ denote the
kernel of the morphism of complexes
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Then the cone ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of the admissible monomorphism
${\mathfrak{R}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
maps naturally onto
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with the cone
isomorphic to the total complex of a short exact sequence of complexes of
${\mathcal{C}}$-contramodules termwise induced from a short exact sequence of
cotorsion $A$-modules. As a morphism of graded ${\mathcal{C}}$-contramodules,
the composition
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
factorizes through the graded ${\mathcal{C}}$-contramodule
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The latter
being a projective graded object of
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$,
it follows that the composition
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is homotopic to zero by Lemma A.2.2. ∎

###### Proof of Theorem B.5.1.

First of all, we notice that the functor
${\mathfrak{Q}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}$ acting between the
full subcategories of left ${}_{B}{\mathcal{C}}_{B}$-contramodules induced
from cotorsion $B$-modules and left ${\mathcal{C}}$-contramodules induced from
cotorsion $A$-modules in
${}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$
and
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
is fully faithful. Indeed, one has

$\displaystyle\operatorname{Hom}^{{}_{B}{\mathcal{C}}_{B}}(\operatorname{Hom}_{B}({}_{B}{\mathcal{C}}_{B},U),\>\operatorname{Hom}_{B}({}_{B}{\mathcal{C}}_{B},V))\simeq\operatorname{Hom}_{B}(U,\operatorname{Hom}_{B}({}_{B}{\mathcal{C}}_{B},V))$
$\displaystyle\simeq\operatorname{Hom}_{B}({}_{B}{\mathcal{C}}_{B}\otimes_{B}U,\>V)\simeq\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}U,\>V)\simeq\operatorname{Hom}^{\mathcal{C}}(\operatorname{Hom}_{A}({\mathcal{C}},U),\operatorname{Hom}_{A}({\mathcal{C}},V)).$

Furthermore, the composition of functors
${\mathfrak{Q}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}\longmapsto{}^{B}({}^{\mathcal{C}}{\mathfrak{Q}})$
is naturally isomorphic to the identity endofunctor on the full subcategory of
${}_{B}{\mathcal{C}}_{B}$-contramodules induced from cotorsion $B$-modules in
${}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}}$,
as
$\operatorname{Hom}_{A}(B,\operatorname{Hom}_{A}({\mathcal{C}},V))\simeq\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}B,\>V)\simeq\operatorname{Hom}_{B}(B\otimes_{A}{\mathcal{C}}\otimes_{A}B,\>V)$.

Clearly, the functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
acting between the homotopy categories of complexes of
${}_{B}{\mathcal{C}}_{B}$-contramodules termwise induced from complexes of
cotorsion $B$-modules and complexes of ${\mathcal{C}}$-contramodules termwise
induced from cotorsion $A$-modules, is “partially left adjoint” to the functor
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{B}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon\mathsf{Hot}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})\allowbreak\longrightarrow\mathsf{Hot}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$.
By Theorem B.5.3(c), the former functor can be used to construct a left
derived functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\nobreak{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$.
It is easy to see that the functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is left adjoint to the functor
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\nobreak{}^{B}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$.

According to the above, the composition of the adjoint triangulated functors
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{B}({}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\colon{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$
is isomorphic to the identity functor. Hence the functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
is fully faithful, while the functor
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{B}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\colon{\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}({}_{B}{\mathcal{C}}_{B}{\operatorname{\mathsf{--contra}}}^{B\operatorname{\mathsf{--cot}}})$
is a Verdier localization functor.

It remains to show that the triangulated functor
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is essentially surjective. It is straightforward from the constructions that
both functors
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{B}{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and
${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{}^{\mathcal{C}}_{\mathbb{L}}{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
preserve infinite products. So it suffices to prove that the contraderived
category
${\mathsf{D}}^{\mathsf{ctr}}({\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}})$
coincides with its minimal triangulated subcategory containing all complexes
of ${\mathcal{C}}$-contramodules termwise induced from left $A$-modules
obtained by restriction of scalars from cotorsion left $B$-modules and closed
under infinite products. In view of the argument of Lemmas A.3.3–A.3.4, we
only need to check that there is an admissible epimorphism onto any object of
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
from an induced left ${\mathcal{C}}$-contramodule of the desired type.

The contraaction morphism
$\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
being an admissible epimorphism in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{A\operatorname{\mathsf{--cot}}}$
for any $A$-cotorsion left ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, the
problem reduces to constructing an admissible epimorphism in
$A{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ onto any cotorsion left
$A$-module from an $A$-module obtained by restriction of scalars from a
cotorsion left $B$-module. Here we are finally using the assumption that $B$
is a _faithfully_ flat left $A$-module. According to [50, Section 7.4.1], the
latter is equivalent to the ring homomorphism $A\longrightarrow B$ being
injective and its cokernel being a flat left $A$-module. Assuming that, the
morphism
$\operatorname{Hom}_{A}(B,P)\longrightarrow\operatorname{Hom}_{A}(A,P)=P$
induced by the map $A\longrightarrow B$ is an admissible epimorphism in
$A{\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$ for any cotorsion left
$A$-module $P$ by Lemma 1.3.3(a). It remains to say that the left $B$-module
$\operatorname{Hom}_{A}(B,P)$ is cotorsion by Lemma 1.3.5(a). ∎

##  Appendix C Affine Noetherian Formal Schemes

This appendix is a continuation of [54, Appendix B]. Its goal is to construct
the derived co-contra correspondence between quasi-coherent torsion sheaves
and contraherent cosheaves of contramodules on an affine Noetherian formal
scheme with a dualizing complex. The affineness condition allows to speak of
(co)sheaves in terms of (contra)modules over a ring, simplifying the
exposition.

###  C.1. Torsion modules and contramodules

Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Given
an $R$-module $E$, we denote by ${}_{(n)}E\subset E$ its $R$-submodule
consisting of all the elements annihilated by $I^{n}$. An $R$-module
${\mathcal{M}}$ is said to be _$I$ -torsion_ if ${\mathcal{M}}=\bigcup_{n\ge
1}\,{}_{(n)}{\mathcal{M}}$, i. e., for any element $m\in{\mathcal{M}}$ there
exists an integer $n\ge 1$ such that $I^{n}m=0$ in ${\mathcal{M}}$. An
$R$-module ${\mathfrak{P}}$ is called an _$(R,I)$ -contramodule_ if
$\operatorname{Ext}^{*}_{R}(R[s^{-1}],{\mathfrak{P}})=0$ for all $s\in I$, or
equivalently, the system of equations $q_{n}=p_{n}+sq_{n+1}$, $n\ge 0$, is
uniquely solvable in $q_{n}\in{\mathfrak{P}}$ for any fixed sequence
$p_{n}\in{\mathfrak{P}}$ (cf. the beginning of Section 1.1).

Just as the category of $I$-torsion $R$-modules, the category of
$(R,I)$-contramodules is abelian and depends only on the $I$-adic completion
of the ring $R$ [54, Theorem B.1.1] (cf. the ending part of Section 1.3
above). We denote the category of $I$-torsion $R$-modules by
$(R,I){\operatorname{\mathsf{--tors}}}$ and the category of
$(R,I)$-contramodules by $(R,I){\operatorname{\mathsf{--contra}}}$. Both
categories are full subcategories in the abelian category of $R$-modules
$R{\operatorname{\mathsf{--mod}}}$, closed under the kernels and cokernels;
the former subcategory is also closed under infinite direct sums, while the
latter one is closed under infinite products.

Given a sequence of elements $r_{n}\in R$ converging to zero in the $I$-adic
topology of $R$ and a sequence of elements $p_{n}\in{\mathfrak{P}}$ of an
$(R,I)$-contramodule ${\mathfrak{P}}$, the result of the “infinite summation
operation” $\sum_{n}r_{n}p_{n}$ is well-defined as an element of
${\mathfrak{P}}$ [54, Section 1.2]. One can define the _contratensor product_
${\mathfrak{P}}\odot_{(R,I)}{\mathcal{M}}$ of an $(R,I)$-contramodule
${\mathfrak{P}}$ and an $I$-torsion $R$-module ${\mathcal{M}}$ as the quotient
$R$-module of the tensor product
${\mathfrak{P}}\otimes_{\mathbb{Z}}{\mathcal{M}}$ by the relations
$(\sum_{n}r_{n}p_{n})\otimes m=\sum_{n}p_{n}\otimes r_{n}m$ for any sequence
$r_{n}$ converging to zero in $R$, any sequence of elements
$p_{n}\in{\mathfrak{P}}$, and any element $m\in{\mathcal{M}}$. Here all but a
finite number of summands in the right-hand side vanish due to the conditions
imposed on the sequence $r_{n}$ and the module ${\mathcal{M}}$.

In fact, however, for any $(R,I)$-contramodule ${\mathfrak{P}}$ and
$I$-torsion $R$-module ${\mathcal{M}}$ there is a natural isomorphism
${\mathfrak{P}}\odot_{(R,I)}{\mathcal{M}}\simeq{\mathfrak{P}}\otimes_{R}{\mathcal{M}}$.
This can be deduced from the fact that the functor
$(R,I){\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is fully faithful, or alternatively, follows
directly from the observation that any sequence of elements of $I^{n}\subset
R$ converging to zero in the $I$-adic topology of $R$ is a linear combination
of a finite number of sequences converging to zero in $R$ with the
coefficients belonging to $I^{n}$ (cf. [50, proof of Proposition B.9.1]).

Clearly, the tensor product $E\otimes_{R}{\mathcal{M}}$ of any $R$-module $E$
and any $I$-torsion $R$-module ${\mathcal{M}}$ is an $I$-torsion $R$-module.
Similarly, the $R$-module $\operatorname{Hom}_{R}({\mathcal{M}},E)$ of
$R$-linear maps from an $I$-torsion $R$-module ${\mathcal{M}}$ into any
$R$-module $E$ is an $(R,I)$-contramodule. Finally, the $R$-module
$\operatorname{Hom}_{R}(E,{\mathfrak{P}})$ of homomorphisms from any
$R$-module $E$ into an $(R,I)$-contramodule ${\mathfrak{P}}$ is an
$(R,I)$-contramodule [54, Sections 1.5 and B.2].

An $(R,I)$-contramodule ${\mathfrak{F}}$ is said to be _$(R,I)$ -contraflat_
if the functor
${\mathcal{M}}\longmapsto{\mathfrak{F}}\odot_{(R,I)}{\mathcal{M}}$ is exact on
the abelian category of $I$-torsion $R$-modules. According to the above, this
is equivalent to the functor
${\mathcal{M}}\longmapsto{\mathfrak{F}}\otimes_{R}{\mathcal{M}}$ being exact
on the full abelian subcategory $(R,I){\operatorname{\mathsf{--tors}}}\subset
R{\operatorname{\mathsf{--mod}}}$.

By the Artin–Rees lemma, an object
${\mathcal{K}}\in(R,I){\operatorname{\mathsf{--tors}}}$ is injective if and
only if it is injective in $R{\operatorname{\mathsf{--mod}}}$, and if and only
if its submodules ${}_{(n)}{\mathcal{K}}$ of elements annihilated by $I^{n}$
are injective $R/I^{n}$-modules for all $n\ge 1$. By [54, Lemma B.9.2], an
$(R,I)$-contramodule ${\mathfrak{F}}$ is $(R,I)$-contraflat if and only if it
is a flat $R$-module, and if and only if its reductions
${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are flat $R/I^{n}$-modules for all $n\ge
1$. Furthermore, an object
${\mathfrak{F}}\in(R,I){\operatorname{\mathsf{--contra}}}$ is projective if
and only if it is $(R,I)$-contraflat and its reduction
${\mathfrak{F}}/I{\mathfrak{F}}$ is a projective $R/I$-module, and if and only
if all the reductions ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are projective
$R/I^{n}$-modules [54, Corollary B.8.2].

######  Theorem C.1.1.

(a) The coderived category
${\mathsf{D}}^{\mathsf{co}}((R,I){\operatorname{\mathsf{--tors}}})$ of the
abelian category of $I$-torsion $R$-modules is equivalent to the homotopy
category of complexes of injective $I$-torsion $R$-modules.

(b) The contraderived category
${\mathsf{D}}^{\mathsf{ctr}}((R,I){\operatorname{\mathsf{--contra}}})$ of the
abelian category of $(R,I)$-contramodules is equivalent to the contraderived
category of the exact category of $R$-flat (i. e., $(R,I)$-contraflat)
$(R,I)$-contramodules.

(c) Assume that the Noetherian ring $R/I$ has finite Krull dimension. Then the
contraderived category
${\mathsf{D}}^{\mathsf{ctr}}((R,I){\operatorname{\mathsf{--contra}}})$ is
equivalent to the absolute derived category of the exact category of $R$-flat
$(R,I)$-contramodules.

(d) Assume that the Noetherian ring $R/I$ has finite Krull dimension. Then the
contraderived category
${\mathsf{D}}^{\mathsf{ctr}}((R,I){\operatorname{\mathsf{--contra}}})$ is
equivalent to the homotopy category of complexes of projective
$(R,I)$-contramodules.

###### Proof.

Part (a) holds, since there are enough injectives in the abelian category
$(R,I){\operatorname{\mathsf{--tors}}}$ and the class of injective objects is
closed under infinite direct sums. Part (b) is true, because there are enough
$(R,I)$-contraflat (and even projective) objects in
$(R,I){\operatorname{\mathsf{--contra}}}$ and the class of $(R,I)$-contraflat
$(R,I)$-contramodules (or even flat $R$-modules) is closed under infinite
products (see Proposition A.3.1(b)).

Parts (c-d) hold, since in their assumptions any $(R,I)$-contraflat
$(R,I)$-contramodule ${\mathfrak{F}}$ has finite projective dimension in
$(R,I){\operatorname{\mathsf{--contra}}}$. Indeed, consider a projective
resolution ${\mathfrak{P}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
${\mathfrak{F}}$ in $(R,I){\operatorname{\mathsf{--contra}}}$ and apply the
functor $R/I\otimes_{R}{-}$ to it. Being a bounded above exact complex of flat
$R$-modules, the complex
${\mathfrak{P}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{F}}$
will remain exact after taking the tensor product. By Theorem 1.5.6, the
$R/I$-modules of cycles in the complex
$R/I\otimes_{R}{\mathfrak{P}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$
are eventually projective, and it follows that the $(R,I)$-contramodules of
cycles in ${\mathfrak{P}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
eventually projective, too. Now it remains to apply Corollary A.6.2,
Proposition A.5.6, and/or [50, Remark 2.1]. ∎

For noncommutative generalizations of the next two lemmas (as well as the
theorem following them), see Sections C.5 and D.2.

