The following corollary is to be compared with Corollaries 5.2.2(a) and
5.2.9(a).

######  Corollary 4.3.6.

Any colocally flat contraherent cosheaf over a semi-separated Noetherian
scheme is flat.

###### Proof.

Follows from Corollary 4.3.4(c) together with the remarks about flat
contraherent cosheaves over affine Noetherian schemes and the direct images of
flat cosheaves of ${\mathcal{O}}$-modules in Section 3.7. ∎

######  Corollary 4.3.7.

Over a semi-separated Noetherian scheme $X$, the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ is closed with respect to
infinite products in $X{\operatorname{\mathsf{--ctrh}}}$.

###### Proof.

In addition to what has been said in the proof of Corollary 4.2.9, it is also
important here that infinite products of flat modules over a coherent ring are
flat. ∎

######  Corollary 4.3.8.

Let $X$ be a semi-separated Noetherian scheme and $j\colon Y\longrightarrow X$
be an affine open embedding. Then the inverse image functor $j^{!}$ takes
colocally flat contraherent cosheaves to colocally flat contraherent
cosheaves.

###### Proof.

Similar to the proof of Corollary 4.2.10 and based on Corollary 4.3.4(c); the
only difference is that one also has to use Corollary 1.6.5(a). ∎

###  4.4. Projective contraherent cosheaves

Let $X$ be a quasi-compact semi-separated scheme and ${\mathbf{W}}$ be its
affine open covering.

######  Lemma 4.4.1.

(a) The exact category of ${\mathbf{W}}$-locally contraherent cosheaves on $X$
has enough projective objects.

(b) Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then a ${\mathbf{W}}$-locally contraherent
cosheaf on $X$ is projective if and only if it is a direct summand of a direct
sum over $\alpha$ of the direct images of contraherent cosheaves on
$U_{\alpha}$ corresponding to very flat contraadjusted
${\mathcal{O}}(U_{\alpha})$-modules.

###### Proof.

The assertion “if” in part (b) follows from the adjunction of the direct and
inverse image functors for the embeddings $U_{\alpha}\longrightarrow X$
together with the fact that the very flat contraadjusted modules are the
projective objects of the exact categories of contraadjusted modules over
${\mathcal{O}}(U_{\alpha})$ (see Section 1.4).

It remains to show that there exists an admissible epimorphism onto any
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on $X$ from a
direct sum of the direct images of contraherent cosheaves on $U_{\alpha}$
corresponding to very flat contraadjusted modules. The construction is similar
to the one used in the proof of Lemma 4.3.3 and based on Theorem 1.1.1(b). One
picks admissible epimorphisms from contraherent cosheaves
${\mathfrak{F}}_{\alpha}$ of the desired kind on the affine schemes
$U_{\alpha}$ onto the restrictions $j_{\alpha}^{!}{\mathfrak{M}}$ of the
cosheaf ${\mathfrak{M}}$ and considers the corresponding morphism
$\bigoplus_{\alpha}j_{\alpha}{}_{!}{\mathfrak{F}}_{\alpha}\longrightarrow{\mathfrak{M}}$.
∎

######  Corollary 4.4.2.

(a) There are enough projective objects in the exact category
$X{\operatorname{\mathsf{--lcth}}}$ of locally contraherent cosheaves on $X$,
and all these projective objects belong to the full subcategory of
contraherent cosheaves $X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}$.

(b) The full subcategories of projective objects in the three exact categories
$X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\subset
X{\operatorname{\mathsf{--lcth}}}$ coincide. ∎

######  Lemma 4.4.3.

(a) The exact category of locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaves on $X$ has enough projective objects.

(b) Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then a locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ is projective if and only if it is a direct
summand of a direct sum over $\alpha$ of the direct images of locally
cotorsion contraherent cosheaves on $U_{\alpha}$ corresponding to flat
cotorsion ${\mathcal{O}}(U_{\alpha})$-modules.

###### Proof.

Similar to the proof of Lemma 4.4.1 and based on Theorem 1.3.1(b). ∎

######  Corollary 4.4.4.

(a) There are enough projective objects in the exact category
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ of locally cotorsion
locally contraherent cosheaves on $X$, and all these projective objects belong
to the full subcategory of locally cotorsion contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$.

(b) The full subcategories of projective objects in the three exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ coincide. ∎

We denote the additive category of projective (objects in the category of)
contraherent cosheaves on $X$ by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$, and the additive category
of projective (objects in the category of) locally cotorsion contraherent
cosheaves $X$ by
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.

Let us issue a _warning_ that both the terminology and notation are misleading
here: a projective locally cotorsion contraherent cosheaf on $X$ does _not_
have to be a projective contraherent cosheaf. Indeed, a flat cotorsion module
over a commutative ring would not be in general very flat.

On the other hand we notice that, by the definition, both additive categories
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ are
contained in the exact category of colocally flat contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ (and consequently also in
the exact category of colocally projective contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$). Moreover, by the
definition one clearly has
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}=X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\cap
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$. Finally, we notice that
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ is the category of
_projective_ objects in $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$,
while $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is the
category of _injective_ objects in
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ (and there are enough of
both).

A version of part (a) of the following corollary that is valid in a different
generality will be obtained in Section 5.2.

######  Corollary 4.4.5.

Let $X$ be a semi-separated Noetherian scheme. Then

(a) any cosheaf from $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ is
flat;

(b) any cosheaf from
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is flat.

###### Proof.

Follows from Corollary 4.3.6. ∎

More general versions of the next corollary and of part (b) of the previous
one will be obtained in Section 5.1. In both cases, we will see that the semi-
separatedness and quasi-compactness assumptions can be dropped.

######  Corollary 4.4.6.

Over a semi-separated Noetherian scheme $X$, the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ of
projective locally cotorsion contraherent cosheaves is closed under infinite
products in $X{\operatorname{\mathsf{--ctrh}}}$.

###### Proof.

Follows from Corollary 4.3.7. ∎

A more general version of parts (a-b) of the following Corollary will be
proven in the next Section 4.5, while more general versions of part (b-c) will
be also obtained in Section 5.1. In both cases, we will see that the
affineness assumption on the morphism $f$ is unnecessary.

######  Corollary 4.4.7.

Let $f\colon Y\longrightarrow X$ be an affine morphism of quasi-compact semi-
separated schemes. Then

(a) if the morphism $f$ is very flat, then the direct image functor $f_{!}$
takes projective contraherent cosheaves to projective contraherent cosheaves;

(b) if the morphism $f$ is flat, then the direct image functor $f_{!}$ takes
projective locally cotorsion contraherent cosheaves to projective locally
cotorsion contraherent cosheaves;

(c) if the scheme $X$ is Noetherian and the morphism $f$ is an open embedding,
then the inverse image functor $f^{!}$ takes projective locally cotorsion
contraherent cosheaves to projective locally cotorsion contraherent cosheaves.

###### Proof.

Part (a) holds, since in its assumptions the functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$ is “parially left adjoint” to the exact
functor $f^{!}\colon X{\operatorname{\mathsf{--lcth}}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}$. The proof of part (b) is similar
(alternatively, it can be deduced from the facts that the functor $f_{!}$
takes $Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ and
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$). Part (c) follows from
Corollary 4.3.8. ∎

###  4.5. Homology of locally contraherent cosheaves

The functor $\Delta(X,{-})$ of global cosections of locally contraherent
cosheaves on a scheme $X$, which assigns to a cosheaf ${\mathfrak{E}}$ the
abelian group (or even the ${\mathcal{O}}(X)$-module) ${\mathfrak{E}}[X]$, is
right exact as a functor on the exact category of locally contraherent
cosheaves $X{\operatorname{\mathsf{--lcth}}}$ on $X$. In other words, if
$0\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{L}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$ is a short exact sequence of locally contraherent cosheaves on $X$, then
the sequence of abelian groups

$\Delta(X,{\mathfrak{K}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\Delta(X,{\mathfrak{L}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\Delta(X,{\mathfrak{M}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

is exact. Indeed, the procedure recovering the groups of cosections of
cosheaves ${\mathcal{F}}$ on $X$ from their groups of cosections over affine
open subschemes $U\subset X$ subordinate to a particular covering
${\mathbf{W}}$ and the corestriction maps between such groups uses the
operations of the infinite direct sum and the cokernel of a morphism (or in
other words, the nonfiltered inductive limit) only (see (5), (7), or (22)).

Recall that for any $({\mathbf{W}},{\mathbf{T}})$-affine morphism of schemes
$f\colon Y\longrightarrow X$ the functor of direct image $f_{!}$ takes
${\mathbf{T}}$-locally contraherent cosheaves on $Y$ to ${\mathbf{W}}$-locally
contraherent cosheaves on $X$. By the definition, there is a natural
isomorphism of ${\mathcal{O}}(X)$-modules
${\mathfrak{E}}[Y]\simeq(f_{!}{\mathfrak{E}})[X]$ for any cosheaf of
${\mathcal{O}}_{Y}$-modules ${\mathfrak{E}}$.

