# Roos axiom holds for quasi-coherent sheaves

Leonid Positselski  Institute of Mathematics, Czech Academy of Sciences
Žitná 25, 115 67 Prague 1
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We show that the Grothendieck abelian category
$X{\operatorname{\mathsf{--Qcoh}}}$ of quasi-coherent sheaves on a quasi-
compact semi-separated scheme $X$ satisfies the Roos axiom
$\mathrm{AB}4^{*}$-$n$: the derived functors of infinite product have finite
homological dimension in $X{\operatorname{\mathsf{--Qcoh}}}$, not exceeding
the number $n$ of open subschemes in an affine open covering of $X$. The
hereditary complete cotorsion pair (very flat quasi-coherent sheaves,
contraadjusted quasi-coherent sheaves) in the abelian category
$X{\operatorname{\mathsf{--Qcoh}}}$ plays the key role in our arguments.
Simply put, a suitable very flat quasi-coherent sheaf (or alternatively, a
suitable direct sum of locally countably presented flat quasi-coherent
sheaves) on $X$ is a generator of finite projective dimension for the abelian
category $X{\operatorname{\mathsf{--Qcoh}}}$.

###### Contents

  1. 1 Contraadjusted and Very Flat Modules
  2. 2 Very Flat Quasi-Coherent Sheaves
  3. 3 Adjusted Classes
  4. 4 Contraadjusted Quasi-Coherent Sheaves

## Introduction

The condition of (existence and) exactness of infinite direct products in an
abelian category $\mathsf{A}$ is known as “Grothendieck’ axiom
$\mathrm{AB}4^{*}$ ” [6, Section 1.5]. It is well known that the categories of
quasi-coherent sheaves $X{\operatorname{\mathsf{--Qcoh}}}$ over schemes $X$ do
_not_ satisfy $\mathrm{AB}4^{*}$ in general. A counterexample of Keller,
reproduced by Krause in the paper [11, Example 4.9], shows that the infinite
products are not exact in the category of quasi-coherent sheaves over the
projective line $X=\mathbb{P}^{1}_{k}$ over a field $k$. More generally, if
$X$ is a Noetherian scheme with an ample family of line bundles, then the
infinite products are exact in $X{\operatorname{\mathsf{--Qcoh}}}$ if and only
if $X$ is affine [10, Theorem 1.1].

The axiom $\mathrm{AB}4^{*}$-$n$ ($n\geq 0$) was proposed by Roos in his 2006
paper [20, Definition 1.1] as a natural weakening of $\mathrm{AB}4^{*}$. An
abelian category with infinite products and enough injective objects is said
to satisfy $\mathrm{AB}4^{*}$-$n$ if the derived functor of infinite product
$\prod_{\lambda\in\Lambda}^{(i)}A_{\lambda}$ vanishes on all families of
objects $(A_{\lambda}\in\mathsf{A})_{\lambda\in\Lambda}$ for $i>n$. A
characterization of Grothendieck abelian categories satisfying
$\mathrm{AB}4^{*}$-$n$ was obtained in the same paper [20, Theorem 1.3]. It is
clear from that characterization that existence of a generator of projective
dimension $n$ in a Grothendieck category $\mathsf{A}$ implies
$\mathrm{AB}4^{*}$-$n$.

Some examples of Grothendieck categories satisfying $\mathrm{AB}4^{*}$-$n$
were discussed in [20, Section 1], and the minimal possible value of $n$ for a
given category was computed in some cases. In particular, [20, Section 1.3]
was dedicated to the categories of quasi-coherent sheaves on Noetherian
schemes $X$, and contained a full treatment of the case when $X$ is the
complement to the closed point in the spectrum of a Noetherian local ring. As
an application demostrating the utility of the $\mathrm{AB}4^{*}$-$n$
condition, it was shown in [20, Theorem 2.1] that the derived functor
$\varprojlim^{*}$ of inverse limit of sequences of objects in $\mathsf{A}$
(indexed by the natural numbers) has homological dimension at most $n+1$ if
$\mathsf{A}$ is a Grothendieck category satisfying $\mathrm{AB}4^{*}$-$n$.

Further applications of the $\mathrm{AB}4^{*}$-$n$ condition can be found in
the recent preprint [8]. According to [8, Theorems 2.9 and 4.3], certain
classical constructions going back to Cartan–Eilenberg and Spaltenstein allow
to explicitly produce homotopy injective (DG-injective) resolutions of
complexes in an abelian category $\mathsf{A}$ provided that $\mathsf{A}$ has
infinite products, enough injective objects, and satisfies
$\mathrm{AB}4^{*}$-$n$. This allows one to avoid using the more general but
less explicit small object argument.

The notion of a _very flat_ quasi-coherent sheaf on a scheme $X$ was
introduced in an early September 2012 version of the present author’s preprint
[13]. It was immediately clear from [13, Lemma 4.1.1] that, for any quasi-
compact semi-separated scheme $X$, the category of quasi-coherent sheaves
$X{\operatorname{\mathsf{--Qcoh}}}$ has a very flat generator. It was also
easy to see that the projective dimension of any very flat quasi-coherent
sheaf on $X$ does not exceed the minimal number $N$ of open subschemes in an
affine open covering of $X$. But I did not realize the importance of the
property of a Grothendieck category to have a generator of finite projective
dimension.

That I learned from Jan St’ovíček sometime around 2016. In our joint paper
[18], the reference to very flat quasi-coherent sheaves and to [13, Lemma
4.1.1] appeared in this connection in [18, last paragraph of Example 6.4]. A
conversation with Manuel Saorín, who visited us in Prague in July 2024,
convinced me that a paragraph in an example in [18] was not enough to
advertise the fact, which seemed to remain unknown to people working in the
area. This motivated me to write the present paper up.

To finish this introduction, let us point out that our bound on the
homological dimension of derived products in
$X{\operatorname{\mathsf{--Qcoh}}}$ is probably _not_ the best possible one.
Given a quasi-compact semi-separated scheme $X$, denote by $n\geq 0$ the
minimal integer such that $X{\operatorname{\mathsf{--Qcoh}}}$ satisfies
$\mathrm{AB}4^{*}$-$n$. Let $N$ be the minimal number of open subschemes in an
affine open covering $X=\bigcup_{\alpha}U_{\alpha}$ of the scheme $X$. In
particular, when the scheme $X$ is affine, we have $N=1$ and $n=0$ (as the
functors of infinite products are exact in the category of quasi-coherent
sheaves on an affine scheme). So one would naturally expect an inequality
$n\leq N-1$. The example when $X$ is the complement to the closed point in the
spectrum of a Noetherian local ring, treated in [20, Theorem 1.15], suggests
the same answer. Still, we are only able to prove that $n\leq N$ in this
paper. We do _not_ know whether $n=N$ is possible.

