adjoint [1, Theorem 1.66], so it has a left adjoint functor $\Delta\colon
R{\operatorname{\mathsf{--mod}}}\longrightarrow\mathsf{B}$. Set $P=\Delta(R)$,
where $R$ is the free left $R$-module with one generator; then $P$ is a
projective object in $\mathsf{B}$ endowed with a right action of the ring $R$,
that is, with a homomorphism of associative rings
$R\longrightarrow\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$. The
functor $\Theta$ is corepresented by the object $P\in\mathsf{B}$: one has
$\Theta(B)=\operatorname{Hom}_{\mathsf{B}}(P,B)$ for all $B\in\mathsf{B}$,
with the left $R$-module structure on the abelian group
$\operatorname{Hom}_{\mathsf{B}}(P,B)$ coming from the right action of $R$ in
$P$.

The projective object $P$ is a projective generator of $\mathsf{B}$ if and
only if the exact functor $\Theta$ is _conservative_ , i. e., it takes
nonisomorphisms to nonisomorphisms, or equivalently, takes nonzero objects to
nonzero objects. An exact functor between abelian categories is conservative
if and only if it is faithful. Thus conservative exact functors
$\Theta\colon\mathsf{B}\longrightarrow R{\operatorname{\mathsf{--mod}}}$
preserving infinite products and $\kappa$-filtered colimits are indexed by
projective generators $P\in\mathsf{B}$ endowed with an associative ring
homomorphism
$R\longrightarrow\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$. We are
interested in those of such functors $\Theta$ that are not only conservative,
but, actually, fully faithful.

The following theorem is the main result of this section.

######  Theorem 4.6.

Let $\mathsf{B}$ be a locally presentable abelian category with a projective
generator $P$, let $R$ be an associative ring, and let $\theta\colon
R\longrightarrow\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$ be an
associative ring homomorphism. Assume that the related functor
$\Theta=\operatorname{Hom}_{\mathsf{B}}(P,{-})\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is fully faithful. Then

(a) the full subcategory $\Theta(\mathsf{B})\subset
R{\operatorname{\mathsf{--mod}}}$ is the right $0$-perpendicular subcategory
to a set of morphisms in $R{\operatorname{\mathsf{--mod}}}$;

(b) assuming additionally that $\theta$ is injective, the full subcategory
$\Theta(\mathsf{B})\subset R{\operatorname{\mathsf{--mod}}}$ is a right
$1$-perpendicular subcategory (to a set of objects) in
$R{\operatorname{\mathsf{--mod}}}$ if and only if it is closed under
extensions;

(c) assuming that $\theta$ is injective and given an integer $n\geq 1$, the
full subcategory $\Theta(\mathsf{B})\subset R{\operatorname{\mathsf{--mod}}}$
is the right $n$-perpendicular subcategory to a set of objects in
$R{\operatorname{\mathsf{--mod}}}$ whenever the induced maps between the Ext
groups
$\Theta\colon\operatorname{Ext}_{\mathsf{B}}^{i}(B,C)\longrightarrow\operatorname{Ext}_{R}^{i}(\Theta(B),\Theta(C))$
are isomorphisms for all $B$, $C\in\mathsf{B}$ and $i\leq n$;

(d) assuming that $\theta$ is injective, the full subcategory
$\Theta(\mathsf{B})\subset R{\operatorname{\mathsf{--mod}}}$ is the right
$\infty$-perpendicular subcategory to a set of objects in
$R{\operatorname{\mathsf{--mod}}}$ whenever the induced maps
$\Theta\colon\operatorname{Ext}_{\mathsf{B}}^{i}(B,C)\longrightarrow\operatorname{Ext}_{R}^{i}(\Theta(B),\Theta(C))$
are isomorphisms for all $B$, $C\in\mathsf{B}$ and all $0\leq i<\infty$.

###### Proof.

According to Example 1.2(3), the functor
$\operatorname{Hom}_{\mathsf{B}}(P,{-})$ identifies the category $\mathsf{B}$
with the category of modules over the additive monad $\mathbb{T}\colon
X\longmapsto\operatorname{Hom}_{\mathsf{B}}(P,P^{(X)})$. Assume that the
category $\mathsf{B}$ is locally $\kappa$-presentable and the object $P$ is
$\kappa$-presentable; then the monad $\mathbb{T}$ is $\kappa$-accessible.
Identifying $\mathsf{B}$ with $\mathbb{T}{\operatorname{\mathsf{--mod}}}$, the
functor $\Theta$ gets identified with the forgetful functor
$\operatorname{Hom}_{\mathbb{T}}(\mathbb{T}(*),{-})$ discussed in the
beginning of Section 2. In particular, for every set $X$ we have the free
$\mathbb{T}$-module
$\mathbb{T}(X)\in\mathbb{T}{\operatorname{\mathsf{--mod}}}$ corresponding to
the object $P^{(X)}\in\mathsf{B}$, and there is a natural left $R$-module
morphism $R[X]\longrightarrow\mathbb{T}(X)$, as in Proposition 2.1. Denote
this morphism by $\theta_{X}$.

Part (a): we claim that a left $R$-module $C$ belongs to the full subcategory
$\Theta(\mathsf{B})\subset R{\operatorname{\mathsf{--mod}}}$ if and only if
the morphism of abelian groups

$\operatorname{Hom}_{R}(\theta_{Z},C)\colon\operatorname{Hom}_{R}(\mathbb{T}(Z),C)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(R[Z],C)$

is an isomorphism for all sets $Z$ of cardinality less than $\kappa$. Indeed,
if $C\in\Theta(\mathsf{B})$, then

$\operatorname{Hom}_{R}(\mathbb{T}(X),C)\simeq
C^{X}\simeq\operatorname{Hom}_{R}(R[X],C)$

for all sets $X$, since the functor $\Theta$ is fully faithful by assumption.

Conversely, assume that $\operatorname{Hom}_{R}(\theta_{Z},C)$ is an
isomorphism for all sets $Z$ of cardinality less than $\kappa$. Then
$\operatorname{Hom}_{R}(\theta_{X},C)$ is an isomorphism for all sets $X$,
because $R[X]=\varinjlim_{Z\subset X}R[Z]$ and
$\mathbb{T}(X)=\varinjlim_{Z\subset X}\mathbb{T}(Z)$ in the category of left
$R$-modules, where the $\kappa$-filtered colimit is taken over all the subsets
$Z\subset X$ of cardinality less than $\kappa$.

In particular, we have
$\operatorname{Hom}_{R}(\mathbb{T}(C),C)=\operatorname{Hom}_{R}(R[C],C)=C^{C}$,
so there is a natural surjective morphism of left $R$-modules
$\mathbb{T}(C)\longrightarrow C$ corresponding to the identity map
$C\longrightarrow C$. Let $K$ denote the kernel of this morphism; then the
morphism $\operatorname{Hom}_{R}(\theta_{X},K)$ is an isomorphism for all sets
$X$, since the morphisms $\operatorname{Hom}_{R}(\theta_{X},\mathbb{T}(C))$
and $\operatorname{Hom}_{R}(\theta_{X},C)$ are. Hence we also have a natural
surjective morphism of left $R$-modules $\mathbb{T}(K)\longrightarrow K$. Now
$C$ is the cokernel of the composition $\mathbb{T}(K)\longrightarrow
K\longrightarrow\mathbb{T}(C)$, and it follows that $C\in\Theta(\mathsf{B})$,
since $\mathbb{T}(K)$, $\mathbb{T}(C)\in\Theta(\mathsf{B})$ and the full
subcategory $\Theta(\mathsf{B})\subset R{\operatorname{\mathsf{--mod}}}$ is
closed under cokernels.

Part (b): obviously, $\Theta(\mathsf{B})$ is closed under extensions in
$R{\operatorname{\mathsf{--mod}}}$ if and only if $\Theta$ induces
isomorphisms on the groups $\operatorname{Ext}^{1}$. If $\Theta(\mathsf{B})$
is a right $1$-perpendicular subcategory in
$R{\operatorname{\mathsf{--mod}}}$, then $\Theta(\mathsf{B})\subset
R{\operatorname{\mathsf{--mod}}}$ is closed under extensions by Lemma 4.2(b).

Conversely, if $\Theta$ induces isomorphisms on $\operatorname{Ext}^{1}$, then
$\operatorname{Ext}^{1}_{R}(\mathbb{T}(X),C)=0$ for all
$C\in\Theta(\mathsf{B})$, as the free $\mathbb{T}$-modules $\mathbb{T}(X)$ are
projective objects in $\mathbb{T}{\operatorname{\mathsf{--mod}}}$. We claim
that a left $R$-module belongs to the full subcategory
$\Theta(\mathsf{B})\subset R{\operatorname{\mathsf{--mod}}}$ if and only if

$\operatorname{Hom}_{R}(\mathbb{T}(Z)/R[Z],C)=0=\operatorname{Ext}^{1}_{R}(\mathbb{T}(Z)/R[Z],C)$

for all sets $Z$ of cardinality less than $\kappa$, where $\mathbb{T}(Z)/R[Z]$
is the cokernel of the morphism $\theta_{Z}$. Notice that the morphism
$\theta_{Z}$ is injective in our present assumptions (since the homomorphism
$\theta$ is injective).

Indeed, it suffices to consider the long exact sequence

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(\mathbb{T}(Z)/R[Z],C)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(\mathbb{T}(Z),C)\\\ \mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Hom}_{R}(R[Z],C)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}^{1}_{R}(\mathbb{T}(Z)/R[Z],C)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{Ext}^{1}_{R}(\mathbb{T}(Z),C)$

and recall that a left $R$-module $C$ belongs to $\Theta(\mathsf{B})$ if and
only if $\operatorname{Hom}_{R}(\theta_{Z},C)$ is an isomorphism for all sets
$Z$ of the cardinality less than $\kappa$, and that
$\operatorname{Ext}^{1}_{R}(\mathbb{T}(X),C)=0$ for all sets $X$ and all
$C\in\Theta(\mathsf{B})$.

Part (c): for every set $X$, we have
$\operatorname{Ext}^{i}_{R}(\mathbb{T}(X)/R[X],C)=\operatorname{Ext}^{i}_{R}(\mathbb{T}(X),C)$
for all left $R$-modules $C$ and all $i\geq 2$. If the functor $\Theta$
induces isomorphisms of the groups $\operatorname{Ext}^{i}$ for all $i\leq n$,
then one has

$\operatorname{Ext}^{i}_{R}(\mathbb{T}(X)/R[X],C)=\operatorname{Ext}^{i}_{R}(\mathbb{T}(X),C)=\operatorname{Ext}^{i}_{\mathsf{B}}(P^{(X)},\Theta^{-1}(C))=0$

for all $C\in\Theta(\mathsf{B})$ and $2\leq i\leq n$, where
$\Theta^{-1}(C)\in\mathsf{B}$ is an object such that
$\Theta(\Theta^{-1}(C))=C$. Hence
$\Theta(\mathsf{B})=\mathsf{E}^{\perp_{0,1}}=\mathsf{E}^{\perp_{0..n}}$, where
$\mathsf{E}\subset R{\operatorname{\mathsf{--mod}}}$ is the set of all the
left $R$-modules $\mathbb{T}(Z_{\mu})/R[Z_{\mu}]$, where $\mu$ runs over all
the cardinals smaller than $\kappa$ and $Z_{\mu}$ is a set of cardinality
$\mu$.

Part (d) is provable in the same way as part (c). ∎

######  Remark 4.7.

It is clear from the proof of Theorem 4.6(c) that, denoting by
$\mathsf{E}\subset R{\operatorname{\mathsf{--mod}}}$ the set of all left
$R$-modules $\mathbb{T}(Z_{\mu})/R[Z_{\mu}]$ and assuming that
$\Theta(\mathsf{B})=\mathsf{E}^{\perp_{0,1}}$, one has
$\mathsf{E}^{\perp_{0,1}}=\mathsf{E}^{\perp_{0..n}}$ if and only if
$\operatorname{Ext}^{i}_{R}(\mathbb{T}(Z_{\mu}),C)=0$ for all the cardinals
$\mu$ involved, all $C\in\Theta(\mathsf{B})$, and all $1\leq i\leq n$. If we
knew that, moreover, $\operatorname{Ext}^{i}_{R}(\mathbb{T}(X),C)=0$ for all
sets $X$, all objects $C\in\Theta(\mathsf{B})$, and all $1\leq i\leq n$, it
would follow that the functor $\Theta\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ induces isomorphisms of the groups
$\operatorname{Ext}^{i}$ for $i\leq n$.

Nevertheless, there are examples of associative (and even commutative) rings
$R$ with a set $\mathsf{E}\subset R{\operatorname{\mathsf{--mod}}}$ of left
$R$-modules of projective dimension $1$ for which the embedding functor
$\Theta\colon\mathsf{B}\longrightarrow R{\operatorname{\mathsf{--mod}}}$ of
the right $\infty$-perpendicular subcategory
$\mathsf{B}=\mathsf{E}^{\perp_{0,1}}=\mathsf{E}^{\perp_{0..\infty}}$ does
_not_ induce an isomorphism of the groups $\operatorname{Ext}^{i}$ for for
some $i\geq 2$. Such examples can even be found among the embedding functors
$\Theta\colon\mathsf{B}\longrightarrow R{\operatorname{\mathsf{--mod}}}$
corepresented by projective generators $P\in\mathsf{B}$ with
$R=\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$ (see Example 5.2(5)
below).

######  Proposition 4.8.

Let $\mathsf{B}$ be a locally presentable abelian category with a projective
generator $P$, let $R$ be an associative ring, let $\theta\colon
R\longrightarrow\operatorname{Hom}_{\mathsf{B}}(P,P)$ be a ring homomorphism,
and let
$\Theta=\operatorname{Hom}_{\mathsf{B}}(P,{-})\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ be the related exact functor. Then the
induced maps
$\Theta\colon\operatorname{Ext}^{i}_{\mathsf{B}}(B,C)\longrightarrow\operatorname{Ext}^{i}_{R}(\Theta(B),\Theta(C))$
are isomorphisms for all $B$, $C\in\mathsf{B}$ and all $0\leq i<\infty$ if and
only if the induced triangulated functor between the bounded above derived
categories
$\Theta\colon\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(R{\operatorname{\mathsf{--mod}}})$
is fully faithful.

