# K-flatness in Grothendieck categories: Application to quasi-coherent sheaves

Sergio Estrada  S.E. Universidad de Murcia
Departamento de Matemáticas
Campus de Espinardo
Murcia 30100, Spain<EMAIL_ADDRESS>,  James Gillespie  J.G. Ramapo College of
New Jersey
School of Theoretical and Applied Science
505 Ramapo Valley Road
Mahwah, NJ 07430
U.S.A<EMAIL_ADDRESS>http://pages.ramapo.edu/ jgillesp/ and  Sinem
Odabaşı  S.O. Universidad de Murcia
Departamento de Matemáticas
Campus de Espinardo
Murcia 30100, Spain<EMAIL_ADDRESS>
###### Abstract.

Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck
category. We show that K-flat covers exist universally in the category of
chain complexes and that the Verdier quotient of $K(\mathcal{G})$ by the
K-flat complexes is always a well generated triangulated category. Under the
further assumption that $\mathcal{G}$ has a set of $\otimes$-flat generators
we can show more: (i) The category is in recollement with the $\otimes$-pure
derived category and the usual derived category, and (ii) The usual derived
category is the homotopy category of a cofibrantly generated and monoidal
model structure whose cofibrant objects are precisely the K-flat complexes. We
also give a condition guaranteeing that the right orthogonal to K-flat is
precisely the acyclic complexes of $\otimes$-pure injectives. We show this
condition holds for quasi-coherent sheaves over a quasi-compact and
semiseparated scheme.

2020 Mathematics Subject Classification. 18N40, 18G35, 18G25

The first and the third named authors were partly supported by grant
PID2020-113206GB-I00 funded by MCIN/AEI/10.13039/ 501100011033 and by grant
22004/PI/22 funded by Fundación Séneca.

##  1\. Introduction

In [EGO17] we defined and studied the pure derived category of a quasi-compact
and semiseparated scheme. An important point is that there are two natural
notions of purity in the category of quasi-coherent sheaves over such a
scheme. First, there is the categorical purity arising from the fact that this
category is locally finitely presented. The second notion is obtained by
considering the short exact sequences that remain exact upon tensoring with
any quasi-coherent sheaf. These two notions of purity coincide for affine
schemes, but typically they differ. In general, the more natural notion of
purity for quasi-coherent sheaves is the second one, for it is equivalent to
having purity on the stalks and this notion is in agreement with the local
nature of the tensor product and flatness for sheaves. Hence we call this the
_geometric_ purity while the former is the _categorical_ purity. Motivated by
studying the geometric pure derived category of a scheme, we introduced in
[EGO17] the $\otimes$-pure derived category of a general closed symmetric
monoidal Grothendieck category $(\mathcal{G},\otimes)$. Recall that a
Grothendieck category is a cocomplete abelian category $\mathcal{G}$ with a
set of generators and such that direct limits of monomorphisms are again
monomorphisms. In practice, the category $\mathcal{G}$ often possesses a
tensor product $\otimes$ providing the formal structure of a closed symmetric
monoidal category.

Now given any closed symmetric monoidal Grothendieck category
$(\mathcal{G},\otimes)$, all of this structure lifts to chain complexes,
resulting in a closed symmetric monoidal structure
$(\textnormal{Ch}(\mathcal{G}),\otimes)$ on the category of complexes.
Spaltenstein’s notion of a K-flat complex from [Spa88] makes sense in this
general setting. A chain complex $X$ is said to be K-flat if $E\otimes X$ is
an acyclic (i.e. exact) chain complex whenever $E$ is acyclic. In other words,
the functor $-\otimes X$ preserves acyclicity. Given a ring $R$, Emmanouil
studied in [Emm22] the Verdier quotient of the chain homotopy category,
$K(R)$, by the thick class of all K-flat complexes. He shows it to be
equivalent to $K_{ac}(\mathcal{PI})$, the chain homotopy category of all
acyclic complexes of pure-injective $R$-modules. In this paper we prove an
analog of this for quasi-coherent sheaves, using the geometric pure injective
(i.e. $\otimes$-pure injective) quasi-coherent sheaves.

But first we prove some results of interest that hold in the full generality
of any closed symmetric monoidal Grothendieck category
$(\mathcal{G},\otimes)$. Let $\mathcal{KF}$ denote the class of all K-flat
chain complexes. First, Corollary 4.3 shows that $\mathcal{KF}$ is a covering
class in the category $\textnormal{Ch}(\mathcal{G})$ of chain complexes. In
particular, it is a special precovering class: For any complex $Y$ there is a
K-flat complex $X$ and a surjective chain map $p:X\xrightarrow{}Y$ such that
any other chain map $f:X^{\prime}\xrightarrow{}Y$, with $X^{\prime}$ K-flat,
lifts over $p$. The second result concerns the Verdier quotient of
$K(\mathcal{G})$, the chain homotopy category of $\mathcal{G}$, by the class
$\mathcal{KF}$. Theorem 4.2 implies that the Verdier quotient,
$K(\mathcal{G})/\mathcal{KF}$, is always at least a well-generated (if not
compactly generated) triangulated category. In particular, the above results
hold for any scheme $\mathbb{X}$. That is, any chain complex of quasi-coherent
sheaves has a K-flat cover, and $K(\mathbb{X})/\mathcal{KF}$ is always well
generated.

Theorem 4.2 actually constructs a cofibrantly generated abelian model
structure on $\textnormal{Ch}(\mathcal{G})$ whose weak equivalences are
precisely the chain maps whose mapping cone is a K-flat complex. Every complex
is cofibrant in this model structure and the fibrant objects are certain
complexes with $\otimes$-pure injective components. This model was constructed
on chain complexes of modules over a ring $R$ in [Gil23a]. But constructing
the model structure in the present generality requires us to expand upon
techniques from [EGO17] which, in particular, utilize the theory of
$\lambda$-purity from [AR94].

In Section 5, we are able to say more under the assumption that $\mathcal{G}$
possesses a generating set consisting of $\otimes$-flat objects. An object
$F\in\mathcal{G}$ is $\otimes$-flat if $F\otimes-$ is an exact functor. With
the presence of $\otimes$-flat generators for $\mathcal{G}$, we display a
recollement linking $K(\mathcal{G})/\mathcal{KF}$ to the usual derived
category, $\mathcal{D}(\mathcal{G})$, and the $\otimes$-pure derived category,
$\mathcal{D}_{\otimes\textnormal{-pur}}(\mathcal{G})$, which was introduced in
[EGO17]. See Theorem 5.3. Related to this, we see in Theorem 5.4 the existence
of a monoidal model structure whose homotopy category is
$\mathcal{D}(\mathcal{G})$. This model is equivalent to the usual flat model
structure but in some sense it is more natural because every K-flat complex is
cofibrant, not just the ones with $\otimes$-flat components.

Finally, Section 6 focuses on the central example of quasi-coherent sheaves
over a quasi-compact semiseparated scheme $\mathbb{X}$. With the help of a
technical but useful result concerning cotorsion pairs and direct limits
(Theorem 6.3) we imitate the Čech resolution argument from [CET21, Theorem
3.3], to obtain the wanted analog of Emmanouil’s result concerning the Verdier
quotient $K(R)/\mathcal{KF}$: The right Ext-orthogonal to $\mathcal{KF}$ is
precisely the class of all acyclic complexes of geometric pure injective
($\otimes$-pure injective) quasi-coherent sheaves. It follows that the Verdier
quotient $K(\mathbb{X})/\mathcal{KF}$ is equivalent to
$K_{ac}(\mathcal{PI}_{\otimes})$, the chain homotopy category of all acyclic
complexes of $\otimes$-pure injectives; see Theorem 6.6. We also consider the
affine case $\mathbb{X}=\text{Spec}(R)$. Here, taking advantage of having not
only enough flats but utilizing that we also have enough pure-projectives and
that geometric and categorical purity agree in this case, we see that there is
another abelian model structure for $K(R)/\mathcal{KF}$. Its cofibrant objects
are the chain complexes of pure projective modules. The resulting homotopy
category reflects an equivalence of $K(R)/\mathcal{KF}$ with the Verdier
quotient of $K(R)$ by Emmanouil and Kaperonis’ _K-absolutely pure_ complexes,
introduced in [EK22].

##  2\. Purity in symmetric monoidal Grothendieck categories

Throughout this section we let $(\mathcal{G},\otimes)$ be a closed symmetric
monoidal Grothendieck category. Recall that $\mathcal{G}$ is locally
$\lambda$-presentable (in the sense of [AR94]) for some regular cardinal
$\lambda$, and, as described in [EGO17, Section 2], there are two generally
different notions of purity. Let $\mathcal{P}_{\lambda}$ denote the proper
class of $\lambda$-pure short exact sequences in $\mathcal{G}$, and
$\mathcal{P}_{\otimes}$ denote the proper class of $\otimes$-pure short exact
sequences in $\mathcal{G}$. We have the containment
$\mathcal{P}_{\lambda}\subseteq\mathcal{P}_{\otimes}$; see [EGO17, Remark
2.8], and the book [AR94] is the standard reference for locally
$\lambda$-presentable categories and $\lambda$-purity.

We will denote by $\textnormal{Ch}(\mathcal{G})_{\otimes}$ the exact structure
consisting of $\textnormal{Ch}(\mathcal{G})$, the category of
$\mathcal{G}$-chain complexes, along with the componentwise $\otimes$-pure
exact sequences. The associated Yoneda Ext group of all (equivalence classes
of) degreewise $\otimes$-pure short exact sequences will be denoted by
$\operatorname{Ext}^{1}_{\otimes}(X,Y)$. Note that since $X\otimes-$ preserves
direct limits for any object $X\in\mathcal{G}$, any transfinite composition of
$\otimes$-pure monomorphisms is again a $\otimes$-pure monomorphism.

The following result tells us that we can use
$\operatorname{Ext}^{1}_{\otimes}$ to construct complete cotorsion pairs in
$\textnormal{Ch}(\mathcal{G})_{\otimes}$ by the usual method of cogenerating
by a set. To state it, let $\lambda$ be some regular cardinal for which
$\mathcal{G}$ is locally $\lambda$-presentable. Let
$\\{L_{i}\\}_{i\in\Lambda}$ be a set of representatives of all the isomorphism
classes of $\lambda$-presented objects. This is a generating set for the exact
category $(\mathcal{G},\mathcal{P}_{\lambda})$, meaning every object is a
$\lambda$-pure epimorphic image of some direct sum of copies of some $L_{i}$.
Since any $\lambda$-pure epic is also a $\otimes$-pure epic we see that
$\\{L_{i}\\}$ is also a generating set for the exact structure
$(\mathcal{G},\mathcal{P}_{\otimes})$ [EGO17, Prop. 2.9]. By taking all their
$n$-disks ($n\in\mathbb{Z}$), this lifts to a set $\\{D^{n}(L_{i})\\}$ of
generators for the exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$.