### Acknowledgement

I am grateful to Jan St’ovíček and Manuel Saorín for illuminating
conversations. The author is supported by the GAČR project 23-05148S and the
Czech Academy of Sciences (RVO 67985840).

##  1\. Contraadjusted and Very Flat Modules

Let $R$ be a commutative ring. Given an element $s\in R$, we consider the
localization $R[s^{-1}]=S^{-1}R$ of the ring $R$ at the multiplicative subset
$S=\\{1,s,s^{2},s^{3},\dotsc\\}$. So $R[s^{-1}]$ is the ring of functions on
the principal affine open subscheme in $\operatorname{Spec}R$ corresponding to
the element $s$. An $R$-module $C$ is said to be _contraadjusted_ [13, Section
1.1] if $\operatorname{Ext}^{1}_{R}(R[s^{-1}],C)=0$ for every $s\in S$. An
$R$-module $F$ is said to be _very flat_ if
$\operatorname{Ext}^{1}_{R}(F,C)=0$ for every contraadjusted $R$-module $C$.

The word “contraadjusted” is supposed to mean “adjusted for contraherent
cosheaves”. So contraadjusted modules were introduced in [13] as a basic
building element of the contraherent cosheaf theory. The definitions of
contraadjusted and very flat modules given above fall under the general scheme
of a concept known as a _cotorsion pair_ (or _cotorsion theory_) of classes of
modules, or more generally, classes of objects in an abelian or exact
category. The definition of a cotorsion theory is due to Salce [21], and the
main result in this connection is the _Eklof–Trlifaj theorem_ [3, Theorems 2
and 10]. The main reference source for cotorsion pairs in module categories is
the book [5]. An introductory discussion tailored to the situation at hand can
be found in [16, Section 4]; see specifically [16, Theorem 4.5].

In particular, the pair of classes (very flat $R$-modules, cotorsion
$R$-modules) is a complete cotorsion pair in the abelian category of
$R$-modules $R{\operatorname{\mathsf{--Mod}}}$ (generated by the set of
$R$-modules $R[s^{-1}]$, $s\in S$) [13, Theorem 1.1.1]. It is called the _very
flat cotorsion pair_ in $R{\operatorname{\mathsf{--Mod}}}$. As one of the
aspects of the Eklof–Trlifaj theorem, one obtains the following description of
very flat $R$-modules: an $R$-module is very flat if and only if it is a
direct summand of a module admitting a transfinite (ordinal-indexed)
increasing filtration whose successive quotients are $R$-modules of the form
$R[s^{-1}]$, $s\in S$ [13, Corollary 1.1.4]. Any projective $R$-module is very
flat, and any very flat $R$-module is flat.

In addition, one observes that the $R$-module $R[s^{-1}]$ has projective
dimension at most $1$ [13, Section 1.1], [14, proof of Lemma 2.1]. It follows
that any quotient module of a contraadjusted $R$-module is contraadjusted, and
that any very flat $R$-module has projective dimension at most $1$.
Furthermore, the very flat cotorsion pair is _hereditary_ : the kernel of any
surjective morphism of very flat $R$-modules is very flat.

######  Example 1.1.

Let $X$ be an affine scheme and $U\subset X$ be an affine open subscheme in
$X$. Then the ring of functions $S=\mathcal{O}(U)$ is a very flat module over
the ring $R=\mathcal{O}(X)$. This is [13, Lemma 1.2.4].

Given a commutative ring homomorphism $f\colon R\longrightarrow S$, one says
that $f$ is _very flat_ (or equivalently, $S$ is a _very flat $R$-algebra_)
if, for every element $s\in S$, the $R$-module $S[s^{-1}]$ is very flat.
Notice that the condition that the $R$-module $S$ itself be very flat is
strictly weaker and _not_ sufficient [16, Section 6.1].

The _Very Flat Conjecture_ (one of possible formulations) stated that, for any
commutative ring $R$, any finitely presented flat commutative $R$-algebra is
very flat. This was proved in [17, Main Theorem 1.1 and Corollary 9.1]. We
will not use this important and difficult result in this paper.

The main construction below in Section 2 is based on the properties of very
flat modules listed in the following lemmas.

######  Lemma 1.2.

Let $R\longrightarrow S$ be a homomorphism of commutative rings. Then, for any
very flat $R$-module $F$, the $S$-module $S\otimes_{R}F$ is very flat.

###### Proof.

This is [13, Lemma 1.2.2(b)]. ∎

######  Lemma 1.3.

Let $R\longrightarrow S$ be a very flat homomorphism of commutative rings.
Then any very flat $R$-module $G$ is very flat as an $R$-module.

###### Proof.

This is [13, Lemma 1.2.3(b)]. ∎

######  Lemma 1.4.

Let $f_{\alpha}\colon R\longrightarrow S_{\alpha}$ be a finite family of
homomorphisms of commutative rings such that the related family of morphisms
of the spectra
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is an open
covering of $\operatorname{Spec}R$ (in particular,
$\operatorname{Spec}S_{\alpha}\longrightarrow\operatorname{Spec}R$ is the
embedding of an affine open subscheme for every $\alpha$). Then an $R$-module
$F$ is very flat _if and only if_ the $S_{\alpha}$-module
$S_{\alpha}\otimes_{R}F$ is very flat for every $\alpha$.

###### Proof.

This is [13, Lemma 1.2.6(a)]. ∎

A further discussion of Lemmas 1.2–1.4 can be found in the paper [15, Example
2.5].

##  2\. Very Flat Quasi-Coherent Sheaves

Let $X$ be a scheme. A quasi-coherent sheaf $\mathcal{F}$ on $X$ is said to be
_very flat_ [13, Section 1.8] if, for every affine open subscheme $U\subset
X$, the $\mathcal{O}_{X}(U)$-module $\mathcal{F}(U)$ is very flat. In view of
Lemmas 1.2 and 1.4, it suffices to check this condition for affine open
subschemes $U=U_{\alpha}$ belonging to any given affine open covering
$X=\bigcup_{\alpha}U_{\alpha}$ of the scheme $X$. In particular, if $X=V\cup
W$ is a (not necessarily affine) open covering of $X$, then a quasi-coherent
sheaf $\mathcal{F}$ on $X$ is very flat if and only if the quasi-coherent
sheaf $\mathcal{F}|_{V}$ is very flat on $V$ and the quasi-coherent sheaf
$\mathcal{F}|_{W}$ is very flat on $W$.

The class of all very flat quasi-coherent sheaves on a scheme $X$ is closed
under extensions, kernels of surjective morphisms, and infinite direct sums
(because the class of all very flat modules over a commuative ring has these
closure properties).