Now let $X$ be a quasi-compact semi-separated scheme. Then the left derived
functor of the functor of global cosections of locally contraherent cosheaves
on $X$ can be defined in the conventional way using left projective
resolutions in the exact category $X{\operatorname{\mathsf{--lcth}}}$ (see
Lemma 4.4.1 and Corollary 4.4.2). Notice that the derived functors of
$\Delta(X,{-})$ (and in fact, any left derived functors) computed in the exact
category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ for a particular
open covering ${\mathbf{W}}$ and in the whole category
$X{\operatorname{\mathsf{--lcth}}}$ agree. We denote this derived functor by
${\mathbb{L}}_{*}\Delta(X,{-})$. The groups
${\mathbb{L}}_{i}\Delta(X,{\mathfrak{E}})$ are called the _homology groups_ of
a locally contraherent cosheaf ${\mathfrak{E}}$ on the scheme $X$.

Let us point out that the functor $\Delta(U,{-})$ of global cosections of
contraherent cosheaves on an affine scheme $U$ is exact, so the groups
${\mathbb{L}}_{>0}\Delta(U,{\mathfrak{E}})$ vanish when $U$ is affine and
${\mathfrak{E}}$ is contraherent.

By Corollary 4.4.7(a), for any very flat $({\mathbf{W}},{\mathbf{T}})$-affine
morphism of quasi-compact semi-separated schemes $f\colon Y\longrightarrow X$
the exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ takes projective contraherent
cosheaves on $Y$ to projective contraherent cosheaves on $X$. It also makes a
commutative diagram with the restrictions of the functors $\Delta(X,{-})$ and
$\Delta(Y,{-})$ to the categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$. Hence one has
${\mathbb{L}}_{*}\Delta(Y,{\mathfrak{E}})\simeq{\mathbb{L}}_{*}\Delta(X,f_{!}{\mathfrak{E}})$
for any ${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{E}}$ on $Y$.

In particular, the latter assertion applies to the embeddings of affine open
subschemes $j\colon U\longrightarrow X$, so
${\mathbb{L}}_{>0}\Delta(X,j_{!}{\mathfrak{E}})=0$ for all contraherent
cosheaves ${\mathfrak{E}}$ on $U$. Since the derived functor
${\mathbb{L}}_{*}\Delta$ takes short exact sequences of locally contraherent
cosheaves to long exact sequences of abelian groups, it follows from Corollary
4.2.5(c) that ${\mathbb{L}}_{>0}\Delta(X,{\mathfrak{P}})=0$ for any colocally
projective contraherent cosheaf ${\mathfrak{P}}$ on $X$.

Therefore, the derived functor ${\mathbb{L}}_{*}\Delta$ can be computed using
colocally projective left resolutions. Now we also see that the derived
functors ${\mathbb{L}}_{*}\Delta$ defined in the theories of arbitrary (i. e.,
locally contraadjusted) contraherent cosheaves and of locally cotorsion
contraherent cosheaves agree.

Let ${\mathfrak{E}}$ be a ${\mathbf{W}}$-locally contraherent cosheaf on $X$,
and let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering of $X$
subordinate to ${\mathbf{W}}$. Then the contraherent Čech resolution (27) for
${\mathfrak{E}}$ is a colocally projective left resolution of a locally
contraherent cosheaf ${\mathfrak{E}}$, and one can use it to compute the
derived functor ${\mathbb{L}}_{*}(X,{\mathfrak{E}})$. In other words, the
homology of a ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{E}}$ on
a quasi-compact semi-separated scheme $X$ are computed by the homological Čech
complex $C_{*}(\\{U_{\alpha}\\},{\mathfrak{E}})$ (see (22)) related to any
finite affine open covering $X=\bigcup_{\alpha}U_{\alpha}$ subordinate to
${\mathbf{W}}$.

The following result is to be compared with the cohomological criterion of
affineness of schemes [26, Théorème 5.2.1].

######  Corollary 4.5.1.

A locally contraherent cosheaf ${\mathfrak{E}}$ on an affine scheme $U$ is
contraherent if and only if its higher homology
${\mathbb{L}}_{>0}\Delta(X,{\mathfrak{E}})$ vanish.

###### Proof.

See Lemma 3.2.2. ∎

######  Corollary 4.5.2.

Let $R$ be a commutative ring, $I\subset R$ be a nilpotent ideal, and $S=R/I$
be the quotient ring. Let
$i\colon\operatorname{Spec}S\longrightarrow\operatorname{Spec}R$ denote the
corresponding homeomorphic closed embedding of affine schemes. Then a locally
injective locally contraherent cosheaf ${\mathfrak{J}}$ on
$U=\operatorname{Spec}R$ is contraherent if and only if its inverse image
$i^{!}{\mathfrak{J}}$ on $V=\operatorname{Spec}S$ is contraherent.

###### Proof.

Since any morphism into an affine scheme is coaffine, the “only if” assertion
is obvious. To prove the “if”, we apply again Lemma 3.2.2. Assuming that
${\mathfrak{J}}$ is ${\mathbf{W}}$-locally contraherent on $U$ and
$U=\bigcup_{\alpha}U_{\alpha}$ is a finite affine open covering of $U$
subordinate to ${\mathbf{W}}$, the homological Čech complex
$C_{*}(\\{U_{\alpha}\\},{\mathfrak{J}})$ is a finite complex of injective
$R$-modules. Set $V_{\alpha}=i^{-1}(U_{\alpha})\subset V$; then the complex
$C_{*}(\\{V_{\alpha}\\},{\mathfrak{J}})$ is the maximal subcomplex of
$R$-modules in $C_{*}(\\{U_{\alpha}\\},{\mathfrak{J}})$ annihilated by the
action of $I$.

Since the maximal submodule ${}_{I}M\subset M$ annihilated by $I$ is nonzero
for any nonzero $R$-module $M$, one easily proves by induction that a finite
complex $K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of injective
$R$-modules is acyclic at all its terms except perhaps the rightmost one
whenever so is the complex of injective $R/I$-modules
${}_{I}K^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. Notice that this
argument does not seem to apply to the maximal reduced closed subscheme
$\operatorname{Spec}R/J$ of an arbitrary affine scheme $\operatorname{Spec}R$
in general, as the maximal submodule ${}_{J}K\subset K$ annihilated by the
nilradical $J\subset R$ may well be zero even for a nonzero injective
$R$-module $K$ (set $K=\operatorname{Hom}_{k}(F,k)$ in the example from Remark
1.7.5). ∎

The following result is to be compared with Corollaries 3.4.8 and 4.1.13 (for
another comparison, see Corollaries 4.2.10 and 4.3.8).

######  Corollary 4.5.3.

Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes. Then

(a) the functor of direct image of cosheaves of ${\mathcal{O}}$-modules
$f_{!}$ takes colocally projective contraherent cosheaves on $Y$ to colocally
projective contraherent cosheaves on $X$, and induces an exact functor
$f_{!}\colon Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ between these exact
categories;

(b) the functor of direct image of cosheaves of ${\mathcal{O}}$-modules
$f_{!}$ takes colocally projective locally cotorsion contraherent cosheaves on
$Y$ to colocally projective locally cotorsion contraherent cosheaves on $X$,
and induces an exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$ between these
exact categories;

(c) if the morphism $f$ is flat, then the functor of direct image of cosheaves
of ${\mathcal{O}}$-modules $f_{!}$ takes colocally flat contraherent cosheaves
on $Y$ to colocally flat contraherent cosheaves on $X$, and induces and exact
functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ between these exact
categories;

(d) if the morphism $f$ is flat, then the functor of direct image of cosheaves
of ${\mathcal{O}}$-modules $f_{!}$ takes colocally projective locally
injective contraherent cosheaves on $Y$ to colocally projective locally
injective contraherent cosheaves on $X$.

###### Proof.

Part (a): by Corollary 4.2.10(a), the inverse image of a colocally projective
contraherent cosheaf on $Y$ with respect to an affine open embedding $j\colon
V\longrightarrow Y$ is colocally projective. As we have seen above, the global
cosections of colocally projective contraherent cosheaves is an exact functor.
It follows that the functor $f_{!}$ takes short exact sequences in
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ to short exact sequences in
the exact category of cosheaves of ${\mathcal{O}}_{X}$-modules (with the exact
category structure
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\\{X\\}}$ related to the
covering $\\{X\\}$ of the scheme $X$; see Section 3.1).

Since $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ is a full exact
subcategory closed under extensions in $X{\operatorname{\mathsf{--ctrh}}}$,
and the latter exact category is such an exact subcategory in
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\\{X\\}}$, in view of
Corollary 4.2.5(c) it remains to recall that the direct images of colocally
projective contraherent cosheaves with respect to affine morphisms of schemes
are colocally projective (see the remarks in the beginning of Section 4.2).

Part (b) is similar; the proof of part (c) is also similar and based on the
remarks about direct images in the beginning of Section 4.3 together with
Corollary 4.3.8; and to prove part (d) one only needs to recall that the
direct images of locally injective contraherent cosheaves with respect to flat
affine morphisms of schemes are locally injective and use Corollary 4.2.8. ∎

Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes. By the result of Section 3.3 (see (25)), the adjunction isomorphism
(23) holds, in particular, for any colocally projective contraherent cosheaf
${\mathfrak{Q}}$ on $Y$ and any locally injective locally contraherent cosheaf
${\mathfrak{J}}$ on $X$.