######  Lemma 2.1.

Let $\mathcal{S}$ be a set (not a proper class) of chain complexes such that
${}^{\perp}(\mathcal{S}^{\perp})$ contains the generating set
$\\{D^{n}(L_{i})\\}$ for the exact category
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. Then
$({}^{\perp}(\mathcal{S}^{\perp}),\mathcal{S}^{\perp})$ is a complete
cotorsion pair in the exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$.
Moreover, the class ${}^{\perp}(\mathcal{S}^{\perp})$ consists precisely of
direct summands (retracts) of transfinite degreewise $\otimes$-pure extensions
of complexes in $\mathcal{S}\cup\\{D^{n}(L_{i})\\}$.

###### Proof.

This follows from [SŠ11, Corollary 2.15] since
$(\mathcal{G},\mathcal{P}_{\otimes})$, and hence
$\textnormal{Ch}(\mathcal{G})_{\otimes}$, is an efficient exact category, by
[EGO17, Lemma 3.6]. ∎

Let us recall how $\otimes$ lifts to a closed symmetric monoidal structure on
$\textnormal{Ch}(\mathcal{G})$. Given $X,Y\in\textnormal{Ch}(\mathcal{G})$,
their tensor product $X\otimes Y$ is defined by

$(X\otimes Y)_{n}=\bigoplus_{i+j=n}(X_{i}\otimes Y_{j})$

in degree $n$. The boundary map $\delta_{n}$ is induced by the formula
$d^{X}_{i}\otimes 1_{Y_{j}}+(-1_{X_{i}})^{i}\otimes d^{Y}_{j}$. The closed
structure also lifts from $\mathcal{G}$ to $\textnormal{Ch}(\mathcal{G})$, but
we won’t need its description in this paper. The point is that the above
tensor product gives us a symmetric monoidal structure on
$\textnormal{Ch}(\mathcal{G})$ with $X\otimes-$ a left adjoint for each
$X\in\textnormal{Ch}(\mathcal{G})$. If $I$ denotes the unit of the monoidal
structure on $\mathcal{G}$, then $S^{0}(I)$, the trivial complex with $I$
concentrated in degree 0, becomes the unit for the monoidal structure on
$\textnormal{Ch}(\mathcal{G})$.

The following lemma states that the $\otimes$-pure short exact sequences of
chain complexes are precisely the short exact sequences in
$\textnormal{Ch}(\mathcal{G})_{\otimes}$.

######  Lemma 2.2.

$0\xrightarrow{}W\xrightarrow{}X\xrightarrow{}Y\xrightarrow{}0$ is a short
exact sequence in the exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$
if and only if $0\xrightarrow{}C\otimes W\xrightarrow{}C\otimes
X\xrightarrow{}C\otimes Y\xrightarrow{}0$ remains exact in
$\textnormal{Ch}(\mathcal{G})$ for every chain complex $C$.

###### Proof.

($\implies$) Assume
$0\xrightarrow{}W\xrightarrow{}X\xrightarrow{}Y\xrightarrow{}0$ is
$\otimes$-pure exact in each degree, and let
$C\in\textnormal{Ch}(\mathcal{G})$. Then for all pairs of integers $i,j$ we
have short exact sequences in $\mathcal{G}$

$0\xrightarrow{}C_{i}\otimes W_{j}\xrightarrow{}C_{i}\otimes
X_{j}\xrightarrow{}C_{i}\otimes Y_{j}\xrightarrow{}0.$

Since short exact sequences in $\mathcal{G}$ are closed under direct sums, it
follows that for each $n\in\mathbb{Z}$ we have short exact sequences

$0\xrightarrow{}\bigoplus_{i+j=n}C_{i}\otimes
W_{j}\xrightarrow{}\bigoplus_{i+j=n}C_{i}\otimes
X_{j}\xrightarrow{}\bigoplus_{i+j=n}C_{i}\otimes Y_{j}\xrightarrow{}0.$

By definition, this is the degree $n$ component of the tensor products, so we
get a short exact sequence of chain complexes

$0\xrightarrow{}C\otimes W\xrightarrow{}C\otimes X\xrightarrow{}C\otimes
Y\xrightarrow{}0.$

($\impliedby$) Use that $S^{0}(M)\otimes X=M\otimes X$, for any object
$M\in\mathcal{G}$. ∎

For a subcomplex $X\subseteq Y$ we will write $X\subseteq_{\otimes}Y$ to mean
that each inclusion $X_{n}\subseteq Y_{n}$ is a $\otimes$-pure monomorphism.
In other words, the inclusion $X\subseteq Y$ is an admissible monic in the
exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$.

######  Lemma 2.3.

Let $X\subseteq Y\subseteq Z$ be subcomplexes in
$\textnormal{Ch}(\mathcal{G})$. If $X\subseteq_{\otimes}Z$ and
$Y/X\subseteq_{\otimes}Z/X$, then also $Y\subseteq_{\otimes}Z$.

###### Proof.

We have a comutative diagram in $\textnormal{Ch}(\mathcal{G})$ with exact rows
and with each vertical arrow a monomorphism.

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>X@>{}>{}>Y@>{}>{}>Y/X@>{}>{}>0\\\
\Big{\|}@V{}V{}V@V{}V{}V\\\ 0@>{}>{}>X@>{}>{}>Z@>{}>{}>Z/X@>{}>{}>0\\\
\end{CD}$

We will use the characterization in Lemma 2.2. Since $X\subseteq_{\otimes}Z$,
applying $W\otimes-$ for any complex $W$ yields another commutative diagram

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>W\otimes X@>{}>{}>W\otimes
Y@>{}>{}>W\otimes Y/X@>{}>{}>0\\\ \Big{\|}@V{}V{}V@V{}V{}V\\\
0@>{}>{}>W\otimes X@>{}>{}>W\otimes Z@>{}>{}>W\otimes Z/X@>{}>{}>0\\\
\end{CD}$

with exact rows. Now since $Y/X\subseteq_{\otimes}Z/X$, the rightmost vertical
morphism is monic. So by the snake lemma, the middle vertical morphism is
monic which means $Y\subseteq_{\otimes}Z$. ∎

##  3\. $\otimes$-acyclicity of chain complexes

Throughout this section we again let $(\mathcal{G},\otimes)$ be a closed
symmetric monoidal Grothendieck category and we let $\lambda$ denote a regular
cardinal for which $\mathcal{G}$ is locally $\lambda$-presentable. We note
that $\textnormal{Ch}(\mathcal{G})$ is then locally $\lambda$-presentable too.

Many of the results we would like to prove for K-flat complexes are actually
special cases of a more general phenomenon. In particular, K-flat complexes
are $\mathcal{C}$-acyclic in the following sense, by taking $\mathcal{C}$ to
be the class of all exact complexes.

######  Definition 3.1.

Let $\mathcal{C}$ be any given fixed class of chain complexes in
$\textnormal{Ch}(\mathcal{G})$. We will say that a chain complex $X$ is
_$\mathcal{C}$ -acyclic_ if $C\otimes X$ is acyclic for all $C\in\mathcal{C}$.
We let ${}_{\mathcal{C}}\mathcal{W}$ denote the class of all
$\mathcal{C}$-acyclic complexes $X$.

In addition to K-flat complexes, the pure acyclic complexes (see Section 4)
are also a special case. Note too that the $\\{S^{0}(I)\\}$-acyclic complexes
are precisely the usual acyclic (exact) complexes.

######  Lemma 3.2.

Let $P\subseteq X$ be a $\lambda$-pure subcomplex of
$X\in\textnormal{Ch}(\mathcal{G})$. If $X$ is acyclic (just exact not
necessarily pure acyclic), then both $P$ and $X/P$ are acyclic too. In fact,
if $X$ is $\mathcal{C}$-acyclic, then $P$ and $X/P$ are also
$\mathcal{C}$-acyclic.

###### Proof.

We generalize ideas inside the proof of [EGO17, Prop. 3.11]. In particular, if
$L$ is any $\lambda$-presented object of $\mathcal{G}$, then $S^{n}(L)$ and
$D^{n}(L)$ are $\lambda$-presented in $\textnormal{Ch}(\mathcal{G})$. So by
applying
$\operatorname{Hom}_{\textnormal{Ch}(\mathcal{G})}(S^{n}(L),-)\cong\operatorname{Hom}_{\mathcal{G}}(L,Z_{n}(-))$
and
$\operatorname{Hom}_{\textnormal{Ch}(\mathcal{G})}(D^{n}(L),-)\cong\operatorname{Hom}_{\mathcal{G}}(L,(-)_{n})$
we find that the rows of the commutative diagram must be $\lambda$-pure exact
sequences:

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>Z_{n}P@>{}>{}>Z_{n}X@>{}>{}>Z_{n}(X/P)@>{}>{}>0\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
0@>{}>{}>P_{n}@>{}>{}>X_{n}@>{}>{}>(X/P)_{n}@>{}>{}>0\\\ \end{CD}$

In particular they are usual short exact sequences and so applying the snake
lemma we deduce that
$0\xrightarrow{}B_{n-1}P\xrightarrow{}B_{n-1}X_{n}\xrightarrow{}B_{n-1}(X/P)\xrightarrow{}0$
is a short exact sequence. But then applying the snake lemma again to

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>B_{n}P@>{}>{}>B_{n}X@>{}>{}>B_{n}(X/P)@>{}>{}>0\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\
0@>{}>{}>Z_{n}P@>{}>{}>Z_{n}X@>{}>{}>Z_{n}(X/P)@>{}>{}>0\\\ \end{CD}$

produces a short exact sequence in homology
$0\xrightarrow{}H_{n}P\xrightarrow{}H_{n}X\xrightarrow{}H_{n}(X/P)\xrightarrow{}0$.
We conclude that $P$ and $X/P$ are each acyclic whenever $X$ is acylic.

We now show that if $X$ is $\mathcal{C}$-acyclic, then $P$ and $X/P$ are also
$\mathcal{C}$-acyclic. Since
$0\xrightarrow{}P\xrightarrow{}X\xrightarrow{}X/P\xrightarrow{}0$ is a
$\lambda$-pure exact sequence it is characterized as a $\lambda$-directed
colimit of splitting short exact sequences; see [AR94, 2.30, page 86] and
[Gil16a, Prop. 6.5]. For any complex $C$ we have that $C\otimes-$ preserves
direct limits, and since direct limits are exact it follows that
$0\xrightarrow{}C\otimes P\xrightarrow{}C\otimes X\xrightarrow{}C\otimes
X/P\xrightarrow{}0$ is also a $\lambda$-directed colimit of splitting short
exact sequences. Therefore it too is a $\lambda$-pure exact sequence. So
whenever $C\otimes X$ is acyclic, it follows from what we already proved above
that both $C\otimes P$ and $C\otimes X/P$ are also acyclic. In particular, for
any class of complexes $\mathcal{C}$, both $P$ and $X/P$ are
$\mathcal{C}$-acyclic whenever $X$ is $\mathcal{C}$-acyclic. ∎

######  Proposition 3.3.

Let $\mathcal{C}$ be any class of chain complexes. The class
${}_{\mathcal{C}}\mathcal{W}$ of all $\mathcal{C}$-acyclic complexes satisfies
the following properties.