For any morphism of schemes $f\colon Y\longrightarrow X$, the inverse image
functor $f^{*}\colon
X{\operatorname{\mathsf{--Qcoh}}}\allowbreak\longrightarrow
Y{\operatorname{\mathsf{--Qcoh}}}$ takes very flat quasi-coherent sheaves on
$X$ to very flat quasi-coherent sheaves on $Y$ (by Lemma 1.2).

A morphism of schemes $f\colon Y\longrightarrow X$ is said to be _very flat_
if, for any pair of affine open subschemes $U\subset X$ and $V\subset Y$ such
that $f(V)\subset U$, the $\mathcal{O}_{X}(U)$-module $\mathcal{O}_{Y}(V)$ is
very flat [13, Section 1.8]. A morphism of affine schemes
$\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ is very flat if and
only if the related morphism of commutative rings $R\longrightarrow S$ is very
flat (in the sense of the definition in Section 1).

Recall that a morphism of schemes $f\colon Y\longrightarrow X$ is said to be
_affine_ if, for any affine open subscheme $U\subset X$, the open subscheme
$f^{-1}(U)\subset Y$ is affine. For any affine morphism of schemes $f\colon
Y\longrightarrow X$, the direct image functor $f_{*}\colon
Y{\operatorname{\mathsf{--Qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--Qcoh}}}$ is exact. For any very flat affine morphism
of schemes $f\colon Y\longrightarrow X$, the direct image functor $f_{*}$
takes very flat quasi-coherent sheaves on $Y$ to very flat quasi-coherent
sheaves on $X$ (by Lemma 1.3).

In particular, for any open subscheme $W$ in a scheme $X$, the open embedding
morphism $j\colon W\longrightarrow X$ is very flat (by Example 1.1).
Therefore, if the morphism $j$ is affine, then the quasi-coherent sheaf
$j_{*}\mathcal{G}$ on $X$ is very flat for any very flat quasi-coherent sheaf
$\mathcal{G}$ on $W$.

The following lemma is a very flat version of [2, Lemma A.1] and a part of
[13, Lemma 4.1.1].

######  Lemma 2.1.

On any quasi-compact semi-separated scheme $X$, any quasi-coherent sheaf is a
quotient sheaf of a very flat quasi-coherent sheaf.

###### Proof.

Let $\mathcal{M}$ be a quasi-coherent sheaf on $X$. Arguing by induction,
suppose that we have already constructed a surjective morphism of quasi-
coherent sheaves $\mathcal{H}\longrightarrow\mathcal{M}$ on $X$ such that, for
a certain open subscheme $V\subset X$, the restriction $\mathcal{H}|_{V}$ is a
very flat quasi-coherent sheaf on $V$. Let $U\subset X$ be an affine open
subscheme in $X$; denote its open embedding morphism by $j\colon
U\longrightarrow X$. Since $X$ is a semi-separated scheme by assumption and
$U$ is an affine scheme, the morphism $j$ is affine.

Consider the restriction/inverse image $\mathcal{H}|_{U}=j^{*}\mathcal{H}$ on
$U$. Any $\mathcal{O}(U)$-module is a quotient module of a very flat
$\mathcal{O}(U)$-module; so we have a short exact sequence of quasi-coherent
sheaves $0\longrightarrow\mathcal{K}\longrightarrow\mathcal{G}\longrightarrow
j^{*}\mathcal{H}\longrightarrow 0$ on $U$ with a very flat quasi-coherent
sheaf $\mathcal{G}$. The direct image functor $j_{*}\colon
U{\operatorname{\mathsf{--Qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--Qcoh}}}$ is exact (since the morphism $j$ is
affine); so we have a short exact sequence of direct images $0\longrightarrow
j_{*}\mathcal{K}\longrightarrow j_{*}\mathcal{G}\longrightarrow
j_{*}j^{*}\mathcal{H}\longrightarrow 0$ in
$X{\operatorname{\mathsf{--Qcoh}}}$.

Consider the adjunction morphism of quasi-coherent sheaves
$\mathcal{H}\longrightarrow j_{*}j^{*}\mathcal{H}$ on $X$. Let
$0\longrightarrow
j_{*}\mathcal{K}\longrightarrow\mathcal{L}\longrightarrow\mathcal{H}\longrightarrow
0$ be the pullback of the short exact sequence $0\longrightarrow
j_{*}\mathcal{K}\longrightarrow j_{*}\mathcal{G}\longrightarrow
j_{*}j^{*}\mathcal{H}\longrightarrow 0$ with respect to the morphism
$\mathcal{H}\longrightarrow j_{*}j^{*}\mathcal{H}$ in
$X{\operatorname{\mathsf{--Qcoh}}}$. Put $W=U\cup V\subset X$; so $W$ is an
open subscheme in $X$. We claim that the quasi-coherent sheaf
$\mathcal{L}|_{W}$ is very flat on $W$.

Indeed, it suffices to check that the quasi-coherent sheaf $\mathcal{L}|_{U}$
is very flat on $U$ and the quasi-coherent sheaf $\mathcal{L}|_{V}$ is very
flat on $V$. The exact functor $j^{*}\colon
X{\operatorname{\mathsf{--Qcoh}}}\longrightarrow
U{\operatorname{\mathsf{--Qcoh}}}$ takes the morphism
$\mathcal{H}\longrightarrow j_{*}j^{*}\mathcal{H}$ to the identity map
$j^{*}\mathcal{H}\longrightarrow j^{*}\mathcal{H}$; so we have
$j^{*}\mathcal{L}\simeq j^{*}j_{*}\mathcal{G}\simeq\mathcal{G}$. The quasi-
coherent sheaf $\mathcal{G}$ on $U$ is very flat by construction. This proves
that the quasi-coherent sheaf $\mathcal{L}|_{U}=j^{*}\mathcal{L}$ is very
flat.

To prove that the quasi-coherent sheaf $\mathcal{L}|_{V}$ is very flat, denote
by $h\colon V\longrightarrow X$ the open embedding morphism. Then we have a
short exact sequence $0\longrightarrow h^{*}j_{*}\mathcal{K}\longrightarrow
h^{*}\mathcal{L}\longrightarrow h^{*}\mathcal{H}\longrightarrow 0$ of quasi-
coherent sheaves on $V$. As the quasi-coherent sheaf
$\mathcal{H}|_{V}=h^{*}\mathcal{H}$ is very flat by assumption, and the class
of very flat quasi-coherent sheaves on $V$ is closed under extensions, it
suffices to check that the quasi-coherent sheaf $h^{*}j_{*}\mathcal{K}$ on $V$
is very flat.