If the morphism $f$ is flat then, according to (26), the adjunction
isomorphism

(49)
$\operatorname{Hom}^{X}(f_{!}{\mathfrak{Q}},{\mathfrak{M}})\simeq\operatorname{Hom}^{Y}({\mathfrak{Q}},f^{!}{\mathfrak{M}})$

holds for any colocally projective contraherent cosheaf ${\mathfrak{Q}}$ on
$Y$ and any locally cotorsion locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$. When the morphism $f$ is also affine, the restrictions of $f_{!}$ and
$f^{!}$ form an adjoint pair of functors between the exact categories
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$. In
addition, these functors take the additive categories
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ into one
another.

If the morphism $f$ is very flat, the same adjunction isomorphism (49) holds
for any colocally projective contraherent cosheaf ${\mathfrak{Q}}$ on $Y$ and
any locally contraherent cosheaf ${\mathfrak{E}}$ on $X$. When the morphism
$f$ is also affine, the restrictions of $f_{!}$ and $f^{!}$ form an adjoint
pair of functors between the exact categories
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.

######  Corollary 4.5.4.

Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes. Then

(a) if the morphism $f$ is very flat, then the direct image functor
$f_{!}\colon Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ takes projective
contraherent cosheaves to projective contraherent cosheaves;

(b) if the morphism $f$ is flat, then the direct image functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$ takes
projective locally cotorsion contraherent cosheaves to projective locally
cotorsion contraherent cosheaves.

###### Proof.

Follows from the above partial adjunctions (49) between the exact functors
$f_{!}$ and $f^{!}$. Part (b) can be also deduced from Corollary 4.5.3(b-c). ∎

###  4.6. The “naïve” co-contra correspondence

Let $X$ be a quasi-compact semi-separated scheme and ${\mathbf{W}}$ be its
open covering. We refer to Section A.1 for the definitions of the derived
categories mentioned below.

Recall the definition of the left homological dimension
$\operatorname{ld}_{{\mathsf{F}}/{\mathsf{E}}}E$ of an object $E$ of an exact
category ${\mathsf{E}}$ with respect to a full exact subcategory
${\mathsf{F}}\subset{\mathsf{E}}$, given (under a specific set of assumptions
on ${\mathsf{F}}$ and ${\mathsf{E}}$) in Section A.5. The _right homological
dimension with respect to an exact subcategory ${\mathsf{F}}$_ (or the right
${\mathsf{F}}$-homological dimension)
$\operatorname{rd}_{{\mathsf{F}}/{\mathsf{E}}}E$ is defined in the dual way
(and under the dual set of assumptions).

######  Lemma 4.6.1.

(a) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering,
then the right homological dimension of any quasi-coherent sheaf on $X$ with
respect to the exact subcategory of contraadjusted quasi-coherent sheaves
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ (is well-defined and) does not exceed $N$.

(b) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering
subordinate to ${\mathbf{W}}$, then the left homological dimension of any
${\mathbf{W}}$-locally contraherent cosheaf on $X$ with respect to the exact
subcategory of colocally projective contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ does not exceed $N-1$.
Consequently, the same bound holds for the left homological dimension of any
object of $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ with respect to the
exact subcategory $X{\operatorname{\mathsf{--ctrh}}}$.

###### Proof.

Part (a): first of all, the assumptions about the pair of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ making the right homological dimension
well-defined hold by Corollaries 4.1.2(c) and 4.1.4(b).

Furthermore, the right homological dimension of any module over a commutative
ring $R$ with respect to the exact category of contraadjusted $R$-modules
$R{\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}\subset
R{\operatorname{\mathsf{--mod}}}$ does not exceed $1$. It follows easily that
the homological dimension of any quasi-coherent sheaf of the form
$j_{*}{\mathcal{G}}$, where $j\colon U\longrightarrow X$ is an affine open
subscheme, with respect to the exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ does not exceed $1$, either. Now any quasi-
coherent sheaf ${\mathcal{F}}$ on $X$ has a Čech resolution (12) of length
$N-1$ by finite direct sums of quasi-coherent sheaves of the above form. It
remains to use the dual version of Corollary A.5.5(a).

Part (b): the conditions on the exact categories
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ making the left homological
dimension well-defined hold by Corollaries 4.2.3(b) and 4.2.5(b). The pair of
exact categories $X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ satisfies the same assumptions
for the reasons explained in Section 3.2. It remains to recall the resolution
(27). ∎

######  Lemma 4.6.2.

Let $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then

(a) the left homological dimension of any locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on $X$ with respect to the exact
subcategory of colocally projective locally cotorsion contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ does not exceed
$N-1$. Consequently, the same bound holds for the left homological dimension
of any object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ with respect
to the exact subcategory $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$;

(b) the left homological dimension of any locally injective
${\mathbf{W}}$-locally contraherent cosheaf on $X$ with respect to the exact
subcategory of colocally projective locally injective contraherent cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ does not exceed
$N-1$. Consequently, the same bound holds for the left homological dimension
of any object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ with respect
to the exact subcategory $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}$.

###### Proof.

Similar to the proof of Lemma 4.6.1(b). ∎

######  Corollary 4.6.3.

(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is an equivalence of
triangulated categories.

(b) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}$ is an equivalence of triangulated
categories.

The reason why most unbounded derived categories aren’t mentioned in part (b)
is because one needs a uniform restriction on the extension of locality of
locally contraherent cosheaves in order to work simultaneously with infinite
collections of these. In particular, infinite products exist in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, but not necessarily in
$X{\operatorname{\mathsf{--lcth}}}$, so the contraderived category of the
latter exact category is not well-defined.

###### Proof of Corollary 4.6.3.

Part (a) follows from Proposition A.5.6 together with Lemma 4.6.1(b). Part (b)
in the case $\star={\mathsf{b}}$ is obtained from part (a) by passing to the
inductive limit over refinements of coverings, while in the case $\star=-$ it
is provided by Proposition A.3.1(a). ∎

The following two corollaries are similar to the previous one. The only
difference in the proofs is that Lemma 4.6.2 is being used in place of Lemma
4.6.1(b).

######  Corollary 4.6.4.

(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\allowbreak\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is an
equivalence of triangulated categories.

(b) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ is an equivalence of
triangulated categories. ∎

######  Corollary 4.6.5.

(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, the triangulated
functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\allowbreak\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ is an
equivalence of triangulated categories.

(b) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$ is an equivalence of
triangulated categories. ∎

The next theorem is the main result of this section.

######  Theorem 4.6.6.

For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$ there is a natural equivalence of
triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$.
These equivalences of derived categories form commutative diagrams with the
natural functors
${\mathsf{D}}^{\mathsf{b}}\longrightarrow{\mathsf{D}}^{\pm}\longrightarrow{\mathsf{D}}$,
${\mathsf{D}}^{\mathsf{b}}\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}\longrightarrow{\mathsf{D}}^{\mathsf{abs}}$,
${\mathsf{D}}^{{\mathsf{abs}}\pm}\longrightarrow{\mathsf{D}}^{\pm}$,
${\mathsf{D}}^{\mathsf{abs}}\longrightarrow{\mathsf{D}}$ between different
versions of derived categories of the same exact category.

Notice that Theorem 4.6.6 does not say anything about the coderived and
contraderived categories ${\mathsf{D}}^{\mathsf{co}}$ and
${\mathsf{D}}^{\mathsf{ctr}}$ of quasi-coherent sheaves and contraherent
cosheaves (neither does Corollary 4.6.3 mention the coderived categories). The
reason is that infinite products are not exact in the abelian category of
quasi-coherent sheaves and infinite direct sums may not exist in the exact
category of contraherent cosheaves. So only the coderived category
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ and the
contraderived category
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$ are well-
defined. Comparing these two requires a different approach; the entire Section
5 will be devoted to that.

###### Proof of Theorem 4.6.6.

By Proposition A.5.6 and its dual version, together with Lemma 4.6.1, the
functors
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$
induced by the corresponding embedings of exact categories are all
equivalences of triangulated categories. Hence it suffices to construct a
natural equivalence of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ in order to prove all
assertions of Theorem.

According to Sections 2.5 and 2.6, there are natural functors

$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})\colon
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--ctrh}}}$

and

${\mathcal{O}}_{X}\odot_{X}{-}\colon X{\operatorname{\mathsf{--lcth}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--qcoh}}}$

related by the adjunction isomorphism (20), which holds for those objects for
which the former functor is defined. So it remains to prove the following
lemma. ∎

######  Lemma 4.6.7.

On a quasi-compact semi-separated scheme $X$, the functor
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ takes
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, the functor
${\mathcal{O}}_{X}\odot_{X}{-}$ takes
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$, and the restrictions of
these functors to these subcategories are mutually inverse equivalences of
exact categories.

###### Proof.

Obviously, on an affine scheme $U$ the functor
$\operatorname{\mathfrak{Hom}}_{U}({\mathcal{O}}_{U},{-})$ takes a
contraadjusted quasi-coherent sheaf ${\mathcal{Q}}$ with the contraadjusted
${\mathcal{O}}(U)$-module of global sections ${\mathcal{Q}}(U)$ to the
contraherent cosheaf ${\mathfrak{Q}}$ with the contraadjusted
${\mathcal{O}}(U)$-module of global cosections
${\mathfrak{Q}}[U]={\mathcal{Q}}(U)$. Furthermore, if $j\colon
U\longrightarrow X$ is the embedding of an affine open subscheme, then by the
formula (44) of Section 3.8 there is a natural isomorphism
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},j_{*}{\mathcal{Q}})\simeq
j_{!}{\mathfrak{Q}}$ of contraherent cosheaves on $X$.