  1. (1)

${}_{\mathcal{C}}\mathcal{W}$ is closed under direct sums, direct summands
(retracts), and direct limits. Also, ${}_{\mathcal{C}}\mathcal{W}$ contains
all contractible complexes.

  2. (2)

${}_{\mathcal{C}}\mathcal{W}$ is thick in
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. That is, it satisfies the 2 out of 3
property with respect to short exact sequences of chain complexes that are
$\otimes$-pure exact in each degree.

  3. (3)

${}_{\mathcal{C}}\mathcal{W}$ is closed under transfinite extensions in
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. That is, if $X$ has a filtration
$X\cong\varinjlim_{\alpha<\beta}X_{\alpha}$ where each
$X_{\alpha}\xrightarrow{}X_{\alpha+1}$ is a degreewise $\otimes$-pure
monomorphism, and $X_{0}$ and each $X_{\alpha+1}/X_{\alpha}$ are
$\mathcal{C}$-acyclic, then $X$ too is $\mathcal{C}$-acyclic.

###### Proof.

Since we are in a Grothendieck category the class of all acyclic complexes is
closed under direct sums, direct summands, and direct limits. For any chain
complex $C$, the functor $C\otimes-$ preserves direct sums, direct summands,
and direct limits. So ${}_{\mathcal{C}}\mathcal{W}$ is closed under these
operations. As for the contractible complexes being $\mathcal{C}$-acyclic,
recall that any contractible complex $X$ must take the form
$X\cong\bigoplus_{n\in\mathbb{Z}}D^{n}(M_{n})$ for some $\mathcal{G}$-objects
$\\{M_{n}\\}_{n\in\mathbb{Z}}$. So then we have $C\otimes
X\cong\bigoplus_{n\in\mathbb{Z}}(C\otimes D^{n}(M_{n}))$, and each $C\otimes
D^{n}(M_{n})$ can be shown to be acyclic. For example, see [Wei94, Exercise
1.2.5].

Now let $C\in\mathcal{C}$. By Lemma 2.2 we get a short exact sequence of chain
complexes

$0\xrightarrow{}C\otimes W\xrightarrow{}C\otimes X\xrightarrow{}C\otimes
Y\xrightarrow{}0.$

So then if 2 out of 3 of these are acyclic, so is the third. Therefore
${}_{\mathcal{C}}\mathcal{W}$ satisfies the 2 out of 3 property on short exact
sequences in $\textnormal{Ch}(\mathcal{G})_{\otimes}$.

So now it is easy to see that ${}_{\mathcal{C}}\mathcal{W}$ is closed under
transfinite extensions in $\textnormal{Ch}(\mathcal{G})_{\otimes}$. This
follows from the fact that we have shown ${}_{\mathcal{C}}\mathcal{W}$ to be
closed under extensions in $\textnormal{Ch}(\mathcal{G})_{\otimes}$, and under
direct limits. ∎

######  Theorem 3.4.

Let $\mathcal{C}$ be any class of $\mathcal{G}$-chain complexes and
${}_{\mathcal{C}}\mathcal{W}$ the class of all $\mathcal{C}$-acyclic
complexes. Set $\mathcal{F}:={}_{\mathcal{C}}\mathcal{W}^{\perp}$, defined in
the exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$. That is,

$\mathcal{F}=\\{\,F\in\textnormal{Ch}(\mathcal{G})\,|\,\operatorname{Ext}^{1}_{\otimes}(X,F)=0\
\forall\,X\in\,{}_{\mathcal{C}}\mathcal{W}\\}.$

Then $({}_{\mathcal{C}}\mathcal{W},\mathcal{F})$ is a complete cotorsion pair,
cogenerated by a set, in the exact category
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. It satisfies the following
properties:

  1. (1)

${}_{\mathcal{C}}\mathcal{W}$ is thick in
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. In particular the cotorsion pair is
hereditary.

  2. (2)

$F\in\mathcal{F}$ if and only if each $F_{n}$ is $\otimes$-pure injective and
every chain map $X\xrightarrow{}F$ is null homotopic whenever
$X\in{}_{\mathcal{C}}\mathcal{W}$.

  3. (3)

${}_{\mathcal{C}}\mathcal{W}\cap\mathcal{F}$ equals the class of all injective
objects in the exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$. These
are precisely the contractible complexes with $\otimes$-pure-injective
components.

###### Proof.

Since $\textnormal{Ch}(\mathcal{G})$ is locally $\lambda$-presentable there
exists, up to isomorphism, only a set (as opposed to a proper class) of
$\gamma$-presentable chain complexes for each regular cardinal $\gamma$
([AR94, Corollary 1.69]). Moreover, by [AR94, Theorem 2.33] there exist
arbitrarily large regular cardinals $\gamma\rhd\lambda$ such that every
$\gamma$-presentable subcomplex $S\subseteq X$ in
$\textnormal{Ch}(\mathcal{G})$ is contained in a $\lambda$-pure subcomplex
$P\subseteq X$, where $P$ is $\gamma$-presentable. So we choose such a
$\gamma$ and let $\mathcal{S}$ be a set of isomorphism representatives for all
$\gamma$-presentable $\mathcal{C}$-acyclic chain complexes. Lemma 2.1 applies
because $\\{D^{n}(L_{i})\\}\subseteq\mathcal{S}$. (Each $D^{n}(L_{i})$ is
$\mathcal{C}$-acylic and $\gamma$-presentable, since $\gamma>\lambda$.) So we
obtain a complete cotorsion pair
$({}^{\perp}(\mathcal{S}^{\perp}),\mathcal{S}^{\perp})$ in the exact category
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. Moreover, the class
${}^{\perp}(\mathcal{S}^{\perp})$ consists precisely of direct summands
(retracts) of transfinite degreewise $\otimes$-pure extensions of complexes in
$\mathcal{S}=\mathcal{S}\cup\\{D^{n}(L_{i})\\}$.

We will show ${}^{\perp}(\mathcal{S}^{\perp})={}_{\mathcal{C}}\mathcal{W}$.
Since $\mathcal{S}\subseteq{}_{\mathcal{C}}\mathcal{W}$, the containment
${}^{\perp}(\mathcal{S}^{\perp})\subseteq{}_{\mathcal{C}}\mathcal{W}$ follows
from Proposition 3.3. On the other hand, we will show
${}_{\mathcal{C}}\mathcal{W}\subseteq{}^{\perp}(\mathcal{S}^{\perp})$ by
showing that every $X\in{}_{\mathcal{C}}\mathcal{W}$ is a transfinite
degreewise $\otimes$-pure extension of complexes in $\mathcal{S}$.

But given $X\in{}_{\mathcal{C}}\mathcal{W}$, we may use transfinite induction
to construct a filtration of $X$ by $\mathcal{C}$-acyclic subcomplexes
$P_{\alpha}\subseteq_{\otimes}X$ with $P_{0}\in\mathcal{S}$ and each
$P_{\alpha+1}/P_{\alpha}\in\mathcal{S}$. (Recall that $\subseteq_{\otimes}$
denotes a degreewise $\otimes$-pure subcomplex.) For $i=0$, we let $P_{0}$ be
a nonzero $\gamma$-presentable and $\lambda$-pure subcomplex of $X$. Being a
$\lambda$-pure subcomplex of the $\mathcal{C}$-acyclic $X$, we note that
$P_{0}$ must also be $\mathcal{C}$-acyclic, by Lemma 3.2. Moreover,
$P_{0}\subseteq_{\otimes}X$, since $\lambda$-pure subcomplexes are degreewise
$\lambda$-pure, hence degreewise $\otimes$-pure. Having defined a
$\mathcal{C}$-acyclic and degreewise $\otimes$-pure subcomplex
$P_{\alpha}\subseteq_{\otimes}X$, and assuming that $P_{\alpha}\neq X$, we let
$P_{\alpha+1}/P_{\alpha}\subseteq_{\otimes}X/P_{\alpha}$ be a nonzero
$\gamma$-presentable and $\lambda$-pure subcomplex of $X/P_{\alpha}$. Since
$P_{\alpha}$ and $X$ are both $\mathcal{C}$-acyclic, Proposition 3.3(2)
assures us that $X/P_{\alpha}$ is also in ${}_{\mathcal{C}}\mathcal{W}$. So
then $P_{\alpha+1}/P_{\alpha}$ is $\mathcal{C}$-acyclic as well by Lemma 3.2.
Since both $P_{\alpha+1}/P_{\alpha}\subseteq_{\otimes}X/P_{\alpha}$ and
$P_{\alpha}\subseteq_{\otimes}X$, we infer from Lemma 2.3 that
$P_{\alpha+1}\subseteq_{\otimes}X$ too. We want to show that $P_{\alpha+1}$ is
$\mathcal{C}$-acyclic too. For this we first note
$(X/P_{\alpha})/(P_{\alpha+1}/P_{\alpha})\cong X/P_{\alpha+1}$ must be in
${}_{\mathcal{C}}\mathcal{W}$, by Lemma 3.2 (or even by Proposition 3.3(2)).
It then follows from Proposition 3.3(2) that $P_{\alpha+1}$ is
$\mathcal{C}$-acyclic. For the limit ordinal step, we define
$P_{\beta}=\bigcup_{\alpha<\beta}P_{\alpha}$; this is a colimit of degreewise
$\otimes$-pure subcomplexes of $X$, so is also a degreewise $\otimes$-pure
subcomplex of $X$. Of course $P_{\beta}$ must also be $\mathcal{C}$-acyclic by
Proposition 3.3(1). This process will eventually stop when $P_{\alpha}=X$, at
which point we have written $X$ as a transfinite extension in
$\textnormal{Ch}(\mathcal{G})_{\otimes}$ of complexes in $\mathcal{S}$. So
${}_{\mathcal{C}}\mathcal{W}\subseteq{}^{\perp}(\mathcal{S}^{\perp})$.

This proves
$({}_{\mathcal{C}}\mathcal{W},\mathcal{F})=({}^{\perp}(\mathcal{S}^{\perp}),\mathcal{S}^{\perp})$
is a complete cotorsion pair in $\textnormal{Ch}(\mathcal{G})_{\otimes}$. We
showed in Proposition 3.3 that ${}_{\mathcal{C}}\mathcal{W}$ is thick in
$\textnormal{Ch}(\mathcal{G})_{\otimes}$.