Consider the intersection $U\cap V$, and denote by $j^{\prime}\colon U\cap
V\longrightarrow V$ and $h^{\prime}\colon U\cap V\longrightarrow U$ its open
embedding morphisms. So the morphism $j^{\prime}$ is affine as a base change
of an affine morphism of schemes. We have an (obvious) base change isomorphism
$h^{*}j_{*}\mathcal{K}\simeq j^{\prime}_{*}h^{\prime}{}^{*}\mathcal{K}$ of
quasi-coherent sheaves on $V$. Furthermore, the two compositions $h^{\prime}j$
and $j^{\prime}h$ are one and the same morphism $U\cap V\longrightarrow X$; so
we have an isomorphism $h^{\prime}{}^{*}j^{*}\mathcal{H}\simeq
j^{\prime}{}^{*}h^{*}\mathcal{H}$ of quasi-coherent sheaves on $U\cap V$.

Applying the functor $h^{\prime}{}^{*}\colon
U{\operatorname{\mathsf{--Qcoh}}}\longrightarrow(U\cap\nobreak
V){\operatorname{\mathsf{--Qcoh}}}$ to the short exact sequence
$0\longrightarrow\mathcal{K}\longrightarrow\mathcal{G}\longrightarrow
j^{*}\mathcal{H}\ \longrightarrow 0$ in $U{\operatorname{\mathsf{--Qcoh}}}$,
we obtain a short exact sequence $0\longrightarrow
h^{\prime}{}^{*}\mathcal{K}\longrightarrow
h^{\prime}{}^{*}\mathcal{G}\longrightarrow
h^{\prime}{}^{*}j^{*}\mathcal{H}\longrightarrow 0$ in $(U\cap\nobreak
V){\operatorname{\mathsf{--Qcoh}}}$. The quasi-coherent sheaf
$h^{\prime}{}^{*}\mathcal{G}$ on $U\cap V$ is very flat, since the quasi-
coherent sheaf $\mathcal{G}$ on $U$ is very flat. The quasi-coherent sheaf
$h^{\prime}{}^{*}j^{*}\mathcal{H}\simeq j^{\prime}{}^{*}h^{*}\mathcal{H}$ on
$U\cap V$ is very flat, since the quasi-coherent sheaf
$h^{*}\mathcal{H}=\mathcal{H}|_{V}$ on $V$ is very flat.

As the class of very flat quasi-coherent sheaves on $U\cap V$ is closed under
kernels of surjective morphisms, we can conclude from the short exact sequence
$0\longrightarrow h^{\prime}{}^{*}\mathcal{K}\longrightarrow
h^{\prime}{}^{*}\mathcal{G}\longrightarrow
h^{\prime}{}^{*}j^{*}\mathcal{H}\longrightarrow 0$ that the quasi-coherent
sheaf $h^{\prime}{}^{*}\mathcal{K}$ on $U\cap V$ is very flat. Finally, since
the morphism $j^{\prime}\colon U\cap V\longrightarrow V$ is affine and very
flat, we arrive to the conclusion that the quasi-coherent sheaf
$h^{*}j_{*}\mathcal{K}\simeq j^{\prime}_{*}h^{\prime}{}^{*}\mathcal{K}$ on $V$
is very flat. We have proved that the quasi-coherent sheaf $\mathcal{L}|_{W}$
is very flat on $W=U\cup V$.

The composition
$\mathcal{L}\longrightarrow\mathcal{H}\longrightarrow\mathcal{M}$ of
surjective morphisms of quasi-coherent sheaves on $X$ is surjective. Having
started with a surjective morphism of quasi-coherent sheaves
$\mathcal{H}\longrightarrow\mathcal{M}$ such that the quasi-coherent sheaf
$\mathcal{H}|_{V}$ is very flat on an open subscheme $V\subset X$, and given
an affine open subscheme $U\subset X$, we have constructed an surjective
morphism of quasi-coherent sheaves $\mathcal{L}\longrightarrow\mathcal{M}$
such that the quasi-coherent sheaf $\mathcal{L}|_{W}$ is very flat on the open
subscheme $W=U\cup V$.

Starting with $V=\varnothing$ and $\mathcal{H}=\mathcal{M}$ as the induction
base, and proceeding in this way by adjoining affine open subschemes
$U=U_{\alpha}$ from a given finite affine open covering
$X=\bigcup_{\alpha}U_{\alpha}$ of the scheme $X$, we arrive to the desired
surjective morphism onto $\mathcal{M}$ from a very flat quasi-coherent sheaf
on $X$. ∎

######  Corollary 2.2.

For any quasi-compact semi-separated scheme $X$, there exists a very flat
quasi-coherent sheaf $\mathcal{F}$ on $X$ such that $\mathcal{F}$ is a
generator of the Grothendieck category $X{\operatorname{\mathsf{--Qcoh}}}$.

###### Proof.

The category $X{\operatorname{\mathsf{--Qcoh}}}$ of quasi-coherent sheaves on
any scheme $X$ has a generator by Gabber’s theorem [1, Lemma 2.1.7], [9,
Proposition Tag 077P]. More specifically, for any quasi-compact quasi-
separated scheme $X$, the category $X{\operatorname{\mathsf{--Qcoh}}}$ is
locally finitely presentable by Grothendieck’s theorem [7, 0.5.2.5 and
Corollaire I.6.9.12]. So, in the situation at hand with $X$ a quasi-compact
semi-separated scheme, let $\mathcal{M}$ be a generator of
$X{\operatorname{\mathsf{--Qcoh}}}$. By Lemma 2.1, there exists a very flat
quasi-coherent sheaf $\mathcal{F}$ on $X$ together with a surjective morphism
of quasi-coherent sheaves $\mathcal{F}\longrightarrow\mathcal{M}$. Then
$\mathcal{F}$ is a desired very flat generator of
$X{\operatorname{\mathsf{--Qcoh}}}$. ∎

Given an object $F$ in an abelian category $\mathsf{A}$, one has that $F$ has
_projective dimension $\leq d$_ if
$\operatorname{Ext}_{\mathsf{A}}^{n}(F,A)=0$ for all objects $A\in\mathsf{A}$
and all integers $n>d$. Notice that this definition does _not_ presume
existence of enough projective objects in $\mathsf{A}$.

######  Lemma 2.3.

Let $X$ be a quasi-compact semi-separated scheme with a finite affine open
covering $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$. Then any very flat quasi-
coherent sheaf $\mathcal{F}$ on $X$ has projective dimension at most $N$ (as
an object of $X{\operatorname{\mathsf{--Qcoh}}}$).