###### Proof.

Clearly, an exact functor between abelian categories
$\mathsf{B}\longrightarrow\mathsf{A}$ induces isomorphisms on all the Ext
groups if and only if the induced triangulated functor between the bounded
derived categories
$\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{A})$
is fully faithful. According to Proposition 6.5 below, the functor
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$ is fully
faithful if and only if the functor
$\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{A})$
is, provided that there are enough projective objects in the category
$\mathsf{A}$ and the exact functor $\mathsf{B}\longrightarrow\mathsf{A}$ has a
left adjoint $\Delta\colon\mathsf{A}\longrightarrow\mathsf{B}$.

In the situation at hand, obviously there are enough projectives in
$R{\operatorname{\mathsf{--mod}}}$, so it remains to recall from the
discussion before Theorem 4.6 that the functor $\Theta$ has a left adjoint.
Alternatively, one can construct the functor $\Delta$ explicitly, as the
unique right exact functor taking the free left $R$-module $R[X]$ to the
object $P^{(X)}\in\mathsf{B}$ for all sets $X$ and acting on morphisms between
free left $R$-modules in the natural way (determined by the ring homomorphism
$\theta$). ∎

Now let us restrict to the particular case when the ring $R$ coincides with
the endomorphism ring $\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$ and
$\theta$ is the identity isomorphism. Let $\mathsf{B}$ be a locally
presentable abelian category with a projective generator $P$. We will say that
the projective generator $P\in\mathsf{B}$ is _$0$ -good_ if the conservative
exact functor
$\Theta_{P}=\operatorname{Hom}_{\mathsf{B}}(P,{-})\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$, where
$R=\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$, is fully faithful.

A projective generator $P\in\mathsf{B}$ is $0$-good if and only if the full
subcategory on one object $\\{P\\}\subset\mathsf{B}$ is _dense_ in the
category $\mathsf{B}$ [1, Proposition 1.26(i)] (cf. [1, Remark 1.23 and
Example 1.24(4)]). Such full subcategories were called “left adequate” in the
terminology of the paper [14].

Furthermore, we will say that a projective generator $P\in\mathsf{B}$ is _$1$
-good_ if the functor $\Theta_{P}$ is fully faithful and the full subcategory
$\Theta_{P}(\mathsf{B})\subset R{\operatorname{\mathsf{--mod}}}$ is closed
under extensions. More generally, for any $n\geq 0$, we will say that a
projective generator $P\in\mathsf{B}$ is _$n$ -good_ if the functor
$\Theta_{P}$ induces isomorphisms on the groups $\operatorname{Ext}^{i}$ for
all $0\leq i\leq n$.

Finally, we will say that a projective generator $P\in\mathsf{B}$ is _good_
(or _$\infty$ -good_) if it is $n$-good for all $n\geq 0$. According to
Proposition 4.8, a projective generator $P\in\mathsf{B}$ is good if and only
if the functor $\Theta_{P}$ induces a fully faithful functor between the
bounded above derived categories

$\mathsf{D}^{-}(\mathsf{B})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathsf{D}^{-}(R{\operatorname{\mathsf{--mod}}}).$

Clearly, any $(n+1)$-good projective generator, $n\geq 0$, is at the same time
$n$-good.

The following corollary summarizes the assertions of Theorem 4.6 in the case
$\theta=\operatorname{id}$.

######  Corollary 4.9.

Let $\mathsf{B}$ be a locally presentable abelian category and
$P\in\mathsf{B}$ be a projective generator. Then

(a) if $P$ is $0$-good, then the full subcategory
$\Theta_{P}(\mathsf{B})\subset R{\operatorname{\mathsf{--mod}}}$ is an
abelian, exactly embedded right $0$-perpendicular subcategory to a set of
morphisms in $R{\operatorname{\mathsf{--mod}}}$;

(b) assume that $P$ is $0$-good; then $P$ is $1$-good if and only if
$\Theta_{P}(\mathsf{B})\subset R{\operatorname{\mathsf{--mod}}}$ is a right
$1$-perpendicular subcategory (to a set of objects) in
$R{\operatorname{\mathsf{--mod}}}$;

(c) if $P$ is $n$-good for some $n\geq 1$, then $\Theta_{P}(\mathsf{B})\subset
R{\operatorname{\mathsf{--mod}}}$ is a right $n$-perpendicular subcategory to
a set of objects in $R{\operatorname{\mathsf{--mod}}}$;

(d) if $P$ is good, then $\Theta_{P}(\mathsf{B})\subset
R{\operatorname{\mathsf{--mod}}}$ is a right $\infty$-perpendicular
subcategory to a set of objects in $R{\operatorname{\mathsf{--mod}}}$. ∎

For any cardinal $\lambda$, we denote its successor cardinar by $\lambda^{+}$.
The proof of the following theorem will be given in Section 6.

######  Theorem 4.10.

Let $\mathsf{B}$ be an abelian category with a projective generator $P$ and
$\lambda$ be a cardinal such that the category $\mathsf{B}$ is locally
$\lambda^{+}$-presentable and the object $P\in\mathsf{B}$ is
$\lambda^{+}$-presentable. Then $Q=P^{(\lambda)}$ is a good projective
generator of $\mathsf{B}$.

The following corollary shows that the classes of abelian categories embodied
by the (abelian and exactly embedded) right $n$-perpendicular subcategories to
sets of objects/morphisms in the categories of modules over associative rings
are the same for all integers $n\geq 0$ and for $n=\infty$.

######  Corollary 4.11.

The following classes of abelian categories (viewed as abstract categories
irrespectively of a particular embedding into a category of modules) coincide
with each other, and with the class of all locally presentable abelian
categories with a projective generator:

  * •

abelian, exactly and accessibly embedded full subcategories in the categories
of modules over associative rings, closed under infinite products;

  * •

abelian, exactly embedded right $0$-perpendicular subcategories to sets of
morphisms in the categories of modules over associative rings;

  * •

abelian, exactly embedded right $1$-perpendicular subcategories to sets of
objects in the categories of modules over associative rings;

  * •

right $n$-perpendicular subcategories to sets of objects in the categories of
modules over associative rings (where $n\geq 2$ is some fixed integer);

  * •

right $\infty$-perpendicular subcategories to sets of objects in the
categories of modules over associative rings.

###### Proof.

Any right $(n+1)$-perpendicular subcategory to a set of objects in
$R{\operatorname{\mathsf{--mod}}}$ is at the same time a right
$n$-perpendicular subcategory to a set of objects/morphisms in
$R{\operatorname{\mathsf{--mod}}}$ by Lemma 4.1. Any right $2$-perpendicular
subcategory in an abelian category is abelian and exactly embedded by Lemma
4.2.

Any right $0$-perpendicular subcategory to a set of morphisms in
$R{\operatorname{\mathsf{--mod}}}$ is accessibly embedded and closed under
infinite products (in fact, arbitrary limits) by [1, Observation 1.34 and
Proposition 1.35]. Any abelian, exactly and accessively embedded full
subcategory in $R{\operatorname{\mathsf{--mod}}}$ is locally presentable and
reflective by [1, Theorem and Corollary 2.48] (see also [1, Theorem 1.39] and
Lemma 4.4 above). Any abelian, exactly embedded, reflective full subcategory
in $R{\operatorname{\mathsf{--mod}}}$ has a projective generator by Lemma 4.5.
This proves that all the itemized classes of categories consist of locally
presentable abelian categories with a projective generator.

Conversely, if $\mathsf{B}$ is a locally $\kappa$-presentable abelian category
with a $\kappa$-presentable projective generator $P$, and $\lambda$ is a
cardinal such that $\lambda^{+}\geq\kappa$, then $Q=P^{(\lambda)}$ is a good
projective generator of $\mathsf{B}$ by Theorem 4.10. Set
$S=\operatorname{Hom}_{\mathsf{B}}(Q,Q)^{\mathrm{op}}$; then the functor
$\Theta_{Q}\colon\mathsf{B}\longrightarrow S{\operatorname{\mathsf{--mod}}}$
is fully faithful and its image is the right $\infty$-perpendicular
subcategory to a set of objects in $S{\operatorname{\mathsf{--mod}}}$ by
Corollary 4.9(d). ∎

Under Vopěnka’s principle, the set-theoretical conditions (“accessibly
embedded”, “to sets of”) in Corollary 4.11 can be dropped. The related classes
of abelian categories (that is, the classes of all abelian, exactly embedded
right $n$-perpendicular subcategories to classes of objects or morphisms in
the categories of modules over associative rings, for various values of $n$)
will still be the same, and coincide with the class of all locally presentable
abelian categories with a projective generator. This is provable using the
above arguments together with [1, Corollary 6.24].

##  5\. Examples of Good Projective Generators

The aim of this section is to provide some examples to the theory developed in
Section 4, mostly based on Examples 1.2–1.3 and examples from Sections 2–3.

######  Examples 5.1.

(1) Let $\mathfrak{R}$ be a complete, separated topological associative ring
with a base of neighborhoods of zero formed by open right ideals, $R$ be an
associative ring, and $\theta\colon R\longrightarrow\mathfrak{R}$ be an
associative ring homomorphism such that the forgetful functor
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is fully faithful. Then the forgetful
functor
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\longrightarrow\mathfrak{R}{\operatorname{\mathsf{--mod}}}$
is also fully faithful. So, by the definition, the free left
$\mathfrak{R}$-contramodule with one generator
$P=\mathfrak{R}=\mathfrak{R}[[*]]$ is a $0$-good projective generator of
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}$. This includes all the
topological rings satisfying the assumptions of Theorem 3.1, and in
particular, all the topological rings from Examples 3.3 and 3.6 (and those
topological rings from Examples 3.7 which satisfy the conditions mentioned
there).

See Examples 5.2(6-8), 5.3(4), 5.4(2), and 5.5(2) for further discussion.

(2) Let $\mathfrak{R}$ be a complete, separated topological associative ring
with a base of neighborhoods of zero formed by open right ideals, $R$ be an
associative ring, and $\theta\colon R\longrightarrow\mathfrak{R}$ be an
associative ring homomorphism such that the related forgetful functor
$\Theta\colon\mathfrak{R}{\operatorname{\mathsf{--contra}}}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is fully faithful. Then the argument from
the proof of Theorem 4.6 shows how to distinguish the objects of the full
subcategory $\mathfrak{R}{\operatorname{\mathsf{--contra}}}$ among the objects
of the ambient category $R{\operatorname{\mathsf{--mod}}}$.

Specifically, let $Y$ be a set of the cardinality greater or equal to the
cardinality of a base of neighborhoods of zero in $\mathfrak{R}$. Denote by
$\theta_{Y}$ the natural $R$-module morphism
$R[Y]\longrightarrow\mathfrak{R}[[Y]]$. Then a left $R$-module $C$ belongs to
the full subcategory
$\Theta(\mathfrak{R}{\operatorname{\mathsf{--contra}}})\subset
R{\operatorname{\mathsf{--mod}}}$ if and only if the morphism of abelian
groups
$\operatorname{Hom}_{R}(\theta_{Y},C)\colon\operatorname{Hom}_{R}(\mathfrak{R}[[Y]],C)\longrightarrow\operatorname{Hom}_{R}(R[Y],C)=C^{Y}$
is an isomorphism. (Notice that if $Z$ is a subset in $Y$ and the map
$\operatorname{Hom}_{R}(\theta_{Y},C)$ is an isomorphism, then the map
$\operatorname{Hom}_{R}(\theta_{Z},C)$ is also an isomorphism, because the
left $R$-module morphism $\theta_{Z}\colon
R[Z]\longrightarrow\mathfrak{R}[[Z]]$ is a direct summand of the morphism
$\theta_{Y}$.)

In particular, in the context of Theorem 3.1, it suffices to take a countable
set $Y$.

######  Examples 5.2.

(1) Let $R$ be a Noetherian commutative ring and $I\subset R$ be a finitely
generated ideal; denote by $\mathfrak{R}=\varprojlim_{n}R/I^{n}$ the $I$-adic
completion of the ring $R$. Let $\mathsf{B}$ be the locally
$\aleph_{1}$-presentable abelian category
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\simeq
R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$ (see
Examples 2.2) and $P=\mathfrak{R}=\Delta_{I}(R)$ be its natural projective
generator. Then the forgetful/natural embedding functor
$\Theta\colon\mathsf{B}\longrightarrow R{\operatorname{\mathsf{--mod}}}$ is
corepresented by the projective generator $P\in\mathsf{B}$ with the natural
action of the ring $R$ in it (provided by the natural ring homomorphism
$\theta\colon
R\longrightarrow\mathfrak{R}=\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$).

According to [22, Theorem B.8.1], the functor $\Theta$ induces isomorphisms of
the groups $\operatorname{Ext}^{i}$ for all $0\leq i<\infty$. Moreover, one
can show using [22, Proposiitons B.9.1 and B.10.1] that the triangulated
functor between the bounded above derived categories
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(R{\operatorname{\mathsf{--mod}}})$
induced by exact functor $\Theta$ is fully faithful.

In particular, $\mathfrak{R}$ is also a Noetherian commutative ring; so one
can replace $R$ with $\mathfrak{R}$ and consider the case $R=\mathfrak{R}$.
Then we have $\Theta=\Theta_{P}$, so we can conclude that $P=\mathfrak{R}$ is
a good projective generator of $\mathsf{B}$.

(2) More generally, let $R$ be a commutative ring and $I\subset R$ be a weakly
proregular finitely generated ideal. We keep the notation
$\mathfrak{R}=\varprojlim_{n}R/I^{n}$,
$\mathsf{B}=\mathfrak{R}{\operatorname{\mathsf{--contra}}}=R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$,
and $P=\mathfrak{R}=\Delta_{I}(R)$ (see Example 2.2(3)). As in (1),
$\mathsf{B}$ is a locally $\aleph_{1}$-presentable abelian category, $P$ is an
$\aleph_{1}$-presentable projective generator of $\mathsf{B}$, and the
forgetful/natural embedding functor $\Theta\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is corepresented by $P$.

According to [25, Theorem 2.9], the triangulated functor between the unbounded
derived categories
$\mathsf{D}(\mathsf{B})\longrightarrow\mathsf{D}(R{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$ is fully faithful. Hence the functor $\Theta$ induces
isomorphisms of the groups $\operatorname{Ext}^{i}$ for all $0\leq i<\infty$.

By [25, Lemma 5.3(b) or 5.4(b)], the ideal
$\mathfrak{R}I=\varprojlim_{n}I/I^{n}$ in the ring $\mathfrak{R}$ is also
weakly proregular; so one can replace $R$ with $\mathfrak{R}$ and consider the
case $R=\mathfrak{R}$. Then $\Theta=\Theta_{P}$, so we can conclude that
$P=\mathfrak{R}$ is a good projective generator of $\mathsf{B}$.