Analogously, the functor ${\mathcal{O}}_{U}\odot_{U}{-}$ takes a contraherent
cosheaf ${\mathfrak{Q}}$ with the contraadjusted ${\mathcal{O}}(U)$-module of
global cosections ${\mathfrak{Q}}[U]$ to the contraadjusted quasi-coherent
sheaf ${\mathcal{Q}}$ with the ${\mathcal{O}}(U)$-module of global sections
${\mathcal{Q}}(U)$ on $U$. If an embedding of affine open subscheme $j\colon
U\longrightarrow X$ is given, then by the formula (46) there is a natural
isomorphism ${\mathcal{O}}_{X}\odot_{X}j_{!}{\mathfrak{Q}}\simeq
j_{*}{\mathcal{Q}}$ of quasi-coherent sheaves on $X$.

By Corollary 4.1.4(c), any sheaf from
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ is a direct summand of a
finitely iterated extension of the direct images of contraadjusted quasi-
coherent sheaves from affine open subschemes of $X$. It is clear from the
definition of the functor
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ that it preserves
exactness of short sequences of contraadjusted quasi-coherent cosheaves; hence
it preserves, in particular, such iterated extensions.

By Corollary 4.2.5(c), any cosheaf from
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ is a direct summand of a
finitely iterated extension of the direct images of contraherent cosheaves
from affine open subschemes of $X$. Let us show that the functor
${\mathcal{O}}_{X}\odot_{X}{-}$ preserves exactness of short sequences of
colocally projective contraherent cosheaves on $X$, and therefore, in
particular, preserves such extensions. Indeed, the adjunction isomorphism

$\operatorname{Hom}_{X}({\mathcal{O}}_{X}\odot_{X}{\mathfrak{P}},\>{\mathcal{F}})\simeq\operatorname{Hom}^{X}({\mathfrak{P}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{F}}))$

holds for any contraherent cosheaf ${\mathfrak{P}}$ and contraadjusted quasi-
coherent sheaf ${\mathcal{F}}$. Besides, the contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{J}})$ is
locally injective for any injective quasi-coherent sheaf ${\mathcal{J}}$ on
$X$. By Corollary 4.2.3(a), it follows that the functor
${\mathfrak{P}}\longmapsto\operatorname{Hom}_{X}({\mathcal{O}}_{X}\odot_{X}{\mathfrak{P}},\>{\mathcal{J}})$
preserves exactness of short sequences of colocally projective contraherent
cosheaves on $X$, and consequently so does the functor
${\mathfrak{P}}\longmapsto{\mathcal{O}}_{X}\odot_{X}{\mathfrak{P}}$.

Now one can easily deduce that the adjunction morphisms

${\mathfrak{P}}\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},\>{\mathcal{O}}_{X}\odot_{X}{\mathfrak{P}})\quad\text{and}\quad{\mathcal{O}}_{X}\odot_{X}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{\mathcal{F}})\longrightarrow{\mathcal{F}}$

are isomorphisms for any colocally projective contraherent cosheaf
${\mathfrak{P}}$ and contraadjusted quasi-coherent sheaf ${\mathcal{F}}$, as a
morphism of finitely filtered objects inducing an isomorphism of the
associated graded objects is also itself an isomorphism. The proof of Lemma,
and hence also of Theorem 4.6.6, is finished. ∎

Given an exact category ${\mathsf{E}}$, let
$\mathsf{Hot}^{\star}({\mathsf{E}})$ denote the homotopy category
$\mathsf{Hot}({\mathsf{E}})$ if $\star={\mathsf{abs}}$, ${\mathsf{co}}$,
${\mathsf{ctr}}$, or $\empt$; the category $\mathsf{Hot}^{+}({\mathsf{E}})$,
if $\star={\mathsf{abs}}+$ or $+$; the category
$\mathsf{Hot}^{-}({\mathsf{E}})$, if $\star={\mathsf{abs}}-$ or $-$; and the
category $\mathsf{Hot}^{\mathsf{b}}({\mathsf{E}})$ if $\star={\mathsf{b}}$.
The following two corollaries provide, essentially, several restricted
versions of Theorem 4.6.6.

######  Corollary 4.6.8.

(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, there is a natural equivalence of
triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$.

(b) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{ctr}}$, or ${\mathsf{abs}}$, there is a natural
equivalence of triangulated categories
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$.

###### Proof.

By Proposition A.5.6 together with Lemma 4.6.2, the functors
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}})\allowbreak\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
induced by the corresponding embeddings of exact categories are equivalences
of triangulated categories. Hence it suffices to show that the equivalence of
exact categories from Lemma 4.6.7 identifies
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$. The former
of these assertions follows from Corollaries 4.1.11(c) and 4.2.6(c), while the
latter one is obtained from Corollary 4.2.8 together with the fact that any
injective quasi-coherent sheaf on $X$ is a direct summand of a finite direct
sum of the direct images of injective quasi-coherent sheaves from open
embeddings $U_{\alpha}\longrightarrow X$ forming a covering. ∎

######  Lemma 4.6.9.

Let $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering.
Then

(a) the right homological dimension of any very flat quasi-coherent sheaf on
$X$ with respect to the exact subcategory of contraadjusted very flat quasi-
coherent sheaves $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ does not exceed $N$;

(b) the right homological dimension of any flat quasi-coherent sheaf on $X$
with respect to the exact subcategory of contraadjusted flat quasi-coherent
sheaves $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ does not exceed $N$.

###### Proof.

The right homological dimension is well-defined due to Corollary 4.1.4(b), so
it remains to apply Lemma 4.6.1(a) and the dual version of Corollary A.5.3. ∎

######  Corollary 4.6.10.

(a) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{co}}$, or ${\mathsf{abs}}$, there is a natural
equivalence of triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\simeq\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$.

(b) For any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, there is a natural equivalence of
triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}})$.

(c) There is a natural equivalence of triangulated categories
${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}^{+}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$.

###### Proof.

Part (a): assuming $\star\neq{\mathsf{co}}$, by Lemma 4.6.9(a) together with
the dual version of Proposition A.5.6 the triangulated functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$
is an equivalence of categories. In particular, we have proven that
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})={\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$,
so the previous assertion holds for $\star={\mathsf{co}}$ as well. Hence it
suffices to show that the equivalence of exact categories from Lemma 4.6.7
identifies $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$. This follows from Lemmas
4.1.5 and 4.4.1(b). The proof of part (b) is similar: in view of Lemma
4.6.9(b), the triangulated functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is an equivalence of categories, and it remains to show that the equivalence
of exact categories from Lemma 4.6.7 identifies
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ with
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$. Here one applies Lemma
4.1.6 and Corollary 4.3.4(c).

To prove part (c), notice that the triangulated functor
$\mathsf{Hot}^{+}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is an equivalence of categories by Corollary 4.1.11(b) and the dual version of
Proposition A.3.1(a). So it remains to show that the equivalence of exact
categories $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ identifies
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ with
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. This
follows from Lemmas 4.1.12 and 4.4.3(b). Alternatively, one can prove that the
functor
${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}})$
is an equivalence of categories by applying directly the dual version of
Proposition A.3.1(a) together with Corollaries 4.3.4(a) and 4.3.2(b). ∎

###  4.7. Homotopy locally injective complexes

Let $X$ be a quasi-compact semi-separated scheme and ${\mathbf{W}}$ be its
open covering. The goal of this section is to construct a full subcategory in
the homotopy category
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
that would be equivalent to the unbounded derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$. The
significance of this construction is best illustrated using the duality-
analogy between the contraherent cosheaves and the quasi-coherent sheaves.

As usually, the notation
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ refers to the
unbounded derived category of the exact category of flat quasi-coherent
sheaves on $X$. The full triangulated subcategory
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}\subset{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
of _homotopy flat complexes_ of flat quasi-coherent sheaves on $X$ is defined
as the minimal triangulated subcategory in
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ containing the
objects of $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ and closed under
infinite direct sums (cf. Section A.4). The following result is essentially
due to Spaltenstein [60].

######  Theorem 4.7.1.

(a) The composition of natural triangulated functors
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}\allowbreak\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories.

(b) A complex of flat quasi-coherent sheaves
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ belongs
to
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}$
if and only if its tensor product
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with any acyclic complex of quasi-coherent sheaves
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ is also
an acyclic complex of quasi-coherent sheaves.

###### Proof.

Part (a) is a particular case of Proposition A.4.3. To prove part (b), let us
first show that the tensor product of any complex of quasi-coherent sheaves
and a complex of sheaves acyclic with respect to the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ is acyclic. Indeed, any
complex in an abelian category is a locally stabilizing inductive limit of
finite complexes; so it suffices to notice that the tensor product of any
quasi-coherent sheaf with a complex acyclic with respect to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ is acyclic. Hence the class
of all complexes of flat quasi-coherent sheaves satisfying the condition in
part (b) can be viewed as a strictly full triangulated subcategory in
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$.