###### Proof.

Let $\mathcal{F}$ be an arbitrary quasi-coherent sheaf on $X$. Then, by [19,
Theorem 5.3(b)], the projective dimension of $\mathcal{F}$ in
$X{\operatorname{\mathsf{--Qcoh}}}$ does not exceed $N-1$ plus the supremum of
the projective dimensions of the $\mathcal{O}(U_{\alpha})$-modules
$\mathcal{F}(U_{\alpha})$. When $\mathcal{F}$ is a very flat quasi-coherent
sheaf, the $\mathcal{O}(U_{\alpha})$-module $\mathcal{F}(U_{\alpha})$ is very
flat, so its projective dimension does not exceed $1$ (according to Section
1). ∎

##  3\. Adjusted Classes

Let $\mathsf{B}$ be an abelian category with enough injective objects. We will
say that a class of objects $\mathsf{C}\subset\mathsf{B}$ is _coresolving_ if
it satisfies the following conditions:

  1. (i)

the full subcategory $\mathsf{C}$ is closed under extensions in $\mathsf{B}$;

  2. (ii)

the full subcategory $\mathsf{C}$ is closed under the cokernels of
monomorphisms in $\mathsf{B}$;

  3. (iii)

all the injective objects of $\mathsf{C}$ belong to $\mathsf{B}$.

Let $\mathsf{A}$ be another abelian category and
$G\colon\mathsf{B}\longrightarrow\mathsf{A}$ be a left exact functor. We will
denote by $\mathbb{R}^{*}G\colon\mathsf{B}\longrightarrow\mathsf{A}$ the
sequence of right derived functors of the functor $G$ (computed, as usual, in
terms of injective coresoltions of objects of $\mathsf{B}$).

We will say that a coresolving class of objects $\mathsf{C}\subset\mathsf{B}$
is _adjusted to $G$_ if it satisfies the following condition:

  1. (iv)

for any short exact sequence $0\longrightarrow C^{\prime}\longrightarrow
C\longrightarrow C^{\prime\prime}\longrightarrow 0$ in $\mathsf{B}$ with the
objects $C^{\prime}$, $C$, $C^{\prime\prime}$ belonging to $\mathsf{C}$, the
short sequence

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muG(C^{\prime})\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muG(C)\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muG(C^{\prime\prime})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

is exact in $\mathsf{A}$.

######  Lemma 3.1.

Let $\mathsf{B}$ be an abelian category with enough injective objects,
$\mathsf{A}$ be an abelian category,
$G\colon\mathsf{B}\longrightarrow\mathsf{A}$ be a left exact functor, and
$\mathsf{C}\subset\mathsf{B}$ be a coresolving class of objects adjusted to
$G$. Let $B\in\mathsf{B}$ be an object and $0\longrightarrow B\longrightarrow
C^{0}\longrightarrow C^{1}\longrightarrow C^{2}\longrightarrow\dotsb$ be a
coresolution of the object $B$ by objects $C^{i}\in\mathsf{C}$ in the abelian
category $\mathsf{B}$. Then there are natural isomorphisms

$\mathbb{R}^{i}G(B)\simeq
H^{i}G(C^{\text{\smaller\smaller$\scriptstyle\bullet$}}),\qquad i\geq 0$

of objects in $\mathsf{A}$.

###### Proof.

This lemma is very well known. Given an object $C\in\mathsf{C}$, consider its
injective coresolution $0\longrightarrow C\longrightarrow J^{0}\longrightarrow
J^{1}\longrightarrow J^{2}\longrightarrow\dotsb$ in $\mathsf{B}$. Put
$Z^{0}=C$, and denote by $Z^{i}$ the image of the morphism
$J^{i-1}\longrightarrow J^{i}$ for $i\geq 1$. So the long exact sequence
$0\longrightarrow C\longrightarrow
J^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ can be obtained by splicing
the short exact sequences $0\longrightarrow Z^{i}\longrightarrow
J^{i}\longrightarrow Z^{i+1}\longrightarrow 0$ in $\mathsf{B}$. Then, by
(iii), we have $J^{i}\in\mathsf{C}$ for all $i\geq 0$, and by (ii), it follows
by induction that $Z^{i}\in\mathsf{C}$ for all $i\geq 0$. By (iv), the short
sequences $0\longrightarrow G(Z^{i})\longrightarrow G(J^{i})\longrightarrow
G(Z^{i+1})\longrightarrow 0$ are exact in $\mathsf{A}$ for all $i\geq 0$. So
the complex $0\longrightarrow G(C)\longrightarrow
G(J^{\text{\smaller\smaller$\scriptstyle\bullet$}})$ is exact in $\mathsf{A}$,
and we have shown that $\mathbb{R}G^{n}(C)=0$ for all $n>0$. Returning to the
situation at hand, we have $\mathbb{R}G^{n}(C^{i})=0$ for all $i\geq 0$ and
$n>0$, which implies the desired natural isomorphisms of cohomology. ∎

Let $\mathsf{A}$ be an abelian category with infinite products and enough
injective objects. Let $\Lambda$ be a set. We are interested in the case when
$\mathsf{B}=\mathsf{A}^{\Lambda}=\times_{\lambda\in\Lambda}\,\mathsf{A}$ is
the Cartesian product of $\Lambda$ copies of $\mathsf{A}$; so the objects of
$\mathsf{B}$ are all the $\Lambda$-indexed collections
$B=(A_{\lambda}\in\mathsf{A})_{\lambda\in\Lambda}$ of objects of $\mathsf{A}$.
Furthermore, we are interested in the direct product functor
$G=\prod_{\lambda\in\Lambda}\colon\mathsf{B}\longrightarrow\mathsf{A}$; so
$G((A_{\lambda})_{\lambda\in\Lambda})=\prod_{\lambda\in\Lambda}A_{\lambda}\in\mathsf{A}$.
Obviously, there are enough injective objects in $\mathsf{B}$; the injective
objects of $\mathsf{B}$ are precisely all the $\Lambda$-indexed collections of
injective objects in $\mathsf{A}$.

Given a class of objects $\mathsf{F}$ in an abelian category $\mathsf{A}$, we
denote by $\mathsf{F}^{\perp_{\geq 1}}\subset\mathsf{A}$ the class of all
objects $K\in\mathsf{A}$ such that
$\operatorname{Ext}^{n}_{\mathsf{A}}(F,K)=0$ for all $F\in\mathsf{F}$ and
$n\geq 1$. In particular, for a single object $F\in\mathsf{A}$, we put
$F^{\perp_{\geq 1}}=\\{F\\}^{\perp_{\geq 1}}\subset\mathsf{A}$.