(3) Even more generally, let $R$ be a commutative ring and $I\subset R$ be a
finitely generated ideal. Set
$\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$
and $P=\Delta_{I}(R)$ (see Example 2.2(1)). As in (2), $\mathsf{B}$ is a
locally $\aleph_{1}$-presentable abelian category, $P$ is an
$\aleph_{1}$-presentable projective generator of $\mathsf{B}$, and the natural
embedding functor $\Theta\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is corepresented by the projective generator
$P\in\mathsf{B}$ with the natural action of the ring $R$ in it (provided by
the natural ring homomorphism $\theta\colon
R\longrightarrow\Delta_{I}(R)=\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$).

Following the argument in [25, proof of Theorem 2.9] based on [25, Lemma 2.7],
the triangulated functor
$\mathsf{D}(\mathsf{B})\longrightarrow\mathsf{D}(R{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$ is fully faithful provided that
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s_{1},\dotsc,s_{m}),R[X]))=0$
for all sets $X$ and all $i>0$. Conversely, arguing as in [27, second
paragraph of Remark 6.8] (with torsion modules replaced by contramodules), one
can show that
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s_{1},\dotsc,s_{m}),R[X]))=0$
for all $i>0$ whenever the functor
$\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(R{\operatorname{\mathsf{--mod}}})$
is fully faithful. Thus, given any derived category symbol
$\star={\mathsf{b}}$, $+$, $-$, or $\varnothing$, the triangulated functor
$\mathsf{D}^{\star}(\mathsf{B})\longrightarrow\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$ is fully faithful if and only if for all sets $X$ one has
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s_{1},\dotsc,s_{m}),R[X]))=0$
for $i>0$, or equivalently, if and only if
$\varprojlim_{n}H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R[X]))=0$
for $i\geq 1$ and
$\varprojlim_{n}^{1}H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R[X]))=0$
for $i\geq 2$.

Arguing as in [27, Proposition 3.1], one can show that $\Delta_{I}(R)$ is a
commutative ring. All the $I$-contramodule $R$-modules admit a unique natural
extension of their $R$-module structure to a $\Delta_{I}(R)$-module structure.
So one can replace $R$ with $\Delta_{I}(R)$ and $I$ with $\Delta_{I}(R)I$, and
consider the case $R=\Delta_{I}(R)$. Then $\Theta=\Theta_{P}$. So $P$ is
always a $1$-good projective generator of $\mathsf{B}$, since
$\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$
is closed under extensions in $R{\operatorname{\mathsf{--mod}}}$. As we have
seen, $P$ is a good projective generator of $\mathsf{B}$ if and only if
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s_{1},\dotsc,s_{m}),\mskip
1.5mu\Delta_{I}(R)[X]))=0$ for all sets $X$ and all $i>0$.

(4) Let $E_{1}\longleftarrow E_{2}\longleftarrow E_{3}\longleftarrow\dotsb$ be
a projective system of abelian groups. Clearly, one has
$\varprojlim_{n}E_{n}[X]=0$ for all sets $X$ if and only if
$\varprojlim_{n}E_{n}=0$. Furthermore, one has $\varprojlim_{n}^{1}E_{n}[X]=0$
for all sets $X$ if and only if the projective system $E_{n}$ satisfies the
Mittag-Leffler condition [8, Corollary 6]. Finally, one easily checks that a
projective system $(E_{n})$ satisfies both the condition of the vanishing of
the projective limit $\varprojlim_{n}E_{n}=0$ and the Mittag-Leffler condition
if and only if it is pro-zero.

Thus, in the context of (3), the triangulated functor
$\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}})\longrightarrow\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$
is fully faithful if and only if the projective system
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;\allowbreak
s_{1},\dotsc,s_{m}),R))$ is pro-zero for $i\geq 2$ and
$\varprojlim_{n}H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R))=0$.

(5) In particular, let $R$ be the commutative algebra over a field $k$
generated by the elements $s$, $t$, and $x_{i}$, $i\geq 1$, with the relations
$x_{i}x_{j}=0$ and $s^{i}t^{i}x_{i}=0$ for all $i$, $j\geq 1$ (cf. [25,
Example 2.6]). Let $I=(s,t)\subset R$ be the ideal generated by the elements
$s$ and $t$. Then
$H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s,t),R))=0$
for all $n\geq 1$ and
$\varprojlim_{n}H_{2}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s,t),R[X]))=0$
for all sets $X$, while
$\varprojlim_{n}^{1}H_{2}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s,t),R))\neq
0$. So $\Delta_{I}(R[X])=\Lambda_{I}(R[X])=\mathfrak{R}[[X]]$ for all $X$.

Set
$\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}=\mathfrak{R}{\operatorname{\mathsf{--contra}}}$
and $P=\Delta_{I}(R)=\mathfrak{R}$. One has $\bigcap_{n}I^{n}=0$ in $R$, hence
the morphism $\theta\colon R\longrightarrow\mathfrak{R}$ is injective. Thus
the category $\mathsf{B}$ with its projective generator $P$ and the morphism
$\theta$ provide a counterexample to the converse assertions to Theorem
4.6(c-d) promised in Remark 4.7. Denoting by $C$ the $I$-contramodule
$R$-module
$H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s,t),R))=\varprojlim_{n}^{1}H_{2}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s,t),R))$,
the complex of $R$-modules
$\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s,t),R)$
represents a nontrivial extension class in $\operatorname{Ext}^{2}_{R}(P,C)$.

Furthermore, one has
$H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s,t),\mathfrak{R}))=0$
for all $n\geq 1$ and $\varprojlim_{n}\allowbreak
H_{2}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s,t),\mathfrak{R}[X]))=0$
for all sets $X$, while
$\varprojlim_{n}^{1}H_{2}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s,t),\allowbreak\mathfrak{R}[Z]))\neq
0$ for a countable set $Z$. Thus the category $\mathsf{B}$ with its projective
generator $P$ (and the identity isomorphism
$\theta=\operatorname{id}_{\mathfrak{R}}$) provide a counterexample to the
converse assertions to Corollary 4.9(c-d). Setting
$Q=\Delta_{I}(R[Z])=\Delta_{\mathfrak{R}I}(\mathfrak{R}[Z])=\mathfrak{R}[[Z]]$
and denoting by $C$ the $(\mathfrak{R}I)$-contramodule $\mathfrak{R}$-module
$H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s,t),\mathfrak{R}[Z]))=\varprojlim_{n}^{1}H_{2}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s,t),\mathfrak{R}[Z]))$,
the complex of $\mathfrak{R}$-modules
$\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s,t),\mathfrak{R}[Z])$
represents a nontrivial extension class in
$\operatorname{Ext}^{2}_{\mathfrak{R}}(Q,C)$.

(6) Let $R$ be a commutative ring and $I\subset R$ be a finitely generated
ideal. Set $\mathfrak{R}=\varprojlim_{n}R/I^{n}$,
$\mathsf{B}=\mathfrak{R}{\operatorname{\mathsf{--contra}}}$, and
$P=\mathfrak{R}$ (see Examples 1.3(2), 2.2(2-3), and 3.6(2-4)). Then the
abelian category $\mathsf{B}$ is an exactly embedded full subcategory of the
abelian category
$R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$. We
claim that every object of
$R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$ is an
extension of two objects from
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}$.

Indeed, every object of
$R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$ has the
form $\Delta_{I}(C)$ for some $C\in R{\operatorname{\mathsf{--mod}}}$.
According to [26, Lemma 7.5], there is a natural short exact sequence of
$R$-modules $0\longrightarrow
K\longrightarrow\Delta_{I}(C)\longrightarrow\Lambda_{I}(C)\longrightarrow 0$
with an $R$-module $K$ of the form $K=\varprojlim_{n}^{1}D_{n}$, where $D_{n}$
is a sequence of $R$-modules, each of which is annihilated by some power of
the ideal $I$. Now we have
$D_{n}\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$, and it follows that
$K\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$, as the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ is closed under the cokernels and infinite
products. Similarly, we have
$\Lambda_{I}(C)\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$.

So the full subcategory $\mathfrak{R}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ is closed under extensions if and only if it
coincides with
$R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$.
According to (4) and Example 2.2(3)), this holds if and only if the projective
system
$H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R))$
satisfies the Mittag-Leffler condition (where $s_{1}$, …, $s_{m}$ is some set
of generators of the ideal $I$). Furthermore, it follows that, for any derived
category symbol $\star={\mathsf{b}}$, $+$, $-$, or $\varnothing$, the
triangulated functor
$\mathsf{D}^{\star}(\mathfrak{R}{\operatorname{\mathsf{--contra}}})\longrightarrow\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$
is fully faithful if and only if the projective system
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R))$
is pro-zero for all $i\geq 1$, or in other words, the ideal $I\subset R$ is
weakly proregular.

It was mentioned in Example 5.1(1) that the projective generator
$P\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$ is $0$-good. Now we see
that $P$ is $1$-good if and only if the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\subset\mathfrak{R}{\operatorname{\mathsf{--mod}}}$
coincides with
$\mathfrak{R}{\operatorname{\mathsf{--mod}}}_{(\mathfrak{R}I){\operatorname{\mathsf{-ctra}}}}$,
and this holds if and only if the projective system
$H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),\mathfrak{R}))=0$
satisfies the Mittag-Leffler condition. The projective generator $P$ is good
if and only if the ideal $\mathfrak{R}I\subset\mathfrak{R}$ is weakly
proregular.

(7) In particular, let $I=(s)\subset R$ be a principal ideal. Then
$H_{1}(\operatorname{Hom}_{R}(T_{n}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s),R))$
is the submodule of elements annihilated by $s^{n}$ in $R$, and the maps in
the projective system formed by these modules are the multiplications with
(the powers of) $s$. Hence
$\varprojlim_{n}H_{1}(\operatorname{Hom}_{R}(T_{n}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s),R))=0$
if and only if there is no $s$-divisible $s$-torsion in $R$, and the
projective system
$H_{1}(\operatorname{Hom}_{R}(T_{n}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s),R))$
is pro-zero if and only if the $s$-torsion in $R$ is bounded.

Furthermore, the rings $\Delta_{s}(R)$ and $\mathfrak{R}=\Lambda_{s}(R)$ are
$s$-contramodules, so they cannot contain $s$-divisible $s$-torsion. Thus
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}=R{\operatorname{\mathsf{--mod}}}_{s{\operatorname{\mathsf{-ctra}}}}$
if and only if the $s$-torsion in $\Delta_{s}(R)$ is bounded; and
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}=\mathfrak{R}{\operatorname{\mathsf{--mod}}}_{s{\operatorname{\mathsf{-ctra}}}}$
if and only if the $s$-torsion in $\mathfrak{R}$ is bounded. According to [26,
Remark 6.9], the latter two conditions are equivalent. The projective
generator $P=\mathfrak{R}\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$ is
$1$-good if and only if it good, and if and only if these equivalent
conditions hold.

(8) In particular, if $R$ is the commutative $k$-algebra from Example 1.3(6)
and [25, Example 2.6], and $I$ is the principal ideal $I=(s)$, then the
projective generator $P\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$ is
not $1$-good. Indeed, the $s$-torsion in $\mathfrak{R}$ is not bounded, so the
morphism
$\Delta_{I}(\mathfrak{R}[Z])\longrightarrow\Lambda_{I}(\mathfrak{R}[Z])$ has a
nontrivial kernel for a countable set $Z$.

######  Examples 5.3.

(1) Let $R$ be a right Noetherian associative ring and $I\subset R$ be an
ideal generated by a finite set of central elements. Denote by
$\mathfrak{R}=\varprojlim_{n}R/I^{n}$ the $I$-adic completion of the ring $R$.
Let $\mathsf{B}$ be the locally $\aleph_{1}$-presentable abelian category
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\simeq
R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$ (see
Examples 2.3) and $P=\mathfrak{R}=\Delta_{I}(R)$ be its natural projective
generator. Then the forgetful/natural embedding functor
$\Theta\colon\mathsf{B}\longrightarrow R{\operatorname{\mathsf{--mod}}}$ is
corepresented by the projective generator $P\in\mathsf{B}$ with the natural
right action of the ring $R$ in it (provided by the natural ring homomorphism
$\theta\colon
R\longrightarrow\mathfrak{R}=\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$).

Accoding to [23, Corollary C.5.6(b)], the functor $\Theta$ induces
isomorphisms of the groups $\operatorname{Ext}^{i}$ for all $0\leq i<\infty$.
Moreover, one can show using [23, Proposition C.5.5 and Corollary C.5.6(a)]
that the triangulated functor
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(R{\operatorname{\mathsf{--mod}}})$
induced by the exact functor $\Theta$ is fully faithful.

In particular, $\mathfrak{R}$ is also a right Noetherian ring, so one can
replace $R$ with $\mathfrak{R}$ and consider the case $R=\mathfrak{R}$. Then
we have $\Theta=\Theta_{P}$, so we can conclude that $P=\mathfrak{R}$ is a
good projective generator of $\mathsf{B}$.

(2) More generally, let $R$ be an associative ring and $I\subset R$ be the
ideal generated by a weakly proregular finite sequence of central elements
$s_{1}$, …, $s_{m}\in R$, i. e., a finite sequence of central elements such
that the projective system of $R$-$R$-bimodules
$H_{i}(\operatorname{Hom}_{R}(T_{n}(R;s_{1},\dotsc,s_{m}),R))$, $n\geq 1$, is
pro-zero for all $i>0$ (see [35, 20, 25]; cf. Example 2.3(3)). We keep the
notation $\mathfrak{R}=\varprojlim_{n}R/I^{n}$,
$\mathsf{B}=\mathfrak{R}{\operatorname{\mathsf{--contra}}}=R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$,
and $P=\mathfrak{R}=\Delta_{I}(R)$. As above, $\mathsf{B}$ is a locally
$\aleph_{1}$-presentable abelian category, $P$ is an $\aleph_{1}$-presentable
projective generator of $\mathsf{B}$, and the forgetful/natural embedding
functor $\Theta\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is corepresented by $P$.

Arguing as in [25, Section 2] (see [27, Theorem 6.4] for additional details),
one can show that the triangulated functor
$\mathsf{D}(\mathsf{B})\longrightarrow\mathsf{D}(R{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$ is fully faithful. Hence the functor $\Theta$ induces
isomorphisms of the groups $\operatorname{Ext}^{i}$ for all $0\leq i<\infty$.