Now the “only if” assertion easily follows from the facts that the tensor
products of quasi-coherent sheaves preserve infinite direct sums and the
tensor product with a flat quasi-coherent sheaf is an exact functor. In view
of (the proof of) part (a), it suffices to show that any complex
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ satisfying the tensor
product condition of part (b) and acyclic with respect to
$X{\operatorname{\mathsf{--qcoh}}}$ is also acyclic with respect to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ in order to prove “if”.

Notice that the tensor product of a bounded above complex of flat quasi-
coherent sheaves and an acyclic complex of quasi-coherent sheaves is an
acyclic complex. Since any quasi-coherent sheaf ${\mathcal{M}}$ over $X$ has a
flat left resolution, it follows that the complex of quasi-coherent sheaves
${\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is acyclic. One easily concludes that the complex
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is acyclic with
respect to $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$. ∎

The full triangulated subcategory
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$
of _homotopy locally injective complexes_ of localy injective
${\mathbf{W}}$-locally contraherent cosheaves on $X$ is defined as the minimal
full triangulated subcategory in
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
containing the objects of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ and closed
under infinite products. Given a complex of quasi-coherent sheaves
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and a complex
of ${\mathbf{W}}$-locally contraherent cosheaves
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ such
that the ${\mathbf{W}}$-locally contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}}^{i},{\mathfrak{P}}^{j})$ is
defined for all $i$, $j\in{\mathbb{Z}}$ (see Sections 2.4 and 3.6), we define
the complex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
as the total complex of the bicomplex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}}^{i},{\mathfrak{P}}^{j})$
constructed by taking infinite products of ${\mathbf{W}}$-locally contraherent
cosheaves along the diagonals.

######  Theorem 4.7.2.

(a) The composition of natural triangulated functors
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}\allowbreak\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories.

(b) A complex of locally injective ${\mathbf{W}}$-locally contraherent
cosheaves ${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on
$X$ belongs to
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
if and only if the complex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ into it
from any acyclic complex of quasi-coherent sheaves
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is an acyclic
complex in the exact category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$
(or, at one’s choice, in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$.

###### Proof.

Part (a): the argument goes along the lines of the proof of Theorem 4.7.1(a),
but Proposition A.4.3 is not directly applicable, the category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ being not
abelian; so there are some complications. First of all, we will need another
definition. The full triangulated subcategory
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$
of homotopy locally injective complexes of colocally projective locally
injective contraherent cosheaves on $X$ is defined as the minimal full
triangulated subcategory containing the objects of
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ and closed
under infinite products.

It was shown in Section 4.6 that the natural functor
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories. Analogously one shows (using, e.
g., Corollary A.5.3) that the natural functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
is an equivalence of categories, as are the similar functors
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
and
$\mathsf{Hot}^{+}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$.
Therefore, the equivalence
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
identifies the subcategories generated by bounded or bounded below complexes.
Thus the natural functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$
is also an equivalence of triangulated categories, and it remains to show that
the functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})$
is an equivalence of categories.

We will show that any complex over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ admits a quasi-isomorphism
with respect to the exact category
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ into a complex belonging to
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}$.
In particular, by the dual version of [51, Lemma 1.6] applied to the homotopy
category $\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})$ with
the full triangulated subcategory
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$
and the thick subcategory of complexes acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ it will follow that the
category ${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})$ is
equivalent to the localization of
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$
by the thick subcategory of complexes acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$. By the dual version of
Corollary A.4.2, the latter subcategory is semiorthogonal to
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}$
in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$.
In view of the same construction of a quasi-isomorphism with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, these two subcategories
form a semiorthogonal decomposition of
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$,
which implies the desired assertion.

######  Lemma 4.7.3.

There exists a positive integer $d$ such that for any complex
${\mathfrak{P}}^{0}\longrightarrow\dotsb\longrightarrow{\mathfrak{P}}^{d+1}$
over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ there exists a complex
${\mathfrak{Q}}^{0}\longrightarrow\dotsb\longrightarrow{\mathfrak{Q}}^{d+1}$
over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ together with a
morphism of complexes
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
such that ${\mathfrak{Q}}^{0}=0$, while the morphisms of contraherent
cosheaves ${\mathfrak{P}}^{d}\longrightarrow{\mathfrak{Q}}^{d}$ and
${\mathfrak{P}}^{d+1}\longrightarrow{\mathfrak{Q}}^{d+1}$ are isomorphisms.

###### Proof.

In view of Lemma 4.6.7, it suffices to prove the assertion of Lemma for a
complex
${\mathcal{P}}^{0}\longrightarrow\dotsb\longrightarrow{\mathcal{P}}^{d+1}$
over the category $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$. Set
${\mathcal{Q}}^{0}=0$. Consider the quasi-coherent sheaf
${\mathcal{R}}^{1}=\operatorname{coker}({\mathcal{P}}^{0}\to{\mathcal{P}}^{1})$
and embed it into a contraadjusted quasi-coherent sheaf ${\mathcal{Q}}^{1}$.
Denote by ${\mathcal{R}}^{2}$ the fibered coproduct of the quasi-coherent
sheaves ${\mathcal{Q}}^{1}$ and ${\mathcal{P}}^{2}$ over ${\mathcal{R}}^{1}$,
embed it into a contraadjusted quasi-coherent sheaf ${\mathcal{Q}}^{2}$, and
proceed by applying the dual version of the construction of Lemma A.3.2(a) up
to producing the quasi-coherent sheaves ${}{\mathcal{Q}}^{d-2}$ and
${\mathcal{R}}^{d-1}$.

The sequence
$0\longrightarrow{\mathcal{R}}^{1}\longrightarrow{\mathcal{Q}}^{1}\oplus{\mathcal{P}}^{2}\longrightarrow{\mathcal{Q}}^{2}\oplus{\mathcal{P}}^{3}\longrightarrow\dotsb\longrightarrow{\mathcal{Q}}^{d-2}\oplus{\mathcal{P}}^{d-1}\longrightarrow{\mathcal{R}}^{d-1}\longrightarrow
0$ is a right resolution of the quasi-coherent sheaf ${\mathcal{R}}^{1}$, all
of whose terms, except perhaps the rightmost one, are contraadjusted quasi-
coherent sheaves. By Lemma 4.6.1(a) and the dual version of Corollary A.5.2,
for $d$ large enough the quasi-coherent sheaf ${\mathcal{R}}^{d-1}$ will be
contraadjusted. It remains to set ${\mathcal{Q}}^{d-1}={\mathcal{R}}^{d-1}$,
${\mathcal{Q}}^{d}={\mathcal{P}}^{d}$, and
${\mathcal{Q}}^{d+1}={\mathcal{P}}^{d+1}$. ∎

Now let ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
complex over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$. For each
fragment of $d+2$ consequtive terms
${\mathfrak{P}}^{i-d-1}\longrightarrow\dotsb\longrightarrow{\mathfrak{P}}^{i}$
in ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ we
construct the corresponding complex
${}^{(i)}{\mathfrak{Q}}^{i-d-1}\longrightarrow\dotsb\longrightarrow{}^{(i)}{\mathfrak{Q}}^{i}$
as in Lemma 4.7.3. Pick an admissible monomorphism
${}^{(i)}{\mathfrak{Q}}^{i-d}\longrightarrow{}^{(i)}{\mathfrak{I}}^{i-d}$ from
a colocally projective contraherent cosheaf ${}^{(i)}{\mathfrak{Q}}^{i-d}$
into a colocally projective locally injective contraherent cosheaf
${}^{(i)}{\mathfrak{I}}^{i-d}$ on $X$ (see Corollaries 4.2.5(b) and 4.2.3(a)).

Proceeding with the dual version of the construction of Lemma A.3.2(a) (see
also the above proof of Lemma 4.7.3), we obtain a termwise admissible
monomorphism with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ from the complex
${}^{(i)}{\mathfrak{Q}}^{i-d}\longrightarrow\dotsb\longrightarrow{}^{(i)}{\mathfrak{Q}}^{i}$
into a complex
${}^{(i)}{\mathfrak{I}}^{i-d}\longrightarrow\dotsb\longrightarrow{}^{(i)}{\mathfrak{I}}^{i}$
over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ such
that the cone of this morphism is quasi-isomorphic to an object of
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ placed in the cohomological
degree $i$.

Set ${}^{(i)}{\mathfrak{I}}^{j}=0$ for $j$ outside of the segment $[i-d,\>i]$.
We obtain a finite complex
${}^{(i)}{\mathfrak{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ endowed with
a morphism of complexes
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{(i)}{\mathfrak{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with the following property. For any affine open subscheme $U\subset X$, the
induced morphism of cohomology modules
$H^{i}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U])\longrightarrow
H^{i}({\mathfrak{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U])$ is
injective.

Denote by
${}^{0}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the
direct product of all the complexes
${}^{(i)}{\mathfrak{I}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. The
morphism of complexes
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{0}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over the exact category $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ is
a termwise admissible monomorphism. Consider the corresponding complex of
cokernels and apply the same procedure to it, constructing a termwise acyclic
complex of complexes
$0\longrightarrow{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{0}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{1}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\dotsb$
over the exact category $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ in
which all the complexes
${}^{i}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
infinite products of finite complexes over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.