######  Lemma 3.2.

Let $\mathsf{A}$ be an abelian category with infinite products and enough
injective objects, and let $\Lambda$ be an infinite set. Let $F\in\mathsf{A}$
be a generator of $\mathsf{A}$. Consider the class $\mathsf{K}=F^{\perp_{\geq
1}}\subset\mathsf{A}$ of all objects $K\in\mathsf{A}$ such that
$\operatorname{Ext}^{n}_{\mathsf{A}}(F,K)=0$ for all $n\geq 1$. Let
$\mathsf{C}=\mathsf{K}^{\Lambda}\subset\mathsf{A}^{\Lambda}=\mathsf{B}$ be the
class of all $\Lambda$-indexed collections of objects from $\mathsf{K}$. Then
$\mathsf{C}$ is a coresolving class of objects in $\mathsf{B}$ adjusted to the
direct product functor
$G=\prod_{\lambda\in\Lambda}\colon\mathsf{B}\longrightarrow\mathsf{A}$.

###### Proof.

One can easily see that $\mathsf{K}$ is a coresolving class of objects in
$\mathsf{A}$, and it follows that $\mathsf{C}$ is a coresolving class of
objects in $\mathsf{B}$. (This holds for the right $\operatorname{Ext}^{\geq
1}$-orthogonal class $\mathsf{K}=\mathsf{F}^{\perp_{\geq 1}}$ to an arbitrary
class of objects $\mathsf{F}\subset\mathsf{A}$.) To show that the class
$\mathsf{C}$ is adjusted to the direct product functor $G$, we argue as in
[19, Lemma 8.3].

Let $0\longrightarrow K_{\lambda}\longrightarrow L_{\lambda}\longrightarrow
M_{\lambda}\longrightarrow 0$ be a $\Lambda$-indexed family of short exact
sequences in $\mathsf{A}$ with $K_{\lambda}\in\mathsf{K}$ for all
$\lambda\in\Lambda$. In order to show that
$0\longrightarrow\prod_{\lambda\in\Lambda}K_{\lambda}\longrightarrow\prod_{\lambda\in\Lambda}L_{\lambda}\longrightarrow\prod_{\lambda\in\Lambda}M_{\lambda}\longrightarrow
0$ is a short exact sequence in $\mathsf{A}$, we need to check that
$\prod_{\lambda\in\Lambda}L_{\lambda}\longrightarrow\prod_{\lambda\in\Lambda}M_{\lambda}$
is an epimorphism. As $F$ is a generator of $\mathsf{A}$, it suffices to check
that every morphism $F\longrightarrow\prod_{\lambda\in\Lambda}M_{\lambda}$ can
be lifted to a morphism
$F\longrightarrow\prod_{\lambda\in\Lambda}L_{\lambda}$. Now the datum of a
morphism $F\longrightarrow\prod_{\lambda\in\Lambda}M_{\lambda}$ is equivalent
to the datum of a family of morphisms $F\longrightarrow M_{\lambda}$,
$\lambda\in\Lambda$. Since
$\operatorname{Ext}^{1}_{\mathsf{A}}(F,K_{\lambda})=0$ by assumption, any
morphism $F\longrightarrow M_{\lambda}$ can be lifted to a morphism
$F\longrightarrow L_{\lambda}$. Any family of such liftings defines the
desired morphism $F\longrightarrow\prod_{\lambda\in\Lambda}L_{\lambda}$. ∎

Let $\mathsf{B}$ be an abelian category and $\mathsf{C}\subset\mathsf{B}$ be a
coresolving class of objects in $\mathsf{B}$. One says that the _coresolution
dimension_ of an object $B\in\mathsf{B}$ with respect to the coresolving class
$\mathsf{C}\subset\mathsf{B}$ does not exceed an integer $d\geq 0$ if there
exists an exact sequence $0\longrightarrow B\longrightarrow
C^{0}\longrightarrow C^{1}\longrightarrow\dotsb\longrightarrow
C^{d}\longrightarrow 0$ in $\mathsf{B}$ with $C^{i}\in\mathsf{C}$ for all
$0\leq i\leq d$. One can show that the $\mathsf{C}$-coresolution dimension of
an object $B\in\mathsf{B}$ does not depend on the choice of a coresolution of
$B$ by objects from $\mathsf{C}$, in an obvious sense [23, Lemma 2.1], [22,
Proposition 2.3], [13, Corollary A.5.2].

######  Lemma 3.3.

Let $\mathsf{A}$ be an abelian category and $\mathsf{F}\subset\mathsf{A}$ be a
class of (some) objects of projective dimension $\leq d$ in $\mathsf{A}$. Then
every object of $\mathsf{A}$ has coresolution dimension at most $d$ with
respect to the coresolving class $\mathsf{K}=\mathsf{F}^{\perp_{\geq
1}}\subset\mathsf{A}$ of all objects $K\in\mathsf{A}$ such that
$\operatorname{Ext}^{n}_{\mathsf{A}}(F,K)=0$ for all $n\geq 1$.

###### Proof.

This is straightforward. ∎

######  Corollary 3.4.

Let $\mathsf{A}$ be an abelian category with infinite products and enough
injective objects, and let $\Lambda$ be an infinite set. Let $F\in\mathsf{A}$
be a generator of $\mathsf{A}$; assume that the projective dimension of $F$ in
$\mathsf{A}$ does not exceed $d$. Consider the direct product functor
$G=\prod_{\lambda\in\Lambda}\colon\mathsf{B}=\mathsf{A}^{\Lambda}\longrightarrow\mathsf{A}$.
Then one has $\mathbb{R}^{i}G=0$ for all $i>d$. So the abelian category
$\mathsf{A}$ satisfies Roos’ axiom $\mathrm{AB}4^{*}$-$d$.

###### Proof.

This is a weak version of a resulf of Roos [20, Theorem 1.3]. As the proof in
[20] is somewhat complicated, we offer a simple alternative proof of the weak
version stated in the corollary.

Put $\mathsf{K}=F^{\perp_{\geq 1}}\subset\mathsf{A}$ and
$\mathsf{C}=\mathsf{K}^{\Lambda}\subset\mathsf{B}$. Then the coresolving class
$\mathsf{C}$ is adjusted to the functor $G$ by Lemma 3.2, and the coresolution
dimension of any object of $\mathsf{B}$ with respect to the class $\mathsf{C}$
does not exceed $d$ by Lemma 3.3. The desired assertion follows by virtue of
Lemma 3.1. ∎

Now we can prove the main theorem of this paper.

######  Theorem 3.5.

Let $X$ be a quasi-compact semi-separated scheme with an affine open covering
$X=\bigcup_{\alpha=1}^{N}U_{\alpha}$. Then the Grothendieck abelian category
of quasi-coherent sheaves $X{\operatorname{\mathsf{--Qcoh}}}$ satisfies the
axiom $\mathrm{AB}4^{*}$-$N$ from [20, Definition 1.1].