######  Lemma C.1.2.

(a) For any $R$-module $M$, injective $R$-module $J$, and $n\ge 1$, there is a
natural isomorphism of $R/I^{n}$-modules
$\operatorname{Hom}_{R}(M,J)/I^{n}\operatorname{Hom}_{R}(M,J)\simeq\operatorname{Hom}_{R/I^{n}}({}_{(n)}M,\>{}_{(n)}J)$.

(b) For any $R$-module $M$, flat $R$-module $F$, and $n\ge 1$, there is a
natural isomorphism of $R/I^{n}$-modules
${}_{(n)}(M\otimes_{R}F)\simeq\,{}_{(n)}M\otimes_{R/I^{n}}F/I^{n}F$.

###### Proof.

For any finitely generated $R$-module $E$ there are natural isomorphisms
$\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(E,M),J)\simeq
E\otimes_{R}\operatorname{Hom}_{R}(M,J)$ and
$\operatorname{Hom}_{R}(E,\>M\otimes_{R}F)\simeq\operatorname{Hom}_{R}(E,M)\allowbreak\otimes_{R}F$.
It remains to take $E=R/I^{n}$. ∎

A finite complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of injective
$I$-torsion $R$-modules is said to be a _dualizing complex_ for the pair
$(R,I)$ if for every integer $n\ge 1$ the complex of $R/I^{n}$-modules
${}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for the commutative Noetherian ring $R/I^{n}$. If a finite
complex of injective $R$-modules
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a dualizing complex for
the ring $R$, then its subcomplex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=\bigcup_{n\ge
0}\,{}_{(n)}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ consisting of
all the elements annihilated by some powers of $I\subset R$ is a dualizing
complex for $(R,I)$.

######  Lemma C.1.3.

Let ${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a finite
complex of injective $I$-torsion $R$-modules such that the complex of
$R/I$-modules
${}_{(1)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for the ring $R/I$. Then
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a dualizing
complex for the pair $(R,I)$.

###### Proof.

Clearly,
${}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
finite complex of injective $R/I^{n}$-modules. Since any $R/I^{n}$-module $K$
with finitely generated $R/I$-module ${}_{(1)}K$ is finitely generated itself,
and the functors $\operatorname{Ext}^{i}_{R/I^{n}}(R/I,{-})$ take finitely
generated $R/I^{n}$-modules to finitely generated $R/I$-modules, one can check
by induction that the $R/I^{n}$-modules of cohomology of the complex
${}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
finitely generated.

It remains to show that the natural map
$R/I^{n}\longrightarrow\operatorname{Hom}_{R/I^{n}}({}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism. Both the left-hand and the right-hand sides are finite
complexes of flat (or, equivalently, projective) $R/I^{n}$-modules. By Lemma
C.1.2(a), the functor $P\longmapsto P/IP$ transforms this morphism into the
morphism
$R/I\longrightarrow\operatorname{Hom}_{R/I}({}_{(1)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}_{(1)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
which is a quasi-isomorphism by assumption. Hence the desired assertion
follows by Nakayama’s lemma. ∎

######  Theorem C.1.4.

The data of a dualizing complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for a pair
$(R,I)$ with a Noetherian commutative ring $R$ and an ideal $I\subset R$
induces an equivalence of triangulated categories
${\mathsf{D}}^{\mathsf{co}}((R,I){\operatorname{\mathsf{--tors}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}((R,I){\operatorname{\mathsf{--contra}}})$,
which is provided by the derived functors
${\mathbb{R}}\operatorname{Hom}_{R}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}^{\mathbb{L}}{-}$.

###### Proof.

The constructions of the derived functors are based on Theorem C.1.1(a,c).
Applying the functor
$\operatorname{Hom}_{R}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
to a complex of injective $I$-torsion $R$-modules produces a complex of
$R$-flat $(R,I)$-contramodules. Applying the functor
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}{-}$
to a complex of $R$-flat $(R,I)$-contramodules produces a complex of injective
$I$-torsion $R$-modules, and for any (absolutely) acyclic complex of flat
$R$-modules $F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, the complex of
injective ($I$-torsion) $R$-modules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}F^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is contractible.

It remains to show that the morphism of finite complexes
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}\operatorname{Hom}_{R}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})\longrightarrow{\mathcal{J}}$
is a quasi-isomorphism for any $I$-torsion $R$-module ${\mathcal{J}}$, and the
morphism of finite complexes
${\mathfrak{F}}\longrightarrow\operatorname{Hom}_{R}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{R}{\mathfrak{F}})$
is a quasi-isomorphism for any $(R,I)$-contramodule ${\mathfrak{F}}$. Both
assertions follow from Lemma C.1.2, which implies that the morphisms become
quasi-isomorphisms after applying the functors
${\mathcal{K}}\longmapsto{}_{(n)}{\mathcal{K}}$ and
${\mathfrak{P}}\longmapsto{\mathfrak{P}}/I^{n}{\mathfrak{P}}$. In the latter
situation, one also has to use the assertion that the natural morphism
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/I^{n}{\mathfrak{P}}$
is an isomorphism for any $R$-flat $(R,I)$-contramodule ${\mathfrak{P}}$ (see
[54, proof of Lemma B.9.2]). ∎

###  C.2. Contraadjusted and cotorsion contramodules

Let $R$ be a Noetherian commutative ring and $s\in R$ be an element. In the
spirit of the definitions from Sections 1.1 and C.1, let us say that an
$R$-module $P$ is _$s$ -contraadjusted_ if
$\operatorname{Ext}_{R}^{1}(R[s^{-1}],P)=0$, and that $P$ is an _$s$
-contramodule_ if $\operatorname{Ext}_{R}^{*}(R[s^{-1}],P)=0$. The property of
an $R$-module $P$ to be $s$-contraadjusted or an $s$-contramodule only depends
on the abelian group $P$ with the operator $s\colon P\longrightarrow P$.

An $R$-module $P$ is an $s$-contramodule if and only if it is an
$(R,(s))$-contramodule, where $(s)\subset R$ denotes the principal ideal
generated by $s\in R$. More generally, given an ideal $I\subset R$, an
$R$-module ${\mathfrak{P}}$ is an $(R,I)$-contramodule if and only if it is an
$s$-contramodule for every $s\in I$, and it suffices to check this condition
for any given set of generators of the ideal $I$ [54, proof of Theorem B.1.1].

Recall that any quotient $R$-module of an $s$-contraadjusted $R$-module is
$s$-contraadjusted. Given an $R$-module $L$, let us denote by $L(s)\subset L$
the image of the morphism $\operatorname{Hom}_{R}(R[s^{-1}],L)\longrightarrow
L$ induced by the localization map $R\longrightarrow R[s^{-1}]$. Equivalently,
the $R$-submodule $L(s)\subset L$ can be defined as the maximal $s$-divisible
$R$-submodule in $R$, i. e., the sum of all $R$-submodules (or even all
$s$-invariant abelian subgroups) in $L$ in which the element $s$ acts
surjectively.

Therefore, if $P=L/L(s)$ denotes the corresponding quotient $R$-module, then
one has $P(s)=0$. Notice also that one has
$\operatorname{Hom}_{R}(R[s^{-1}],P)=0$ for any $R$-module $P$ for which
$P(s)=0$. It follows that the $R$-quotient module $L/L(s)$ is an
$s$-contramodule whenever an $R$-module $L$ is $s$-contraadjusted.

Now let $s$, $t\in R$ be two elements; suppose that an $R$-module $L$ is a
$t$-contramodule. Then the $R$-module $\operatorname{Hom}_{R}(R[s^{-1}],L)$ is
also a $t$-contramodule [54, Section B.2], as is the image $L(s)$ of the
morphism of $t$-contramodules
$\operatorname{Hom}_{R}(R[s^{-1}],L)\longrightarrow L$. Hence the quotient
$R$-module $L/L(s)$ is also a $t$-contramodule. Assuming additionally that the
$R$-module $L$ was $s$-contraadjusted, the quotient module $L/L(s)$ is both a
$t$\- and an $s$-contramodule (i. e., it is an $(R,(t,s))$-contramodule).

Recall that an $R$-module $L$ is said to be _contraadjusted_ if it is
$s$-contraadjusted for every element $s\in R$. Given a contraadjusted
$R$-module $L$ and an ideal $I\subset R$, one can apply the above construction
of a quotient $R$-module successively for all the generators of the ideal $I$.
Proceeding in this way, one in a finite number of steps (equal to the number
of generators of $I$) obtains the unique maximal quotient $R$-module
${\mathfrak{P}}$ of the $R$-module $L$ such that ${\mathfrak{P}}$ is an
$s$-contramodule for every $s\in I$, i. e., ${\mathfrak{P}}$ is an
$(R,I)$-contramodule. We denote this quotient module by
${\mathfrak{P}}=L/L(I)$.

######  Lemma C.2.1.

Let $R$ be a Noetherian commutative ring, $s\in R$ be an element, $F$ be a
flat $R$-module, and $M$ be a finitely generated $R$-module. Then the natural
$R$-module homomorphism
$\operatorname{Hom}_{R}(R[s^{-1}],\>M\otimes_{R}F)\longrightarrow
M\otimes_{R}F$ is injective. In other words, one has
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>M\otimes_{R}F)=0$.

###### Proof.

Consider the sequence of annihilator submodules of the elements $s^{m}\in R$,
$m\ge 1$, in the $R$-module $M$. This is an increasing sequence of
$R$-submodules in $M$. Let $N$ be the maximal submodule in this sequence and
$n$ be a positive integer such that $N$ is the annihilator of $s^{n}$ in $M$.
Then any element of the $R$-module $M\otimes_{R}F$ annihilated by $s^{n+1}$ is
also annihilated by $s^{n}$.

Indeed, the element $s$ acts by an injective endomorphism of an $R$-module
$M/N$; hence so does the element $s^{n+1}$. Since $F$ is flat, it follows that
$s^{n+1}$ must act injectively in the tensor product
$(M/N)\otimes_{R}F\simeq(M\otimes_{R}F)/(N\otimes_{R}F)$. Hence any element of
$M\otimes_{R}F$ annihilated by $s^{n+1}$ belongs to $N\otimes_{R}F$ and is
therefore annihilated by $s^{n}$. ∎

######  Lemma C.2.2.

Let $R$ be a Noetherian commutative ring, $s\in R$ be an element, and
$I\subset R$ be an ideal. Then

(a) whenever an $R$-module $L$ is $s$-contraadjusted, the $R$-modules $L(s)$
and $L/L(s)$ are also $s$-contraadjusted (and $L/L(s)$ is even an
$s$-contramodule);

(b) whenever an $R$-module $L$ is contraadjusted, the $R$-modules $L(s)$ and
$L/L(s)$ are also contraadjusted;

(c) whenever an $R$-module $L$ is contraadjusted, the $R$-modules $L(I)$ and
$L/L(I)$ are also contraadjusted (and $L/L(I)$ is in addition an
$(R,I)$-contramodule);

(d) whenever an $R$-module $L$ is cotorsion and
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>L)=0$, the $R$-modules $L(s)$ and
$L/L(s)$ are also cotorsion.

###### Proof.

The parenthesized assertions in (a) and (c) have been explained above. Since
the class of cotorsion $R$-modules is closed under cokernes of injective
morphisms, while the classes of contraadjusted and $s$-contraadjusted
$R$-modules are even closed under quotients, it suffices to check the
assertions related to the $R$-modules $L(s)$ and $L(I)$. Furthermore, there is
a natural surjective morphism of $R$-modules
$\operatorname{Hom}_{R}(R[s^{-1}],L)\longrightarrow L(s)$, which in the
assumptions of (d) is an isomorphism.

Now part (d) is a particular case of Lemma 1.3.2(a), part (b) is provided by
Lemma 1.2.1(b), and part (a) follows from the similar claim that the
$R$-module $\operatorname{Hom}_{R}(R[s^{-1}],P)$ is $s$-contraadjusted for any
$s$-contraadjusted $R$-module $P$. Finally, (b) implies (c) in view of the
above recursive construction of the $R$-module $L(I)$ and the fact that the
class of constraadjusted $R$-modules is closed under extensions. ∎

######  Corollary C.2.3.

Let $R$ be a Noetherian commutative ring, $s\in R$ be an element, and
$I\subset R$ be an ideal. In this setting

(a) if $F$ is an $s$-contraadjusted flat $R$-module, then the $R$-modules
$F(s)$ and $F/F(s)$ are also flat and $s$-contraadjusted (and $F/F(s)$ is even
an $s$-contramodule);

(b) if $F$ is a contraadjusted flat $R$-module, then $F(I)$ is a
contraadjusted flat $R$-module and $F/F(I)$ is an $R$-flat $R$-contraadjusted
$(R,I)$-contramodule;

(c) if $F$ is a flat cotorsion $R$-module, then $F(I)$ is a flat cotorsion
$R$-module and $F/F(I)$ is an $R$-flat $R$-cotorsion $(R,I)$-contramodule.

###### Proof.

Part (a): in order to prove that the $R$-module $F/F(s)$ is flat, let us check
that the map $M\otimes_{R}F(s)\longrightarrow M\otimes_{R}F$ induced by the
natural embedding $F(s)\longrightarrow F$ is injective for any finitely
generated $R$-module $M$. By Lemma C.2.1, one has
$F(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],F)\subset F$ and
$(M\otimes_{R}F)(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],\>M\otimes_{R}F)\subset
M\otimes_{R}F$. It remains to point out that the natural morphism
$M\otimes_{R}\operatorname{Hom}_{R}(R[s^{-1}],F)\longrightarrow\operatorname{Hom}_{R}(R[s^{-1}],\>M\otimes_{R}F)$
is an isomorphism by Lemma 1.6.2.

Part (b): the above recursive construction of the $(R,I)$-contramodule
$F/F(I)$ together with part (a) imply the assertion that $F/F(I)$ is a flat
$R$-module. Now it remains to use Lemma C.2.2(c).