For (2). Since $D^{n}(M)$ is $\mathcal{C}$-acyclic for any
$\mathcal{G}$-object $M$, the isomorphism

$\operatorname{Ext}^{1}_{\otimes}(D^{n}(M),Y)\cong\operatorname{Ext}^{1}_{\mathcal{P}_{\otimes}}(M,Y_{n})$

implies that each $Y_{n}$ is $\otimes$-pure injective whenever
$F\in\mathcal{F}$. From this we deduce $F\in\mathcal{F}$ if and only if each
$F_{n}$ is $\otimes$-pure injective and every chain map $X\xrightarrow{}F$ is
null homotopic whenever $X\in{}_{\mathcal{C}}\mathcal{W}$.

For (3), it follows from (2) that
$X\in{}_{\mathcal{C}}\mathcal{W}\cap\mathcal{F}$ if and only if $X$ has
$\otimes$-pure injective components and $X\xrightarrow{1_{X}}X$ is null
homotopic. As in [EGO17, Section 3.2], this class coincides with the class
$\widetilde{\mathcal{PI}}_{\otimes}$ of injective objects relative to
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. ∎

In the language of abelian model structures we have shown the following.

######  Corollary 3.5.

Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck
category. Let $\mathcal{C}$ be any class of $\mathcal{G}$-chain complexes and
${}_{\mathcal{C}}\mathcal{W}$ the class of all $\mathcal{C}$-acyclic
complexes. Then $(All,{}_{\mathcal{C}}\mathcal{W},\mathcal{F})$ is a Hovey
triple with respect to the exact structure
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. The corresponding model structure is
cofibrantly generated.

The existence of $\mathcal{C}$-acyclic covers is also rather immediate.

######  Corollary 3.6.

Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck
category. Then any chain complex $X\in\textnormal{Ch}(\mathcal{G})$ has a
$\mathcal{C}$-acyclic cover.

###### Proof.

By the theorem, every complex has a special $\mathcal{C}$-acyclic precover and
by Proposition 3.3, the class ${}_{\mathcal{C}}\mathcal{W}$ of all
$\mathcal{C}$-acyclic complexes is closed under direct limits. It follows from
a general result, for example see [EGO20, Prop. 3.9], that
${}_{\mathcal{C}}\mathcal{W}$ is a covering class. ∎

##  4\. K-flat covers and the K-flat derived category

In this section we consider the case of K-flat and pure acyclic complexes.
Again, $(\mathcal{G},\otimes)$ denotes any closed symmetric monoidal
Grothendieck category throughout.

######  Definition 4.1.

A chain complex $X$ is _K-flat_ if $E\otimes X$ is acyclic for all acyclic
chain complexes $E$. We denote the class of all K-flat complexes by
$\mathcal{KF}$.

Building on Theorem 3.4 and Corollary 3.5 we have our main application: A
cofibrantly generated model for the Verdier quotient
$K(\mathcal{G})/\mathcal{KF}$. One might refer to this as the _K-flat derived
category_ of $(\mathcal{G},\otimes)$.

######  Theorem 4.2.

Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck
category. Then the class $\mathcal{KF}$ of all K-flat complexes is a thick
subcategory of $K(\mathcal{G})$. The Verdier quotient,
$K(\mathcal{G})/\mathcal{KF}$, is a well generated triangulated category and
is equivalent to the homotopy category associated to a Hovey triple
$(All,\mathcal{KF},\mathcal{I})$ on the exact category
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. This model structure is cofibrantly
generated and satisfies the following:

  1. (1)

$\mathcal{I}$ is the class of all complexes with $\otimes$-pure injective
components and such that all chain maps $X\xrightarrow{}I$ are null homotopic
whenever $X$ is K-flat.

  2. (2)

A chain map $f:X\xrightarrow{}Y$ is a weak equivalence in this model structure
if and only if its mapping cone is K-flat, equivalently, if and only if
$E\otimes X\xrightarrow{E\otimes f}E\otimes Y$ is a homology isomorphism for
all acyclic complexes $E$.

  3. (3)

We have a triangle equivalence, $K(\mathcal{G})/\mathcal{KF}\cong
K(\mathcal{I})$, where $K(\mathcal{I})$ is the strictly full subcategory of
$K(\mathcal{G})$ generated by $\mathcal{I}$.

###### Proof.

It follows from Theorem 3.4 and Corollary 3.5. Arguments similar to those in
[Gil23a, Theorem 4.4] and [Gil23b, Theorem 4.1(ii)] will show that the weak
equivalences are as described. ∎

Of course we also have the following special case of Corollary 3.6.

######  Corollary 4.3.

Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck
category. Then the class $\mathcal{KF}$ of all K-flat complexes is a covering
class.

In particular, for any scheme $\mathbb{X}$, any chain complex of quasi-
coherent sheaves has a K-flat cover.

A special case of K-flat complexes are the pure acyclic complexes.

######  Definition 4.4.

A chain complex $X$ is called $\otimes$-pure acyclic (or simply pure acyclic)
if $C\otimes X$ is acyclic for _all_ complexes
$C\in\textnormal{Ch}(\mathcal{G})$.

We prove in Proposition 4.6 that it is enough to only require that $M\otimes
X:=S^{0}(M)\otimes X$ is acyclic for all objects $M\in\mathcal{G}$. Our proof
will use the following lemma. Here $\mathcal{S}$ denotes a set which
cogenerates the $\otimes$-pure injective cotorsion pair
$(\mathcal{G},\mathcal{PI}_{\otimes})$. Such a set must exist; see [EGO17,
Lemma 3.6 and Prop. 3.7].

######  Lemma 4.5.

Let $\mathcal{S}$ be a set cogenerating the $\otimes$-pure injective cotorsion
pair $(\mathcal{G},\mathcal{PI}_{\otimes})$ in the exact category
$(\mathcal{G},\mathcal{P}_{\otimes})$. Then the injective cotorsion pair
$(\textnormal{Ch}(\mathcal{G}),\widetilde{\mathcal{PI}}_{\otimes})$ in the
exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$ (see [EGO17, Prop.
3.11]) is cogenerated by the set

$\mathcal{X}:=\\{S^{n}(M)\,|\,M\in\mathcal{S},n\in\mathbb{Z}\\}.$

Consequently, every chain complex $X\in\textnormal{Ch}(\mathcal{G})$ is a
direct summand of a transfinite degreewise $\otimes$-pure extension (i.e. a
transfinite extension in the exact category
$\textnormal{Ch}(\mathcal{G})_{\otimes}$) of complexes in the set

$\mathcal{X}=\\{S^{n}(M)\,|\,M\in\mathcal{S},n\in\mathbb{Z}\\}\cup\\{D^{n}(L_{i})\\}$

where $\\{L_{i}\\}$ is a set of generators for the pure exact structure
$\mathcal{P}_{\otimes}$.

###### Proof.

Suppose $Y\in\mathcal{X}^{\perp}$, the right orthogonal in the exact category
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. For each $L_{i}\in\\{L_{i}\\}$ we
have an obvious short exact sequence

$0\xrightarrow{}S^{n-1}(L_{i})\xrightarrow{}D^{n}(L_{i})\xrightarrow{}S^{n}(L_{i})\xrightarrow{}0$

in the exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$. Applying
$\operatorname{Hom}_{\textnormal{Ch}(\mathcal{G})}(-,Y)$ we obtain an exact
sequence of abelian groups

$\operatorname{Hom}_{\textnormal{Ch}(\mathcal{G})}(D^{n}(L_{i}),Y)\xrightarrow{}\operatorname{Hom}_{\textnormal{Ch}(\mathcal{G})}(S^{n-1}(L_{i}),Y)\xrightarrow{}\operatorname{Ext}^{1}_{\otimes}(S^{n}(L_{i}),Y)=0.$

It follows that
$\operatorname{Hom}_{\mathcal{G}}(L_{i},Y_{n})\xrightarrow{\operatorname{Hom}_{\mathcal{G}}(L_{i},d_{n})}\operatorname{Hom}_{\mathcal{G}}(L_{i},Z_{n-1}Y)$
is an epimorphism. Since $\\{L_{i}\\}$ is a set of generators for the pure
exact structure $\mathcal{P}_{\otimes}$, it means each
$0\xrightarrow{}Z_{n}Y\xrightarrow{}Y_{n}\xrightarrow{d_{n}}Z_{n-1}Y\xrightarrow{}0$
is a $\otimes$-pure exact sequence. Being a $\otimes$-pure exact complex, $Y$
must be exact in the ordinary sense, and so we have a standard isomorphism of
Yoneda Ext groups

$\operatorname{Ext}^{1}_{\textnormal{Ch}(\mathcal{G})}(S^{n}(M),Y)\cong\operatorname{Ext}^{1}_{\mathcal{G}}(M,Z_{n}Y)$

for any object $M\in\mathcal{G}$. For example, there is proof of this in
[Gil08, Lemma 4.2]. Following the proof there, one can check that, since $Y$
is $\otimes$-pure exact, the isomorphism restricts to an isomorphism of the
subgroups

$\operatorname{Ext}^{1}_{\otimes}(S^{n}(M),Y)\cong\operatorname{Ext}^{1}_{\mathcal{P}_{\otimes}}(M,Z_{n}Y).$

In particular, we have

$\operatorname{Ext}^{1}_{\mathcal{P}_{\otimes}}(M,Z_{n}Y)\cong\operatorname{Ext}^{1}_{\otimes}(S^{n}(M),Y)=0$

for each $M\in\mathcal{S}$. This proves $Z_{n}Y$ is $\otimes$-pure injective,
and thus $Y\in\widetilde{\mathcal{PI}}_{\otimes}$. ∎

######  Proposition 4.6.

A complex $X$ is pure acyclic (in the strong sense of Definition 4.4) if and
only if $M\otimes X:=S^{0}(M)\otimes X$ is acyclic for all objects
$M\in\mathcal{G}$.