###### Proof.

By Corollary 2.2, there exists a very flat quasi-coherent sheaf $\mathcal{F}$
on $X$ such that $\mathcal{F}$ is a generator of
$X{\operatorname{\mathsf{--Qcoh}}}$. Lemma 2.3 tells us that the projective
dimension of the object $\mathcal{F}\in X{\operatorname{\mathsf{--Qcoh}}}$
does not exceed $N$. So Corollary 3.4 is applicable for
$\mathsf{A}=X{\operatorname{\mathsf{--Qcoh}}}$ and $d=N$. ∎

######  Remark 3.6.

The proof of Theorem 3.5 above is based on the notion of a very flat quasi-
coherent sheaf. Alternatively, there is a similar argument proving the same
theorem based on the notion of a _locally countably presented_ flat quasi-
coherent sheaf. A quasi-coherent sheaf $\mathcal{M}$ on a scheme $X$ is said
to be _locally countably presented_ [19, Section 2] if the
$\mathcal{O}_{X}(U)$-module $\mathcal{M}(U)$ is countably presented for every
affine open subscheme $U\subset X$. It suffices to check this condition for
affine open subschemes $U=U_{\alpha}$ forming a given affine open covering
$X=\bigcup_{\alpha}U_{\alpha}$ of a scheme $X$ [19, Lemma 2.1]. By [19,
Theorem 2.4 or 3.5], any flat quasi-coherent sheaf on a quasi-compact quasi-
separated (or more generally, countably quasi-compact and countably quasi-
separated) scheme $X$ is a direct limit (in fact, an $\aleph_{1}$-direct
limit) of locally countably presented flat quasi-coherent sheaves.

For a quasi-compact semi-separated scheme $X$, it is known since [12, Section
2.4] (see [2, Lemma A.1] for a perhaps clearer argument) that any quasi-
coherent sheaf $\mathcal{M}$ on $X$ is a quotient sheaf of a flat quasi-
coherent sheaf $\mathcal{G}$. Then $\mathcal{G}$ is a direct limit of locally
countably presented flat quasi-coherent sheaves. So $\mathcal{M}$ is a
quotient sheaf of an infinite direct sum of locally finitely presented flat
quasi-coherent sheaves. Take $\mathcal{M}$ to be a generator of
$X{\operatorname{\mathsf{--Qcoh}}}$. Any countably presented flat module over
a ring has projective dimension at most $1$ [5, Corollary 2.23]. If $X$
covered by $N$ open affines, then it follows by virtue of [19, Theorem 5.3]
that any locally countably presented flat quasi-coherent sheaf on $X$ has
projective dimension at most $N$ [19, Corollary 5.6]. Hence the same applies
to any infinite direct sum of locally countably presented flat quasi-coherent
sheaves on $X$. Once again, we have constructed a generator of projective
dimension at most $N$ for $X{\operatorname{\mathsf{--Qcoh}}}$, and it remains
to refer to Corollary 3.4 in order to finish the proof of the theorem.

######  Question 3.7.

Is the dimension bound provided by Theorem 3.5 sharp? Specifically, does there
exist a quasi-compact semi-separated scheme $X$ covered by $N$ open affines
such that the axiom $\mathrm{AB}4^{*}$-$n$ does not hold for $X$ and $n=N-1$ ?
Cf. the discussion of the affine scheme case ($N=1$ and $n=0$) in the
introduction.

One possible approach to checking the axiom $\mathrm{AB}^{*}$-$n$ for $n=N-1$
would be to show that the class of _dilute quasi-coherent sheaves_ [12,
Section 4.1], [13, Section 4.2] is adjusted to infinite products in
$X{\operatorname{\mathsf{--Qcoh}}}$. We do _not_ know whether this is true.
Notice that the coresolution dimension of any quasi-coherent sheaf on $X$ with
respect to the coresolving subcategory of dilute quasi-coherent sheaves does
not exceed $N-1$ [13, Lemma 4.7.1(a)]; so it would be sufficient in view of
Lemma 3.1.

##  4\. Contraadjusted Quasi-Coherent Sheaves

Given a quasi-compact semi-separated scheme $X$, the discussion in Section (3)
provides a coresolving class
$\mathsf{K}\subset\mathsf{A}=X{\operatorname{\mathsf{--Qcoh}}}$ such that all
the objects of $\mathsf{A}$ have finite (uniformly bounded)
$\mathsf{C}$-coresolution dimensions and the class
$\mathsf{C}=\mathsf{K}^{\Lambda}\subset\mathsf{A}^{\Lambda}=\mathsf{B}$ is
adjusted to the functor of direct product
$G=\prod_{\lambda\in\Lambda}\colon\mathsf{B}\longrightarrow\mathsf{A}$. But
this construction is not natural, in that it depends on the choice of a very
flat generator $\mathcal{F}\in X{\operatorname{\mathsf{--Qcoh}}}$.

In this section we discuss a straightforward modification or particular case
of the construction of Section 3 providing a natural coresolving class in
$X{\operatorname{\mathsf{--Qcoh}}}$ adjusted to infinite products and such
that the coresolution dimensions are finite. This class is called the class of
_contraadjusted_ quasi-coherent sheaves.

Let $X$ be a scheme. We will denote by $\operatorname{Ext}_{X}^{*}({-},{-})$
the functor $\operatorname{Ext}^{*}$ in the abelian category of quasi-coherent
sheaves $X{\operatorname{\mathsf{--Qcoh}}}$.

A quasi-coherent sheaf $\mathcal{C}$ on $X$ is said to be _contraadjusted_
[13, Section 2.5] if $\operatorname{Ext}^{1}_{X}(\mathcal{F},\mathcal{C})=0$
for every very flat quasi-coherent sheaf $\mathcal{F}$ on $X$. For example,
any cotorsion quasi-coherent sheaf on $X$ (in the sense of [4]) is
contraadjusted, since any very flat quasi-coherent sheaf is flat. A quasi-
coherent sheaf on an affine scheme $U$ is contraadjusted if and only if it
corresponds to a contraadjusted $\mathcal{O}(U)$-module.

By the definition, the class of contraadjusted quasi-coherent sheaves is
closed under extensions and direct summands in
$X{\operatorname{\mathsf{--Qcoh}}}$. For any quasi-compact quasi-separated
morphism of schemes $f\colon Y\longrightarrow X$, the direct image functor
$f_{*}\colon Y{\operatorname{\mathsf{--Qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--Qcoh}}}$ takes contraadjusted quasi-coherent sheaves
on $Y$ to contraadjusted quasi-coherent sheaves on $X$ [13, Section 2.5], [15,
Lemma 1.7(c)].