Part (c) can be proven in the way similar to part (b), using the fact that the
class of cotorsion $R$-modules is closed under extensions and Lemma C.2.2(d).
Alternatively, (c) can be deduced from Theorem 1.3.8. Indeed, if
$F\simeq\prod_{\mathfrak{p}}{\mathfrak{F}}_{\mathfrak{p}}$, where
${\mathfrak{p}}\subset R$ are prime ideals and ${\mathfrak{F}}_{\mathfrak{p}}$
are $(R,{\mathfrak{p}})$-contramodules, then one easily checks that $F/F(I)$
is the product of ${\mathfrak{F}}_{\mathfrak{p}}$ over the prime ideals
${\mathfrak{p}}$ containing $I$, while $F(I)$ is the product of
${\mathfrak{F}}_{\mathfrak{q}}$ over all the other prime ideals
${\mathfrak{q}}$. ∎

######  Lemma C.2.4.

(a) Let $0\longrightarrow P\longrightarrow K\longrightarrow F\longrightarrow
0$ be a short exact sequence of $R$-modules, where $P$ is an $s$-contramodule,
$K$ is a contraadjusted $R$-module, and $F$ is a flat $R$-module. Then
$0\longrightarrow P\longrightarrow K/K(s)\longrightarrow F/F(s)\longrightarrow
0$ is a short exact sequence of $R$-modules in which $K/K(s)$ is a
contraadjusted $R$-module, $F/F(s)$ is a flat $R$-module, and all the three
modules are $s$-contramodules.

(b) Let $0\longrightarrow P\longrightarrow K\longrightarrow F\longrightarrow
0$ be a short exact sequence of $R$-modules, where $P$ is an $s$-contramodule,
$K$ is a cotorsion $R$-module, and $F$ is a flat $R$-module. Then
$0\longrightarrow P\longrightarrow K/K(s)\longrightarrow F/F(s)\longrightarrow
0$ is a short exact sequence of $R$-modules in which $K/K(s)$ is a cotorsion
$R$-module, $F/F(s)$ is a flat $R$-module, and all the three modules are
$s$-contramodules.

###### Proof.

Part (a): first of all let us show that the morphism of $R$-modules
$K\longrightarrow F$ restricts to an isomorphism of their submodules
$K(s)\longrightarrow F(s)$. Indeed, we have
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>F)=0$ by Lemma C.2.1 and
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>P)\subset\operatorname{Hom}_{R}(R[s^{-1}],P)=0$
by the definition of an $s$-contramodule, hence
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>\allowbreak K)=0$. Therefore, there are
isomorphisms $K(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],K)$ and
$F(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],F)$. Using the condition that $P$
is an $s$-contramodule, that is $\operatorname{Ext}^{*}(R[s^{-1}],P)=0$ again,
we conclude that $K(s)\simeq F(s)$.

It follows that the sequence $0\longrightarrow P\longrightarrow
K/K(s)\longrightarrow F/F(s)\longrightarrow 0$ is exact. The $R$-module $K$
being contraadjusted by assumption, its quotient $R$-module $F$ is
contraadjusted, too. Now the quotient module $K/K(s)$ is contraadjusted by
Lemma C.2.2(b), the quotient module $F/F(s)$ is flat by Corollary C.2.3(a),
and both are $s$-contramodules by Lemma C.2.2(a).

In part (b), it only remains to prove that $K/K(s)$ is a cotorsion $R$-module.
This follows from the above argument and Lemma C.2.2(d). ∎

######  Corollary C.2.5.

(a) Let $0\longrightarrow P\longrightarrow K\longrightarrow F\longrightarrow
0$ be a short exact sequence of $R$-modules, where $P$ is an
$(R,I)$-contramodule, $K$ is a contraadjusted $R$-module, and $F$ is a flat
$R$-module. Then $0\longrightarrow P\longrightarrow K/K(I)\longrightarrow
F/F(I)\longrightarrow 0$ is a short exact sequence of $(R,I)$-contramodules in
which $K/K(I)$ is a contraadjusted $R$-module and $F/F(I)$ is a flat
$R$-module.

(b) Let $0\longrightarrow P\longrightarrow K\longrightarrow F\longrightarrow
0$ be a short exact sequence of $R$-modules, where $P$ is an
$(R,I)$-contramodule, $K$ is a cotorsion $R$-module, and $F$ is a flat
$R$-module. Then $0\longrightarrow P\longrightarrow K/K(I)\longrightarrow
F/F(I)\longrightarrow 0$ is a short exact sequence of $(R,I)$-contramodules in
which $K/K(I)$ is a cotorsion $R$-module and $F/F(I)$ is a flat $R$-module.

###### Proof.

Follows by recursion from Lemma C.2.4. ∎

######  Lemma C.2.6.

(a) Let $0\longrightarrow K\longrightarrow F\longrightarrow P\longrightarrow
0$ be a short exact sequence of $R$-modules, where $K$ is a contraadjusted
$R$-module, $F$ is a flat $R$-module, and $P$ is an $s$-contramodule. Then
$0\longrightarrow K/K(s)\longrightarrow F/F(s)\longrightarrow P\longrightarrow
0$ is a short exact sequence of $R$-modules in which $K/K(s)$ is a
contraadjusted $R$-module, $F/F(s)$ is a flat $R$-module, and all the three
modules are $s$-contramodules.

(b) Let $0\longrightarrow K\longrightarrow F\longrightarrow P\longrightarrow
0$ be a short exact sequence of $R$-modules, where $K$ is a cotorsion
$R$-module, $F$ is a flat $R$-module, and $P$ is an $s$-contramodule. Then
$0\longrightarrow K/K(s)\longrightarrow F/F(s)\longrightarrow P\longrightarrow
0$ is a short exact sequence of $R$-modules in which $K/K(s)$ is a cotorsion
$R$-module, $F/F(s)$ is a flat $R$-module, and all the three modules are
$s$-contramodules.

###### Proof.

Part (a): first we prove that the morphism of $R$-modules $K\longrightarrow F$
restricts to an isomorphism of their submodules $K(s)\longrightarrow F(s)$.
Indeed,
$\operatorname{Hom}_{R}(R[s^{-1}]/R,\>K)\subset\operatorname{Hom}_{R}(R[s^{-1}]/R,\>F)=0$
by Lemma C.2.1, hence $K(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],K)$ and
$F(s)\simeq\operatorname{Hom}_{R}(R[s^{-1}],F)$. Since
$\operatorname{Hom}_{R}(R[s^{-1}],P)=0$ by assumption, we conclude that
$K(s)\simeq F(s)$, and it follows that the sequence $0\longrightarrow
K/K(s)\longrightarrow F/F(s)\longrightarrow P\longrightarrow 0$ is exact.

The $R$-module $F$, being an extension of $s$-contraadjusted $R$-modules $K$
and $P$, is $s$-contraadjusted, too. The rest of the argument coincides with
the respective part of the proof of Lemma C.2.4, and so does the proof of part
(b). ∎

######  Corollary C.2.7.

(a) Let $0\longrightarrow K\longrightarrow F\longrightarrow P\longrightarrow
0$ be a short exact sequence of $R$-modules, where $K$ is a contraadjusted
$R$-module, $F$ is a flat $R$-module, and $P$ is an $(R,I)$-contramodule. Then
$0\longrightarrow K/K(I)\longrightarrow F/F(I)\longrightarrow P\longrightarrow
0$ is a short exact sequence of $(R,I)$-contramodules in which $K/K(I)$ is a
contraadjusted $R$-module and $F/F(I)$ is a flat $R$-module.

(b) Let $0\longrightarrow K\longrightarrow F\longrightarrow P\longrightarrow
0$ be a short exact sequence of $R$-modules, where $K$ is a cotorsion
$R$-module, $F$ is a flat $R$-module, and $P$ is an $(R,I)$-contramodule. Then
$0\longrightarrow K/K(I)\longrightarrow F/F(I)\longrightarrow P\longrightarrow
0$ is a short exact sequence of $(R,I)$-contramodules in which $K/K(I)$ is a
cotorsion $R$-module and $F/F(I)$ is a flat $R$-module.

###### Proof.

Follows by recursion from Lemma C.2.6. ∎

Recall that, according to [54, Theorem B.8.1], the $\operatorname{Ext}$
groups/modules computed in the abelian categories
$R{\operatorname{\mathsf{--mod}}}$ and
$(R,I){\operatorname{\mathsf{--contra}}}$ agree.

######  Corollary C.2.8.

Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Then

(a) any $(R,I)$-contramodule can be embedded into an $R$-cotorsion
$(R,I)$-contramodule in such a way that the quotient $(R,I)$-contramodule is
$R$-flat;

(b) any $(R,I)$-contramodule admits a surjective morphism onto it from an
$R$-flat $(R,I)$-contramodule such that the kernel is an $R$-cotorsion
$(R,I)$-contramodule;

(c) an $(R,I)$-contramodule ${\mathfrak{Q}}$ is $R$-cotorsion if and only if
one has $\operatorname{Ext}_{R}^{1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for any
$R$-flat $(R,I)$-contramodule ${\mathfrak{F}}$;

(d) an $(R,I)$-contramodule ${\mathfrak{F}}$ is $R$-flat if and only if one
has $\operatorname{Ext}_{R}^{1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for any
$R$-cotorsion $(R,I)$-contramodule ${\mathfrak{Q}}$.

###### Proof.

Parts (a-b) follow from Theorem 1.3.1 together with Corollaries C.2.5(b) and
C.2.7(b). Part (c) is deduced from (a) and part (d) deduced from (b) easily. ∎

Let us denote by $(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}}$ the
full exact subcategory of $R$-contraadjusted $(R,I)$-contramodules and by
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}$ the full exact
subcategory of $R$-cotorsion $(R,I)$-contramodules in the abelian category
$(R,I){\operatorname{\mathsf{--contra}}}$.

Notice that the full subcategory
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}\subset(R,I){\operatorname{\mathsf{--contra}}}$
depends only on the $I$-adic completion of the ring $R$, as one can see from
Corollary C.2.8(c). The similar assertion for the full subcategory
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}\subset(R,I){\operatorname{\mathsf{--contra}}}$
will follow from the results of Section C.3 below.

######  Theorem C.2.9.

(a) Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal.
Then for any symbol $\star={\mathsf{b}}$, $+$, $-$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}((R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}})\longrightarrow{\mathsf{D}}^{\star}((R,I){\operatorname{\mathsf{--contra}}})$
induced by the embedding of exact categories
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}}\longrightarrow(R,I){\operatorname{\mathsf{--contra}}}$
is an equivalence of triangulated categories.

(b) Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal such
that the quotient ring $R/I$ has finite Krull dimension. Then for any symbol
$\star={\mathsf{b}}$, $+$, $-$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$,
${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated functor
${\mathsf{D}}^{\star}((R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}})\longrightarrow{\mathsf{D}}^{\star}((R,I){\operatorname{\mathsf{--contra}}})$
induced by the embedding of exact categories
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}\longrightarrow(R,I){\operatorname{\mathsf{--contra}}}$
is an equivalence of triangulated categories.

###### Proof.

Part (a) follows from Corollary C.2.5(a) or C.2.8(a) together with the
opposite version of Proposition A.5.6. To prove part (b) in the similar way,
one needs to know that in its assumptions any $(R,I)$-contramodule has finite
right homological dimension with respect to the exact subcategory
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}\subset(R,I){\operatorname{\mathsf{--contra}}}$.
In view of Corollary C.2.8(c), this follows from the fact that any $R$-flat
$(R,I)$-contramodule has finite projective dimension in
$(R,I){\operatorname{\mathsf{--contra}}}$, which was established in the proof
of Theorem C.1.1 (using [54, Corollary B.8.2]). ∎

######  Remark C.2.10.

Let $X=\operatorname{Spec}R$ be the Noetherian affine scheme corresponding to
the ring $R$ and $U=X\setminus Z$ be the open complement to the closed
subscheme $Z=\operatorname{Spec}R/I\subset X$. Denote by $j\colon
U\longrightarrow X$ the natural open embedding morphism. Then the exact
category $(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}}$ is
equivalent to the full exact subcategory in the exact category
$X{\operatorname{\mathsf{--ctrh}}}$ of contraherent cosheaves on $X$
consisting of all the contraherent cosheaves ${\mathfrak{P}}\in
X{\operatorname{\mathsf{--ctrh}}}$ with vanishing restrictions
$j^{!}{\mathfrak{P}}={\mathfrak{P}}|_{U}$ to the open subscheme $U$.
Similarly, the exact category
$(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}$ is equivalent to the
full exact subcategory in the exact category
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ consisting of all the
(locally) cotorsion contraherent cosheaves ${\mathfrak{P}}$ on $X$ for which
$j^{!}{\mathfrak{P}}=0$. Indeed, a contraadjusted $R$-module $P$ is an
$s$-contramodule if and only if the corresponding contraherent cosheaf
${\mathfrak{P}}=\widecheck{P}$ on $X$ vanishes in the restriction to
$\operatorname{Spec}R[s^{-1}]\subset\operatorname{Spec}R$ (see Sections
2.2–2.4 for the definitions and notation). In particular, when the ring $R/I$
is Artinian, the exact category $\ker(j^{!}\colon
X{\operatorname{\mathsf{--ctrh}}}\to
U{\operatorname{\mathsf{--ctrh}}})=\ker(j^{!}\colon
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\to
U{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$ is abelian and equivalent
to the abelian category
$(R,I){\operatorname{\mathsf{--contra}}}=(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cta}}=(R,I){\operatorname{\mathsf{--contra}}}^{\mathsf{cot}}$.

###  C.3. Very flat contramodules

Unlike the flatness, cotorsion, and contraadjustness properties of
$(R,I)$-contramodules, their very flatness property is _not_ defined in terms
of the similar property of $R$-modules. Instead, it is described in terms of
the reductions modulo $I^{n}$ and the very flatness properties of
$R/I^{n}$-modules.

For a (straightforward) generalization of the next lemma, see Lemma D.1.7.

######  Lemma C.3.1.

(a) Let $F$ be a flat $R$-module and ${\mathfrak{Q}}$ be an
$(R,I)$-contramodule such that the $R/I$-module $F/IF$ is very flat, the
natural $(R,I)$-contramodule morphism
${\mathfrak{Q}}\longrightarrow\varprojlim_{n}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$
is an isomorphism, and the $R/I$-module ${\mathfrak{Q}}/I{\mathfrak{Q}}$ is
contraadjusted. Then one has
$\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{Q}})=0$.

(b) Let $F$ be a flat $R$-module and ${\mathfrak{Q}}$ be an $R$-flat
$(R,I)$-contramodule such that the $R/I$-module
${\mathfrak{Q}}/I{\mathfrak{Q}}$ is cotorsion. Then one has
$\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{Q}})=0$.