Using the appropriate generalization of [25, Lemma 5.3(b)], one can show that
the sequence of central elements $s_{1}$, …, $s_{m}$ is also weakly proregular
in the ring $\mathfrak{R}$. So one can replace $R$ with $\mathfrak{R}$ and
consider the case $R=\mathfrak{R}$. Then $\Theta=\Theta_{P}$, and one can
conclude that $P=\mathfrak{R}$ is a good projective generator of $\mathsf{B}$.

(3) Even more generally, let $R$ be an associative ring and $I\subset R$ be
the ideal generated by a finite set of central elements $s_{1}$, …, $s_{m}\in
R$. Set
$\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$
and $P=\Delta_{I}(R)$ (see Example 2.3(1)). As above, $\mathsf{B}$ is a
locally $\aleph_{1}$-presentable abelian category, $P$ is an
$\aleph_{1}$-presentable projective generator of $\mathsf{B}$, and the natural
embedding functor $\Theta\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is corepresented by the projective generator
$P\in\mathsf{B}$ with the natural right action of the ring $R$ in it (provided
by the natural ring homomorphism $\theta\colon
R\longrightarrow\Delta_{I}(R)=\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$).

Arguing as in Example 5.2(3), one shows that the triangulated functor
$\mathsf{D}^{\star}(\mathsf{B})\longrightarrow\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$, where $\star={\mathsf{b}}$, $+$, $-$, or $\varnothing$,
is fully faithful if and only if
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s_{1},\dotsc,s_{m}),R[X]))=0$
for all sets $X$ and all $i>0$, or equivalently, if and only if
$\varprojlim_{n}H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R[X]))=0$
for $i\geq 1$ and
$\varprojlim_{n}^{1}H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R[X]))=0$
for $i\geq 2$. As explained in Example 5.2(4), this holds if and only if the
projective system
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),\allowbreak
R))$ is pro-zero for $i\geq 2$ and
$\varprojlim_{n}H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R))=0$.

Furthermore, all the $I$-contramodule left $R$-modules admit a unique natural
extension of their left $R$-module structure to a left $\Delta_{I}(R$)-module
structure. Central elements in $R$ remain central in $\Delta_{I}(R)$. So one
can replace $R$ with $\Delta_{I}(R$) and $I$ with $\Delta_{I}(R)I$, and
consider the case $R=\Delta_{I}(R)$; then $\Theta=\Theta_{P}$. Hence $P$ is
always a $1$-good projective generator of $\mathsf{B}$, as
$\mathsf{B}=\mathfrak{R}{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$
is closed under extensions in $R{\operatorname{\mathsf{--mod}}}$. As we have
seen, $P$ is a good projective generator of $\mathsf{B}$ if and only if
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}(R;s_{1},\dotsc,s_{m}),\mskip
1.5mu\Delta_{I}(R)[X]))=0$ for all sets $X$ and all $i>0$, and if and only if
the projective system
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),\Delta_{I}(R)))$
is pro-zero for $i\geq 2$ and
$\varprojlim_{n}H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),\Delta_{I}(R))=0$.

(4) Let $R$ be an associative ring and $I\subset R$ be an ideal generated by a
finite set of central elements. Set $\mathfrak{R}=\varprojlim_{n}R/I^{n}$,
$\mathsf{B}=\mathfrak{R}{\operatorname{\mathsf{--contra}}}$, and
$P=\mathfrak{R}$ (see Examples 1.3(2), 2.3(3), and 3.6(2-4)).

Then the abelian category $\mathsf{B}$ is an exactly embedded full subcategory
of the abelian category
$\mathfrak{R}{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$.
In the same way as in Example 5.2(6), one shows that every object of
$R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$ is an
extension of two objects from
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}$. Hence the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ is closed under extensions if and only if it
coincides with
$R{\operatorname{\mathsf{--mod}}}_{I{\operatorname{\mathsf{-ctra}}}}$, which
holds if and only if the projective system
$H_{1}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R))$
satisfies the Mittag-Leffler condition. Furthermore, for any derived category
symbol $\star={\mathsf{b}}$, $+$, $-$, or $\varnothing$, the functor
$\mathsf{D}^{\star}(\mathfrak{R}{\operatorname{\mathsf{--contra}}})\longrightarrow\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$
is fully faithful if and only if the projective system
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),R))$
is pro-zero for all $i\geq 1$.

According to Example 5.1(1), the projective generator
$P\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$ is $0$-good. Now we see
that $P$ is $1$-good if and only if the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\subset\mathfrak{R}{\operatorname{\mathsf{--mod}}}$
coincides with
$\mathfrak{R}{\operatorname{\mathsf{--mod}}}_{(\mathfrak{R}I){\operatorname{\mathsf{-ctra}}}}$,
and $P$ is good if and only if the projective system
$H_{i}(\operatorname{Hom}_{R}(T^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{n}(R;s_{1},\dotsc,s_{m}),\mathfrak{R}))$
is pro-zero for all $i\geq 1$.

In particular, when $I=(s)$ is a principal ideal, one shows, arguing as in
Example 5.2(7), that
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}=R{\operatorname{\mathsf{--mod}}}_{s{\operatorname{\mathsf{-ctra}}}}$
if and only if the $s$-torsion in $\Delta_{s}(R)$ is bounded, if and only if
the $s$-torsion in $\mathfrak{R}$ is bounded, and if and only if
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}=\mathfrak{R}{\operatorname{\mathsf{--mod}}}_{s{\operatorname{\mathsf{-ctra}}}}$.
These are the necessary and sufficient conditions for the projective generator
$P\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$ to be $1$-good or, which
is equivalent in this case, good.

######  Examples 5.4.

(1) Let $R$ be a commutative ring and $S\subset R$ be a multiplicative subset
such that $\operatorname{pd}_{R}S^{-1}R\leq 1$. Set
$\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$
and $P=\Delta_{S}(R)$ (see Example 2.4(1)). Then $\mathsf{B}$ is a locally
presentable abelian category, $P\in\mathsf{B}$ is a projective generator, and
the natural embedding functor $\Theta\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is corepresented by the projective generator
$P$ with the natural action of the ring $R$ in it (provided by the natural
ring homomorphism $\theta\colon
R\longrightarrow\Delta_{S}(R)=\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$).

According to [27, first paragraph of Remark 6.8], the triangulated functor
$\mathsf{D}(\mathsf{B})\longrightarrow\mathsf{D}(R{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$ is fully faithful provided that there is no
$S$-h-divisible $S$-torsion in $R$. Conversely, arguing as in [27, second
paragraph of Remark 6.8] (with torsion modules replaced by contramodules), one
shows that there is no $S$-h-divisible $S$-torsion in $R$ whenever the functor
$\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(R{\operatorname{\mathsf{--mod}}})$
is fully faithful. Thus, for any derived category symbol $\star={\mathsf{b}}$,
$+$, $-$, or $\varnothing$, the triangulated functor
$\mathsf{D}^{\star}(\mathsf{B})\longrightarrow\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$ is fully faithful if and only if there is no
$S$-h-divisible $S$-torsion in the $R$-module $R$.

According to [27, Proposition 3.1], $\Delta_{S}(R)$ is a commutative ring.
Since $S^{-1}R$ is a flat $R$-module, the condition that
$\operatorname{pd}_{R}S^{-1}R\leq\nobreak 1$ implies that
$\operatorname{pd}_{\Delta_{S}(R)}S^{-1}\Delta_{S}(R)\leq\nobreak 1$. All the
$S$-contramodule $R$-modules admit a unique natural extension of their
$R$-module structure to a $\Delta_{S}(R)$-module structure. So one can replace
$R$ with $\Delta_{S}(R)$ and $S$ with its image in $\Delta_{S}(R)$, and
consider the case $R=\Delta_{S}(R)$, so that $\Theta=\Theta_{P}$. Then the
$R$-module $R$ is $S$-h-reduced, and therefore has no $S$-h-divisible
$S$-torsion. Thus $P$ is always a good projective generator of $\mathsf{B}$.

(2) Let $R$ be a commutative ring and $S\subset R$ be a countable
multiplicative subset. Set $\mathfrak{R}=\varprojlim_{s\in S}R/sR$,
$\mathsf{B}=\mathfrak{R}{\operatorname{\mathsf{--contra}}}$, and
$P=\mathfrak{R}$ (see Examples 2.4(2) and 3.7(2)). Then the abelian category
$\mathsf{B}$ is an exactly embedded full subcategory of the abelian category
$R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$. Both of
these are locally $\aleph_{1}$-presentable abelian categories.

We claim that every object of
$R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$ is an
extension of two objects from $\mathfrak{R}{\operatorname{\mathsf{--contra}}}$
(cf. Example 5.2(6)). Indeed, every object of
$R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$ has the
form $\Delta_{S}(C)$ for some $C\in R{\operatorname{\mathsf{--mod}}}$. The
functor $\Delta_{S}$ can be computed as
$\Delta_{S}(C)=\operatorname{Ext}^{1}_{R}(K^{\text{\smaller\smaller$\scriptstyle\bullet$}},C)$
(see Example 2.4(1)).

Let $s_{1}$, $s_{2}$, $s_{3}$, … be a sequence of elements of $S$ containing
every element of $S$ infinitely many times. Set $t_{0}=1$ and
$t_{n}=s_{1}s_{2}\dotsm s_{n}$ for all $n\geq 1$. Then the two-term complex
$K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is the inductive limit of
the sequence of two-term complexes $R\overset{t_{n}}{\longrightarrow}R$ (with
the morphism of complexes
$(R\overset{t_{n-1}}{\longrightarrow}R)\longrightarrow(R\overset{t_{n}}{\longrightarrow}R)$
acting by $1$ on the leftmost terms of the complexes and by $s_{n}$ on the
rightmost terms). Using this description of
$K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, one can compute that for
any $R$-module $C$ there is a short exact sequence of $R$-modules
$0\longrightarrow\varprojlim_{n}^{1}{}_{t_{n}}C\longrightarrow\Delta_{S}(C)\longrightarrow\varprojlim_{n}C/t_{n}C=\Lambda_{S}(C)\longrightarrow
0$, where ${}_{t}M\subset M$ denotes the submodule of all elements annihilated
by an element $t\in R$ in an $R$-module $M$, and the maps
${}_{t_{n+1}}C\longrightarrow{}_{t_{n}}C$ are the multiplications with $s_{n}$
(see [32, Lemma 2.2]). Since any $R$-module annihilated by an element from $S$
is an $\mathfrak{R}$-contramodule, and the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ is closed under the kernels, cokernels, and
infinite products, it follows that both $\Lambda_{S}(C)$ and the kernel of the
surjective morphism $\Delta_{S}(C)\longrightarrow\Lambda_{S}(C)$ belong to
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}$.

So the full subcategory $\mathfrak{R}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ is closed under extensions if and only if it
coincides with
$R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$.
According to Example 2.4(3) and [8, Corollary 6] (cf. Example 5.2(4)), this
holds if and only if the projective system $({}_{t_{n}}R)_{n\geq 0}$ satisfies
the Mittag-Leffler condition. Furthermore, can easily see that
$\varprojlim_{n}{}_{t_{n}}R=0$ if and only if there is no $S$-h-divisible
$S$-torsion in $R$; and the projective system $({}_{t_{n}}R)_{n\geq 0}$ is
pro-zero if and only if the $S$-torsion in $R$ is bounded (which means that
there exists $t\in S$ such that $sr=0$, $s\in S$, $r\in R$, implies $tr=0$).
Comparing these observations with (1), we conclude that the functor
$\mathsf{D}^{\star}(\mathfrak{R}{\operatorname{\mathsf{--contra}}})\longrightarrow\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$
is fully faithful if and only if the $S$-torsion in $R$ is bounded.

Arguing as in Example 5.2(7) and using an appropriate version of [26, Remark
6.9], one can show that
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}=R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$
if and only if the $S$-torsion in $\Delta_{S}(R)$ is bounded, if and only if
the $S$-torsion in $\mathfrak{R}$ is bounded, and if and only if
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}=\mathfrak{R}{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$.
According to Example 5.1(1), the projective generator
$P\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$ is $0$-good. Now we see
that $P\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$ is $1$-good if and
only if it is good, and if and only if these equivalent conditions hold.

######  Examples 5.5.

(1) Let $R$ be an associative ring and $S\subset R$ be a multiplicative subset
of central elements in $R$ such that $\operatorname{pd}_{R}S^{-1}R\leq 1$. Set
$\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$
and $P=\Delta_{S}(R)$ (see Example 2.5(1)). Then $\mathsf{B}$ is a locally
presentable abelian category, $P\in\mathsf{B}$ is a projective generator, and
the natural embedding functor $\Theta\colon\mathsf{B}\longrightarrow
R{\operatorname{\mathsf{--mod}}}$ is corepresented by the projective generator
$P$ with the natural right action of the ring $R$ in it (provided by the
natural ring homomorphism $\theta\colon
R\longrightarrow\Delta_{S}(R)=\operatorname{Hom}_{\mathsf{B}}(P,P)^{\mathrm{op}}$).

Arguing as in Example 5.4(1), one shows that, for any derived category symbol
$\star={\mathsf{b}}$, $+$, $-$, or $\varnothing$, the triangulated functor
$\mathsf{D}^{\star}(\mathsf{B})\longrightarrow\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$ is fully faithful if and only if there are is no
$S$-h-divisible $S$-torsion in the left $R$-module $R$.

Furthermore, central elements in $R$ remain central in $\Delta_{S}(R)$. Since
$S^{-1}R$ is a flat left $R$-module, the condition that
$\operatorname{pd}_{R}S^{-1}R\leq\nobreak 1$ implies that
$\operatorname{pd}_{\Delta_{S}(R)}S^{-1}\Delta_{S}(R)\leq\nobreak 1$. All the
$S$-contramodule left $R$-modules admit a unique natural extension of their
left $R$-module structure to a left $\Delta_{S}(R)$-module structure. So one
can replace $R$ with $\Delta_{S}(R)$ and $S$ with its image in
$\Delta_{S}(R)$, and consider the case $R=\Delta_{S}(R)$, so that
$\Theta=\Theta_{P}$. Then the left $R$-module $R$ is $S$-h-reduced (i. e.,
there are no nonzero morphisms into it from left $S^{-1}R$-modules), and
therefore has no $S$-h-divisible $S$-torsion. Thus $P$ is always a good
projective generator of $\mathsf{B}$.