The following two corollaries say that the pair of classes (very flat quasi-
coherent sheaves, contraadjusted quasi-coherent sheaves) is a hereditary
complete cotorsion pair in $X{\operatorname{\mathsf{--Qcoh}}}$ for a quasi-
compact semi-separated scheme $X$.

######  Corollary 4.1.

Let $X$ be a quasi-compact semi-separated scheme. Then

(a) any quasi-coherent sheaf on $X$ is a quotient sheaf of a very flat quasi-
coherent sheaf by a subsheaf that is a contraadjusted quasi-coherent sheaf;

(b) any quasi-coherent sheaf on $X$ can be embedded as subsheaf into a
contraadjusted quasi-coherent sheaf in such a way that the quotient sheaf is a
very flat quasi-coherent sheaf.

######  Corollary 4.2.

Let $X$ be a quasi-compact semi-separated scheme. Then

(a) the class of contraadjusted quasi-coherent sheaves on $X$ is closed under
cokernels of injective morphisms;

(b) for any very flat quasi-coherent sheaf $\mathcal{F}$ and any
contraadjusted quasi-coherent sheaf $\mathcal{C}$ on $X$, one has
$\operatorname{Ext}_{X}^{n}(\mathcal{F},\mathcal{C})=0$ for all $n\geq 1$.

The third corollary provides an additional description of the class of
contraadjusted quasi-coherent sheaves on $X$.

######  Corollary 4.3.

Let $X$ be a quasi-compact semi-separated scheme with a finite affine open
covering $X=\bigcup_{\alpha}U_{\alpha}$. Denote by $j_{\alpha}\colon
U_{\alpha}\longrightarrow X$ the open embedding morphisms. Then a quasi-
coherent sheaf $\mathcal{C}$ on $X$ is contraadjusted if and only if
$\mathcal{C}$ is a direct summand of a finitely iterated extension in
$X{\operatorname{\mathsf{--Qcoh}}}$ of the direct images
$j_{\alpha}{}_{*}\mathcal{C}_{\alpha}$ of contraadjusted quasi-coherent
sheaves $\mathcal{C}_{\alpha}$ on $U_{\alpha}$.

###### Proof of Corollaries 4.1, 4.2, and 4.3.

Corollary 4.1 is [13, Corollary 4.1.4(a–b)]. Corollary 4.2 is [13, Corollary
4.1.2(b–c)]. Corollary 4.3 is [13, Corollary 4.1.4(c)].

Let us provide a little discussion with more details. To prove Corollary
4.1(a), one just needs to follow the proof of Lemma 2.1 above carefully and
look into what one can say about the kernel of the surjective morphism of
quasi-coherent sheaves that the construction from the proof of the lemma
provides. In the course of the proof of the lemma, when choosing a
presentation of the quasi-coherent sheaf $j^{*}\mathcal{H}$ on $U$ as a
quotient of a very flat quasi-coherent sheaf $\mathcal{G}$, one needs to do it
in such a way that the kernel $\mathcal{K}$ is a contraadjusted quasi-coherent
sheaf on $U$ (which is possible by [13, Theorem 1.1.1(b)]). One will see that
not only the kernel of the resulting surjective morphism from a very flat
quasi-coherent sheaf on $X$ is contraadjusted, but also it has the form of a
finitely iterated extension described in Corollary 4.3. The rest of the proof
of all the three corollaries is easier and more standard (in particular,
Corollary 4.2 follows directly from the assertion of Lemma 2.1).

The results of Corollaries 4.1–4.3 form a special case of a general
construction of gluing of hereditary complete cotorsion pairs from an affine
open covering of a quasi-compact semi-separated scheme $X$, as described in
[13, Section B.4]. For a version of this construction for principal affine
open coverings of affine schemes, see [15, Proposition 4.3]. ∎

Corollary 4.2(a) implies that the class of contraadjusted quasi-coherent
sheaves is coresolving in $X{\operatorname{\mathsf{--Qcoh}}}$ when $X$ is a
quasi-compact semi-separated scheme. The coresolution dimension of a quasi-
coherent sheaf $\mathcal{M}$ on $X$ with respect to the coresolving class of
contraadjusted quasi-coherent sheaves $\mathsf{K}$ in
$X{\operatorname{\mathsf{--Qcoh}}}$ is called the _contraadjusted dimension_
of $\mathcal{M}$.

######  Lemma 4.4.

Let $X$ be a quasi-compact semi-separated scheme with an affine open covering
$X=\bigcup_{\alpha=1}^{n}U_{\alpha}$. Then the contraadjusted dimension of any
quasi-coherent sheaf on $X$ does not exceed $N$.

###### Proof.

This is [13, Lemma 4.7.1(b)]. The assertion is easily provable by comparing
Lemma 2.3 with Lemma 3.3 (for $d=N$ and the class $\mathsf{F}$ of all very
flat quasi-coherent sheaves in
$\mathsf{A}=X{\operatorname{\mathsf{--Qcoh}}}$). One also has to keep in mind
Corollary 4.2(b). ∎

######  Lemma 4.5.

Let $X$ be a quasi-compact semi-separated scheme and $\Lambda$ be a set. Put
$\mathsf{A}=X{\operatorname{\mathsf{--Qcoh}}}$, and consider the direct
product functor
$G=\prod_{\lambda\in\Lambda}\colon\mathsf{B}=\mathsf{A}^{\Lambda}\longrightarrow\mathsf{A}$.
Let $\mathsf{K}\subset\mathsf{A}$ be the class of all contraadjusted quasi-
coherent sheaves on $X$. Then the class
$\mathsf{C}=\mathsf{K}^{\Lambda}\subset\mathsf{B}$ of all $\Lambda$-indexed
collections of contraadjusted quasi-coherent sheaves on $X$ is adjusted to the
functor $G$.

In other words, let
$0\longrightarrow\mathcal{K}_{\lambda}\longrightarrow\mathcal{L}_{\lambda}\longrightarrow\mathcal{M}_{\lambda}\longrightarrow
0$ be a $\Lambda$-indexed collection of short exact sequences of quasi-
coherent sheaves on $X$ with contraadjusted quasi-coherent sheaves
$\mathcal{K}_{\lambda}$, $\lambda\in\Lambda$. Then the short sequence of
direct products

$0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{\lambda\in\Lambda}\mathcal{K}_{\lambda}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{\lambda\in\Lambda}\mathcal{L}_{\lambda}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\prod\nolimits_{\lambda\in\Lambda}\mathcal{M}_{\lambda}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

is exact in $X{\operatorname{\mathsf{--Qcoh}}}$.

###### Proof.

The argument from the proof of Lemma 3.2 is applicable in view of Corollary
2.2. ∎

Comparing Lemma 3.1 with Lemmas 4.4 and 4.5, we obtain another (slightly
differently structured) proof of Theorem 3.5.

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