###### Proof.

Part (a): by (the proof of) Lemma 1.6.8(a), the $R/I^{n}$-modules
${\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$ and the $R/I$-modules
$I^{n-1}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$ are contraadjusted, while by part
(b) of the same lemma the $R/I^{n}$-modules $F/I^{n}F$ are very flat. Now we
follow the argument in the proof of [54, Proposition B.10.1]. The $R$-module
$F$ being flat by assumption, one has
$\operatorname{Ext}^{i}_{R}(F,\>{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})\simeq\operatorname{Ext}^{i}_{R/I^{n}}(F/I^{n}F,\>{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})=0$
for all $i>0$ and any $n\ge 1$. The natural map
$\operatorname{Hom}_{R}(F,\>{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})\longrightarrow\operatorname{Hom}_{R}(F,\>{\mathfrak{Q}}/I^{n-1}{\mathfrak{Q}})$
is surjective, since
$\operatorname{Ext}^{1}_{R/I^{n}}(F/I^{n}F,\>I^{n-1}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})=0$.
By [54, Lemma B.10.3], we have
$\operatorname{Ext}_{R}^{i}(F,\>\varprojlim_{n}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})\simeq\varprojlim_{n}^{i}\operatorname{Hom}_{R}(F,\>{\mathfrak{Q}}/I^{n}{\mathfrak{Q}})=0$
for all $i>0$.

Part (b): by (the proof of) Lemma 1.6.8(c), the $R/I^{n}$-modules
${\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$ and the $R/I$-modules
$I^{n-1}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$ are cotorsion, while by [54, proof
of Lemma B.9.2] the natural map
${\mathfrak{Q}}\longrightarrow\varprojlim_{n}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$
is an isomorphism. The argument continues exactly the same as in part (a). ∎

For generalizations of respective parts of the following corollary, see Lemma
D.3.3 and Corollary D.4.5.

######  Corollary C.3.2.

(a) For any contraadjusted $R$-module $Q$, the $R/I$-module $Q/IQ$ is
contraadjusted. If the natural map $Q\longrightarrow\varprojlim_{n}Q/I^{n}Q$
is an isomorphism for an $R$-module $Q$ and the $R/I$-module $Q/IQ$ is
contraadjusted, then the $R$-module $Q$ is contraadjusted.

(b) A flat $(R,I)$-contramodule ${\mathfrak{Q}}$ is $R$-cotorsion if and only
if the $R/I$-module ${\mathfrak{Q}}/I{\mathfrak{Q}}$ is cotorsion.

###### Proof.

Part (a): the first assertion is Lemma 1.6.6(b). Since the $R/I$-module $F/IF$
is very flat for any very flat $R$-module $F$ (see Lemma 1.2.2(b)), the second
assertion follows from Lemma C.3.1(a). Part (b) is similarly a consequence of
Lemma 1.6.7(a) and Lemma C.3.1(b). ∎

For a generalization of the next corollary, see Corollary D.3.8.

######  Corollary C.3.3.

Let $F$ be a flat $R$-module for which the $R/I$-module $F/IF$ is very flat.
Then $\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{P}})=0$ for any
$R$-contraadjusted $(R,I)$-contramodule ${\mathfrak{P}}$.

###### Proof.

By Corollary C.2.7(a) or C.2.8(b), there exists a short exact sequence of
$(R,I)$-contramodules
$0\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{P}}\longrightarrow
0$, where the $R$-module ${\mathfrak{Q}}$ is contraadjusted and the $R$-module
${\mathfrak{G}}$ is flat. Then the $R$-module ${\mathfrak{G}}$ is also
contraadjusted. By Corollary C.3.2(a), it follows that so are the
$R/I$-modules ${\mathfrak{Q}}/I{\mathfrak{Q}}$ and
${\mathfrak{G}}/I{\mathfrak{G}}$. Furthermore, according to the proof of [54,
Lemma B.9.2], one has
${\mathfrak{G}}\simeq\varprojlim_{n}{\mathfrak{G}}/I^{n}{\mathfrak{G}}$. The
$(R,I)$-contramodule ${\mathfrak{Q}}$ being a subcontramodule of
${\mathfrak{G}}$, it follows that $\bigcap_{n}I^{n}{\mathfrak{Q}}=0$, hence
also ${\mathfrak{Q}}\simeq\varprojlim_{n}{\mathfrak{Q}}/I^{n}{\mathfrak{Q}}$.
Now it remains to apply Lemma C.3.1(a) to (the $R$-module $F$ and) the
$(R,I)$-contramodules ${\mathfrak{Q}}$ and ${\mathfrak{G}}$. ∎

######  Lemma C.3.4.

(a) Any $(R,I)$-contramodule ${\mathfrak{M}}$ can be included in a short exact
sequence of $(R,I)$-contramodules
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$, where the $R/I^{n}$-modules ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are very
flat, while the $R$-module ${\mathfrak{P}}$ is contraadjusted.

(b) Any $(R,I)$-contramodule ${\mathfrak{M}}$ can be included in a short exact
sequence of $(R,I)$-contramodules
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$, where the $R/I^{n}$-modules ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are very
flat, while the $R$-module ${\mathfrak{P}}$ is contraadjusted.

###### Proof.

First of all, let us show that any $R$-flat $(R,I)$-contramodule
${\mathfrak{G}}$ can be included in a short exact sequence of
$(R,I)$-contramodules
$0\longrightarrow{\mathfrak{E}}\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{G}}\longrightarrow
0$, where the $R/I^{n}$-modules ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ are very
flat, while the $R$-module ${\mathfrak{E}}$ is (flat and) contraadjusted. By
Theorem 1.1.1(b), there exists a short exact sequence of $R$-modules
$0\longrightarrow E\longrightarrow
F\longrightarrow{\mathfrak{G}}\longrightarrow 0$ such that the $R$-module $F$
is very flat, while the $R$-module $E$ is contraadjusted. Then the $R$-module
$E$ is also flat.

The short sequence of $R/I^{n}$-modules $0\longrightarrow
E/I^{n}E\longrightarrow
F/I^{n}F\longrightarrow{\mathfrak{G}}/I^{n}{\mathfrak{G}}\longrightarrow 0$ is
exact for every $n\ge 1$. Set ${\mathfrak{E}}=\varprojlim_{n}E/I^{n}E$ and
${\mathfrak{F}}=\varprojlim_{n}F/I^{n}F$; recall that the natural map
${\mathfrak{G}}\longrightarrow\varprojlim_{n}{\mathfrak{G}}/I^{n}{\mathfrak{G}}$
is an isomorphism according to the proof of [54, Lemma B.9.2]. Passing to the
projective limit, we obtain a short exact sequence of $(R,I)$-contramodules
$0\longrightarrow{\mathfrak{E}}\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{G}}\longrightarrow
0$. The $R/I^{n}$-modules ${\mathfrak{F}}/I^{n}{\mathfrak{F}}\simeq F/I^{n}F$
are very flat by Lemma 1.2.2(b), and the $R$-module ${\mathfrak{E}}$ is
contraadjusted by Corollary C.3.2(a). Now it is easy to obtain part (a) from
Corollary C.2.8(a) and part (b) from Corollary C.2.8(b). ∎

Let us call an $(R,I)$-contramodule ${\mathfrak{F}}$ _very flat_ if the
functor $\operatorname{Hom}_{R}({\mathfrak{F}},{-})$ takes short exact
sequences of $R$-contraadjusted $(R,I)$-contramodules to short exact sequences
of abelian groups. The next corollary says that this definition is equivalent
to the more familiar formulation is terms of the $\operatorname{Ext}^{1}$
vanishing (cf. the definitions of very flat modules and contraadjusted sheaves
in Sections 1.1 and 2.5).

######  Corollary C.3.5.

Let $R$ be a Noetherian commutative ring and $I\subset R$ be an ideal. Then

(a) any $(R,I)$-contramodule can be embedded into an $R$-contraadjusted
$(R,I)$-contramodule in such a way that the quotient $(R,I)$-contramodule is
very flat;

(b) any $(R,I)$-contramodule admits a surjective morphism onto it from an very
flat $(R,I)$-contramodule such that the kernel is an $R$-contraadjusted
$(R,I)$-contramodule;

(c) an $(R,I)$-contramodule ${\mathfrak{Q}}$ is $R$-contraadjusted if and only
if $\operatorname{Ext}_{R}^{1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for any very
flat $(R,I)$-contramodule ${\mathfrak{F}}$;

(d) an $(R,I)$-contramodule ${\mathfrak{F}}$ is very flat if and only if
$\operatorname{Ext}_{R}^{1}({\mathfrak{F}},{\mathfrak{Q}})=0$ for any
$R$-contraadjusted $(R,I)$-contramodule ${\mathfrak{Q}}$;

(e) an $(R,I)$-contramodule ${\mathfrak{F}}$ is very flat if and only if the
$R/I^{n}$-module ${\mathfrak{F}}/I^{n}{\mathfrak{F}}$ is very flat for every
$n\ge 1$.

###### Proof.

The “if” assertion in part (e) follows Corollary C.3.3 and [54, Lemma B.9.2].
Parts (a-b) are provided by the respective parts of Lemma C.3.4 together with
the “if” assertion in (e). The “if” assertions in parts (c) and (d) are
deduced from parts (a) and (b), respectively; and to prove the “only if”, one
only needs to know that any $(R,I)$-contramodule can be embedded into an
$R$-contraadjusted one. Finally, the “only if” assertion in part (e) follows
from the construction in Lemma C.3.4 on which the proof of part (b) is based.
∎

###  C.4. Affine geometry of $(R,I)$-contramodules

All rings in this section are presumed to be commutative and Noetherian. Let
$f\colon R\longrightarrow S$ be a ring homomorphism, $I$ be an ideal in $R$,
and $J$ be an ideal in $S$. We denote by $Sf(I)$ the extension of the ideal
$I\subset R$ in the ring $S$.

######  Lemma C.4.1.

(a) If $f(I)\subset J$, then any $(S,J)$-contramodule is also an
$(R,I)$-contramodule in the $R$-module structure obtained by restriction of
scalars via $f$.

(b) If $J\subset Sf(I)$, then an $S$-module is an $(S,J)$-contramodule
whenever it is an $(R,I)$-contramodule in the $R$-module structure obtained by
restriction of scalars via $f$.

###### Proof.

Part (a) holds, since an $S$-module $Q$ is an $(S,J)$-contramodule if and only
if the system of equations $q_{n}=p_{n}+tq_{n+1}$, $n\ge 0$, is uniquesly
solvable in $q_{n}\in Q$ for any fixed sequence $p_{n}\in Q$ and any $t\in J$.
Part (b) is true, because it suffices to check the previous condition for the
elements $t$ belonging to any given set of generators of the ideal $J\subset
S$ only [54, Sections B.1 and B.7]. ∎

######  Lemma C.4.2.

Assume that $J\subset Sf(I)$, and let ${\mathfrak{P}}$ be an
$(R,I)$-contramodule. Then

(a) the $S$-module $\operatorname{Hom}_{R}(S,{\mathfrak{P}})$ is an
$(S,J)$-contramodule;

(b) the $S$-module $S\otimes_{R}{\mathfrak{P}}$ is an $(S,J)$-contramodule
whenever $f$ is a finite morphism.

###### Proof.

Both assertions follow from Lemma C.4.1(b). Indeed, the $R$-module
$\operatorname{Hom}_{R}(M,{\mathfrak{P}})$ is an $(R,I)$-contramodule for any
$R$-module $M$, because the contramodule infinite summation operations can be
defined on it (see the beginning of Section C.1), while the $R$-module
$M\otimes_{R}{\mathfrak{P}}$ is an $(R,I)$-contramodule for any finitely
generated $R$-module $M$, being the cokernel of a morphism between two finite
direct sums of copies of the $(R,I)$-contramodule ${\mathfrak{P}}$. ∎

Denote by ${\mathfrak{R}}$ the $I$-adic completion of the ring $R$ and by
${\mathfrak{S}}$ the $J$-adic completion of the ring $S$, both viewed as
topological rings. By [54, Theorem B.1.1], the full subcategories
$(R,I){\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ and
$(S,J){\operatorname{\mathsf{--contra}}}\subset
S{\operatorname{\mathsf{--mod}}}$ in the categories of $R$\- and $S$-modules
are equivalent to the categories
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ and
${\mathfrak{S}}{\operatorname{\mathsf{--contra}}}$ of contramodules over the
topological rings ${\mathfrak{R}}$ and ${\mathfrak{S}}$.

Assume that $f(I)\subset J$; then the ring homomorphism $f\colon
R\longrightarrow S$ induces a continuous homomorphism of topological rings
$\phi\colon{\mathfrak{R}}\longrightarrow{\mathfrak{S}}$. According to [54,
Section 1.8], there is a pair of adjoint functors of “contrarestriction” and
“contraextension” of scalars
$R^{\phi}\colon{\mathfrak{S}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$
and $E^{\phi}\colon
R{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{S}}{\operatorname{\mathsf{--contra}}}$.
While the functor $R^{\phi}$ is easily identified with the functor of
restriction of scalars from Lemma C.4.1, the functor $E^{\phi}$, which is left
adjoint to $R^{\phi}$, is defined by the rules that $E^{\phi}$ is right exact
and takes the free ${\mathfrak{R}}$-contramodule
${\mathfrak{R}}[[X]]=\varprojlim_{n}R/I^{n}[X]$ to the free
${\mathfrak{S}}$-contramodule ${\mathfrak{S}}[[X]]=\varprojlim_{n}S/J^{n}[X]$
for any set $X$.

When $f$ is a finite morphism and $J=Sf(I)$, the functor $E^{\phi}$ is simply
the functor of extension of scalars from Lemma C.4.2(b).

######  Lemma C.4.3.

For any $R$-flat $(R,I)$-contramodule ${\mathfrak{F}}$ there is a natural
isomorphism of $(S,J)$-contramodules
$E^{\phi}({\mathfrak{F}})\simeq\varprojlim_{n}(S\otimes_{R}{\mathfrak{F}})/\allowbreak
J^{n}(S\otimes_{R}{\mathfrak{F}})$. In particular, the functor $E^{\phi}$
takes $R$-flat $(R,I)$-contramodules to $S$-flat $(S,J)$-contramodules and
very flat $(R,I)$-contramodules to very flat $(S,J)$-contramodules.