###### Proof.

For any chain complex $X$, Proposition 3.3 implies (by taking
$\mathcal{C}=\\{X\\}$) that the class ${}_{X}\mathcal{W}$, of all chain
complexes $W$ for which $X\otimes W\cong W\otimes X$ is acyclic, is closed
under direct summands and transfinite extensions in the exact category
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. If we assume $X$ is a complex for
which $M\otimes X:=S^{0}(M)\otimes X$ is acyclic for all objects
$M\in\mathcal{G}$, then certainly $S^{n}(M)\in{}_{X}\mathcal{W}$ for all $n$
and $M\in\mathcal{G}$. It follows easily that also
$D^{n}(M)\in{}_{X}\mathcal{W}$ for all $n$ and $M\in\mathcal{G}$, see for
example [Wei94, Exercise 1.2.5]. But by Lemma 4.5, every complex must be a
direct summand of a transfinite extension in the exact category
$\textnormal{Ch}(\mathcal{G})_{\otimes}$ of such sphere or disk complexes. So
${}_{X}\mathcal{W}$ must be the class of all complexes, which means $X$ is
pure acyclic in the sense of Definition 4.4. ∎

##  5\. Consequences of $\otimes$-flat generators

We continue to let $(\mathcal{G},\otimes)$ denote a closed symmetric monoidal
Grothendieck category, and $\mathcal{KF}$ denote the class of K-flat
complexes. An object $F\in\mathcal{G}$ will be called $\otimes$-flat if the
functor $F\otimes-$ is exact. Note that a sphere complex $S^{n}(F)$ is K-flat
if and only if $F$ is $\otimes$-flat. In this section we look at the
consequences of the assumption that $\mathcal{G}$ possesses a set of
generators, $\\{F_{i}\\}$, with each $F_{i}$ a $\otimes$-flat object. For
short, we will express this by saying _$\mathcal{G}$ has a set of
$\otimes$-flat generators_.

######  Lemma 5.1.

Assume $\mathcal{G}$ has a set of $\otimes$-flat generators. Then every
complex in $\mathcal{KF}^{\perp}$, the right Ext-orthogonal in the exact
category $\textnormal{Ch}(\mathcal{G})_{\otimes}$, must be acyclic.

###### Proof.

Let $\\{F_{i}\\}$ be a set of $\otimes$-flat generators for $\mathcal{G}$. We
already know that any $E\in\mathcal{KF}^{\perp}$ must have $\otimes$-pure
injective components. So for each $F_{i}$, we have

$0=\operatorname{Ext}^{1}_{\otimes}(S^{n}(F_{i}),E)=\operatorname{Ext}^{1}_{dw}(S^{n}(F_{i}),E)\cong
H_{n-1}[\operatorname{Hom}_{\mathcal{G}}(F_{i},E)].$

So each complex $\operatorname{Hom}_{\mathcal{G}}(F_{i},E)$ is acyclic, and
the assumption that $\\{F_{i}\\}$ is a generating set implies $E$ is acyclic.
∎

######  Proposition 5.2.

Assume $\mathcal{G}$ has a set of $\otimes$-flat generators. $X$ is both
acyclic and K-flat if and only if $X$ is pure acyclic.

###### Proof.

$(\impliedby)$ This direction is clear, for if $I$ is the unit of the monoidal
structure, then $S^{0}(I)\otimes X=X$ is acyclic. For $(\implies)$, assume $X$
is both acyclic and K-flat. Let $C\in\textnormal{Ch}(\mathcal{G})$ be
arbitrary. By Theorem 4.2 we can find a degreewise $\otimes$-pure sequence
$0\xrightarrow{}E\xrightarrow{}K\xrightarrow{}C\xrightarrow{}0$ where
$K\in\mathcal{KF}$ and $E\in\mathcal{KF}^{\perp}$. Lemma 2.2 promises we get
another short exact sequence of complexes $0\xrightarrow{}E\otimes
X\xrightarrow{}K\otimes X\xrightarrow{}C\otimes X\xrightarrow{}0$. By the
assumption that $X$ is K-flat, and by Lemma 5.1, we obtain that $E\otimes X$
is acyclic. By the assumption that $X$ is acyclic, and because $K$ is K-flat,
we obtain that $K\otimes X\cong X\otimes K$ is acyclic too. It follows from
the 2 out of 3 property that $C\otimes X$ is also acyclic, proving $X$ is pure
acyclic. ∎

To state the next result we set
$\mathcal{D}_{\textnormal{K-flat}}(\mathcal{G}):=K(\mathcal{G})/\mathcal{KF}$.

######  Theorem 5.3.

Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck
category possessing a set of $\otimes$-flat generators. We have a recollement
of well generated triangulated categories

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101.53261pt\hbox{{\hbox{\kern-101.53261pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathcal{D}_{\textnormal{K-flat}}(\mathcal{G})}$}}}}}{\hbox{\kern-21.95087pt\raise
0.0pt\hbox{\hbox{\kern
3.0pt\raise-2.3889pt\hbox{$\textstyle{\mathcal{D}_{\otimes\textnormal{-pur}}(\mathcal{G})}$}}}}}{\hbox{\kern
56.49956pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathcal{D}(\mathcal{G})}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>5.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
54.05995pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
28.4526pt\raise-5.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
28.4526pt\raise 5.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>
###### Proof.

From Theorem 4.2, we have the K-flat model,
$\mathfrak{M}_{2}=(All,\mathcal{KF},\mathcal{I})$, whose homotopy category is
$\mathcal{D}_{\textnormal{K-flat}}(\mathcal{G})$. Taking $\mathcal{C}$ to be
the class of all chain complexes, Corollary 3.5 provides an abelian model
structure $\mathfrak{M}_{1}=(All,\mathcal{E}_{\otimes},\mathcal{F}_{1})$ on
the exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$. By Proposition
4.6 its homotopy category is precisely
$\mathcal{D}_{\otimes\textnormal{-pur}}(\mathcal{G})$, the $\otimes$-pure
derived category of [EGO17, Theorem A]. Taking $\mathcal{C}=\\{S^{0}(I)\\}$,
where $I$ is the unit of the monoidal structure, Corollary 3.5 provides an
abelian model structure $\mathfrak{M}_{3}=(All,\mathcal{E},\mathcal{F}_{3})$,
where $\mathcal{E}$ is the class of all (usual) acyclic chain complexes. From
the description of the fibrant objects given in Theorem 3.4, the class
$\mathcal{F}_{3}$ must be the class of all K-injective complexes with
$\otimes$-pure injective components. Its homotopy category is
$\mathcal{D}(\mathcal{G})$, the usual derived category of $\mathcal{G}$. These
three abelian model structures are cofibrantly generated and so their homotopy
categories are each well generated triangulated categories. The recollement
follows at once from Proposition 5.2 and [Gil16b, Theorem 4.6] ∎

Next, we note that a monoidal model structure for the usual derived category,
based on the K-flats, always exists when we have a set of $\otimes$-flat
generators. In the following statement, we keep the same notation as in the
proof of Theorem 5.3.

######  Theorem 5.4.

Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck
category possessing a set of $\otimes$-flat generators. Then

$\mathfrak{M}^{Kflat}_{\textnormal{der}}:=(\mathcal{KF},\mathcal{E},\mathcal{F}_{1})$

is a Hovey triple relative to the exact category
$\textnormal{Ch}(\mathcal{G})_{\otimes}$. It corresponds to a cofibrantly
generated model structure for the usual derived category,
$\mathcal{D}(\mathcal{G})$. The model structure is monoidal and satisfies the
monoid axiom as long as the unit $I$ is $\otimes$-flat. Moreover, the model
structure

$\mathfrak{M}_{\otimes\textnormal{-pur}}:=(All,\mathcal{E}_{\otimes},\mathcal{F}_{1})$

for $\mathcal{D}_{\otimes\textnormal{-pur}}(\mathcal{G})$, the $\otimes$-pure
derived category, is also monoidal and satisfies the monoid axiom (even if the
unit $I$ is not $\otimes$-flat).

###### Proof.

We have $\mathcal{KF}\cap\mathcal{E}=\mathcal{E}_{\otimes}$, the class of all
($\otimes$-)pure acyclic complexes, by Proposition 5.2. It follows from this
that we also have $\mathcal{E}\cap\mathcal{F}_{1}=\mathcal{I}$, the right
orthogonal to $\mathcal{KF}$. Indeed
$\mathcal{I}\subseteq\mathcal{E}\cap\mathcal{F}_{1}$ by Lemma 5.1. To show
$\mathcal{E}\cap\mathcal{F}_{1}\subseteq\mathcal{I}$, embed a complex
$X\in\mathcal{E}\cap\mathcal{F}_{1}$ as a $\otimes$-pure subobject of a
complex in $\mathcal{I}$ with K-flat cokernel, and note that the embedding
must be split exact.

It is easy to check that the four monoidal conditions of [Hov02, Theorem 7.2]
hold for both $\mathfrak{M}^{Kflat}_{\textnormal{der}}$ and
$\mathfrak{M}_{\otimes\textnormal{-pur}}$. In particular, all cofibrations are
$\otimes$-pure by Lemma 2.2. And $S^{0}(I)$, the unit for the monoidal
structure on $\textnormal{Ch}(\mathcal{G})$, is K-flat whenever $I$ is
$\otimes$-flat. By associativity of $\otimes$, the class $\mathcal{KF}$ is
closed under $\otimes$ and $\mathcal{E}_{\otimes}$ is a $\otimes$-ideal. The
monoid axiom is verified by the two conditions given in [Hov02, Theorem 7.4].
In particular, pure acyclic complexes are closed under transfinite
compositions of admissible monics in $\textnormal{Ch}(\mathcal{G})_{\otimes}$.
∎

##  6\. The right orthogonal to K-flats

For modules over a ring $R$, Emmanouil shows in [Emm22] that
$\mathcal{KF}^{\perp}$, in $K(R)$, is the class of acyclic complexes of pure
injective $R$-modules. Using the language of cotorsion pairs and Ext-
orthogonals in $\textnormal{Ch}(R)_{\otimes}$ this is described in [Gil23a,
Them 4.4]. In this section we would like to find conditions that allow us to
generalize this result. In particular, we would like this to be true for
complexes of quasi-coherent sheaves. We prove here that this is indeed the
case whenever the underlying scheme is quasi-compact and semiseparated. We
first prove some general results that appear to be useful in other categories,
and for other cotorsion pairs.

###  6.1. General Results

Here we prove some general lemmas, and Theorem 6.3, which can be useful to
characterize the right Ext-orthogonal of a class of chain complexes.

######  Lemma 6.1.

Let $\mathcal{G}$ be a Grothendieck category, considered along with a proper
class $\mathcal{E}$ of short exact sequences that is closed under direct
limits. Assume $\mathcal{F}$ is a class of objects that is closed under direct
summands, $\mathcal{E}$-extensions, and direct limits. Let
$\mathcal{E}_{\mathcal{F}}$ denote the inherited Quillen exact structure on
$\mathcal{F}$. If $(\mathcal{X},\mathcal{Y})$ is a cotorsion pair on the exact
category $(\mathcal{F},\mathcal{E}_{\mathcal{F}})$, and $\mathcal{X}$ is thick
relative to $\mathcal{E}_{\mathcal{F}}$, then $\mathcal{X}$ is closed under
direct limits.

Our proof is merely a verification that the one in [Gil17, Prop. 3.1/3.2]
generalizes to such a cotorsion pair $(\mathcal{X},\mathcal{Y})$.
Alternatively, the result follows from a variation of this type of argument
which is clearly written down in Positselski and Št’ovíček [PS23, Prop. 5.1].