(2) Let $R$ be an associative ring and $S\subset R$ be a countable
multiplicative subset of central elements. Set $\mathfrak{R}=\varprojlim_{s\in
S}R/sR$, $\mathsf{B}=\mathfrak{R}{\operatorname{\mathsf{--contra}}}$, and
$P=\mathfrak{R}$ (see Examples 2.5(2-3) and 3.7(2)). The abelian category
$\mathsf{B}$ is an exactly embedded full subcategory of the abelian category
$R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$. Both
$\mathsf{B}$ and
$R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$ are
locally $\aleph_{1}$-presentable categories. Arguing as in Example 5.4(2), one
shows that every object of
$R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$ is an
extension of two objects from
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}$.

Thus the full subcategory
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}\subset
R{\operatorname{\mathsf{--mod}}}$ is closed under extensions if and only if it
coincides with
$R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$. The
functor
$\mathsf{D}^{\star}(\mathfrak{R}{\operatorname{\mathsf{--contra}}})\longrightarrow\mathsf{D}^{\star}(R{\operatorname{\mathsf{--mod}}})$
is fully faithful if and only if the $S$-torsion in $R$ is bounded.
Furthermore, one has
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}=R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$
if and only if the $S$-torsion in $\Delta_{S}(R)$ is bounded, if and only if
the $S$-torsion in $\mathfrak{R}$ is bounded, and if and only if
$\mathfrak{R}{\operatorname{\mathsf{--contra}}}=\mathfrak{R}{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}$.
The projective generator $P\in\mathfrak{R}{\operatorname{\mathsf{--contra}}}$
is always $0$-good. The previous several equivalent condition are necessary
and sufficient for $P$ to be $1$-good or, which is equivalent in this
situation, good.

The next series of examples explains the reason for the “good projective
generator” terminology.

######  Examples 5.6.

(1) Let $\mathsf{C}$ be an accessible additive category (notice that any such
category contains the images of idempotent endomorphisms of its objects [1,
Observation 2.4]. Let $M\in\mathsf{C}$ be an object. According to Example
1.2(2), there exists a unique abelian category $\mathsf{B}$ with enough
projective objects for which there is an equivalence of additive categories
$\mathsf{B}_{\mathsf{proj}}\simeq\mathsf{Add}_{\mathsf{C}}(M)$. The category
$\mathsf{B}$ comes endowed with a natural projective generator
$P\in\mathsf{B}_{\mathsf{proj}}$ corresponding to the free
$\mathbb{T}_{M}$-module with one generator
$\mathbb{T}_{M}(*)\in\mathbb{T}_{M}{\operatorname{\mathsf{--mod}}}$ and to the
object $M\in\mathsf{Add}_{\mathsf{C}}(M)$. If the object $M\in\mathsf{C}$ is
$\kappa$-presentable, where $\kappa$ is a regular cardinal, then the abelian
category $\mathsf{B}$ is locally $\kappa$-presentable and the object
$P\in\mathsf{B}$ is $\kappa$-presentable.

We will say that an object $M\in\mathsf{C}$ is _$n$ -good_, where $n\geq 0$ is
an integer, if the projective generator $P\in\mathsf{B}$ is $n$-good. In other
words, denoting by $\mathfrak{S}$ the ring
$\operatorname{Hom}_{\mathsf{C}}(M,M)^{\mathrm{op}}$, the object
$M\in\mathsf{C}$ is $n$-good if and only if the functor
$\Theta=\operatorname{Hom}_{\mathsf{B}}(P,{-})\colon\mathsf{B}\longrightarrow\mathfrak{S}{\operatorname{\mathsf{--mod}}}$
induces isomorphisms on all the groups $\operatorname{Ext}^{i}$ with $0\leq
i\leq n$. An object $M\in\mathsf{C}$ is _good_ if it is $n$-good for all
$n\geq 0$, or in other words, if the projective generator $P\in\mathsf{B}$ is
good. According to Proposition 4.8, an object $M$ is good if and only if the
triangulated functor
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathfrak{S}{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$ is fully faithful.

According to Theorem 4.10, if the object $M\in\mathsf{C}$ is
$\kappa$-presentable (or, more generally, $\kappa$-generated), then for any
cardinal $\lambda$ such that $\lambda^{+}\geq\kappa$ the object
$M^{(\lambda)}\in\mathsf{C}$ is good.

(2) In particular, let $R$ be an associative ring and $M$ be a left
$R$-module. Substituting $\mathsf{C}=R{\operatorname{\mathsf{--mod}}}$ into
the definitions in (1), we obtain the definition of what it means for $M$ to
be _$n$ -good_ or _good_. Specifically, it means that the forgetful functor
$\mathfrak{S}{\operatorname{\mathsf{--contra}}}\longrightarrow\mathfrak{S}{\operatorname{\mathsf{--mod}}}$,
where $\mathfrak{S}=\operatorname{Hom}_{R}(M,M)^{\mathrm{op}}$ is the
topological ring from Example 1.3(3), should induce isomorphisms on all the
groups $\operatorname{Ext}^{i}$ with $0\leq i\leq n$, or with $i\geq 0$,
respectively. An $R$-module $M$ is good if and only if the triangulated
functor
$\mathsf{D}^{-}(\mathfrak{S}{\operatorname{\mathsf{--contra}}})\longrightarrow\mathsf{D}^{-}(\mathfrak{S}{\operatorname{\mathsf{--mod}}})$
is fully faithful. If a left $R$-module $M$ admits a set of generators of
cardinality $\lambda$, then the left $R$-module $M^{(\lambda)}$ is good.

(3) Any finitely generated left $R$-module $M$ is good. Indeed, in this case
the monad $\mathbb{T}_{M}$ is isomorphic to the monad $\mathbb{T}_{S}$
associated with the discrete ring
$S=\operatorname{Hom}_{R}(M,M)^{\mathrm{op}}$ as in Example 1.3(1), so the
functor
$\Theta=\operatorname{Hom}_{\mathsf{B}}(P,{-})\colon\mathsf{B}\longrightarrow
S{\operatorname{\mathsf{--mod}}}$ is an equivalence of abelian categories.

(4) Let $R$ be a commutative ring and $S\subset R$ be a multiplicative subset
such that all elements of $S$ are nonzero-divisors in $R$. Assume that
$\operatorname{pd}_{R}S^{-1}R\leq 1$ and set $K=S^{-1}R/R\in
R{\operatorname{\mathsf{--mod}}}$. Then
$\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}=\mathfrak{R}{\operatorname{\mathsf{--contra}}}$,
where $\mathfrak{R}=\Delta_{S}(R)=\Lambda_{S}(R)$ (see Example 2.4(3)), is the
corresponding locally presentable abelian category with enough projective
objects such that
$\mathsf{B}_{\mathsf{proj}}\simeq\mathsf{Add}_{R{\operatorname{\mathsf{--mod}}}}(K)$,
and $P=\mathfrak{R}\in\mathsf{B}$ is its natural projective generator. Indeed,
the functors $Q\longmapsto K\otimes_{R}Q$ and
$N\longmapsto\operatorname{Hom}_{R}(K,N)$ establish an equivalence between the
additive categories $\mathsf{B}_{\mathsf{proj}}\ni Q$ and
$\mathsf{Add}_{R{\operatorname{\mathsf{--mod}}}}(K)\ni N$ assigning the object
$P\in\mathsf{B}_{\mathsf{proj}}$ to the object
$K\in\mathsf{Add}_{R{\operatorname{\mathsf{--mod}}}}(K)$ and the object
$P^{(X)}=\Delta_{S}(R[X])=\Lambda_{S}(R[X])\in\mathsf{B}_{\mathsf{proj}}$ to
the object $K^{(X)}\in\mathsf{Add}_{R{\operatorname{\mathsf{--mod}}}}(K)$ for
all sets $X$ (cf., e. g., [27, Corollary 5.2]). According to Example 5.4(1),
the projective generator $P\in\mathsf{B}$ is good, so the $R$-module
$K=S^{-1}R/R$ is good.

(5) More generally, let $R$ be an associative ring and $S\subset R$ be a
multiplicative subset consisting of some central nonzero-divisors in $R$.
Assume that $\operatorname{pd}_{R}S^{-1}R\leq\nobreak 1$ and set
$K=S^{-1}R/R\in R{\operatorname{\mathsf{--mod}}}$. Then
$\mathsf{B}=R{\operatorname{\mathsf{--mod}}}_{S{\operatorname{\mathsf{-ctra}}}}=\mathfrak{R}{\operatorname{\mathsf{--contra}}}$,
where $\mathfrak{R}=\Delta_{S}(R)=\Lambda_{S}(R)$ (see Example 2.5(3)), is the
corresponding locally presentable abelian category with enough projective
objects such that
$\mathsf{B}_{\mathsf{proj}}\simeq\mathsf{Add}_{R{\operatorname{\mathsf{--mod}}}}(K)$,
and $P=\mathfrak{R}\in\mathsf{B}$ is its natural projective generator. Indeed,
just as in (4), the functors $Q\longmapsto K\otimes_{R}Q$ and
$N\longmapsto\operatorname{Hom}_{R}(K,N)$ (where $K$ is viewed as an
$R$-$R$-bimodule) establish an equivalence between the additive categories
$\mathsf{B}_{\mathsf{proj}}$ and
$\mathsf{Add}_{R{\operatorname{\mathsf{--mod}}}}(K)$ assigning
$P\in\mathsf{B}_{\mathsf{proj}}$ to
$K\in\mathsf{Add}_{R{\operatorname{\mathsf{--mod}}}}(K)$ and
$P^{(X)}=\Delta_{S}(R[X])=\Lambda_{S}(R[X])\in\mathsf{B}_{\mathsf{proj}}$ to
$K^{(X)}\in\mathsf{Add}_{R{\operatorname{\mathsf{--mod}}}}(K)$ for all sets
$X$ (cf. [29]). According to Example 5.5(1), the projective generator
$P\in\mathsf{B}$ is good, so the left $R$-module $K=S^{-1}R/R$ is good.

(6) Let $T$ be a _good $n$-tilting_ left $R$-module in the sense of [4] (where
$n\geq 0$ is an integer). Then, according to [4, Theorem 2.2(2)], the
triangulated functor
$\mathbb{R}\operatorname{Hom}_{R}(T,{-})\colon\mathsf{D}(R{\operatorname{\mathsf{--mod}}})\longrightarrow\mathsf{D}(\mathfrak{S}{\operatorname{\mathsf{--mod}}})$
is fully faithful.

On the other hand, by [10, Proposition 2.3] or [3, Theorem 4.5] (cf. the
discussion in [33, Proposition 7.13]), the triangulated functor
$\mathbb{R}\operatorname{Hom}_{R}(T,{-})\colon\mathsf{D}(R{\operatorname{\mathsf{--mod}}})\longrightarrow\mathsf{D}(\mathfrak{S}{\operatorname{\mathsf{--contra}}})$
is an equivalence of triangulated categories. (Notice that, by [3, Proposition
4.3] and our Examples 1.2(2) and 1.3(3), the heart
$\mathsf{B}\subset\mathsf{D}^{\mathsf{b}}(R{\operatorname{\mathsf{--mod}}})$
of the tilting t-structure associated with $T$ is equivalent to the abelian
category $\mathfrak{S}{\operatorname{\mathsf{--contra}}}$ [33, Corollary
7.2].)

Thus the triangulated functor
$\mathsf{D}(\mathfrak{S}{\operatorname{\mathsf{--contra}}})\longrightarrow\mathsf{D}(\mathfrak{S}{\operatorname{\mathsf{--mod}}})$
induced by the forgetful functor
$\mathfrak{S}{\operatorname{\mathsf{--contra}}}\longrightarrow\mathfrak{S}{\operatorname{\mathsf{--mod}}}$
is fully faithful, so $T$ is a good left $R$-module in the sense of our
definition.

##  6\. $\kappa$-Flat Modules and the Fully Faithful Triangulated Functor

The aim of this section is to prove Theorem 4.10. For this purpose, we develop
the theory of $\kappa$-flat modules over associative rings, generalizing the
Govorov–Lazard characterization of flat modules to arbitrary regular
cardinals.

Let $R$ be an associative ring and $\kappa$ be a regular cardinal.

######  Theorem 6.1.

The following conditions on a left $R$-module $F$ are equivalent:

(a) every morphism into $F$ from a $\kappa$-presentable left $R$-module
factorizes through a projective left $R$-module;

(b) every morphism into $F$ from a $\kappa$-presentable left $R$-module
factorizes through a free left $R$-module with less than $\kappa$ generators;

(c) $F$ is the colimit of a $\kappa$-filtered diagram of projective left
$R$-modules;

(d) the cocone formed by all the morphisms into $F$ from projective left
$R$-modules with less than $\kappa$ generators (and all the morphisms between
the latter forming commutative triangles with the morphism into $F$) is a
$\kappa$-filtered colimit cocone;

(e) the cocone formed by all the morphisms into $F$ from free left $R$-modules
with less than $\kappa$ generators (and all the morphisms between the latter
forming commutative triangles with the morphism into $F$) is a
$\kappa$-filtered colimit cocone.

###### Proof.

(a) $\Longrightarrow$ (b): given a left $R$-module morphism $E\longrightarrow
F$, where $E$ is $\kappa$-presentable, one first factorizes the morphism
$E\longrightarrow F$ through a projective left $R$-module $P$, then replaces
$P$ with a free left $R$-module $P^{\prime}$ in which $P$ is a direct summand,
and finally replaces $P^{\prime}$ with its free submodule $P^{\prime\prime}$
with less than $\kappa$ generators containing the image of the morphism
$E\longrightarrow P^{\prime}$.

(c) $\Longrightarrow$ (a): let $F=\varinjlim_{i}P_{i}$ be a presentation of
$F$ as the colimit of a $\kappa$-filtered diagram of projective left
$R$-modules, and let $E$ be a $\kappa$-presentable left $R$-module. Then
$\operatorname{Hom}_{R}(E,\varinjlim_{i}P_{i})=\varinjlim_{i}\operatorname{Hom}_{R}(E,P_{i})$,
hence any left $R$-module morphism $E\longrightarrow F$ factorizes through one
of the modules $P_{i}$.

(b) $\Longrightarrow$ (d): denote our cocone by $(P_{i}\to F)_{i\in I}$ (where
$I$ is the set of all pairs $i=(P_{i},f_{i})$, with $P_{i}$ a projective left
$R$-module with less than $\kappa$ generators and $f_{i}\colon
P_{i}\longrightarrow F$ a left $R$-module morphism). Since the free left
$R$-module with one generator $R$ is a projective left $R$-module with less
than $\kappa$ generators, and every element of $F$ belongs to the image of
some left $R$-module morphism $R\longrightarrow F$, the natural morphism
$\varinjlim_{i\in I}P_{i}\longrightarrow F$ is surjective.