By Lemma A.3.4 applied to the projective system of quotient complexes of silly
filtration with respect to the left index of the bicomplex
${}^{\text{\smaller\smaller$\scriptstyle\bullet$}}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
the total complex of
${}^{\text{\smaller\smaller$\scriptstyle\bullet$}}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
constructed by taking infinite products along the diagonals belongs to
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})^{\mathsf{lin}}$.
It remains to show that the cone
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the
morphism from ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
to the total complex of
${}^{\text{\smaller\smaller$\scriptstyle\bullet$}}{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is acyclic with respect to $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.

Indeed, by the dual version of Lemma A.4.4, the complex of cosections
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U]$ is an
acyclic complex of ${\mathcal{O}}_{X}(U)$-modules for any affine open
subscheme $U\subset X$. Since the ${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{E}}^{i}[U]$ are contraadjusted and quotient modules of
contraadjusted modules are contraadjusted, the complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U]$ is also
acyclic with respect to the exact category
${\mathcal{O}}(U){\operatorname{\mathsf{--mod}}}^{\mathsf{cta}}$.

Since the functor
$\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),{-})$
preserves exactness of short sequences of contraadjusted
${\mathcal{O}}_{X}(U)$-modules for any pair of embedded affine open subschemes
$V\subset U\subset X$, one easily concludes that the rules
$U\longmapsto\operatorname{coker}({\mathfrak{E}}^{i-1}[U]\to{\mathfrak{E}}^{i}[U])$
define contraherent cosheaves on $X$. Hence the complex of contrahent
cosheaves ${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
acyclic over $X{\operatorname{\mathsf{--ctrh}}}$. Since the contraherent
cosheaves ${\mathfrak{E}}^{i}$ belong to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, this complex is also
acyclic over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ by Lemma
4.6.1(b) and Corollary A.5.2.

Part (a) is proven; let us prove part (b). The argument is similar to the
proof of Theorem 4.7.1(b). First we show that
$\operatorname{\mathfrak{Cohom}}_{X}$ from any complex of quasi-coherent
sheaves into a complex of locally contraherent cosheaves acyclic with respect
to $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ is acyclic
with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. Indeed, any
complex of quasi-coherent sheaves is a locally stabilizing inductive limit of
a sequence of finite complexes. So it remains to recall that
$\operatorname{\mathfrak{Cohom}}_{X}$ from a quasi-coherent sheaf into a
complex acyclic with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ is a complex
acyclic with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$, and use Lemma
A.3.4 again.

Hence the class of all complexes over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ satisfying the
$\operatorname{\mathfrak{Cohom}}$ condition in part (b) can be viewed as a
strictly full triangulated subcategory in
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$.
Now the “only if” assertion follows from the preservation of infinite products
in the second argument by the functor $\operatorname{\mathfrak{Cohom}}_{X}$
and its exactness as a functor on the category
$X{\operatorname{\mathsf{--qcoh}}}$ for any fixed locally injective locally
contraherent cosheaf in the second argument. In view of (the proof of) part
(a), in order to prove “if” it suffices to show that any complex
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ satisfying the
$\operatorname{\mathfrak{Cohom}}$ condition in (b) and acyclic with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is also acyclic with respect
to $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$.

Notice that the $\operatorname{\mathfrak{Cohom}}_{X}$ from a bounded above
complex of very flat quasi-coherent sheaves into an acyclic complex of
${\mathbf{W}}$-locally contraherent cosheaves is an acyclic complex of
${\mathbf{W}}$-locally contraherent cosheaves. Since any quasi-coherent sheaf
${\mathcal{M}}$ on $X$ has a very flat left resolution (see Lemma 4.1.1), it
follows that the complex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic with respect to $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.

Now let $U\subset X$ be an affine open subscheme subordinate to
${\mathbf{W}}$, let $N$ be an ${\mathcal{O}}(U)$-module, viewed also as a
quasi-coherent sheaf on $U$, and ${\mathcal{M}}$ be any quasi-coherent
extension (e. g., the direct image) of $N$ to $X$. Then acyclicity of the
complex
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
with respect to the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ implies, in particular,
exactness of the complex of ${\mathcal{O}}_{X}(U)$-modules
$\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}(N,{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U])$.
Since this holds for all ${\mathcal{O}}_{X}(U)$-modules $N$, it follows that
all the ${\mathcal{O}}_{X}(U)$-modules of cocycles in the acyclic complex of
${\mathcal{O}}_{X}(U)$-modules
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[U]$ are
injective. ∎

The following lemma will be needed in Section 4.8.

######  Lemma 4.7.4.

Let ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a
complex over the exact category
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ be a complex
over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ belonging
to
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$.
Then the natural morphism of graded abelian groups
$H^{*}\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq\operatorname{Hom}_{\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{Hom}_{{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is an isomorphism (in other words, the complex
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
computes the groups of morphisms in the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$).

###### Proof.

Since any complex over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ admits
a quasi-isomorphism into it from a complex over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and any complex over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ acyclic over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is also acyclic over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, it suffices to show that
the complex of abelian groups
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ acyclic with
respect to $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and any complex
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\in{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$.
For a complex ${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
obtained from objects of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ by iterating
the operations of cone and infinite direct sum the latter assertion is obvious
(see Corollary 4.2.3(a)), so it remains to consider the case of a complex
${\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ acyclic with
respect to $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$. In
this case we will show that the complex
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.

Let $i$ be an integer. Applying Lemma 4.7.3 to the fragment
${\mathfrak{P}}^{i-d-1}\longrightarrow\dotsb\longrightarrow{\mathfrak{P}}^{i}$
of the complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, we obtain a
morphism of complexes from
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ to a finite
complex ${\mathfrak{Q}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ such that the morphisms
${\mathfrak{P}}^{i-1}\longrightarrow{\mathfrak{Q}}^{i-1}$ and
${\mathfrak{P}}^{i}\longrightarrow{\mathfrak{Q}}^{i}$ are isomorphisms. The
cocone of this morphism splits naturally into a direct sum of two complexes
concentrated in cohomological degrees $\le i$ and $\ge i$, respectively. We
are interested in the former complex. Its subcomplex of silly truncation
${\mathfrak{R}}(j,i)^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a
finite complex over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$
concentrated in the cohomological degrees between $j$ and $i$ and endowed with
a morphism of complexes
${\mathfrak{R}}(j,i)^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$,
which is a termwise isomorphism in the degrees between $j$ and $i-d$.

The complex ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
a termwise stabilizing inductive limit of the sequence of complexes
${\mathfrak{R}}(j,i)^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ as the
degree $j$ decreases, while the degree $i$ increases (fast enough). It remains
to recall that the functor $\operatorname{Hom}^{X}$ from a colocally
projective contraherent cosheaf takes acyclic complexes over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ to acyclic complexes of
abelian groups, and, e. g., use Lemma A.3.4 once again. ∎

###  4.8. Derived functors of direct and inverse image

For the rest of Section 4, the upper index $\star$ in the notation for derived
and homotopy categories stands for one of the symbols ${\mathsf{b}}$, $+$,
$-$, $\empt$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, ${\mathsf{co}}$,
${\mathsf{ctr}}$, or ${\mathsf{abs}}$.

Let $f\colon Y\longrightarrow X$ be a morphism of quasi-compact semi-separated
schemes. Then for any symbol $\star\neq{\mathsf{ctr}}$ the right derived
functor of direct image

(50)
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$

is constructed in the following way. By Lemma 4.6.1(a) together with the dual
version of Proposition A.5.6, the natural functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories (as is the similar functor for
sheaves over $X$). By Corollary 4.1.13(a), the restriction of the functor of
direct image $f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ provides an exact functor
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$. Now the derived functor
${\mathbb{R}}f_{*}$ is defined by by restricting the functor of direct image
$f_{*}\colon\mathsf{Hot}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})$
to the full subcategory of complexes of contraadjusted quasi-coherent sheaves
on $Y$ (with the appropriate boundedness conditions).

For any symbol $\star\neq{\mathsf{co}}$, the left derived functor of direct
image

(51)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$

is constructed in the following way. By Lemma 4.6.1(b) (for the covering
$\\{Y\\}$ of the scheme $Y$) together with Proposition A.5.6, the natural
functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})$
is an equivalence of triangulated categories (as is the similar functor for
cosheaves over $X$). By Corollary 4.5.3(a), there is an exact functor of
direct image $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$. The derived functor
${\mathbb{L}}f_{!}$ is defined as the induced functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}})$.

Similarly one defines the left derived functor of direct image

(52)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}).$

######  Theorem 4.8.1.

For any symbol $\star\neq{\mathsf{co}}$, ${\mathsf{ctr}}$, the equivalences of
categories
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$
from Theorem 4.6.6 transform the right derived functor ${\mathbb{R}}f_{*}$
(50) into the left derived functor ${\mathbb{L}}f_{!}$ (51).

###### Proof.

It suffices to show that the equivalences of exact categories
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ and
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ from Lemma 4.6.7 transform
the functor $f_{*}$ into the functor $f_{!}$. The isomorphism (44) of Section
3.8 proves as much. ∎

Let $f\colon Y\longrightarrow X$ be a morphism of schemes. Let ${\mathbf{W}}$
and ${\mathbf{T}}$ be open coverings of the schemes $X$ and $Y$, respectively,
for which the morphism $f$ is $({\mathbf{W}},{\mathbf{T}})$-coaffine.
According to Section 3.3, there is an exact functor of inverse image
$f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}$; for a flat
morphism $f$, there is also an exact functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$, and for a very
flat morphism $f$, an exact functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$.