###### Proof.

The argument from [Gil17, Prop. 3.1/3.2] generalizes to this setting. Indeed
the proof there analyzes a short exact sequence

(1)
$0\xrightarrow{}K\xrightarrow{\subseteq}\bigoplus_{i<\lambda}X_{i}\xrightarrow{}\varinjlim_{i<\lambda}X_{i}\xrightarrow{}0$

and $K$ is shown to be a directed union of subobjects $K_{J}$, each of which
is (isomorphic to) a direct summand of
$\bigoplus_{i<\lambda}X_{i}\in\mathcal{X}$. In particular, each inclusion
$K_{J}\subseteq\bigoplus_{i<\lambda}X_{i}$ is an admissible
$\mathcal{E}_{\mathcal{F}}$-monic since it is a split monic. We would like to
verify that the reduction in the proof that expresses
$K=\bigcup_{J\in\mathcal{S}}K_{J}$ as a direct union of a well-ordered
continuous (smooth) chain (via [AR94, Lemma1.6/ Corollary1.7/ Remark]), allows
us to still conclude $K\in\mathcal{X}$. This will follow if we are able to
verify the

(2) Claim: $\mathcal{X}$ is closed under direct unions of
$\mathcal{E}_{\mathcal{F}}$-admissible subobjects.

So let $\\{X_{i}\subseteq X\\}_{i\in I}$ be a direct system of subobjects,
indexed by some directed set $(I,\leq)$, with each $X_{i}\in\mathcal{X}$. We
proceed by transfinite induction on the cardinality $\gamma=|I|$ of the
directed indexing set $(I,\leq)$, to show that the direct union
$K:=\bigcup_{i\in I}X_{i}$ must be in the class $\mathcal{X}$.

First, we note that if $\gamma$ is finite, then the direct union
$K:=\bigcup_{i\in I}X_{i}$ must coincide with the particular $X_{i}$
representing the unique maximal element of $I$. So clearly
$K=X_{i}\in\mathcal{X}$ in this case.

For the induction step, let us be given such a direct union $K=\bigcup_{i\in
I}X_{i}$ with $\gamma=|I|$ an infinite cardinal, and suppose that all such
direct unions are in $\mathcal{X}$ whenever they are indexed by a directed set
of cardinality less than $\gamma$. Following the proof of [AR94,
Corollary1.7], we may re-express $K$ as a well-ordered continuous direct
union, $K=\bigcup_{k<\gamma}X_{I_{k}}$, where $\gamma=|I|$ and, by
construction, each $X_{I_{k}}=\bigcup_{i\in I_{k}}X_{i}$ is itself a direct
union of some of the subojects $X_{i}$, over a directed subset $I_{k}\subseteq
I$ of _smaller_ cardinality $|I_{k}|<\gamma$. By the induction hypothesis,
each $X_{I_{k}}\in\mathcal{X}$. Moreover, the inclusion $X_{I_{k}}\subseteq X$
must also be an admissible monic for $\mathcal{E}_{\mathcal{F}}$, since each
inclusion $X_{i}\subseteq X$ is such and by our other assumptions which imply
$\mathcal{E}_{\mathcal{F}}$ is closed under direct limits. It then follows
from [Büh10, Prop. 7.6 (dual)] that each inclusion $X_{I_{k}}\subseteq
X_{I_{k+1}}$ is an admissible monic in $\mathcal{E}_{\mathcal{F}}$. Since
$\mathcal{X}$ is thick relative to $\mathcal{E}_{\mathcal{F}}$, we get that
$X_{I_{k+1}}/X_{I_{k}}\in\mathcal{X}$ for all $k<\gamma$. In other words,
$K=\bigcup_{k<\gamma}X_{I_{k}}$ is a transfinite $\gamma$-extension of all the
objects $X_{I_{k+1}}/X_{I_{k}}\in\mathcal{X}$. So it follows from Eklof’s
Lemma (the version in [SŠ11, Prop. 2.12] applies), that $K\in\mathcal{X}$.

This completes the proof of the above _Claim_. Turning back to the short exact
sequence in (1), we have shown $K\in\mathcal{X}$. Moreover, this sequence must
be in $\mathcal{E}_{\mathcal{F}}$ by our assumptions on direct limits. So,
since $\mathcal{X}$ is thick in $\mathcal{E}_{\mathcal{F}}$, it follows that
$\varinjlim_{i<\lambda}X_{i}\in\mathcal{X}$. ∎

Given an exact category $(\mathcal{A},\mathcal{E})$, let
$\textnormal{Ch}(\mathcal{A})_{\mathcal{E}}$ denote the exact category of
chain complexes along with the inherited degreewise exact structure. Let us
say that a cotorsion pair in $\textnormal{Ch}(\mathcal{A})_{\mathcal{E}}$ is
_suspension closed_ if each class (equivalently, either class) is closed under
$\Sigma$ and $\Sigma^{-1}$. Given a class of objects
$\mathcal{X}\subseteq\mathcal{A}$, we will use the notation
$dw\widetilde{\mathcal{X}}$ to denote the class of all chain complexes that
are degreewise in $\mathcal{X}$.

######  Lemma 6.2.

Let $(\mathcal{X},\mathcal{Y})$ be a complete hereditary cotorsion pair on an
exact category $(\mathcal{A},\mathcal{E})$. Assume
$(\hat{\mathcal{X}},\hat{\mathcal{Y}})$ is a suspension closed complete
cotorsion pair in $\textnormal{Ch}(\mathcal{A})_{\mathcal{E}}$ such that every
complex of $\hat{\mathcal{X}}$ has components in $\mathcal{X}$ and every
complex of $\hat{\mathcal{Y}}$ has components in $\mathcal{Y}$. Then
$(\hat{\mathcal{X}},\hat{\mathcal{Y}})$ restricts to a complete cotorsion pair
$(\hat{\mathcal{X}},\hat{\mathcal{Y}}\cap dw\widetilde{\mathcal{X}})$ in the
exact category $\textnormal{Ch}(\mathcal{X})_{\mathcal{E}_{\mathcal{X}}}$, and
moreover $\hat{\mathcal{X}}$ is thick in
$\textnormal{Ch}(\mathcal{X})_{\mathcal{E}_{\mathcal{X}}}$.

###### Proof.

The key step is to show that ${}^{\perp}[\hat{\mathcal{Y}}\cap
dw\widetilde{\mathcal{X}}]$, taken in
$\textnormal{Ch}(\mathcal{X})_{\mathcal{E}_{\mathcal{X}}}$, is contained
within $\hat{\mathcal{X}}$. So let $X\in\textnormal{Ch}(\mathcal{X})$ be in
${}^{\perp}[\hat{\mathcal{Y}}\cap dw\widetilde{\mathcal{X}}]$. Using that
$(\hat{\mathcal{X}},\hat{\mathcal{Y}})$ is a complete cotorsion pair, we may
write a short exact sequence
$0\xrightarrow{}Y\xrightarrow{}X^{\prime}\xrightarrow{}X\xrightarrow{}0$ where
$Y\in\hat{\mathcal{Y}}$ and $X^{\prime}\in\hat{\mathcal{X}}$. Since all
$X_{n},X^{\prime}_{n}\in\mathcal{X}$ and $(\mathcal{X},\mathcal{Y})$ is
hereditary, we get $Y_{n}\in\mathcal{X}$. So $Y\in\hat{\mathcal{Y}}\cap
dw\widetilde{\mathcal{X}}$ and the sequence splits and we conclude
$X\in\hat{\mathcal{X}}$. Since $(\hat{\mathcal{X}},\hat{\mathcal{Y}})$ is
suspension closed and degreewise
$\operatorname{Ext}^{1}_{\mathcal{E}}$-orthogonal, we see that
$X\in\hat{\mathcal{X}}$ if and only if $\mathit{Hom}(X,Y)$ is acyclic for all
$Y\in\hat{\mathcal{Y}}$. Moreover, given a short exact sequence
$0\xrightarrow{}X\xrightarrow{}X^{\prime}\xrightarrow{}X^{\prime\prime}\xrightarrow{}0$
in $\textnormal{Ch}(\mathcal{X})_{\mathcal{E}_{\mathcal{X}}}$, then

$0\xrightarrow{}\mathit{Hom}(X,Y)\xrightarrow{}\mathit{Hom}(X^{\prime},Y)\xrightarrow{}\mathit{Hom}(X^{\prime\prime},Y)\xrightarrow{}0$

is a short exact sequence of chain complexes of abelian groups for all
$Y\in\hat{\mathcal{Y}}$. So $\hat{\mathcal{X}}$ has the 2 out of 3 property
relative to the exact category
$\textnormal{Ch}(\mathcal{X})_{\mathcal{E}_{\mathcal{X}}}$. ∎

The next theorem will also use the following notation: (i) for a given class
of objects $\mathcal{D}$ in an additive category with direct limits, we let
$\varinjlim\mathcal{D}$ denote the class of all objects that equal a direct
limit of some directed system of objects in $\mathcal{D}$, and (ii) assuming
we are in an exact category with $\mathcal{E}$ denoting the class of all short
exact sequences, we let $\langle\mathcal{D}\rangle_{\mathcal{E}}$, or just
$\langle\mathcal{D}\rangle$ if the context is clear, denote the class of all
objects $X$ possessing either a finite resolution, or a finite coresolutions,
by objects in $\mathcal{D}$.

######  Theorem 6.3.

Let $\mathcal{G}$ be a Grothendieck category, considered along with a proper
class $\mathcal{E}$ of short exact sequences that is closed under direct
limits. Suppose $(\mathcal{X},\mathcal{Y})$ is a complete hereditary cotorsion
pair on $(\mathcal{G},\mathcal{E})$, cogenerated by a (generating) set, and
that $\mathcal{X}$ is closed under direct limits in $\mathcal{G}$. Then:

  1. (1)

$({}^{\perp}dw\widetilde{\mathcal{Y}},dw\widetilde{\mathcal{Y}})$ is a
complete hereditary cotorsion pair in the exact category
$\textnormal{Ch}(\mathcal{G})_{\mathcal{E}}$. The class
${}^{\perp}dw\widetilde{\mathcal{Y}}$ is closed under direct limits, and it is
thick relative to the exact subcategory
$\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}$.

  2. (2)

Let $\mathcal{T}$ be the class of all contractible complexes with components
in $\mathcal{X}$. Then for any complex
$X\in\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}}$,
we have

$\operatorname{Ext}^{1}_{\textnormal{Ch}(\mathcal{G})_{\mathcal{E}}}(X,Y)=\operatorname{Ext}^{1}_{dw}(X,Y)=0$

for all $Y\in dw\widetilde{\mathcal{Y}}$. Equivalently, any chain map
$f:X\xrightarrow{}Y$ is null homotopic whenever
$X\in\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}}$
and $Y\in dw\widetilde{\mathcal{Y}}$.