###### Proof.

Since
$F\longmapsto(S\otimes_{R}F)/J^{n}(S\otimes_{R}F)=(S\otimes_{R}F)/(J^{n}\otimes_{R}F)=S/J^{n}\otimes_{R}F$
is an exact functor on the category of flat $R$-modules $F$, so is the functor
$F\longmapsto\varprojlim_{n}(S\otimes_{R}F)/J^{n}(S\otimes_{R}F)$. Hence it
suffices to compute this functor for free ${\mathfrak{R}}$-contramodules, and
indeed we have
$S/J^{n}\otimes_{R}{\mathfrak{R}}[[X]]=S/J^{n}\otimes_{R/I^{n}}(R/I^{n}\otimes_{R}{\mathfrak{R}}[[X]])=S/J^{n}\otimes_{R/I^{n}}R/I^{n}[X]=S/J^{n}[X]$
and
$\varprojlim_{n}S/J^{n}\otimes_{R}{\mathfrak{R}}[[X]]=\varprojlim_{n}S/J^{n}[X]={\mathfrak{S}}[[X]]$,
as desired. The remaining assertions follow by the way of [54, Lemma B.9.2]
and the above Corollary C.3.5(e) (see also Lemma D.1.2 below). ∎

######  Lemma C.4.4.

(a) If the map $\bar{f}\colon R/I\longrightarrow S/J$ is surjective, then the
functor $E^{\phi}$ takes $R$-contraadjusted $(R,I)$-contramodules to
$S$-contraadjusted $(S,J)$-contramodules.

(b) If the morphism $\bar{f}\colon R/I\longrightarrow S/J$ is finite, then the
functor $E^{\phi}$ takes $R$-flat $R$-cotorsion $(R,I)$-contramodules to
$S$-flat $S$-cotorsion $(S,J)$-contramodules.

###### Proof.

Part (a): by Corollary C.2.8(b) or Corollary C.3.5(b), any $R$-contraadjusted
$(R,I)$-contramodule is a quotient $(R,I)$-contramodule of an $R$-flat
$R$-contraadjusted $(R,I)$-contramodule. The functor $E^{\phi}$ being right
exact and the class of contraadjusted $S$-modules being closed under quotents,
it suffices to show that the $S$-module $E^{\phi}({\mathfrak{P}})$ is
contraadjusted for any $R$-flat $R$-contraadjusted $(R,I)$-contramodule
${\mathfrak{P}}$. Then, by Lemma C.4.3, one has
$E^{\phi}({\mathfrak{P}})=\varprojlim_{n}E^{\phi}({\mathfrak{P}})/J^{n}E^{\phi}({\mathfrak{P}})$
and
$E^{\phi}({\mathfrak{P}})/JE^{\phi}({\mathfrak{P}})\allowbreak=S/J\otimes_{R}{\mathfrak{P}}$,
so it remains to apply Corollary C.3.2(a) together with Lemma 1.6.6(b). Part
(b) follows directly from Lemma C.4.3, Corollary C.3.2(b), and Lemma 1.6.7(a)
in the similar way. ∎

We keep assuming that $f(I)\subset J$. Let $g\colon R\longrightarrow T$ be
another ring homomorphism and $K=Tg(I)\subset T$ be the extension of the ideal
$I\subset R$ in the ring $T$. Suppose that the commutative ring
$H=S\otimes_{R}T$ is Noetherian, denote by $f^{\prime}\colon T\longrightarrow
H$ and $g^{\prime}\colon S\longrightarrow H$ the related ring homomophisms,
and set $L=Hg^{\prime}(J)\subset H$.

Let ${\mathfrak{T}}$ and ${\mathfrak{H}}$ denote the adic completions of the
rings $T$ and $H$ with respect to the ideals $K$ and $L$, and let
$\psi\colon{\mathfrak{R}}\longrightarrow{\mathfrak{T}}$,
$\phi^{\prime}\colon{\mathfrak{T}}\longrightarrow{\mathfrak{H}}$, and
$\psi^{\prime}\colon{\mathfrak{S}}\longrightarrow{\mathfrak{H}}$ be the
induced homomorphisms of topological rings.

######  Lemma C.4.5.

(a) For any $(T,K)$-contramodule ${\mathfrak{N}}$ there is a natural
isomorphism of $(S,J)$-contramodules $E^{\phi}R^{\psi}({\mathfrak{N}})\simeq
R^{\psi^{\prime}}E^{\phi^{\prime}}({\mathfrak{N}})$.

(b) Assume that the ring homomorphism $g\colon R\longrightarrow T$ induces an
open embedding of affine schemes
$\operatorname{Spec}T\longrightarrow\operatorname{Spec}R$, while the morphism
$\bar{f}\colon R/I\longrightarrow S/J$ is finite. Then for any $R$-flat
$R$-contraadjusted $(R,I)$-contramodule ${\mathfrak{P}}$ there is a natural
isomorphism of $(H,L)$-contramodules
$E^{\phi^{\prime}}(\operatorname{Hom}_{R}(T,{\mathfrak{P}}))\simeq\operatorname{Hom}_{S}(H,E^{\phi}({\mathfrak{P}}))$.

###### Proof.

Part (a): the functor of contraextension of scalars
$E^{\phi}\colon(R,I){\operatorname{\mathsf{--contra}}}\longrightarrow(S,J){\operatorname{\mathsf{--contra}}}$
is left adjoint to the functor of (contra)restriction of scalars
$R^{\phi}\colon(S,J)\allowbreak{\operatorname{\mathsf{--contra}}}\longrightarrow(R,I){\operatorname{\mathsf{--contra}}}$,
while the functor
$R^{\psi}\colon(T,K){\operatorname{\mathsf{--contra}}}\longrightarrow(R,I){\operatorname{\mathsf{--contra}}}$
is left adjoint to the functor of coextension of scalars taking an
$(R,I)$-contramodule ${\mathfrak{P}}$ to the $(T,K)$-contramodule
$\operatorname{Hom}_{R}(T,{\mathfrak{P}})$. To obtain the desired isomorphism
of functors, one can start with the obvious functorial isomorphism of
$(T,K)$-contramodules
$\operatorname{Hom}_{R}(T,R^{\phi}({\mathfrak{Q}}))\simeq
R^{\phi^{\prime}}\operatorname{Hom}_{S}(H,{\mathfrak{Q}})$ for any
$(S,J)$-contramodule ${\mathfrak{Q}}$, and then pass to the left adjoint
functors.

Part (b): the $T$-module $\operatorname{Hom}_{R}(T,{\mathfrak{P}})$ being flat
by Corollary 1.6.5(a), one has
$E^{\phi^{\prime}}(\operatorname{Hom}_{R}(T,{\mathfrak{P}}))\simeq\varprojlim_{n}\mskip
1.5muH/L^{n}\otimes_{T}\operatorname{Hom}_{R}(T,{\mathfrak{P}})\simeq\varprojlim_{n}\mskip
1.5muS/J^{n}\otimes_{R}\operatorname{Hom}_{R}(T,{\mathfrak{P}})$ and
$\operatorname{Hom}_{S}(H,E^{\phi}({\mathfrak{P}}))\simeq\operatorname{Hom}_{R}(T,E^{\phi}({\mathfrak{P}}))\simeq\operatorname{Hom}_{R}(T,\>\varprojlim_{n}\mskip
1.5muS/J^{n}\otimes_{R}{\mathfrak{P}})\simeq\varprojlim_{n}\operatorname{Hom}_{R}(T,\>S/J^{n}\otimes_{R}{\mathfrak{P}})$.
Finally, one has
$S/J^{n}\otimes_{R}\operatorname{Hom}_{R}(T,{\mathfrak{P}})\simeq\operatorname{Hom}_{R}(T,\>S/J^{n}\allowbreak\otimes_{R}{\mathfrak{P}})$
by Corollary 1.6.3(c), since the $R$-modules $S/J^{n}$ are finitely generated.
∎

######  Lemma C.4.6.

Let $R\longrightarrow S_{\alpha}$ be a collection of morphisms of commutative
rings for which the corresponding collection of morphisms of affine schemes
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is a finite
open covering. Let $I$ be an ideal in $R$ and $J_{\alpha}$ be its extensions
in $S_{\alpha}$. Then a contraadjusted $R$-module $P$ is an
$(R,I)$-contramodule if and only if the $S_{\alpha}$-modules
$\operatorname{Hom}_{R}(S_{\alpha},P)$ are
$(S_{\alpha},J_{\alpha})$-contramodules for all $\alpha$.

###### Proof.

The “only if” is a particular case of Lemma C.4.2(a), and to prove the “if”
one can use the Čech sequence (2) together with Lemma C.4.1(a) and the facts
that the class of $(R,I)$-contramodules is preserved by the functors
$\operatorname{Hom}_{R}$ from any $R$-module as well as the passages to the
cokernels of $R$-module morphisms (and, actually, kernels and extensions,
too). ∎

###  C.5. Noncommutative Noetherian rings

The aim of this section is to generalize the main results of [54, Appendix B]
and the above Section C.1 to noncommutative rings. Our exposition is somewhat
sketchy with details of the arguments omitted when they are essentially the
same as in the commutative case. For a further generalization, see Sections
D.1–D.2 below.

Let $R$ be a right Noetherian associative ring, and let $m\subset R$ be an
ideal generated by central elements in $R$. Denote by
${\mathfrak{R}}=\varprojlim_{n}R/m^{n}$ the $m$-adic completion of the ring
$R$, viewed as a topological ring in the projective limit ($=$ $m$-adic)
topology. We refer to [50, Remark A.3] and [54, Section 1.2] for the
definitions of the abelian category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ of _left
${\mathfrak{R}}$-contramodules_ and the forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ (see also the beginning of Section D.1).

######  Theorem C.5.1.

The forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is fully faithful. If $s_{1}$, …, $s_{k}$
are some set of central generators of the ideal $m\subset R$, then the image
of the forgetful functor consists precisely of those left $R$-modules $P$ for
which one has $\operatorname{Ext}^{*}_{R}(R[s_{j}^{-1}],P)=0$ for all $1\le
j\le k$.

In other words, extending the terminology of Section C.2 to the noncommutative
situation, one can say that a left $R$-module is an
${\mathfrak{R}}$-contramodule if and only if it is an $s_{j}$-contramodule for
every $j$.

###### Proof.

The argument follows the proof of [54, Theorem B.1.1]. To show that
$\operatorname{Ext}_{R}^{*}(R[s^{-1}],{\mathfrak{P}})=0$ for any left
${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, one can simply
(contra)restrict the scalars to the subring $K\subset R$ generated by $s$ over
${\mathbb{Z}}$ in $R$, completed adically at the ideal $m\cap K$ or
$(s)\subset K$, reducing to the commutative case, where the quoted theorem
from [54, Appendix B] is directly applicable (cf. [54, Section B.2]).

To prove the fully faithfullness and the sufficiency of condition on an
$R$-module $P$, consider the ring of formal power series
${\mathfrak{T}}=R[[t_{1},\dotsc,t_{k}]]$ with central variables $t_{j}$ and
endow it with the adic topology of the ideal generated by $t_{1}$ …, $t_{k}$.
There is a natural surjective open homomorphism of topological rings
${\mathfrak{T}}\longrightarrow{\mathfrak{R}}$ forming a commutative diagram
with the ring homomorphisms $R\longrightarrow{\mathfrak{T}}$ and
$R\longrightarrow{\mathfrak{R}}$ and taking $t_{j}$ to $s_{j}$. Consider also
the polynomial ring $T=R[t_{1},\dotsc,t_{k}]$; there are natural ring
homomorphisms $R\longrightarrow T\longrightarrow{\mathfrak{T}}$. The argument
is based on two lemmas.

######  Lemma C.5.2.

The kernel ${\mathfrak{J}}$ of the ring homomorphism
${\mathfrak{T}}\longrightarrow{\mathfrak{R}}$ is generated by the central
elements $t_{j}-s_{j}$ as an ideal in an abstract, nontopological ring
${\mathfrak{T}}$. Moreover, any family of elements in ${\mathfrak{J}}$
converging to zero in the topology of ${\mathfrak{T}}$ can be presented as a
linear combination of $k$ families of elements in ${\mathfrak{T}}$, converging
to zero in the topology of ${\mathfrak{T}}$, with the coefficients
$t_{j}-s_{j}$.

###### Proof.

The proof is similar to that in [54, Sections B.3–B.4]; the only difference is
that one has to use the noncommutative versions of Hilbert basis theorem [23,
Theorem 1.9 and Exercise 1ZA(c)] and Artin–Rees lemma [23, Theorem 13.3]. ∎

######  Lemma C.5.3.

The forgetful functor
${\mathfrak{T}}{\operatorname{\mathsf{--contra}}}\longrightarrow
T{\operatorname{\mathsf{--mod}}}$ identifies the category of left
contramodules over the topological ring ${\mathfrak{T}}$ with the full
subcategory in the category of left $T$-modules consisting of all those
modules which are $t_{j}$-contramodules for all $j$.

###### Proof.

This assertion is true for any associative ring $R$; the argument is the same
as in [54, Sections B.5–B.6]. ∎

The proof of the theorem finishes similarly to [54, Section B.7]. The category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ is identified with the full
subcategory in ${\mathfrak{T}}{\operatorname{\mathsf{--contra}}}$ consisting
of those left ${\mathfrak{T}}$-contramodules ${\mathfrak{P}}$ in which the
elements $t_{j}-s_{j}$ act by zero. The latter category coincides with the
category of $T$-modules in which the variables $t_{j}$ act the same as the
elements $s_{j}$ and which are also $t_{j}$-contramodules for all $j$. ∎

######  Proposition C.5.4.

A left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ is a flat $R$-module if
and only if the $R/m^{n}$-module ${\mathfrak{P}}/m^{n}{\mathfrak{P}}$ is flat
for every $n\ge 1$. The natural map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/m^{n}{\mathfrak{P}}$
is an isomorphism in this case.

###### Proof.

The same as in [54, Lemma B.9.2]. One computes the functor $M\longmapsto
M\otimes_{R}{\mathfrak{P}}$ on the category of finitely generated right
$R$-modules $M$ and uses (the noncommutative version of) the Artin–Rees lemma
for such modules $M$. ∎

######  Proposition C.5.5.

For any flat $R$-module $F$ such that the $R/m$-module $F/mF$ is projective,
and any $R$-contramodule ${\mathfrak{Q}}$, one has
$\operatorname{Ext}_{R}^{>0}(F,{\mathfrak{Q}})=0$.