Furthermore, given two morphisms $f_{i}\colon P_{i}\longrightarrow F$ and
$f_{j}\colon P_{j}\longrightarrow F$ in the cocone, and two elements
$p^{\prime}\in P_{i}$ and $p^{\prime\prime}\in P_{j}$ such that
$f_{i}(p^{\prime})=f_{j}(p^{\prime\prime})\in F$, there exist left $R$-module
morphisms $R\longrightarrow P_{i}$ and $R\longrightarrow P_{j}$ taking the
unit element $1\in R$ to $p^{\prime}$ and $p^{\prime\prime}$, respectively.
Setting $P_{k}=R$ and denoting by $f_{k}\colon P_{k}\longrightarrow F$ the
morphism taking $1$ to $f_{i}(p^{\prime})=f_{j}(p^{\prime\prime})$, we have
morphisms $k\to i$ and $k\to j$ in the diagram $I$. If $I$ is filtered, there
exists a morphism $f_{l}\colon P_{l}\longrightarrow F$ in the cocone and
morphisms $i\to l$, $j\to l$ in the diagram $I$ such that the square diagram
$k\to i\to l$, $k\to j\to l$ is commutative. It follows that images of the
elements $p^{\prime}\in P_{i}$ and $p^{\prime\prime}\in P_{j}$ coincide in
$P_{l}$, and hence also in $\varinjlim_{i\in I}P_{i}$. Thus the morphism
$\varinjlim_{i\in I}P_{i}\longrightarrow F$ is an isomorphism whenever $I$ is
filtered.

It remains to show that the diagram $I$ is $\kappa$-filtered whenever (b)
holds. Any collection of less than $\kappa$ morphisms $L_{i}\longrightarrow
F$, where $L_{i}$ are projective left $R$-modules with less than $\kappa$
generators, factorizes through the morphism
$M=\bigoplus_{i}L_{i}\longrightarrow F$, and $M$ is also a projective left
$R$-module with less than $\kappa$ generators.

Now let $h_{i}\colon L\longrightarrow M$ be a collection of less than $\kappa$
morphisms between two projective left $R$-modules $L$ and $M$ with less than
$\kappa$ generators each. Then the coequalizer $E$ of the whole system of
morphisms $h_{i}\colon L\longrightarrow M$ is a $\kappa$-presentable left
$R$-module. Given left $R$-module morphisms $f\colon L\longrightarrow F$ and
$g\colon M\longrightarrow F$ forming commutative triangles with all the
morphisms $h_{i}$, we have the induced left $R$-module morphism
$E\longrightarrow F$. Assuming (b), the latter morphism factorizes through a
projective left $R$-module $P$ with less than $\kappa$ generators. Hence we
have a morphism $M\longrightarrow P$ in the diagram $I$ whose compositions
with all the morphisms $h_{i}\colon L\longrightarrow M$ are equal to one and
the same morphism $L\longrightarrow P$.

The proof of (b) $\Longrightarrow$ (e) is similar to (b) $\Longrightarrow$
(d). The implications (d) $\Longrightarrow$ (c), (e) $\Longrightarrow$ (c),
and (b) $\Longrightarrow$ (a) are obvious. ∎

A left $R$-module $F$ is said to be _$\kappa$ -flat_ if it satisfies one of
the equivalent conditions of Theorem 6.1.

######  Lemma 6.2.

(a) The class of all $\kappa$-flat left $R$-modules is closed under
extensions, kernels of surjective morphisms, and $\kappa$-filtered colimits.

(b) Any short exact sequence of $\kappa$-flat left $S$-modules is a
$\kappa$-filtered colimit of split short exact sequences of $\kappa$-flat left
$S$-modules.

###### Proof.

If $G=\varinjlim_{i}F_{i}$ is the colimit of a $\kappa$-filtered diagram of
modules and $E$ is a $\kappa$-presentable module, then any morphism
$E\longrightarrow G$ factorizes through one of the modules $F_{i}$. If, in
addition, any morphism $E\longrightarrow F_{i}$ factorizes through a
projective module, then any morphism $E\longrightarrow G$ also factorizes
through a projective module.

Let $0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$ be
a short exact sequence of left $R$-modules. Suppose that the left $R$-modules
$G$ and $H$ are $\kappa$-flat. Let $E$ be a $\kappa$-presentable left
$R$-module and $E\longrightarrow F$ be a left $R$-module morphism. Then the
composiiton $E\longrightarrow F\longrightarrow G$ factorizes through a free
left $R$-module $P$ with less than $\kappa$ generators. Now the composition
$P\longrightarrow G\longrightarrow H$ factorizes through the cokernel $C=P/E$
of the morphism $E\longrightarrow P$. Since the left $R$-module $C$ is also
$\kappa$-presentable, the morphism $C\longrightarrow H$ factorizes through a
projective left $R$-module $Q$.

Pick an arbitrary lifting $Q\longrightarrow G$ of the morphism
$Q\longrightarrow H$ and subtract the composition $P\longrightarrow
C\longrightarrow Q\longrightarrow G$ from the morphism $P\longrightarrow G$
that we have. The resulting morphism $P\longrightarrow G$ is annihilated by
the composition with the morphism $G\longrightarrow H$, and therefore
factorizes through the monomorphism $F\longrightarrow G$. We have obtained a
morphism $P\longrightarrow F$ whose composition with the morphism
$E\longrightarrow P$ is equal to our original morphism $E\longrightarrow F$.
Thus the left $R$-module $F$ is $\kappa$-flat.

Now suppose that the left $R$-modules $F$ and $H$ are $\kappa$-flat. Then $H$
is a $\kappa$-filtered colimit of projective left $R$-modules $Q_{i}$, hence
it follows that the short exact sequence $0\longrightarrow F\longrightarrow
G\longrightarrow H\longrightarrow 0$ is a $\kappa$-filtered colimit of split
short exact sequences $0\longrightarrow F\longrightarrow F\oplus
Q_{i}\longrightarrow Q_{i}\longrightarrow 0$ (proving the assertion (b)). In
particular, the left $R$-module $G$ is a $\kappa$-filtered colimit of left
$R$-modules isomorphic to $F\oplus Q_{i}$. The latter are obviously
$\kappa$-flat; and a $\kappa$-filtered colimit of $\kappa$-flat modules is
$\kappa$-flat, as we have already seen. Thus the left $R$-module $G$ is
$\kappa$-flat. ∎

######  Remark 6.3.

We are not aware of a definition equivalent to our notion of $\kappa$-flatness
appearing anywhere in the previously existing literature, but certainly there
are all kinds of similar or related definitions known for many years.

In particular, there is the classical notion of a _$\kappa$ -free_ module [7].
In the similar spirit (cf. [6]), one can define _$\kappa$ -projective_
modules. Specifically, an $R$-module $M$ is said to be $\kappa$-projective if
it admits a system of submodules $M_{\alpha}$ (which is said to _witness_ the
$\kappa$-projectivity of $M$) such that (1) every $R$-module $M_{\alpha}$ is
projective with less than $\kappa$ generators; (2) every subset of $M$ of the
cardinality less than $\alpha$ is contained in one of the modules
$M_{\alpha}$; and (3) the set of submodules $M_{\alpha}\subset M$ is closed
under unions of well-ordered chains of length smaller than $\kappa$. An
$R$-module $M$ is said to be _$\kappa$ -projective in the weak sense_ if it
has a witnessing system of submodules satisfying the conditions (1) and (2)
[7, Section IV.1].

An $R$-module is $\kappa$-projective in the weak sense if and only if it is
the colimit of a $\kappa$-filtered diagram of projective $R$-modules with less
than $\kappa$ generators _and injective morphisms between them_. Indeed, given
a module that is $\kappa$-projective in the weak sense, all its projective
submodules with less than $\kappa$ generators form such a diagram. Thus
$\kappa$-projectivity is a stronger condition than $\kappa$-flatness. When $R$
is a left hereditary ring, there is no difference between
$\kappa$-projectivity and $\kappa$-projectivity in the weak sense for left
$R$-modules. Assuming additionally that every left ideal in $R$ is generated
by less than $\kappa$ elements (so any $\kappa$-generated left $R$-module is
$\kappa$-presentable), these two conditions are also equivalent to
$\kappa$-flatness. Indeed, let $F$ be a $\kappa$-flat left $R$-module. Given a
subset of cardinality less than $\kappa$ in $F$, denote by $E$ the
$R$-submodule in $F$ generated by this subset. Then $E$ is
$\kappa$-presentable, so the injective morphism $E\longrightarrow F$
factorizes through a projective left $R$-module $P$. It follows that $E$ is a
submodule in $P$, hence a projective module itself.

Another classical concept closely related to $\kappa$-flatness is that of
_$\kappa$ -purity_. A left $R$-module morphism $f\colon P\longrightarrow Q$ is
said to be a _$\kappa$ -pure epimorphism_ if, for any $\kappa$-presentable
left $R$-module $E$, any left $R$-module morphism $E\longrightarrow Q$
factorizes through $f$. If this is the case, $Q$ is said to be a _$\kappa$
-pure quotient_ of $P$ [16, Chapter 7], [2]. One can easily see that an
$R$-module is $\kappa$-flat if and only if it is a $\kappa$-pure quotient of a
projective $R$-module.

Finally, some of the assertions and proofs of our Theorem 6.1 and Lemma 6.2
resemble those of the theory of _modules with support in a subcategory_ (of
finitely presented modules), as developed in [18] (see [18, Propositions
2.1–2]).

The following result, whose nonadditive version goes back to [14, Section
2.2], was rediscovered and discussed in the additive/abelian context in [33,
Theorem 6.6].

######  Theorem 6.4.

Let $\mathsf{B}$ be a locally $\kappa$-presentable abelian category and
$P\in\mathsf{B}$ be a $\kappa$-presentable projective generator of $P$ (where
$\kappa$ is a regular cardinal). Let $\lambda$ be a cardinal such that
$\lambda^{+}\geq\kappa$ and let $Q=P^{(\lambda)}\in\mathsf{B}$ be the
coproduct of $\lambda$ copies of $P$ in $\mathsf{B}$. Denote by $S$ the ring
$\operatorname{Hom}_{\mathsf{B}}(Q,Q)^{\mathrm{op}}$. Then the exact functor
$\Theta=\operatorname{Hom}_{\mathsf{B}}(Q,{-})\colon\mathsf{B}\longrightarrow
S{\operatorname{\mathsf{--mod}}}$ corepresented by $Q$ is fully faithful.

###### Sketch of proof.

Let us identify $\mathsf{B}$ with the category of modules over the
$\kappa$-accessible monad $\mathbb{T}_{P}\colon
X\longmapsto\operatorname{Hom}_{\mathsf{B}}(P,P^{(X)})$ on the category of
sets. Let $Y$ be a set of the cardinality $\lambda$; then the projective
object $P\in\mathsf{B}$ corresponds to the free $\mathbb{T}_{P}$-module with
one generator $\mathbb{T}_{P}(*)$ and the projective object $Q\in\mathsf{B}$
corresponds to the free $\mathbb{T}_{P}$-module with $\lambda$ generators
$\mathbb{T}_{P}(Y)$. The functor
$\Theta\colon\mathbb{T}_{P}{\operatorname{\mathsf{--mod}}}\longrightarrow
S{\operatorname{\mathsf{--mod}}}$ assigns to a $\mathbb{T}_{P}$-module $B$ the
$S$-module
$\Theta(B)=\operatorname{Hom}_{\mathbb{T}_{P}}(\mathbb{T}_{P}(Y),B)=B^{Y}$.

Let $B$ and $C$ be two $\mathbb{T}_{P}$-modules and $g\colon
B^{Y}\longrightarrow C^{Y}$ be an $S$-module morphism. Given an element $y\in
Y$, denote by $\delta_{B}\colon B\longrightarrow B^{Y}$ the diagonal embedding
and by ${\mathrm{pr}}_{y,B}\colon B^{Y}\longrightarrow B$ the projection onto
the component indexed by $y$. Then the composition
$\delta_{B}{\mathrm{pr}}_{y,B}\colon B^{Y}\longrightarrow B^{Y}$ is equal to
the action of the element $s_{y}\in S$ corresponding to the composition of the
natural morphism $P^{(Y)}\longrightarrow P$ with the coproduct injection of
the $y$-indexed component $P\longrightarrow P^{(Y)}$. Thus the morphism $g$
forms a commutative diagram with the maps $\delta_{B}{\mathrm{pr}}_{y,B}$ and
$\delta_{C}{\mathrm{pr}}_{y,C}$,

$\begin{diagram}$

As this holds for all elements $y\in Y$, it follows easily that there is a map
$f\colon B\longrightarrow C$ such that $g=f^{Y}$.

Now, for any $Y$-ary operation $t\in\mathbb{T}_{P}(Y)$ in the monad
$\mathbb{T}_{P}$, the composition of the natural morphism
$P^{(Y)}\longrightarrow P$ with the morphism $t\colon P\longrightarrow
P^{(Y)}$ defines an element $s_{t}\in S$. The action of $s_{t}$ in $B^{Y}$ is
equal to the composition of the $Y$-ary operation $t_{\mathbb{T}_{P}}(B)\colon
B^{Y}\longrightarrow B$ in $B$ (see Section 1) with the diagonal embedding
$\delta_{B}\colon B\longrightarrow B^{Y}$. Commutativity of the diagram

$\begin{diagram}$

means that the map $f\colon B\longrightarrow C$ preserves the operation
$t_{\mathbb{T}_{P}}$ in $B$ and $C$. As this holds for all
$t\in\mathbb{T}_{P}(Y)$ and all operations in the monad $\mathbb{T}_{P}$
depend essentially on at most $\lambda$ arguments, it follows that $f$ is a
morphism of $\mathbb{T}_{P}$-modules. ∎

The argument deducing Theorem 4.10 from Theorem 6.4 is based on a technique
summarized in the following proposition.