###### Proof.

Taking the set of all disks $\\{D^{n}(S)\\}$ as $S$ ranges through a
cogenerating set for $(\mathcal{X},\mathcal{Y})$, then it cogenerates a
complete hereditary cotorsion pair,
$({}^{\perp}dw\widetilde{\mathcal{Y}},dw\widetilde{\mathcal{Y}})$, in
$\textnormal{Ch}(\mathcal{G})_{\mathcal{E}}$. It satisfies the hypotheses of
Lemma 6.2, and so it restricts to a (complete) cotorsion pair,

$({}^{\perp}dw\widetilde{\mathcal{Y}},dw\widetilde{\mathcal{Y}}\cap
dw\widetilde{\mathcal{X}}),$

on the exact subcategory
$\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}$, and
${}^{\perp}dw\widetilde{\mathcal{Y}}$ must be thick in
$\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}$. Moreover, by our
assumptions, $\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}$ is
closed under direct limits. So referring to Lemma 6.1 we may take
$(\mathcal{G},\mathcal{E})$ to be
$\textnormal{Ch}(\mathcal{G})_{\mathcal{E}}$, take
$(\mathcal{F},\mathcal{E}_{\mathcal{F}})$ to be
$\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}$, and take the
cotorsion pair $(\mathcal{X},\mathcal{Y})$ to be
$({}^{\perp}dw\widetilde{\mathcal{Y}},dw\widetilde{\mathcal{Y}}\cap
dw\widetilde{\mathcal{X}})$. We conclude that
${}^{\perp}dw\widetilde{\mathcal{Y}}$ is closed under direct limits, proving
(1).

Now let $\mathcal{T}$ be the class of all contractible complexes with
components in $\mathcal{X}$. Then certainly
$\mathcal{T}\subseteq{}^{\perp}dw\widetilde{\mathcal{Y}}$, so
$\varinjlim\mathcal{T}\subseteq{}^{\perp}dw\widetilde{\mathcal{Y}}$ by what we
just proved. And since ${}^{\perp}dw\widetilde{\mathcal{Y}}$ is thick in
$\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}$, we get that
$\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}}\subseteq{}^{\perp}dw\widetilde{\mathcal{Y}}$.
It means
$\operatorname{Ext}^{1}_{\textnormal{Ch}(\mathcal{G})_{\mathcal{E}}}(X,Y)=0$
for any complex
$X\in\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}}$,
and $Y\in dw\widetilde{\mathcal{Y}}$. ∎

To show the power of the above theorem we show how it may be used to recover
an important result of Bazzoni, Cortés-Izurdiaga, and Estrada from [BCE20,
Theorem 5.3].

######  Example 6.4 (Theorem 5.3 of [BCE20]).

Let $R$ be a ring. Take $(\mathcal{G},\mathcal{E})$ to be the category of
$R$-modules along with the usual abelian exact structure. Take
$(\mathcal{X},\mathcal{Y})=(\mathcal{F},\mathcal{C})$ to be Enochs’ flat
cotorsion pair. It is known that an acyclic complex with flat cycles is a
direct limit of contractible complexes with projective components. Thus
$\widetilde{\mathcal{F}}=\varinjlim\mathcal{T}$. It follows easily that
$\widetilde{\mathcal{F}}=\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathcal{F})_{\mathcal{E}_{\mathcal{F}}}}$.
Therefore,
$(\widetilde{\mathcal{F}},dg\widetilde{\mathcal{C}})=(\widetilde{\mathcal{F}},dw\widetilde{\mathcal{C}})$.

We can also use Theorem 6.3 to recover the result due to Christensen, Estrada,
and Thompson from [CET21, Theorem 3.3], which is the extention of [BCE20,
Theorem 5.3] to quasi-coherent sheaves. See Remark 6.7. As another example, we
have the following motivating result.

######  Corollary 6.5.

Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck
category and $\textnormal{Ch}(\mathcal{G})_{\otimes}$ the associated chain
complex category along with the degreewise $\otimes$-pure exact structure. Let
$\mathcal{PI}_{\otimes}$ denote the class of all $\otimes$-pure injective
objects, $\mathcal{E}_{\otimes}$ the class of all ($\otimes$-)pure acyclic
complexes, and let $\mathcal{T}$ denote the class of all contractible chain
complexes. If
$\mathcal{E}_{\otimes}\subseteq\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathcal{G})_{\otimes}}$,
then $(\mathcal{E}_{\otimes},dw\widetilde{\mathcal{PI}}_{\otimes})$ is a
complete cotorsion pair in $\textnormal{Ch}(\mathcal{G})_{\otimes}$.

###### Proof.

Take $(\mathcal{G},\mathcal{E})=(\mathcal{G},\mathcal{P}_{\otimes})$ to be the
proper class of all $\otimes$-pure short exact sequences. Take
$(\mathcal{X},\mathcal{Y})=(\mathcal{G},\mathcal{PI}_{\otimes})$ to be the
$\otimes$-pure injective cotorsion pair. Then
$\textnormal{Ch}(\mathcal{X})_{\mathcal{E_{\mathcal{X}}}}$ is nothing more
than the exact category $\textnormal{Ch}(\mathcal{G})_{\otimes}$. We get that
${}^{\perp}dw\widetilde{\mathcal{PI}}_{\otimes}$ is closed under direct limits
and is thick in $\textnormal{Ch}(\mathcal{G})_{\otimes}$. It also clearly
contans all contractible complexes. Therefore,
$\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathcal{G})_{\otimes}}\subseteq{}^{\perp}dw\widetilde{\mathcal{PI}}_{\otimes}$.
So if
$\mathcal{E}_{\otimes}\subseteq\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathcal{G})_{\otimes}}$,
then it is easy to see that the complete cotorsion pair
$(\mathcal{E}_{\otimes},\mathcal{F}_{1})$ in
$\textnormal{Ch}(\mathcal{G})_{\otimes}$ coincides with
$({}^{\perp}dw\widetilde{\mathcal{PI}}_{\otimes},dw\widetilde{\mathcal{PI}}_{\otimes})$.
∎

###  6.2. Acyclic complexes of $\otimes$-pure quasi-coherent sheaves

Throughout this section, we assume $\mathbb{X}=(X,\mathcal{O}_{X})$ is a
quasi-compact and semiseparated scheme. We let $\mathbb{X}$-Mod denote the
category of quasi-coherent sheaves on $\mathbb{X}$. It is a closed symmetric
monoidal category under the usual sheaf tensor product and internal hom coming
from applying the _coherator_ to the usual sheafhom. We let
$\textnormal{Ch}(\mathbb{X})$ denote the category of chain complexes of quasi-
coherent sheaves on $\mathbb{X}$. It inherits the standard symmetric monoidal
structure as indicated previously, before Lemma 2.2.

######  Theorem 6.6.

Let $\mathcal{PI}_{\otimes}$ denote the class of all $\otimes$-pure injective
quasi-coherent sheaves on $\mathbb{X}$. Let $\mathcal{E}_{\otimes}$ denote the
class of all ($\otimes$-)pure acyclic complexes in
$\textnormal{Ch}(\mathbb{X})$, and $\mathcal{KF}$ denote the class of all
K-flat complexes of quasi-coherent sheaves. Then:

  1. (1)

$(\mathcal{E}_{\otimes},dw\widetilde{\mathcal{PI}}_{\otimes})$ is a complete
cotorsion pair in the exact category $\textnormal{Ch}(\mathbb{X})_{\otimes}$.

  2. (2)

$(\mathcal{KF},ex\widetilde{\mathcal{PI}}_{\otimes})$ is a complete cotorsion
pair in the exact category $\textnormal{Ch}(\mathbb{X})_{\otimes}$. Here,
$ex\widetilde{\mathcal{PI}}_{\otimes}$ denotes the class of all acyclic (just
exact, not necessarily $\otimes$-pure exact) complexes of $\otimes$-pure
injectives.

Consequently, $\mathcal{D}_{\otimes\text{-pur}}(\mathbb{X})\cong
K(\mathcal{PI}_{\otimes})$, and
$\mathcal{D}_{\textnormal{K-flat}}(\mathbb{X})\cong
K_{ac}(\mathcal{PI}_{\otimes})$, and $\mathcal{D}(\mathbb{X})\cong
K(\mathcal{KF}\cap dw\widetilde{\mathcal{PI}}_{\otimes})$.

###### Proof.

We prove (1) by imitating the argument from [CET21, Theorem 3.3], making the
necessary adjustments. Let $Y$ be a complex of $\otimes$-pure injective quasi-
coherent sheaves. It is enough to show
$\operatorname{Ext}^{1}_{\otimes}(F,Y)=0$ for all $\otimes$-pure acyclic
complexes of quasi-coherent sheaves $F$, as, by [EGO17, Corollary 3.10] (or
Theorem 3.4), the containment $\mathcal{E}_{\otimes}^{\perp}\subseteq
dw\widetilde{\mathcal{PI}}_{\otimes}$ always holds. Let
$\mathcal{U}=\\{U_{0},U_{1},\cdots,U_{d}\\}$ be a semiseparating open affine
covering of $X$ and consider the usual Čech resolution ([Har77, Section
III.4]), extended in a degreewise fashion to chain complexes,

(3)
$0\xrightarrow{}F\xrightarrow{\epsilon}\mathcal{C}^{0}(\mathcal{U},F)\xrightarrow{}\mathcal{C}^{1}(\mathcal{U},F)\xrightarrow{}\cdots\xrightarrow{}\mathcal{C}^{d}(\mathcal{U},F)\xrightarrow{}0.$

Here,
$\mathcal{C}^{p}(\mathcal{U},F):=\bigoplus_{j_{0}<j_{1}<\cdots<j_{p}}i_{*}(\widetilde{F(U_{{j_{0}},{j_{1}},\cdots,{j_{p}}})})$,
where $j_{0}<j_{1}<\cdots<j_{p}$ ranges over sequences of length $p+1$ in
$\\{0,1,\cdots,d\\}$, and $U_{{j_{0}},{j_{1}},\cdots,{j_{p}}}:=U_{j_{0}}\cap
U_{j_{1}}\cap\cdots U_{j_{p}}$, and
$i:U_{{j_{0}},{j_{1}},\cdots,{j_{p}}}\xrightarrow{}X$ denotes the inclusion of
the open affine into the underlying space of $\mathbb{X}$. (We are picturing
each Čech sheaf complex $\mathcal{C}^{p}(\mathcal{U},F)$ in (3) as a vertical
chain complex.) We note that each such $\mathcal{C}^{p}(\mathcal{U},F)$ is
exact because $F$ is exact; see [Mur07, Section 3.1]. In fact, by [EGO17,
Prop. 2.10 and the proceeding remark], for any tuple of indices,
$j_{0}<j_{1}<\cdots<j_{p}$, the complex
$F(U_{{j_{0}},{j_{1}},\cdots,{j_{p}}})$ is a pure acyclic complex of
$\mathcal{O}_{X}(U_{{j_{0}},{j_{1}},\cdots,{j_{p}}})$-modules. As such, each
is isomorphic to a direct limit