###### Proof.

This assertion, provable in the same way as [54, Proposition B.10.1] (see also
Corollary D.1.8(b) below), does not depend on any Noetherianity assumptions. ∎

######  Corollary C.5.6.

(a) A left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is projective if and
only if it is a flat $R$-module and the $R/m$-module
${\mathfrak{F}}/m{\mathfrak{F}}$ is projective.

(b) The forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces isomorphisms of all the Ext groups.

###### Proof.

To prove the “only if” assertion in part (a), it suffices to notice the
isomorphism of $R/m^{n}$-modules
${\mathfrak{R}}[[X]]/m^{n}{\mathfrak{R}}[[X]]\simeq{\mathfrak{R}}/m^{n}[X]$,
which holds for any set $X$ and any $n\ge 1$, and use Proposition C.5.4. The
“if” follows from the fully faithfulness assertion in Theorem C.5.1 and
Proposition C.5.5, and part (b) follows from the same together with part (a).
(Cf. [54, Section B.8] and Section D.1 below.) ∎

######  Corollary C.5.7.

(a) The contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathfrak{R}}{\operatorname{\mathsf{--contra}}})$
of the abelian category of left ${\mathfrak{R}}$-contramodules is equivalent
to the contraderived category of the exact category of $R$-flat left
${\mathfrak{R}}$-contramodules.

(b) Assume that all flat left $R/m$-modules have finite projective dimensions.
Then the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathfrak{R}}{\operatorname{\mathsf{--contra}}})$
is equivalent to the absolute derived category of the exact category of
$R$-flat left ${\mathfrak{R}}$-contramodules.

(c) Assume that all flat left $R/m$-modules have finite projective dimensions.
Then the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathfrak{R}}{\operatorname{\mathsf{--contra}}})$
is equivalent to the homotopy category of complexes of projective left/
${\mathfrak{R}}$-contramodules.

###### Proof.

Clearly, the class of flat left $R$-modules is closed under infinite products
in our assumptions, so it remains to notice that projective left
${\mathfrak{R}}$-contramdodules are $R$-flat and, in the assumption of parts
(b-c), any $R$-flat left ${\mathfrak{R}}$-contramodule has finite projective
dimension in ${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$. The latter
assertions follow from Corollary C.5.6(a). (Cf. Theorems C.1.1 and D.2.1.) ∎

Let $K$ be a commutative ring, $I\subset K$ be a finitely generated ideal, and
$A$ and $B$ be associative $K$-algebras. For any $K$-module $L$, we will
denote by ${}_{I}L\subset L$ the submodule of elements annihilated by $I$ in
$L$.

######  Lemma C.5.8.

(a) For any left $A$-module $M$ and injective left $A$-module $J$ there is a
natural isomorphism of $K/I$-modules
$\operatorname{Hom}_{A}(M,J)/I\operatorname{Hom}_{A}(M,J)\simeq\operatorname{Hom}_{A/IA}({}_{I}M,{}_{I}J)$.

(b) For any right $B$-module $N$ and flat left $B$-module $F$ there is a
natural isomorphism of $K/I$-modules
${}_{I}(N\otimes_{B}F)\simeq{}_{I}N\otimes_{B/IB}F/IF$.

###### Proof.

For any finitely presented $K$-module $E$ there are natural isomorphisms
$\operatorname{Hom}_{A}(\operatorname{Hom}_{K}(E,M),J)\simeq
E\otimes_{K}\operatorname{Hom}_{A}(M,J)$ and
$\operatorname{Hom}_{K}(E,\>N\otimes_{B}F)\simeq\operatorname{Hom}_{K}(E,\allowbreak
N)\otimes_{B}F$. So it remains to take $E=K/I$. (Cf. Lemmas C.1.2 and D.2.3.)
∎

Let $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a finite complex of
$A$-injective and $B$-injective $A$-$B$-bimodules over $K$ (i. e., it is
presumed that the left action of $A$ and the right action of $B$ restrict to
one and the same action of $K$ in
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$). We are using the
definition of a dualizing complex for a pair of noncommutative rings from
Section B.4.

######  Lemma C.5.9.

(a) Assume that the ring $A$ is left coherent and the ring $B$ is right
coherent. Then ${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
dualizing complex for the rings $A/IA$ and $B/IB$ whenever
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is dualizing complex the
rings $A$ and $B$.

(b) Assume that the ideal $I$ is nilpotent, the ring $A$ is left Noetherian,
and the ring $B$ is right Noetherian. Then
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a dualizing complex for
the rings $A$ and $B$ whenever
${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a dualizing
complex for the rings $A/IA$ and $B/IB$.

###### Proof.

Part (a): clearly, ${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
a complex of injective left $A/IA$-modules. The homothety map
$B\longrightarrow\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism of finite complexes of flat left $B$-modules (see Lemma
1.6.1(b)) and therefore remains a quasi-isomorphism after taking the tensor
product with $B/IB$ over $B$ on the left, i. e., reducing modulo $I$. By Lemma
C.5.8(a), it follows that the map
$B/IB\longrightarrow\operatorname{Hom}_{A/IA}({}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism.

A bounded above complex of left modules over a left coherent ring is quasi-
isomorphic to a bounded above complex of finitely generated projective modules
if and only if its cohomology modules are finitely presented. To show that the
$A/IA$-modules of cohomology of the complex
${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are finitely
presented, we notice that the complex of left $A/IA$-modules
${}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is quasi-isomorphic
to the complex
$\operatorname{Hom}_{B^{\mathrm{op}}}(L^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of right $B$-module homomorphisms from a left resolution
$L^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the right $B$-module
$B/IB$ by finitely generated projective $B$-modules into the complex
$D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Locally in the
cohomological grading, the complex
$\operatorname{Hom}_{B^{\mathrm{op}}}(L^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a finitely iterated cone of morphisms between shifts of copies of the
complex $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, so its $A$-modules
of cohomology are finitely presented. It remains to point out that an
$A/IA$-module is finitely presented if and only if it is finitely presented as
an $A$-module.

Part (b): whenever the ideal $I$ is nilpotent, a morphism of finite complexes
of flat $B$-modules is a quasi-isomorphism if and only if it becomes one after
taking the tensor products with $B/IB$ over $B$. This, together with the above
argument, proves that the homothety map
$B\longrightarrow\operatorname{Hom}_{A}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}},D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism. To show that the $A$-modules of cohomology of the
complex $D^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are finitely
generated, one can consider the spectral sequence
$E_{2}^{pq}=\operatorname{Ext}^{p}_{B^{\mathrm{op}}}(B/IB,H^{q}D^{\text{\smaller\smaller$\scriptstyle\bullet$}})\Longrightarrow
H^{p+q}\operatorname{Hom}_{B^{\mathrm{op}}}(B/IB,D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
converging from the finite direct sums of copies of the $A/I$-modules
${}_{I}H^{q}(D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$ (on the page
$E_{1}$) to the cohomology $A/IA$-modules
$H^{n}({}_{I}D^{\text{\smaller\smaller$\scriptstyle\bullet$}})$. Then one
argues by increasing induction in $q$, using the fact that an $A$-module $M$
is finitely generated provided that so is the $A/IA$-module ${}_{I}M$. (Cf.
Lemmas C.1.3 and D.2.4.) ∎

Let $K$ be a commutative Noetherian ring, $I\subset K$ be an ideal, and $A$
and $B$ be associative $K$-algebras such that the ring $A$ is left Noetherian
and the ring $B$ is right Noetherian. A finite complex of $A$-$B$-bimodules
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over $K$ is
said to be a _dualizing complex for $A$ and $B$ over $(K,I)$_ if

  1. (i)

${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a complex of
injective left $A$-modules and a complex of injective right $B$-modules;

  2. (ii)

any element in ${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is annihilated by some power of the ideal $I$;

  3. (iii)

for any (or, equivalently, some) $n\ge 1$, the complex
${}_{(n)}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}={}_{I^{n}}{\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a dualizing complex for the rings $A/I^{n}A$ and $B/I^{n}B$ (cf. Lemma
C.5.9).

A $K$-module (or $A$-module) is said to be _$I$ -torsion_ if every its element
is annihilated by some power of the ideal $I\subset K$. A _left
$(B,I)$-contramodule_ is a left contramodule over the $I$-adic completion of
the ring $B$, or, equivalently, a left $B$-module that is a
$(K,I)$-contramodule in the $K$-module structure obtained by restriction of
scalars (see Theorem C.5.1 and Section C.1).

We denote the abelian category of $I$-torsion left $A$-modules by
$(A,I){\operatorname{\mathsf{--tors}}}$ and the abelian category of left
$(B,I)$-contramodules by $(B,I){\operatorname{\mathsf{--contra}}}$.

######  Theorem C.5.10.

The choice of a dualizing complex
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for the rings
$A$ and $B$ over $(K,I)$ induces an equivalence between the coderived category
of $I$-torsion left $A$-modules
${\mathsf{D}}^{\mathsf{co}}((A,I){\operatorname{\mathsf{--tors}}})$ and the
contraderived category of left $(B,I)$-contramodules
${\mathsf{D}}^{\mathsf{ctr}}((B,I){\operatorname{\mathsf{--contra}}})$. The
equivalence is provided by the derived functors
${\mathbb{R}}\operatorname{Hom}_{A}({\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{B}^{\mathbb{L}}{-}$.

###### Proof.

Notice first of all that an object
${\mathcal{J}}\in(A,I){\operatorname{\mathsf{--tors}}}$ is injective if and
only if the $A/I^{n}A$ modules ${}_{(n)}{\mathcal{J}}={}_{I^{n}}{\mathcal{J}}$
are injective for all $n$ (obviously), and if and only if it is an injective
$A$-module (by the Artin–Rees lemma for centrally generated ideals in
noncommutative Noetherian rings [23, Theorem 13.3]). The dual result for
$B$-flat $(B,I)$-contramodules is provided by Proposition C.5.4.

Furthermore, we identify the coderived category
${\mathsf{D}}^{\mathsf{co}}((A,I){\operatorname{\mathsf{--tors}}})$ with the
homotopy category of (complexes of) injective $I$-torsion $A$-modules (based
on the facts that there are enough injectives and the class of injectives is
closed with respect to infinite direct sums in
$(A,I){\operatorname{\mathsf{--tors}}}$) and the contraderived category
${\mathsf{D}}^{\mathsf{ctr}}((B,I){\operatorname{\mathsf{--contra}}})$ with
the absolute derived category of $B$-flat $(B,I)$-contramodules (based on
Corollary C.5.7(b) and Lemma B.4.2). The rest of the argument is no different
from the proof of Theorem C.1.4 (see also Lemma B.4.1) and based on Lemma
C.5.8. ∎

##  Appendix D Ind-Affine Ind-Schemes

The aim of this appendix is to lay some bits of preparatory groundwork for the
definition of contraherent cosheaves of contramodules on ind-schemes. We
construct enough very flat and contraadjusted contramodules on an ind-affine
ind-scheme represented by a sequence of affine schemes and their closed
embeddings with finitely generated defining ideals, and also enough cotorsion
contramodules on an ind-Noetherian ind-affine ind-scheme of totally finite
Krull dimension. A version of co-contra correspondence for ind-Noetherian ind-
affine ind-schemes with dualizing complexes (and their noncommutative
generalizations) is also worked out.

###  D.1. Flat and projective contramodules

Let $R_{0}\longleftarrow R_{1}\longleftarrow R_{2}\longleftarrow
R_{3}\longleftarrow\dotsb$ be a projective system of associative rings,
indexed by the ordered set of positive integers, with surjective morphisms
between them. Denote by ${\mathfrak{R}}$ the projective limit
$\varprojlim_{n}R_{n}$, viewed as a topological ring in the topology of
projective limit of discrete rings $R_{n}$. Clearly, the ring homomorphisms
${\mathfrak{R}}\longrightarrow R_{n}$ are surjective; let
${\mathfrak{I}}_{n}\subset{\mathfrak{R}}$ denote their kernels. Then the open
ideals ${\mathfrak{I}}_{n}$ form a base of neighborhoods of zero in the
topological ring ${\mathfrak{R}}$.

We are interested in left contramodules over the topological ring
${\mathfrak{R}}$ (see [50, Remark A.3] and [54, Section 1.2] for the
definition). They form an abelian category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$ having enough projective
objects and endowed with an exact and faithful forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$
preserving infinite products. Here
${\mathfrak{R}}{\operatorname{\mathsf{--mod}}}$ denotes the abelian category
of left modules over the ring ${\mathfrak{R}}$ (viewed as an abstract ring
without any topology).

The projective ${\mathfrak{R}}$-contramodules are precisely the direct
summands of the free ${\mathfrak{R}}$-contramodules ${\mathfrak{R}}[[X]]$.
Here $X$ is an arbitrary set of generators, and
${\mathfrak{R}}[[X]]=\varprojlim_{n}R_{n}[X]$ is the set of all maps
$X\longrightarrow{\mathfrak{R}}$ converging to zero in the topology of
${\mathfrak{R}}$.

For any left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ and any closed
two-sided ideal ${\mathfrak{J}}\subset{\mathfrak{R}}$, we will denote by
${\mathfrak{J}}{\times}{\mathfrak{P}}\subset{\mathfrak{P}}$ the image of the
contraaction map
${\mathfrak{J}}[[{\mathfrak{P}}]]\longrightarrow{\mathfrak{P}}$. As usually,
for any left module $M$ over a ring $R$ and any ideal $J\subset R$ the
notation $JM$ (or, as it may be sometimes convenient, $J\mskip 1.5mu{\cdot}M$)
stands for the image of the action map $J\otimes_{R}M\longrightarrow M$. For
an ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, there is a (generally
speaking, proper) inclusion ${\mathfrak{J}}{\mathfrak{P}}={\mathfrak{J}}\mskip
1.5mu{\cdot}\mskip
1.5mu{\mathfrak{P}}\subset{\mathfrak{J}}{\times}{\mathfrak{P}}$. Of course,
${\mathfrak{J}}{\times}{\mathfrak{P}}$ is an ${\mathfrak{R}}$-subcontramodule
in ${\mathfrak{P}}$, while ${\mathfrak{J}}{\mathfrak{P}}$ is in general only
an ${\mathfrak{R}}$-submodule.