Let $\mathsf{A}$ be an abelian category with enough projective objects and
$\mathsf{B}\subset\mathsf{A}$ be a full subcategory closed under the kernels
and cokernels in $\mathsf{A}$; so $\mathsf{B}$ is also an abelian category and
the embedding functor $\mathsf{B}\longrightarrow\mathsf{A}$ is exact. Assume
that there exists a functor $\Delta\colon\mathsf{A}\longrightarrow\mathsf{B}$
left adjoint to the fully faithful embedding functor
$\mathsf{B}\longrightarrow\mathsf{A}$. Then the functor $\Delta$ takes
projective objects in $\mathsf{A}$ to projective objects in $\mathsf{B}$, and
it follows easily that there are enough projectives in $\mathsf{B}$ (cf. Lemma
4.5). Let $\mathbb{L}_{n}\Delta\colon\mathsf{A}\longrightarrow\mathsf{B}$,
$n\geq 0$, denote the left derived functor of the right exact functor
$\Delta$.

######  Proposition 6.5.

The following four conditions are equivalent:

(a) $\mathbb{L}_{n}\Delta(P)=0$ for every object
$P\in\mathsf{B}_{\mathsf{proj}}\subset\mathsf{A}$ and all $n\geq 1$;

(b) $\mathbb{L}_{n}\Delta(B)=0$ for every object
$B\in\mathsf{B}\subset\mathsf{A}$ and all $n\geq 1$;

(c) the triangulated functor
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$ induced
by the exact embedding functor $\mathsf{B}\longrightarrow\mathsf{A}$ is fully
faithful;

(d) the triangulated functor
$\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{A})$
induced by the exact embedding functor $\mathsf{B}\longrightarrow\mathsf{A}$
is fully faithful.

###### Proof.

This is an infinite homological dimension version of [27, Theorem 6.4] (see
also [25, proofs of Theorems 1.3 and 2.9]).

(a) $\Longleftrightarrow$ (b): The composition
$\mathsf{B}\longrightarrow\mathsf{A}\longrightarrow\mathsf{B}$ of the
embedding $\mathsf{B}\longrightarrow\mathsf{A}$ with its left adjoint functor
$\Delta\colon\mathsf{A}\longrightarrow\mathsf{B}$ is the identity functor
$\operatorname{Id}_{\mathsf{B}}$, since the embedding functor
$\mathsf{B}\longrightarrow\mathsf{A}$ is fully faithful. Let
$P_{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow B$ be a left
projective resulution of an object $B\in\mathsf{B}$. In the assumption of (a),
since $\mathbb{L}_{n}\Delta(P_{i})=0$ for all $i\geq 0$ and $n\geq 1$, the
complex $\Delta(P_{\text{\smaller\smaller$\scriptstyle\bullet$}})$ computes
the derived functor $\mathbb{L}_{*}\Delta(B)$. However, we have
$\Delta(P_{\text{\smaller\smaller$\scriptstyle\bullet$}})=P_{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
hence $\mathbb{L}_{n}\Delta(B)=0$ for $n\geq 1$.

(b) $\Longleftrightarrow$ (c): Denote by
$\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}}\subset\mathsf{A}$ the full
subcategory of all objects $A\in\mathsf{A}$ such that
$\mathbb{L}_{n}\Delta(A)=\nobreak 0$ for all $n\geq 1$. Then the full
subcategory
$\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}}\subset\mathsf{A}$ is closed
under extensions and the kernels of epimorphisms in $\mathsf{A}$; so
$\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}}$ inherits an exact category
structure from the abelian category $\mathsf{A}$. Furthermore, every object of
$\mathsf{A}$ is the image of an epimorphism from an object of
$\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}}$. It follows that the
triangulated functor
$\mathsf{D}^{-}(\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$
induced by the exact embedding
$\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}}\longrightarrow\mathsf{A}$
is a triangulated equivalence (e. g., by [23, Proposition A.3.1(a)]; or simply
because both the derived categories in question are equivalent to the homotopy
category of bounded above complexes of projective objects in $\mathsf{A}$).

The restriction of the functor $\Delta$ to the exact subcategory
$\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}}\subset\mathsf{A}$ is an
exact functor
$\Delta\colon\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}}\longrightarrow\mathsf{B}$.
Applying the functor $\Delta$ to bounded above complexes of objects from
$\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}}$, one constructs a
triangulated functor

$\mathbb{L}\Delta\colon\mathsf{D}^{-}(\mathsf{A})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\mathsf{D}^{-}(\mathsf{B}),$

which is left adjoint to the triangulated functor
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$ induced
by the embedding $\mathsf{B}\longrightarrow\mathsf{A}$ [21, Lemma 8.3].

Now the functor
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$ is fully
faithful if and only if its composition
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})\longrightarrow\mathsf{D}^{-}(\mathsf{B})$
with its left adjoint functor
$\mathbb{L}\Delta\colon\mathsf{D}^{-}(\mathsf{A})\longrightarrow\mathsf{D}^{-}(\mathsf{B})$
is the identity functor
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{B})$. In the
situation of (b), we have
$\mathsf{B}\subset\mathsf{A}_{\Delta{\operatorname{\mathsf{-adj}}}}$. Since
the composition $\mathsf{B}\longrightarrow\mathsf{A}\longrightarrow\mathsf{B}$
is the identity functor, it follows that so is the composition
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})\longrightarrow\mathsf{D}^{-}(\mathsf{B})$.

Conversely, if
$\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})\longrightarrow\mathsf{D}^{-}(\mathsf{B})$
is the identity functor, we obviously have $\mathbb{L}_{n}\Delta(B)=0$ for all
$B\in\mathsf{B}$ and $n\geq 1$.

The implication (c) $\Longrightarrow$ (d) is obvious.

(d) $\Longrightarrow$ (b): Fix an integer $m\geq 0$, and denote by
$\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})\subset\mathsf{D}^{\mathsf{b}}(\mathsf{B})$
the full subcategory of bounded complexes with the cohomology objects
concentrated in the cohomological degrees $\geq-m$. The embedding functor
$\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{B})$
has a left adjoint, which is the canonical truncation functor
$\tau^{\geq-m}\colon\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})$
(taking a complex $\dotsb\longrightarrow B^{-m-2}\longrightarrow
B^{-m-1}\longrightarrow B^{-m}\longrightarrow B^{-m+1}\longrightarrow\dotsb$
to the complex $\dotsb\longrightarrow 0\longrightarrow 0\longrightarrow
B^{-m}/\operatorname{im}B^{-m-1}\longrightarrow
B^{-m+1}\longrightarrow\dotsb$). Hence the composition of functors
$\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$
also has a left adjoint, which can be computed as the composition of the two
left adjoints $\tau^{\geq-m}\mathbb{L}\Delta$.

On the other hand, the same functor
$\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$
can be obtained as the composition
$\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{A})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$.
In the assumption of (d), the functor
$\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{A})$
induced by $\Theta$ is fully faithful. Since the embedding functors
$\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(\mathsf{B})$
and
$\mathsf{D}^{\mathsf{b}}(\mathsf{A})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$
are fully faithful, too, it follows that so is the composition
$\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})$.

Therefore, the composition of the two adjoint functors
$\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})\longrightarrow\mathsf{D}^{-}(\mathsf{A})\longrightarrow\mathsf{D}^{{\mathsf{b}},\geq-m}(\mathsf{B})$
is isomorphic to the identity functor. In other words, it means that the
adjunction morphism $B\longrightarrow\tau^{\geq-m}\mathbb{L}\Delta(B)$ is an
isomorphism for every $B\in\mathsf{B}$. Since $m\geq 0$ is an arbitrary
integer, it follows that $\mathbb{L}_{n}\Delta(B)=0$ for all $n\geq 0$. ∎

###### Proof of Theorem 4.10.

Denote by $S$ the ring $\operatorname{Hom}_{\mathsf{B}}(Q,Q)^{\mathrm{op}}$.
First of all, we already know from Theorem 6.4 that the functor
$\Theta=\operatorname{Hom}_{\mathsf{B}}(Q,{-})\colon\mathsf{B}\longrightarrow
S{\operatorname{\mathsf{--mod}}}$ is fully faithful. So $Q$ is a $0$-good
projective generator of $\mathsf{B}$.

To prove that $Q$ is a good projective generator, we have to check that the
triangulated functor
$\mathsf{D}^{\mathsf{b}}(\mathsf{B})\longrightarrow\mathsf{D}^{\mathsf{b}}(S{\operatorname{\mathsf{--mod}}})$
induced by $\Theta$ is fully faithful. It was explained in the proof of
Proposition 4.8 that the exact embedding functor
$\Theta\colon\mathsf{B}\longrightarrow S{\operatorname{\mathsf{--mod}}}$ has a
left adjoint functor $\Delta\colon
S{\operatorname{\mathsf{--mod}}}\longrightarrow\mathsf{B}$. According to
Proposition 6.5, it remains to show that the projective objects of the abelian
full subcategory $\mathsf{B}\subset S{\operatorname{\mathsf{--mod}}}$ are
adjusted to $\Delta$, that is $\mathbb{L}_{n}\Delta(\Theta(P))=0$ for all
$P\in\mathsf{B}_{\mathsf{proj}}$ and all $n\geq 1$.

As any projective object in $\mathsf{B}$ is a direct summand of an object of
the form $Q^{(X)}$, where $X$ is some set, it suffices to check that the left
$S$-module $\Theta(Q^{(X)})$ is adjusted to $\Delta$ for all sets $X$. Now,
the functor $\Theta$ preserves $\lambda^{+}$-filtered colimits (since the
object $Q\in\mathsf{B}$ is $\lambda^{+}$-presentable). Hence we have
$\Theta(Q^{(X)})=\varinjlim_{Z}\Theta(Q^{(Z)})$, where the colimit is taken
over all the subsets $Z\subset X$ of cardinality not exceeding $\lambda$.

Furthermore, $Q^{(Z)}\simeq Q$ in $\mathsf{B}$ for all nonempty sets $Z$ of
cardinality $\leq\nobreak\lambda$, so the left $S$-module $\Theta(Q^{(Z)})$ is
isomorphic to $S$. By Lemma 6.2(a), it follows that the left $S$-module
$\Theta(Q^{(X)})$ is $\lambda^{+}$-flat for all sets $X$, as a
$\lambda^{+}$-filtered colimit of free left $S$-modules. So the functor
$\Theta$ takes the projective objects of $\mathsf{B}$ to $\lambda^{+}$-flat
left $S$-modules (this observation improves upon the result of [33, Lemma
6.9], where it was noticed that $\Theta$ takes the projective objects of
$\mathsf{B}$ to flat left $S$-modules).

Let us show that all the $\lambda^{+}$-flat left $S$-modules are adjusted to
$\Delta$, that is $\mathbb{L}_{n}\Delta(F)=\nobreak 0$ for all
$\lambda^{+}$-flat left $S$-modules $F$ and all $n\geq 1$. Since the
projective left $S$-modules are $\lambda^{+}$-flat and the class of all
$\lambda^{+}$-flat left $S$-modules is closed under the kernels of surjective
morphisms, we only need to check that the functor $\Delta$ preserves exactness
of short exact sequences of $\lambda^{+}$-flat left $S$-modules.

By Lemma 6.2(b), every short exact sequence of $\lambda^{+}$-flat $S$-modules
is a $\lambda^{+}$-filtered colimit of split short exact sequences. It remains
to point out that the functor $\Delta$ preserves all colimits, and that
$\lambda^{+}$-filtered colimits are exact in $\mathsf{B}$ (e. g., because they
are preserved by the conservative exact functor $\Theta$ and exact in
$R{\operatorname{\mathsf{--mod}}}$; cf. [1, Proposition 1.59] and [30,
Definition 2.1]). ∎

##  7\. Nonabelian Subcategories Defined by Perpendicularity Conditions

The aim of this section is to show what can happen if one relaxes the
conditions in the definitions of right $n$-perpendicular subcategories in
Section 4. We start with one positive assertion before proceeding to present
various counterexamples.

######  Lemma 7.1.

Let $\mathsf{A}$ be a locally presentable abelian category and
$\mathsf{B}\subset\mathsf{A}$ be a right $n$-perpendicular subcategory to a
set of objects or morphisms in $\mathsf{A}$, where $n\geq 0$. Then
$\mathsf{B}$ is an additive category with kernels, cokernels, infinite direct
sums and products. The kernels and products in $\mathsf{B}$ coincide with
those computed in $\mathsf{A}$, while the cokernels and coproducts in
$\mathsf{B}$ can be obtained by applying the reflector
$\Delta\colon\mathsf{A}\longrightarrow\mathsf{B}$ to the cokernels and
coproducts computed in $\mathsf{A}$.

###### Proof.

In view of Lemma 4.1(b), it suffices to consider the case $n=0$. Then, by [1,
Theorem 1.39] (cf. the discussion in the proof of Lemma 4.4), the category
$\mathsf{B}$ is locally presentable and reflective as a full subcategory in
$\mathsf{A}$. By the definition and by [1, Corollary 1.28], any locally
presentable category is complete and cocomplete. It remains to observe that
the full subcategory $\mathsf{B}\subset\mathsf{A}$ is closed under limits
(Lemma 4.2(a)), while the reflector
$\Delta\colon\mathsf{A}\longrightarrow\mathsf{B}$, being a left adjoint
functor, preserves colimits. Before we finish, let us recall that, when $n\geq
2$, the cokernels in $\mathsf{B}$ also coincide with those computed in
$\mathsf{A}$ by Lemma 4.2(c). ∎

Our counterexamples are produced by the following construction inspired by
Example 4.3. Let $R\longrightarrow S$ be a homomorphism of associative rings.
Denote by $\mathsf{A}$ the abelian category whose objects are triples
$(N,M,f)$, where $N$ is a left $S$-module, $M$ is a left $R$-module, and
$f\colon N\longrightarrow M$ is an $R$-module morphism. The morphisms
$(N^{\prime},M^{\prime},f^{\prime})\longrightarrow(N^{\prime\prime},M^{\prime\prime},f^{\prime\prime})$
in $\mathsf{A}$ are pairs consisting of an $S$-module morphism
$N^{\prime}\longrightarrow N^{\prime\prime}$ and an $R$-module morphism
$M^{\prime}\longrightarrow M^{\prime\prime}$ forming a commutative square with
$f^{\prime}$ and $f^{\prime\prime}$. One can easily interpret $\mathsf{A}$ as
the category of modules $A=T{\operatorname{\mathsf{--mod}}}$ over an
appropriate matrix ring $T$.

Denote by $E$ the object $(S,0,0)\in\mathsf{A}$, where $S$ is viewed as a free
let $S$-module with one generator.