$F(U_{{j_{0}},{j_{1}},\cdots,{j_{p}}})\cong\varinjlim_{\lambda\in\Lambda}W_{\lambda}^{U_{{j_{0}},{j_{1}},\cdots,{j_{p}}}},$

of contractible chain complexes,
$W_{\lambda}^{U_{{j_{0}},{j_{1}},\cdots,{j_{p}}}}$, of
$\mathcal{O}_{X}(U_{{j_{0}},{j_{1}},\cdots,{j_{p}}})$-modules, by [Emm16,
Prop. 2.2]. The direct image functor $i_{*}$ preserves direct limits and
contractible complexes, so it follows that each
$i_{*}(\widetilde{F(U_{{j_{0}},{j_{1}},\cdots,{j_{p}}})})$ is a direct limit
of contractible complexes of quasi-coherent sheaves. Letting $\mathcal{T}$
denote the class of all contractible complexes of quasi-coherent sheaves, it
means that each
$i_{*}(\widetilde{F(U_{{j_{0}},{j_{1}},\cdots,{j_{p}}})})\in\varinjlim\mathcal{T}$.
It follows that each Čech sheaf complex
$\mathcal{C}^{p}(\mathcal{U},F)\in\varinjlim\mathcal{T}$ too. As for the
horizontal (degreewise) exactness of the sequence in (3), by construction it
is, upon restriction to each open affine $U_{i}$, degreewise split exact. So
it follows from [EGO17, Prop. 2.10] that the sequence of complexes in (3) is
exact relative to the exact structure $\textnormal{Ch}(\mathbb{X})_{\otimes}$.
This means that
$F\in\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathbb{X})_{\otimes}}$,
and turning to Corollary 6.5, we conclude that
$(\mathcal{E}_{\otimes},dw\widetilde{\mathcal{PI}}_{\otimes})$ is a complete
cotorsion pair in $\textnormal{Ch}(\mathbb{X})_{\otimes}$.

The second statement follows readily: See the first paragraph of the proof of
Theorem 5.4. In fact, we have now shown
$\mathfrak{M}^{Kflat}_{\textnormal{der}}:=(\mathcal{KF},\mathcal{E},\mathcal{F}_{1})=(\mathcal{KF},\mathcal{E},dw\widetilde{\mathcal{PI}}_{\otimes})$
is a Hovey triple. So $ex\widetilde{\mathcal{PI}}_{\otimes}=\mathcal{E}\cap
dw\widetilde{\mathcal{PI}}_{\otimes}=\mathcal{KF}^{\perp}$, in
$\textnormal{Ch}(\mathbb{X})_{\otimes}$. ∎

######  Remark 6.7.

Continuing as in Example 6.4, the extension of [BCE20, Theorem 5.3] to
complexes of quasi-coherent sheaves was made in [CET21, Theorem 3.3] (see also
[PS23, Theorem 7.2]). We note that the argument in the above proof is just a
slight modification of the Čech resolution from their proof. In fact, if the
complex $F$ in our proof above has flat components, then the argument is
identical to theirs and shows
$F\in\langle\varinjlim\mathcal{T}\rangle_{\textnormal{Ch}(\mathcal{F})}$,
where $\mathcal{T}$ is the class of all contractible complexes of flat quasi-
coherent sheaves. That is, we recover [CET21, Theorem 3.3] from Theorem 6.3
applied to the flat cotorsion pair $(\mathcal{F},\mathcal{C})$ in
$\mathbb{X}$-Mod in a manner analogous to Example 6.4 and the above proof of
Theorem 6.6.

###  6.3. The Affine Case: Equivalence of the Verdier quotients by K-flats and
K-absolutely pures

Here we let $R$ be a ring and $\textnormal{Ch}(R)$ denote the category of
chain complexes of $R$-modules. $R$ need not be commutative so we consider,
say, left $R$-modules. Note that our model structure,
$\mathfrak{M}_{\mathcal{KF}}=(All,\mathcal{KF},ex\widetilde{\mathcal{PI}})$ on
$\textnormal{Ch}(R)_{\otimes}$, is _injective_ in the sense that it is abelian
and every object is cofibrant. It is interesting to note that there is another
model structure on $\textnormal{Ch}(R)_{\otimes}$, having complexes of pure-
projectives as the cofibrant objects, but whose homotopy category is also the
K-flat derived category,
$\mathcal{D}_{\textnormal{K-flat}}(R):=K(R)/\mathcal{KF}$. Moreover, this
model shows that $\mathcal{D}_{\textnormal{K-flat}}(R)$ is equivalent to the
Verdier quotient of $K(R)$ by Emmanouil and Kaperonis’ _K-absolutely pure_
complexes, introduced in [EK22].

Throughout this section we use the following notation for classes of chain
complexes in $\textnormal{Ch}(R)$:

  * •

$\mathcal{E}$ denotes the class of all exact (acyclic) complexes.

  * •

$\mathcal{E}_{\otimes}$ denotes the class of all pure acyclic complexes.

  * •

$dw\widetilde{\mathcal{PI}}$ denotes the class of all complexes that are
degreewise pure injective, that is, each component is a pure injective
$R$-module.

  * •

$dw\widetilde{\mathcal{PP}}$ denotes the class of all complexes that are
degreewise pure projective, that is, each component is a pure projective
$R$-module.

  * •

$\mathcal{KF}$ denotes the class of all K-flat complexes.

  * •

$\mathcal{KA}$ denotes the class of all K-absolutely pure complexes in the
sense of [EK22]. (This class was denoted by $\mathcal{V}$ in [Gil23b].)

  * •

$\mathcal{F}$ denotes the class of all K-injective complexes with pure
injective components; complexes in $\mathcal{F}$ were called _DG-pure
injective_ in [Gil23a].

  * •

$\mathcal{C}$ denotes the class of all K-projective complexes with pure
projective components; complexes in $\mathcal{C}$ were called _DG-pure
projective_ in [Gil23b].

The key to the following result is that, by [Emm19, Corollary 3.4], K-flat
complexes with pure projective components must also be K-projective.
Similarly, K-absolutely pure complexes with pure injective components must be
K-injective by [EK22, Prop. 2.3]. The proof of the next result will include
simple cotorsion theoretic proofs of these two facts.

######  Proposition 6.8.

Let $R$ be a ring, and $\textnormal{Ch}(R)_{\otimes}\
(=\textnormal{Ch}(R)_{pur})$ be the exact category of chain complexes along
with the degreewise pure exact structure.

  1. (1)

$(dw\widetilde{\mathcal{PP}},\mathcal{KF},\mathcal{E})$ is a cofibrantly
generated abelian model structure on $\textnormal{Ch}(R)_{\otimes}$. This
provides a triangulated equivalence of the Verdier quotient

$K(R)/\mathcal{KF}\cong K_{ac}(\mathcal{PP}),$

onto the full subcategory $K_{ac}(\mathcal{PP})\subseteq K(R)$ generated by
the class of all acyclic complexes of pure projectives.

  2. (2)

$(\mathcal{E},\mathcal{KA},dw\widetilde{\mathcal{PI}})$ is a cofibrantly
generated abelian model structure on $\textnormal{Ch}(R)_{\otimes}$. This
provides a triangulated equivalence of the Verdier quotient

$K(R)/\mathcal{KA}\cong K_{ac}(\mathcal{PI}),$

onto the full subcategory $K_{ac}(\mathcal{PI})\subseteq K(R)$ generated by
the class of all acyclic complexes of pure injectives.

Consequently, the Verdier quotient $K(R)/\mathcal{KF}$ is equivalent to the
Verdier quotient $K(R)/\mathcal{KA}$.

###### Proof.

For (1), we already have that
$(dw\widetilde{\mathcal{PP}},\mathcal{KF}\cap\mathcal{E})=(dw\widetilde{\mathcal{PP}},\mathcal{E}_{\otimes})$
is a complete cotorsion pair in $\textnormal{Ch}(R)_{\otimes}$, and
cogenerated by a set; see [Gil23b, Cor. 4.2]. Also,
$(\mathcal{C},\mathcal{E})$ is a complete cotorsion pair in
$\textnormal{Ch}(R)_{\otimes}$, and cogenerated by a set, by [Gil23b, Prop.
6.3]. So we only need to show Emmanouil’s relation
$dw\widetilde{\mathcal{PP}}\cap\mathcal{KF}=\mathcal{C}$.

$(\supseteq)$ This is easy to see.

$(\subseteq)$ Let $X\in dw\widetilde{\mathcal{PP}}\cap\mathcal{KF}$. Using
completeness of $(\mathcal{C},\mathcal{E})$, write a degreewise pure short
exact sequence $0\xrightarrow{}E\xrightarrow{}P\xrightarrow{}X\xrightarrow{}0$
with $E\in\mathcal{E}$ and $P\in\mathcal{C}$. Since
$\mathcal{C}\subseteq\mathcal{KF}$, we have $P\in\mathcal{KF}$. Since
$\mathcal{KF}$ is thick in $\textnormal{Ch}(R)_{\otimes}$, we have
$E\in\mathcal{KF}\cap\mathcal{E}=\mathcal{E}_{\otimes}$. The short exact
sequence represents an element of $\operatorname{Ext}^{1}_{\otimes}(X,E)$. But
this Ext group must vanish since
$(dw\widetilde{\mathcal{PP}},\mathcal{E}_{\otimes})$ is a cotorsion pair in
$\textnormal{Ch}(R)_{\otimes}$. Hence the short exact sequence must split,
making $X$ is a direct summand of $P$, thus forcing $X$ to also be in
$\mathcal{C}$. This completes the proof of (1).

The proof of (2) is dual: The dual results concerning complete cotorsion pairs
were already shown in [Gil23a], and $\mathcal{F}=\mathcal{KA}\cap
dw\widetilde{\mathcal{PI}}$ follows by a straightforward dual of the above
argument.

Finally, we observe that

$K_{ac}(\mathcal{PI})\cong K(R)/\mathcal{KF}\cong K_{ac}(\mathcal{PP})\cong
K(R)/\mathcal{KA}\cong K_{ac}(\mathcal{PI})$

The first equivalence is from [Emm22], the second equivalence comes from what
we just showed in (1), the third equivalence is from [EK22, Prop. 2.3], and
the last equivalence follows from statement (2). ∎

## Acknowledgements

This paper was advanced during a visit of the second author to the Univeridad
de Murcia. He thanks his coauthors and his host, Sergio Estrada, for his
gracious hospitality.

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