######  Lemma D.1.1.

For any left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, the natural map
to the projective limit
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/({\mathfrak{I}}_{n}{\times}{\mathfrak{P}})$
is surjective.

###### Proof.

This assertion is true for any topological ring ${\mathfrak{R}}$ with a base
of topology consisting of open right ideals, and any decreasing sequence of
closed abelian subgroups
${\mathfrak{R}}\supset{\mathfrak{I}}_{0}\supset{\mathfrak{I}}_{1}\supset{\mathfrak{I}}_{2}\supset\dotsb$
converging to zero in the topology of ${\mathfrak{R}}$ (meaning that for any
neighborhood of zero ${\mathfrak{U}}\subset{\mathfrak{R}}$ there exists $n\ge
0$ such that ${\mathfrak{I}}_{n}\subset{\mathfrak{U}}$). Indeed, it suffices
to show that for any sequence of elements
$p_{n}\in{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$, $n\ge 0$, there exists an
element $p\in{\mathfrak{P}}$ such that
$p-(p_{0}+\dotsb+p_{n})\in{\mathfrak{I}}_{n+1}{\times}{\mathfrak{P}}$ for all
$n\ge 0$.

Now each element $p_{n}$ can be obtained as an infinite linear combination of
elements of ${\mathfrak{P}}$ with the family of coefficients converging to
zero in ${\mathfrak{I}}_{n}$. The countable sum of such expressions over all
$n\ge 0$ is again an infinite linear combination of elements of
${\mathfrak{P}}$ with the coefficients converging to zero in ${\mathfrak{R}}$.
The value of the latter, understood in the sense of the contramodule infinite
summation operations in ${\mathfrak{P}}$, provides the desired element
$p\in{\mathfrak{P}}$. Another (and perhaps more illuminating) argument can be
found in [50, Lemma A.2.3] (while counterexamples showing that the map in
question may not be injective are provided in [50, Section A.1]). ∎

######  Lemma D.1.2.

Let $P_{0}\longleftarrow P_{1}\longleftarrow P_{2}\longleftarrow\dotsb$ be a
projective system of left $R_{n}$-modules in which the morphism
$P_{n+1}\longrightarrow P_{n}$ identifies $P_{n}$ with
$R_{n}\otimes_{R_{n+1}}P_{n+1}$ for every $n\ge 0$. Let ${\mathfrak{P}}$
denote the ${\mathfrak{R}}$-contramodule $\varprojlim_{n}P_{n}$. Then the
natural map ${\mathfrak{P}}\longrightarrow P_{n}$ identifies $P_{n}$ with
${\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$. Conversely, for any
left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ the projective system
$P_{n}={\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$ satisfies the
above condition.

###### Proof.

Since $P_{n}$ is an $R_{n}$-module and ${\mathfrak{P}}\longrightarrow P_{n}$
is a morphism of ${\mathfrak{R}}$-contramodules, the kernel of this morphism
contains ${\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$. To prove the inverse
inclusion, consider an element $p\in{\mathfrak{P}}$ belonging to the kernel of
the morphism ${\mathfrak{P}}\longrightarrow P_{n}$. The image of the element
$p$ in $P_{n+1}$ belongs to $({\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1})P_{n+1}$
(where ${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}$ is an ideal in $R_{n+1}$).
Let us decompose this image accordingy into a finite linear combination of
elements of $P_{n+1}$ with coefficients from
${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}$, lift all the entering elements of
$P_{n+1}$ to ${\mathfrak{P}}$ and all the elements of
${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}$ to ${\mathfrak{I}}_{n}$, and
subtract from $p$ the corresponding finite linear combination of elements of
${\mathfrak{P}}$ with coefficients in ${\mathfrak{I}}_{n}$.

The image of the resulting element $p^{\prime}\in{\mathfrak{P}}$ in $P_{n+1}$
vanishes, so its image in $P_{n+2}$ belongs to
$({\mathfrak{I}}_{n+1}/{\mathfrak{I}}_{n+2})P_{n+2}$. Continuing this process
indefinitely, we obtain an expression of the original element $p$ in the form
of a countable linear combination of elements from ${\mathfrak{P}}$ with the
coefficient sequence converging to zero in ${\mathfrak{I}}_{n}$. This proves
the first assertion; the second one is straightforward. ∎

We will say that a left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is
_flat_ if the map
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
is an isomorphism and the $R_{n}$-modules
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ are flat for all
$n$.

######  Lemma D.1.3.

If ${\mathfrak{G}}\longrightarrow{\mathfrak{F}}$ is a surjective morphism of
flat ${\mathfrak{R}}$-contramodules then its kernel ${\mathfrak{H}}$ is also a
flat ${\mathfrak{R}}$-contramodule. Moreover, the sequences of $R_{n}$-modules
$0\longrightarrow{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}\longrightarrow{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}\longrightarrow
0$ are exact.

###### Proof.

Clearly, for any short exact sequence of ${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ there are short exact sequences of $R_{n}$-modules
$0\longrightarrow{\mathfrak{H}}/{\mathfrak{H}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{G}})\longrightarrow{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}\longrightarrow
0$ (because the map
${\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
induced by a surjective map ${\mathfrak{G}}\longrightarrow{\mathfrak{F}}$ is
surjective). If the $R_{n}$-module
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ is flat, then the
tensor product with $R_{n-1}$ over $R_{n}$ transforms this sequence into the
similar sequence corresponding to the ideal
${\mathfrak{I}}_{n-1}\subset{\mathfrak{R}}$.

On the other hand, if the maps
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
and
${\mathfrak{G}}\longrightarrow\varprojlim_{n}{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$
are isomorphisms, then the passage to the projective limit of the above short
exact sequences allows to conclude that the map
${\mathfrak{H}}\longrightarrow\varprojlim_{n}{\mathfrak{H}}/{\mathfrak{H}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{G}})$
is an isomorphism. By Lemma D.1.2, it follows from these observations that
${\mathfrak{H}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{G}})={\mathfrak{I}}_{n}{\times}{\mathfrak{H}}$,
so the sequences
$0\longrightarrow{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}\longrightarrow{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}\longrightarrow
0$ are exact. Finally, now if the $R_{n}$-modules
${\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$ are also flat, then
so are the $R_{n}$-modules
${\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}$. ∎

######  Lemma D.1.4.

If
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ is a short exact sequence of ${\mathfrak{R}}$-contramodules and the
${\mathfrak{R}}$-contramodules ${\mathfrak{H}}$ and ${\mathfrak{F}}$ are flat,
then so is the ${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$.

###### Proof.

In view of the proof of Lemma D.1.3, we only need to show that the map
${\mathfrak{G}}\longrightarrow\varprojlim_{n}{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$
is an isomorphism. Choose a termwise surjective map onto the short exact
sequence
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ from a short exact sequence of free ${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{W}}\longrightarrow{\mathfrak{V}}\longrightarrow{\mathfrak{U}}\longrightarrow
0$ (e. g., ${\mathfrak{W}}={\mathfrak{R}}[[{\mathfrak{H}}]]$,
${\mathfrak{U}}={\mathfrak{R}}[[{\mathfrak{F}}]]$ or
${\mathfrak{R}}[[{\mathfrak{G}}]]$, and
${\mathfrak{V}}={\mathfrak{W}}\oplus{\mathfrak{U}}$). Let
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{L}}\longrightarrow{\mathfrak{K}}\longrightarrow
0$ be the corresponding short exact sequence of kernels.

Passing to the projective limit of short exact sequences
$0\longrightarrow{\mathfrak{L}}/{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow{\mathfrak{V}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\longrightarrow{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow
0$, we obtain a short exact sequence
$0\longrightarrow\varprojlim_{n}{\mathfrak{L}}/{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow\varprojlim_{n}{\mathfrak{V}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\longrightarrow\varprojlim{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}\longrightarrow
0$. The ${\mathfrak{R}}$-contramodule ${\mathfrak{V}}$ being free, the
intersection
$\bigcap_{n}{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\subset{\mathfrak{V}}$
vanishes; so both maps
${\mathfrak{V}}\longrightarrow\varprojlim_{n}{\mathfrak{V}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}$
and
${\mathfrak{L}}\longrightarrow\varprojlim_{n}{\mathfrak{L}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{L}}$
are isomorphisms, while the map
${\mathfrak{L}}\longrightarrow\varprojlim_{n}{\mathfrak{L}}/{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})$
is, at least, injective. It remains to show that the latter map is surjective;
equivalently, it means vanishing of the derived projective limit
$\varprojlim^{1}_{n}{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})$.

Similarly, the assumptions that the maps
${\mathfrak{H}}\longrightarrow\varprojlim_{n}{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}$
and
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
are isomorphisms are equivalently expressed as the vanishing of derived
projective limits
$\varprojlim^{1}_{n}{\mathfrak{M}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{W}})$
and
$\varprojlim^{1}_{n}{\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})$.

The short sequences
$0\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{W}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{U}}\longrightarrow
0$ are exact, because the short sequence
$0\longrightarrow{\mathfrak{W}}\longrightarrow{\mathfrak{V}}\longrightarrow{\mathfrak{U}}\longrightarrow
0$ splits. Passing to the fibered product of two short exact sequences
$0\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{W}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{V}}\longrightarrow{\mathfrak{I}}_{n}{\times}{\mathfrak{U}}\longrightarrow
0$ and
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{L}}\longrightarrow{\mathfrak{K}}\longrightarrow
0$ over the short exact sequence
$0\longrightarrow{\mathfrak{W}}\longrightarrow{\mathfrak{V}}\longrightarrow{\mathfrak{U}}\longrightarrow
0$ (into which both of them are embedded), we obtain an exact sequence
$0\longrightarrow{\mathfrak{M}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{W}})\longrightarrow{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow{\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})$.

The ${\mathfrak{R}}$-contramodules ${\mathfrak{U}}$ and ${\mathfrak{F}}$ being
flat, by Lemma D.1.3 we have
${\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})={\mathfrak{I}}_{n}{\times}{\mathfrak{K}}$.
It follows that the map
${\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow{\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})$
is surjective, so the whole sequence
$0\longrightarrow{\mathfrak{M}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{W}})\longrightarrow{\mathfrak{L}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{V}})\longrightarrow{\mathfrak{K}}\cap({\mathfrak{I}}_{n}{\times}{\mathfrak{U}})\longrightarrow
0$ is exact. Applying the right exact functor $\varinjlim^{1}_{n}$ (which is,
in particular, exact in the middle), we obtain the desired vanishing. ∎

######  Lemma D.1.5.

Any projective ${\mathfrak{R}}$-contramodule is flat.

###### Proof.

It suffices to consider the case of a free left ${\mathfrak{R}}$-contramodule
${\mathfrak{R}}[[X]]$. For any set $X$ and any closed ideal
${\mathfrak{J}}\subset{\mathfrak{R}}$ one has
${\mathfrak{J}}{\times}({\mathfrak{R}}[[X]])={\mathfrak{J}}[[X]]$ and
${\mathfrak{R}}[[X]]/{\mathfrak{J}}[[X]]\simeq({\mathfrak{R}}/{\mathfrak{J}})[[X]]$.
So in particular
${\mathfrak{R}}[[X]]/({\mathfrak{I}}_{n}{\times}{\mathfrak{R}}[[X]])=R_{n}[X]$
is a flat (and even free and projective) left $R_{n}$-module and the natural
${\mathfrak{R}}$-contramodule morphism
${\mathfrak{R}}[[X]]\longrightarrow\varprojlim_{n}{\mathfrak{R}}[[X]]/({\mathfrak{I}}_{n}{\times}{\mathfrak{R}}[[X]])$
is an isomorphism. ∎

Given two ${\mathfrak{R}}$-contramodules ${\mathfrak{P}}$ and
${\mathfrak{Q}}$, we will denote by
$\operatorname{Hom}^{\mathfrak{R}}({\mathfrak{P}},{\mathfrak{Q}})$ and
$\operatorname{Ext}^{{\mathfrak{R}},*}({\mathfrak{P}},{\mathfrak{Q}})$ the
$\operatorname{Hom}$ and $\operatorname{Ext}$ groups in the abelian category
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}$.

######  Lemma D.1.6.

Let ${\mathfrak{F}}$ be a flat left ${\mathfrak{R}}$-contramodule and
${\mathfrak{P}}$ be a left ${\mathfrak{R}}$-contramodule for which the natural
map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is an isomorphism. Set
$F_{n}={\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ and
$P_{n}={\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$, and assume
that one has
$\operatorname{Ext}^{1}_{R_{0}}(F_{0},P_{0})=0=\operatorname{Ext}^{1}_{R_{n+1}}(F_{n+1},\>\allowbreak\ker(P_{n+1}\to
P_{n}))$ for all $n\ge 0$. Then
$\operatorname{Ext}^{{\mathfrak{R}},1}({\mathfrak{F}},{\mathfrak{P}})=0$.

###### Proof.

Let ${\mathfrak{G}}\longrightarrow{\mathfrak{F}}$ be a surjective morphism
onto ${\mathfrak{F}}$ from a projective ${\mathfrak{R}}$-contramodule
${\mathfrak{G}}$ with the kernel ${\mathfrak{H}}$. Set
$G_{n}={\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$ and similarly
for $H_{n}$. By Lemmas D.1.5 and D.1.3, the ${\mathfrak{R}}$-contramodules
${\mathfrak{G}}$ and ${\mathfrak{H}}$ are flat, and the short sequences
$0\longrightarrow H_{n}\longrightarrow G_{n}\longrightarrow
F_{n}\longrightarrow 0$ are exact. Let us show that any
${\mathfrak{R}}$-contramodule morphism
${\mathfrak{H}}\longrightarrow{\mathfrak{P}}$ can be extended to an
${\mathfrak{R}}$-contramodule morphism
${\mathfrak{G}}\longrightarrow{\mathfrak{P}}$.

The data of a morphism of ${\mathfrak{R}}$-contramodules
${\mathfrak{H}}\longrightarrow{\mathfrak{P}}$ is equivalent to that of a
morphism of projective systems of $R_{n}$-modules $H_{n}\longrightarrow
P_{n}$. Let us construct by induction an extension of this morphism to a
morphism of projective systems $G_{n}\longrightarrow P_{n}$ for $n\ge 0$.
Since $\operatorname{Ext}_{R_{0}}^{1}(F_{0},P_{0})=0$, the case of $n=0$ is
clear. Assuming that the morphism $G_{n}\longrightarrow P_{n}$ has been
obtained already, we will proceed to construct a compatible morphism
$G_{n+1}\longrightarrow P_{n+1}$.