######  Lemma 7.2.

Let $L=(N,M,f)$ be an object of $\mathsf{A}$. Then the following Ext
computations hold:

(a)
$\operatorname{Hom}_{\mathsf{A}}(E,L)=\ker(N\to\operatorname{Hom}_{R}(S,M))$,

(b)
$\operatorname{Ext}_{\mathsf{A}}^{1}(E,L)=\operatorname{coker}(N\to\operatorname{Hom}_{R}(S,M))$,

(c)
$\operatorname{Ext}_{\mathsf{A}}^{i}(E,L)=\operatorname{Ext}_{R}^{i-1}(S,M)$
for $i\geq 2$,

where the $S$-module morphism $N\longrightarrow\operatorname{Hom}_{R}(S,M)$ is
induced by the $R$-module morphism $N\longrightarrow M$.

###### Proof.

There is a short exact sequence

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(0,S,0)\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(S,S,\operatorname{id}_{S})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu(S,0,0)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

in the category $\mathsf{A}$, with a projective object
$(S,S,\operatorname{id}_{S})\in\mathsf{A}$. Computing
$\operatorname{Hom}_{\mathsf{A}}(E,L)$ and
$\operatorname{Ext}^{1}_{\mathsf{A}}(E,L)$ in terms of this one-step
resolution of the object $E$ produces the natural isomorphisms (a-b).
Furthermore, let $\dotsb\longrightarrow P_{1}\longrightarrow
P_{0}\longrightarrow S\longrightarrow 0$ be a projective resolution of the
left $R$-module $S$. Then the following is a projective resolution of the
object $E\in\mathsf{A}$:

$\dotsb\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(0,P_{1},0)\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(0,P_{0},0)\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu(S,S,\operatorname{id}_{S})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu(S,0,0)\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0.$

Computing $\operatorname{Ext}^{i}_{\mathsf{A}}(E,L)$ in terms of this
resolution produces the natural isomorphism (c) as well. ∎

Given a left $R$-module $G$ and an integer $n\geq 0$, we denote by
$G^{\perp_{1..n}}\subset R{\operatorname{\mathsf{--mod}}}$ the full
subcategory in $R{\operatorname{\mathsf{--mod}}}$ formed by all the
$R$-modules $M$ such that $\operatorname{Ext}_{R}^{i}(G,M)=0$ for all $1\leq
i\leq n$. So, in particular, we have
$G^{\perp_{1..0}}=R{\operatorname{\mathsf{--mod}}}$. Similarly, we denote by
$E^{\perp_{0..n}}\subset\mathsf{A}$ the full subcategory formed by all the
objects $L\in\mathsf{A}$ such that
$\operatorname{Ext}_{\mathsf{A}}^{i}(E,L)=0$ for all $0\leq i\leq n$. For
$n=\infty$, the notation $G^{\perp_{1..\infty}}\subset
R{\operatorname{\mathsf{--mod}}}$ and $E^{\perp_{0..\infty}}\subset\mathsf{A}$
also has the obvious meaning.

######  Corollary 7.3.

The functor assigning to an $R$-module $M$ the object
$(\operatorname{Hom}_{R}(S,M),\allowbreak M,f_{M})\in\mathsf{A}$, where
$f_{M}\colon\operatorname{Hom}_{R}(S,M)\longrightarrow M$ is the $R$-module
morphism induced by the ring homomorphism $R\longrightarrow S$, provides an
equivalence between the full subcategory $E^{\perp_{0..1}}\subset\mathsf{A}$
and the abelian category $R{\operatorname{\mathsf{--mod}}}$. For every $n\geq
1$, this equivalence identifies the full subcategories
$E^{\perp_{0..n+1}}\subset\mathsf{A}$ and $S^{\perp_{1..n}}\subset
R{\operatorname{\mathsf{--mod}}}$ (where $S$ is viewed as a left $R$-module),
providing a category equivalence between them.

###### Proof.

The first assertion follows from Lemma 7.2(a-b), and the second one from Lemma
7.2(c). ∎

######  Example 7.4.

Let $R$ be an associative ring and $G$ be a left $R$-module admitting a right
$R$-module structure which makes it an $R$-$R$-module. Set $S=R\oplus G$ to be
the trivial ring extension of $R$ by $G$ (so the product of any two elements
of $G$ is zero in $S$, while the products of two elements of $R$ and of one
element from $R$ and one element from $G$ in $S$ are defined using the ring
structure on $R$ and the $R$-$R$-bimodule structure on $G$). Then the above
construction provides an associative ring $T$ with a left $T$-module $E$ such
that, by Corollary 7.3, the full subcategory $E^{\perp_{0..1}}\subset
T{\operatorname{\mathsf{--mod}}}$ is equivalent to
$R{\operatorname{\mathsf{--mod}}}$, and this equivalence restricts to
equivalences between the full subcategories $E^{\perp_{0..n+1}}\subset
T{\operatorname{\mathsf{--mod}}}$ and $G^{\perp_{1..n}}\subset
R{\operatorname{\mathsf{--mod}}}$ for all $n\geq 1$.

Notice that, by Lemma 7.2, the projective dimensions of the left $R$-module
$G$ and the left $T$-module $E$ are related by the rule
$\operatorname{pd}_{T}E=\operatorname{pd}_{R}G+1$.

######  Examples 7.5.

(1) In the context of Example 7.4, let $R$ be a commutative ring and set
$G=\bigoplus_{s\in R}R[s^{-1}]$. Then $\operatorname{pd}_{R}G\leq 1$ and the
orthogonal class $G^{\perp_{1..1}}=G^{\perp_{1..\infty}}\subset
R{\operatorname{\mathsf{--mod}}}$ is called the class of all _contraadjusted_
$R$-modules [23, Section 1.1], [26, Sections 2 and 8]. The full subcategory
$G^{\perp_{1..1}}\subset R{\operatorname{\mathsf{--mod}}}$ is equivalent to
the category of all contraherent cosheaves on $\operatorname{Spec}R$ (cf.
Example 4.3).

Set $R=\mathbb{Z}$ to be the ring on integers, and consider the abelian group
$A=\mathbb{Z}_{p}^{(\omega)}$, that is, the direct sum of a countable family
of copies of the group of $p$-adic integers (where $p$ is a fixed prime
number). Set $B=\mathbb{Q}\otimes_{\mathbb{Z}}A$ and
$C=\mathbb{Q}/\mathbb{Z}\otimes_{\mathbb{Z}}A=\operatorname{coker}(A\to B)$.
Being divisible ( $=$ injective) abelian groups, $B$ and $C$ are obviously
contraadjusted. We claim that the surjective morphism $B\longrightarrow C$ has
_no_ kernel in the full subcategory of contraadjusted abelian groups
$G^{\perp_{1..1}}\subset\mathbb{Z}{\operatorname{\mathsf{--mod}}}$.

Indeed, suppose that $K\longrightarrow B$ is such a kernel. Since the class of
contraadjusted modules is closed under the passages to arbitrary quotient
modules, the image of the morphism $K\longrightarrow B$ is also
contraadjusted. Hence the morphism $K\longrightarrow B$ has to be injective,
and we can consider $K$ as a subgroup in $B$. Since the composition
$K\longrightarrow B\longrightarrow C$ has to vanish, we have $K\subset A$.
Now, we have $A=\mathbb{Z}_{p}^{(\omega)}$, and every summand
$\mathbb{Z}_{p}\subset A$ in this direct sum is a contraadjusted abelian
group. Hence $K$ contains every one of these direct summands, so it follows
that $K=A$. However, $A$ is _not_ a contraadjusted abelian group, as
$\operatorname{Ext}^{1}_{R}(\mathbb{Z}[p^{-1}],A)\neq 0$ [26, Section 12].

Thus we have constructed an associative ring $T$ and a left $T$-module $E$ of
projective dimension $2$ such that in the full subcategory
$\mathsf{B}=E^{\perp_{0..2}}=E^{\perp_{0..\infty}}\subset
T{\operatorname{\mathsf{--mod}}}$ there are morphisms which have _no kernel_.
Let us emphasize what this means: not only the full subcategory $\mathsf{B}$
is not closed under kernels in $T{\operatorname{\mathsf{--mod}}}$, but some
morphisms in $\mathsf{B}$ do not even have any kernel internal to
$\mathsf{B}$.

(2) Again in the context of Example 7.4, let $R$ be a commutative ring and set
$G=\bigoplus_{I}R/I$, where the direct sum is taken over all the ideals
$I\subset R$. Then, by Baer’s criterion, the orthogonal class
$G^{\perp_{1..1}}=G^{\perp_{1..\infty}}\subset
R{\operatorname{\mathsf{--mod}}}$ is the class of all injective $R$-modules.
Choose $R$ to be a commutative ring of global dimension $3$, and let $A$ be a
$R$-module of projective dimension $3$. Choose exact sequences of $R$-modules
$0\longrightarrow A\longrightarrow J\longrightarrow B\longrightarrow 0$ and
$0\longrightarrow B\longrightarrow H\longrightarrow C\longrightarrow 0$, where
$J$ and $H$ are injective $R$-modules. We claim that the morphism
$J\longrightarrow H$ has _no_ cokernel in the full subcategory of injective
modules $G^{\perp_{1..1}}\subset R{\operatorname{\mathsf{--mod}}}$.

Indeed, suppose that $H\longrightarrow K$ is such a cokernel. Then the
composition $J\longrightarrow H\longrightarrow K$ vanishes, so we have an
$R$-module morphism $C\longrightarrow K$. Furthermore, for every injective
$R$-module $K^{\prime}$ any any $R$-module morphism $C\longrightarrow
K^{\prime}$ there exists a unique $R$-module morphism $K\longrightarrow
K^{\prime}$ making the triangle diagram $C\longrightarrow K\longrightarrow
K^{\prime}$ commutative. Choosing $K^{\prime}$ to be an injective $R$-module
such that there is an injective $R$-module morphism $C\longrightarrow
K^{\prime}$, we see that the morphism $C\longrightarrow K$ is injective.
Choosing $K^{\prime}$ to be an injective $R$-module such that there is an
injective $R$-module morphism $K/C\longrightarrow K^{\prime}$, we conclude
that the morphism $C\longrightarrow K$ is an isomorphism. Hence $C$ is an
injective $R$-module, which contradicts the assumption that
$\operatorname{pd}_{R}A=3$.

Thus we have constructed an associative ring $T$ and a left $T$-module $E$ of
projective dimension $4$ such that
$E^{\perp_{0..2}}=E^{\perp_{0..4}}=E^{\perp_{0..\infty}}\subset
T{\operatorname{\mathsf{--mod}}}$ and in the full subcategory
$\mathsf{B}=E^{\perp_{0..2}}\subset T{\operatorname{\mathsf{--mod}}}$ there
are morphisms which have _no cokernel_. As in (1), this means that not only
the full subcategory $\mathsf{B}$ is not closed under cokernels in
$T{\operatorname{\mathsf{--mod}}}$, but some morphisms in $\mathsf{B}$ do not
even have any cokernel internal to $\mathsf{B}$.

######  Example 7.6.

This example is a naïve version of the category of contraherent cosheaves on
the projective line $\mathbb{P}^{1}_{k}$ (where $k$ is a field). Consider two
polynomial rings $R^{\prime}=k[x]$ and $R^{\prime\prime}=k[x^{-1}]$ in the
variables $x$ and $x^{-1}$, and the ring of Laurent polynomials
$S=k[x,x^{-1}]$ containing $R^{\prime}$ and $R^{\prime\prime}$ as subrings.
Denote by $\mathsf{C}$ the abelian category whose objects are quintuples
$(N,M^{\prime},M^{\prime\prime},f^{\prime},f^{\prime\prime})$, where $N$ is an
$S$-module, $M^{\prime}$ is an $R^{\prime}$-module, $M^{\prime\prime}$ is an
$R^{\prime\prime}$-module, $f^{\prime}\colon N\longrightarrow M^{\prime}$ is
an $R^{\prime}$-module morphism, and $f^{\prime\prime}\colon N\longrightarrow
M^{\prime\prime}$ is an $R^{\prime\prime}$-module morphism. Morphisms in the
category $\mathsf{C}$ are defined in the obvious way (similar to the
construction of the category $\mathsf{A}$ above). It is not difficult to
construct a matrix ring $U$ such that
$\mathsf{C}=U{\operatorname{\mathsf{--mod}}}$.

Consider the two objects $E^{\prime}=(S,0,S,0,\operatorname{id}_{S})$ and
$E^{\prime\prime}=(S,S,0,\operatorname{id}_{S},0)\in\mathsf{C}$, and set
$E=E^{\prime}\oplus E^{\prime\prime}$. Then $E$ is an object of projective
dimension $2$ in $\mathsf{C}$, and the full subcategory
$E^{\perp_{0..1}}\subset\mathsf{C}$ is equivalent to the category of all pairs
of modules $M^{\prime}$ over $R^{\prime}$ and $M^{\prime\prime}$ over
$R^{\prime\prime}$ endowed with an isomorphism of $S$-modules
$\operatorname{Hom}_{R^{\prime}}(S,M^{\prime})\simeq\operatorname{Hom}_{R^{\prime\prime}}(S,M^{\prime\prime})$.
The full subcategory $E^{\perp_{0..2}}\subset E^{\perp_{0..1}}$ is defined by
the additional conditions
$\operatorname{Ext}^{1}_{R^{\prime}}(S,M^{\prime})=0$ and
$\operatorname{Ext}^{1}_{R^{\prime\prime}}(S,M^{\prime\prime})=0$ (cf. Example
4.3).

By Lemma 7.1, the full subcategory $\mathsf{B}=E^{\perp_{0..1}}\subset
U{\operatorname{\mathsf{--mod}}}$ is an additive category with kernels and
cokernels (and, also, infinite direct sums and products). Let us show that the
category $\mathsf{B}$ is _not_ abelian. Indeed, consider the two objects
$A=(0,R^{\prime},R^{\prime\prime},0,0)$ and
$C=(k,k,k,\operatorname{id}_{k},\operatorname{id}_{k})$, where $x$ acts in $k$
by the identity operator. Then there is a morphism $g\colon A\longrightarrow
C$ in $\mathsf{B}$ whose components $R^{\prime}\longrightarrow k$ and
$R^{\prime\prime}\longrightarrow k$ take $x^{n}$ to $1$ for all
$n\in\mathbb{Z}$. Now, the cokernel of $g$ vanishes, so $C$ is the kernel of
the cokernel of $g$. On the other hand, the kernel of $g$ is the morphism
$h\colon A\longrightarrow A$ with the components $1-x\colon
R^{\prime}\longrightarrow R^{\prime}$ and $1-x^{-1}\colon
R^{\prime\prime}\longrightarrow R^{\prime\prime}$. The cokernel of the kernel
of $g$ is isomorphic to the direct sum of two copies of $C$.

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