For a quasi-compact semi-separated scheme $X$, it follows from Corollary
4.1.11(a) and Proposition A.3.1(a) that the natural functor
${\mathsf{D}}^{-}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of categories. Similarly, it follows from Corollary 4.2.5(a)
and the dual version of Proposition A.3.1(a) that the natural functor
${\mathsf{D}}^{+}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories. This allows to define, for any
morphism $f\colon Y\longrightarrow X$ into a quasi-compact semi-separated
scheme $X$ and coverings ${\mathbf{W}}$, ${\mathbf{T}}$ as above, the derived
functors of inverse image

(53)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--qcoh}}})$

and

${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}})$

by applying the functors $f^{*}$ and $f^{!}$ to (appropriately bounded)
complexes of flat sheaves and locally injective cosheaves. When both schemes
are quasi-compact and semi-separated, one can take into account the
equivalences of categories from Corollary 4.6.3(a) in order to produce the
right derived functor

(54)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--ctrh}}}),$

which clearly does not depend on the choice of the coverings ${\mathbf{W}}$
and ${\mathbf{T}}$.

According to Section 4.7, the natural functors
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalences of categories for any quasi-compact semi-separated scheme $X$
with an open covering ${\mathbf{W}}$. This allows to define, for any morphism
$f\colon Y\longrightarrow X$ into a quasi-compact semi-separated scheme $X$
and coverings ${\mathbf{W}}$, ${\mathbf{T}}$ as above, the derived functors of
inverse image

(55)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$

and

${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}})$

by applying the functors
$f^{*}\colon\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow\mathsf{Hot}(Y{\operatorname{\mathsf{--qcoh}}})$
and
$f^{!}\colon\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\allowbreak\longrightarrow\mathsf{Hot}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}})$
to homotopy flat complexes of flat quasi-coherent sheaves and homotopy locally
injective complexes of locally injective ${\mathbf{W}}$-locally contraherent
cosheaves, respectively. Of course, this construction is well-known for quasi-
coherent sheaves [60, 45]; we discuss here the sheaf and cosheaf situations
together in order to emphasize the duality-analogy between them.

When both schemes are quasi-compact and semi-separated, one can use the
equivalences of categories from Corollary 4.6.3(a) in order to obtain the
right derived functor

(56)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$

which does not depend on the choice of the coverings ${\mathbf{W}}$ and
${\mathbf{T}}$. Notice also that the restriction of the functor $f^{*}$ takes
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}$
into
$\mathsf{Hot}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})^{\mathsf{fl}}$
and the restriction of the functor $f^{!}$ takes
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})^{\mathsf{lin}}$
into
$\mathsf{Hot}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}})^{\mathsf{lin}}$.

It is easy to see that for any morphism of quasi-compact semi-separated
schemes $f\colon Y\longrightarrow X$ the functor ${\mathbb{L}}f^{*}$ (53) is
left adjoint to the functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{-}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{-}(X{\operatorname{\mathsf{--qcoh}}})$
(50) and the functor ${\mathbb{R}}f^{!}$ (54) is right adjoint to the functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{+}(Y{\operatorname{\mathsf{--ctrh}}})\allowbreak\longrightarrow{\mathsf{D}}^{+}(X{\operatorname{\mathsf{--ctrh}}})$
(51). Essentially, one uses the partial adjunctions on the level of exact
categories together with the fact that the derived functor constructions are
indeed those of the “left” and “right” derived functors, as stated (cf. [50,
Lemma 8.3]).

Similarly, one concludes from the construction in the proof of Theorem 4.7.2
that the functor ${\mathbb{L}}f^{*}$ (55) is left adjoint to the functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
(50). And in order to show that the functor ${\mathbb{R}}f^{!}$ (56) is right
adjoint to the functor
${\mathbb{L}}f_{!}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$
(51), one can use Lemma 4.7.4. So we have obtained a new proof of the
following classical result [29, 45].

######  Corollary 4.8.2.

For any morphism of quasi-compact semi-separated schemes $f\colon
Y\longrightarrow X$, the derived direct image functor
${\mathbb{R}}f_{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
has a right adjoint functor
$f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}})$.

###### Proof.

We have (more or less) explicitly constructed the functor $f^{!}$ as the right
derived functor
${\mathbb{R}}f^{!}\colon{\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--ctrh}}})$
(56) of the exact functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$ between exact subcategories
of the exact categories of locally contraherent cosheaves on $X$ and $Y$. The
above construction of the functor $f^{!}$ for bounded below complexes (54) is
particularly explicit. In either case, the construction is based on the
identification of the functor ${\mathbb{R}}f_{*}$ of derived direct image of
quasi-coherent sheaves with the functor ${\mathbb{L}}f_{!}$ of derived direct
image of contraherent cosheaves, which is provided by Theorems 4.6.6 and
4.8.1. ∎

###  4.9. Finite flat and locally injective dimension

A quasi-coherent sheaf ${\mathcal{F}}$ on a scheme $X$ is said to have _flat
dimension not exceeding $d$_ if the flat dimension of the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{F}}(U)$ does not exceed $d$ for any
affine open subscheme $U\subset X$. If a quasi-coherent sheaf ${\mathcal{F}}$
on $X$ admits a left resolution by flat quasi-coherent sheaves (e. g., $X$ is
quasi-compact and semi-separated), then the flat dimension of ${\mathcal{F}}$
is equal to the minimal length of such resolution.

The property of a quasi-coherent sheaf to have flat dimension not exceeding
$d$ is local, since so is the property of a quasi-coherent sheaf to be flat.
Quasi-coherent sheaves of finite flat dimension form a full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ closed under extensions and kernels of
surjective morphisms; the full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$ of quasi-coherent sheaves of
flat dimension not exceeding $d$ is closed under the same operations, and also
under infinite direct sums.

Let us say that a quasi-coherent sheaf ${\mathcal{F}}$ on a scheme $X$ has
_very flat dimension not exceeding $d$_ if the very flat dimension of the
${\mathcal{O}}_{X}(U)$-module ${\mathcal{F}}(U)$ does not exceed $d$ for any
affine open subscheme $U\subset X$ (see Section 1.5 for the definition). Over
a quasi-compact semi-separated scheme $X$, a quasi-coherent sheaf has very
flat dimension $\le d$ if and only if it admits a very flat left resolution of
length $\le\nobreak d$.

Since the property of a quasi-coherent sheaf to be very flat is local, so is
its property to have flat dimension not exceeding $d$ (cf. Lemma 1.5.4).
Quasi-coherent sheaves of very flat dimension $\le\nobreak d$ form a full
subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}$ closed under extensions, kernels of
surjective morphisms, and infinite direct sums. We denote the inductive limit
of the exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ as
$d\to\infty$ by $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}}$.

A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on a scheme $X$
is said to have _locally injective dimension not exceeding $d$_ if the
injective dimension of the ${\mathcal{O}}_{X}(U)$-module ${\mathfrak{P}}[U]$
does not exceed $d$ for any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$. Over a quasi-compact semi-separated scheme $X$, a
${\mathbf{W}}$-locally contraherent cosheaf has locally injective dimension
$\le d$ if and only if it admits a locally injective right resolution of
length $\le d$ in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.

The property of a ${\mathbf{W}}$-locally contraherent cosheaf to have locally
injective dimension not exceeding $d$ is local and refinements of the covering
${\mathbf{W}}$ do not change it. ${\mathbf{W}}$-locally contraherent cosheaves
of finite locally injective dimension form a full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$ closed under
extensions and cokernels of admissible monomorphisms; the full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$ of quasi-
coherent sheaves of locally injective dimension not exceeding $d$ is closed
under the same operations, and also under infinite products. We set
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{\mathsf{flid}}$
and
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{{\operatorname{\mathsf{flid--}}}d}$.

For the rest of the section, let $X$ be a quasi-compact semi-separated scheme
with an open covering ${\mathbf{W}}$.

######  Corollary 4.9.1.

(a) For any symbol $\star\neq{\mathsf{ctr}}$ and any (finite) integer $d\ge
0$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ is an
equivalence of triangulated categories.

(b) For any symbol $\star\neq{\mathsf{ctr}}$ and any (finite) integer $d\ge
0$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ is an
equivalence of triangulated categories.

(c) For any symbol $\star\neq{\mathsf{co}}$ and any (finite) integer $d\ge 0$,
the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
is an equivalence of triangulated categories.

###### Proof.

Parts (a-b) follow from Proposition A.5.6, while part (c) follows from the
dual version of the same. ∎

######  Corollary 4.9.2.

(a) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$ is an equivalence of
triangulated categories.

(b) For any symbol $\star={\mathsf{b}}$ or $-$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}}$ is an equivalence of
triangulated categories.

(c) For any symbol $\star={\mathsf{b}}$ or $+$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$ is an
equivalence of triangulated categories.

###### Proof.

The assertions concerning the case $\star={\mathsf{b}}$ follow from the
respective assertions of Corollary 4.9.1 by passage to the inductive limit as
$d\to\infty$. The assertions concerning the case $\star=-$ in parts (a-b)
follow from Proposition A.3.1(a), while the assertion about $\star=+$ in part
(c) follows from the dual version of it. ∎

######  Lemma 4.9.3.

If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering
subordinate to ${\mathbf{W}}$, then the left homological dimension of any
object of the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
with respect to the full exact subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$ does
not exceed $N-1$.