Since $G_{n+1}$ is a projective $R_{n+1}$-module, the composition
$G_{n+1}\longrightarrow G_{n}\longrightarrow P_{n}$ can be lifted to an
$R_{n+1}$-module morphism $G_{n+1}\longrightarrow P_{n+1}$. (Notice that what
is actually used here is the vanishing of
$\operatorname{Ext}^{1}_{R_{n+1}}(G_{n+1},\>\ker(P_{n+1}\to P_{n}))$.) The
composition of a morphism so obtained with the embedding
$H_{n+1}\longrightarrow G_{n+1}$ differs from the given map
$H_{n+1}\longrightarrow P_{n+1}$ by an $R_{n+1}$-module morphism
$H_{n+1}\longrightarrow\ker(P_{n+1}\to P_{n})$. Given that
$\operatorname{Ext}^{1}_{R_{n+1}}(F_{n+1},\>\ker(P_{n+1}\to P_{n}))=0$, the
latter map can be extended from $H_{n+1}$ to $G_{n+1}$ and added to the
previously constructed map $G_{n+1}\longrightarrow P_{n+1}$. ∎

Suppose $R$ is an associative ring endowed with a descending sequence of two-
sided ideals $R\supset I_{0}\supset I_{1}\supset I_{2}\supset\dotsb$ such that
the projective system of quotient rings $R/I_{n}$ is isomorphic to our
original projective system $R_{0}\longleftarrow R_{1}\longleftarrow
R_{2}\longleftarrow\dotsb$. For example, one can always take
$R={\mathfrak{R}}$ and $I_{n}={\mathfrak{I}}_{n}$; sometimes there may be
other suitable choices of a ring $R$ with the ideals $I_{n}$ as well.

Then there is a natural ring homomorphism $R\longrightarrow{\mathfrak{R}}$
inducing the isomorphisms $R/I_{n}\simeq{\mathfrak{R}}/{\mathfrak{I}}_{n}$.
The restriction of scalars provides a forgetful functor
${\mathfrak{R}}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$, which is exact and faithful, and preserves
infinite products. Given two $R$-modules $L$ and $M$, we denote, as usually,
by $\operatorname{Hom}_{R}(L,M)$ and $\operatorname{Ext}_{R}^{*}(L,M)$ the Hom
and Ext groups in the abelian category of $R$-modules.

######  Lemma D.1.7.

Let $F$ be a flat left $R$-module and ${\mathfrak{P}}$ be a left
${\mathfrak{R}}$-contramodule for which the natural map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is an isomorphism. Set
$P_{n}={\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$, and assume
that that one has
$\operatorname{Ext}^{1}_{R_{0}}(F/I_{0}F,\>P_{0})=0=\operatorname{Ext}^{1}_{R_{n+1}}(F/I_{n+1}F,\>\allowbreak\ker(P_{n+1}\to\nobreak
P_{n}))$ for all $n\ge 0$. Then
$\operatorname{Ext}^{1}_{R}(F,{\mathfrak{P}})=0$.

###### Proof.

The proof is the same as in Lemma C.3.1. ∎

######  Corollary D.1.8.

(a) A left ${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is projective if and
only if the map
${\mathfrak{F}}\longrightarrow\varprojlim_{n}{\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$
is an isomorphism and the $R_{n}$-modules
${\mathfrak{F}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{F}}$ are projective for
all $n\ge 0$.

(b) For any flat left $R$-module $F$ such that the $R_{n}$-modules $F/I_{n}F$
are projective for all $n\ge 0$, and any left ${\mathfrak{R}}$-contramodule
${\mathfrak{Q}}$, one has
$\operatorname{Hom}^{\mathfrak{R}}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{Q}})\simeq\operatorname{Hom}_{R}(F,{\mathfrak{Q}})$
and
$\operatorname{Ext}^{{\mathfrak{R}},>0}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{Q}})=\operatorname{Ext}_{R}^{>0}(F,{\mathfrak{Q}})=0$.

###### Proof.

Part (a): the “only if” assertion holds by (the proof of) Lemma D.1.5. To
prove the “if”, consider a short exact sequence of
${\mathfrak{R}}$-contramodules
$0\longrightarrow{\mathfrak{H}}\longrightarrow{\mathfrak{G}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ with a projective ${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$. The
natural map
${\mathfrak{H}}\longrightarrow\varprojlim_{n}{\mathfrak{H}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{H}}$
being an isomorphism because the map
${\mathfrak{G}}\longrightarrow\varprojlim_{n}{\mathfrak{G}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{G}}$
is (or by Lemma D.1.3), one has
$\operatorname{Ext}^{{\mathfrak{R}},1}({\mathfrak{F}},{\mathfrak{H}})=0$ by
Lemma D.1.6, hence the short exact sequence splits and the
${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is a direct summand of the
${\mathfrak{R}}$-contramodule ${\mathfrak{G}}$.

Part (b): a natural map
$\operatorname{Hom}^{\mathfrak{R}}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{Q}})\longrightarrow\operatorname{Hom}_{R}(F,{\mathfrak{Q}})$
for any $R$-module $F$ and ${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ is
induced by the $R$-module morphism $F\longrightarrow\varprojlim_{n}F/I_{n}F$.
For an ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$ such that the map
${\mathfrak{P}}\longrightarrow\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
is an isomorphism, one has
$\operatorname{Hom}_{R}(F,{\mathfrak{P}})\simeq\varprojlim_{n}\operatorname{Hom}_{R_{n}}(F/I_{n}F,\>{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}})$,
which is isomorphic to
$\operatorname{Hom}^{\mathfrak{R}}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{P}})$
by Lemma D.1.2.

When $F$ is also a flat $R$-module with projective $R_{n}$-modules $F/I_{n}F$,
one has $\operatorname{Ext}^{>0}_{R}(F,{\mathfrak{P}})=0$ by Lemma D.1.7 and
$\operatorname{Ext}^{{\mathfrak{R}},>0}(\varprojlim_{n}F/I_{n}F,\>{\mathfrak{P}})=0$
by part (a). Now to prove the assertion of part (b) in the general case, it
suffices to present an ${\mathfrak{R}}$-contramodule ${\mathfrak{Q}}$ as the
cokernel of an injective morphism of ${\mathfrak{R}}$-contramodules
${\mathfrak{K}}\longrightarrow{\mathfrak{P}}$ with
${\mathfrak{P}}=\varprojlim_{n}{\mathfrak{P}}/{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}$
(and, consequently, the same for ${\mathfrak{K}}$). ∎

When the ideals ${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}\subset R_{n+1}$ are
nilpotent (i. e., for every $n\ge 0$ there exists $N_{n}\ge 1$ such that
$({\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1})^{N_{n}}=0$), it follows from
Corollary D.1.8(a) and [54, Lemma B.10.2] that a left
${\mathfrak{R}}$-contramodule ${\mathfrak{F}}$ is projective if and only if it
is flat and the $R_{0}$-module
${\mathfrak{F}}/{\mathfrak{I}}_{0}{\times}{\mathfrak{F}}$ is projective.
Similarly, it suffices to require that the $R_{0}$-module $F/I_{0}F$ be
projective in Corollary D.1.8(b) in this case.

###  D.2. Co-contra correspondence

In this section we consider a pair of projective systems $R_{0}\longleftarrow
R_{1}\longleftarrow R_{2}\longleftarrow\dotsb$ and $S_{0}\longleftarrow
S_{1}\longleftarrow S_{2}\longleftarrow\dotsb$ of associative rings and
surjective morphisms between them. We assume that the rings $S_{n}$ are left
Noetherian, the rings $R_{n}$ are right coherent, and the kernels
${\mathfrak{I}}_{n}/{\mathfrak{I}}_{n+1}\subset R_{n+1}$ of the ring
homomorphisms $R_{n+1}\longrightarrow R_{n}$ are finitely generated as right
ideals.

Set ${\mathfrak{S}}=\varprojlim_{n}S_{n}$, and denote by
${\mathfrak{J}}_{n}\subset{\mathfrak{S}}$ the kernels of the natural
surjective ring homomorphisms ${\mathfrak{S}}\longrightarrow S_{n}$. Given a
left ${\mathfrak{S}}$-module $M$, we denote by ${}_{S_{n}}\\!M\subset M$ its
submodule consisting of all the elements annihilated by ${\mathfrak{J}}_{n}$;
so ${}_{S_{n}}\\!M$ is the maximal left $S_{n}$-submodule in $M$. A left
${\mathfrak{S}}$-module ${\mathcal{M}}$ is said to be _discrete_ if every its
element is annihilated by the ideal ${\mathfrak{J}}_{n}$ for $n$ large enough.
In other words, a left ${\mathfrak{S}}$-module ${\mathcal{M}}$ is discrete if
its increasing filtration
${}_{S_{0}}{\mathcal{M}}\subset{}_{S_{1}}{\mathcal{M}}\subset{}_{S_{2}}{\mathcal{M}}\subset\dotsb$
is exhaustive, or equivalently, if the left action map
${\mathfrak{S}}\times{\mathcal{M}}\longrightarrow{\mathcal{M}}$ is continuous
in the projective limit topology of ${\mathfrak{S}}$ and the discrete topology
of ${\mathcal{M}}$. We denote the full abelian subcategory of discrete left
${\mathfrak{S}}$-modules by
${\mathfrak{S}}{\operatorname{\mathsf{--discr}}}\subset{\mathfrak{S}}{\operatorname{\mathsf{--mod}}}$.

######  Theorem D.2.1.

(a) The coderived category
${\mathsf{D}}^{\mathsf{co}}({\mathfrak{S}}{\operatorname{\mathsf{--discr}}})$
of the abelian category of discrete left ${\mathfrak{S}}$-modules is
equivalent to the homotopy category of complexes of injective discrete left
${\mathfrak{S}}$-modules.

(b) The contraderived category
${\mathsf{D}}^{\mathsf{ctr}}({\mathfrak{R}}{\operatorname{\mathsf{--contra}}})$
of the abelian category of left ${\mathfrak{R}}$-contramodules is equivalent
to the contraderived category of the exact category of flat left
${\mathfrak{R}}$-contramodules.

###### Proof.

Part (a) holds, because the category
${\mathfrak{S}}{\operatorname{\mathsf{--discr}}}$, being a locally Noetherian
Grothendieck abelian category, has enough injective objects, and injectivity
in ${\mathfrak{S}}{\operatorname{\mathsf{--discr}}}$ is preserved by infinite
direct sums.

Since any left ${\mathfrak{R}}$-contramodule is a quotient contramodule of a
flat (and even projective) one, and the class of flat left
${\mathfrak{R}}$-contramodules is closed under extensions and the passage to
the kernels of surjective morphisms (see Lemmas D.1.3–D.1.5), in order to
prove the assertion (b) it only remains to show that the class of flat left
${\mathfrak{R}}$-contramodules is preserved by infinite products (see
Proposition A.3.1(b)). The latter follows from the definition of flatness for
left ${\mathfrak{R}}$-contramodules, the coherence condition on the rings
$R_{n}$, and the next lemma. ∎

######  Lemma D.2.2.

For any family of left ${\mathfrak{R}}$-contramodules
${\mathfrak{P}}_{\alpha}$ and any $n\ge 0$, the two
${\mathfrak{R}}$-subcontramodules
${\mathfrak{I}}_{n}{\times}\prod_{\alpha}{\mathfrak{P}}_{\alpha}$ and
$\prod_{\alpha}{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}_{\alpha}$ coincide in
$\prod_{\alpha}{\mathfrak{P}}_{\alpha}$.

###### Proof.

The former subcontramodule is obviously contained in the latter one; we have
to prove the converse inclusion. For every $m\ge 0$, pick a finite set of
generators $\bar{r}_{m}^{\gamma}$ of the right ideal
${\mathfrak{I}}_{m}/{\mathfrak{I}}_{m+1}\subset{\mathfrak{R}}/{\mathfrak{I}}_{m+1}$,
and lift them to elements $r_{m}^{\gamma}\in{\mathfrak{R}}$. Then, for any
left ${\mathfrak{R}}$-contramodule ${\mathfrak{P}}$, any element of the
subcontramodule
${\mathfrak{I}}_{n}{\times}{\mathfrak{P}}\subset{\mathfrak{P}}$ can be
expressed in the form $\sum_{m\ge n}^{\gamma}r_{m}^{\gamma}p_{m}^{\gamma}$
with some $p_{m}^{\gamma}\in{\mathfrak{P}}$. In particular, any element $p$ of
the product $\prod_{\alpha}{\mathfrak{I}}_{n}{\times}{\mathfrak{P}}_{\alpha}$
can be presented in the form $p=\big{(}\sum_{m\ge
n}^{\gamma}r_{m}^{\gamma}p_{m}^{\gamma,\alpha}\big{)}_{\alpha}$ with some
elements $p_{m}^{\gamma,\alpha}\in{\mathfrak{P}}_{\alpha}$. Now one has
$\big{(}\sum_{m\ge
n}^{\gamma}r_{m}^{\gamma}p_{m}^{\gamma,\alpha}\big{)}_{\alpha}=\sum_{m\ge
n}^{\gamma}r_{m}^{\gamma}((p_{m}^{\gamma,\alpha})_{\alpha})$, which is an
infinite sum of elements of $\prod_{\alpha}{\mathfrak{P}}_{\alpha}$ with the
coefficient family still converging to zero in
${\mathfrak{I}}_{n}\subset{\mathfrak{R}}$, proving that $p$ belongs to
${\mathfrak{I}}_{n}{\times}\prod_{\alpha}{\mathfrak{P}}_{\alpha}$. (Cf. [54,
Lemmas 1.3.6–1.3.7].) ∎

A right ${\mathfrak{R}}$-module ${\mathcal{N}}$ is said to be discrete if
every its element is annihilated by the ideal
${\mathfrak{I}}_{n}=\ker({\mathfrak{R}}\to R_{n})$ for $n\gg 0$, i. e., in the
notation similar to the above, if
${\mathcal{N}}=\bigcup_{n}{\mathcal{N}}_{R_{n}}$. The abelian category of
discrete right ${\mathfrak{R}}$-modules is denoted by
${\operatorname{\mathsf{discr--}}}{\mathfrak{R}}$. For any associative ring
$S$, any ${\mathfrak{R}}$-discrete $S$-${\mathfrak{R}}$-bimodule