###### Proof.

In view of Corollary A.5.3, it suffices to show that any object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
admits an admissible epimorphism with respect to the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
from an object of
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$. We
will do more and show that the exact sequence (27) is a left resolution of an
object ${\mathfrak{P}}\in
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
by objects of
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$.
Indeed, the functor of inverse image with respect to a very flat
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism $f\colon Y\longrightarrow X$
takes
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
into
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{{\operatorname{\mathsf{flid--}}}d}$,
while the functor of direct image with respect to a flat
$({\mathbf{W}},{\mathbf{T}})$-affine morphism $f$ takes
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{{\operatorname{\mathsf{flid--}}}d}$
into
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$.
The sequence (27) is exact over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$,
since it is exact over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
is closed under admissible monomorphisms in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. ∎

######  Lemma 4.9.4.

(a) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering,
then the homological dimension of the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ does
not exceed $N+d$.

(b) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering
subordinate to ${\mathbf{W}}$, then the homological dimension of the exact
category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
does not exceed $N-1+d$.

###### Proof.

Part (a): in fact, one proves the stronger assertion that
$\operatorname{Ext}^{>N+d}_{X}({\mathcal{F}},{\mathcal{M}})=0$ for any quasi-
coherent sheaf ${\mathcal{M}}$ and any quasi-coherent sheaf of very flat
dimension $\le d$ over $X$ (also, the $\operatorname{Ext}$ groups in the exact
category
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ agree
with those in the abelian category $X{\operatorname{\mathsf{--qcoh}}}$). Since
any object of
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ has a
finite left resolution of length $\le d$ by objects
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$, it suffices to consider
the case of ${\mathcal{F}}\in
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$.

The latter can be dealt with using the Čech resolution (12) of the sheaf
${\mathcal{M}}$ and the adjunction of exact functors $j^{*}$ and $j_{*}$ for
the embedding of an affine open subscheme $j\colon U\longrightarrow X$,
inducing the similar adjunction on the level of $\operatorname{Ext}$ groups
(cf. the proof of Lemma 5.4.1(b) below). Alternatively, the desired assertion
can be deduced from Lemma 4.6.1(a). The proof of part (b) is similar and can
be based either on the Čech resolution (27), or on Lemma 4.6.1(b). ∎

######  Corollary 4.9.5.

(a) The natural triangulated functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})$
and
${\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})$
are equivalences of triangulated categories. In particular, such functors
between the derived categories of the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ are equivalences of
categories.

(b) The natural triangulated functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})$
and
${\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})$
are equivalences of triangulated categories. In particular, such functors
between the derived categories of the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ are
equivalences of categories.

###### Proof.

Follows from the respective parts of Lemma 4.9.4 together with the result of
[50, Remark 2.1]. ∎

As a matter of notational convenience, set the triangulated category
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$ to be
the inductive limit of (the equivalences of categories of)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})$
as $d\to\infty$ for any symbol $\star\neq{\mathsf{ctr}}$. Furthermore, set
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})$ to
be the inductive limit of (the equivalences of categories of)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d})$
as $d\to\infty$. For any morphism $f\colon Y\longrightarrow X$ into a quasi-
compact semi-separated scheme $X$ one constructs the left derived functor

(57)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$

as the functor on the derived categories induced by the exact functor
$f^{*}\colon X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$. The left derived functor

(58)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})$

is constructed in the similar way.

Analogously, set
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
to be the inductive limit of (the equivalences of categories)
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d})$
as $d\to\infty$ for any symbol $\star\neq{\mathsf{co}}$. For any morphism
$f\colon Y\longrightarrow X$ into a quasi-compact semi-separated scheme $X$
and any open coverings ${\mathbf{W}}$ and ${\mathbf{T}}$ of the schemes $Y$
and $X$ for which the morphism $f$ is $({\mathbf{W}},{\mathbf{T}})$-coaffine,
the right derived functor

${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{flid}})$

is constructed as the functor on the derived categories induced by the exact
functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}$.

As usually, we set
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})={\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{\mathsf{flid}})$.
Now Lemma 4.9.3 together with Proposition A.5.6 provide a natural equivalence
of triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$.
For a morphism $f\colon Y\longrightarrow X$ of quasi-compact semi-separated
schemes, such equivalences allow to define the derived functor

(59)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}),$

which clearly does not depend on the choice of the coverings ${\mathbf{W}}$
and ${\mathbf{T}}$.

###  4.10. Morphisms of finite flat dimension

A morphism of schemes $f\colon Y\longrightarrow X$ is said to have _flat
dimension not exceeding $D$_ if for any affine open subschemes $U\subset X$
and $V\subset Y$ such that $f(V)\subset U$ the ${\mathcal{O}}_{X}(U)$-module
${\mathcal{O}}_{Y}(V)$ has flat dimension not exceeding $D$. The morphism $f$
has _very flat dimension not exceeding $D$_ if the similar bound holds for the
very flat dimension of the ${\mathcal{O}}_{X}(U)$-modules
${\mathcal{O}}_{Y}(V)$.

For any morphism $f\colon Y\longrightarrow X$ of finite flat dimension $\le D$
into a quasi-compact semi-separated scheme $X$ and any symbol
$\star\neq{\mathsf{ctr}}$, the left derived functor

(60)
${\mathbb{L}}f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})$

is constructed in the following way. Let us call a quasi-coherent sheaf
${\mathcal{F}}$ on $X$ _adjusted to $f$_ if for any affine open subschemes
$U\subset X$ and $V\subset Y$ such that $f(V)\subset U$ one has
$\operatorname{Tor}^{{\mathcal{O}}_{X}(U)}_{>0}({\mathcal{O}}_{Y}(V),{\mathcal{F}}(U))=0$.
Quasi-coherent sheaves ${\mathcal{F}}$ on $X$ adjusted to $f$ form a full
subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}}\subset
X{\operatorname{\mathsf{--qcoh}}}$ closed under extensions, kernels of
surjective morphisms and infinite direct sums, and such that any quasi-
coherent sheaf on $X$ has a finite left resolution of length $\le D$ by
sheaves from
$X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}}$. By
Proposition A.5.6, it follows that the natural functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories.

The right exact functor $f^{*}\colon
X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$ restricts to an exact functor $f^{*}\colon
X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$. In view of the above equivalence of
categories, the induced functor on the derived categories
$f^{*}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{f{\operatorname{\mathsf{--adj}}}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}})$
provides the desired derived functor ${\mathbb{L}}f^{*}$. For any morphism of
finite flat dimension $f\colon Y\longrightarrow X$ between quasi-compact semi-
separated schemes $Y$ and $X$, the functor ${\mathbb{L}}f^{*}$ is left adjoint
to the functor ${\mathbb{R}}f_{*}$ (50) from Section 4.8 (cf. [53, Section
1.9]).

For any morphism $f\colon Y\longrightarrow X$ of finite very flat dimension
$\le D$ into a quasi-compact semi-separated scheme $X$, any open coverings
${\mathbf{W}}$ and ${\mathbf{T}}$ of the schemes $X$ and $Y$ for which the
morphism $f$ is $({\mathbf{W}},{\mathbf{T}})$-coaffine, and any symbol
$\star\neq{\mathsf{co}}$, the right derived functor

(61)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}})$

is constructed in the following way. Let us call a ${\mathbf{W}}$-locally
contraherent cosheaf ${\mathfrak{P}}$ on $X$ _adjusted to $f$_ if for any
affine open subschemes $U\subset X$ and $V\subset Y$ such that $U$ is
subordinate to ${\mathbf{W}}$ and $f(V)\subset U$ one has
$\operatorname{Ext}_{{\mathcal{O}}_{X}(U)}^{>0}({\mathcal{O}}_{X}(V),{\mathfrak{P}}[U])=0$.
One can easily see that the adjustness condition does not change when
restricted to open subschemes $V$ subordinate to ${\mathbf{T}}$, nor it is
changed by a refinement of the covering ${\mathbf{W}}$.

Locally contraherent cosheaves on $X$ adjusted to $f$ form a full subcategory
$X{\operatorname{\mathsf{--lcth}}}^{f{\operatorname{\mathsf{--adj}}}}\subset
X{\operatorname{\mathsf{--lcth}}}$ closed under extensions and cokernels of
admissible monomorphisms; the category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}}=X{\operatorname{\mathsf{--lcth}}}^{f{\operatorname{\mathsf{--adj}}}}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is also closed under infinite
products in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and such that any
${\mathbf{W}}$-locally contraherent cosheaf on $X$ has a finite right
resolution of length $\le D$ by objects of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}}$.
By the dual version of Proposition A.5.6, the natural functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories. The construction of the exact
functor $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lin}}$ from Section
3.3 extends without any changes to the case of cosheaves from
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}}$,
defining an exact functor

$f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muY{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}.$

Instead of Lemma 1.2.3(a) used in Sections 2.3 and 3.3, one can use Lemma
1.5.5(a) in order to check that the contraadjustness condition is preserved
here.

In view of the above equivalence of triangulated categories, the induced
functor
$f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{f{\operatorname{\mathsf{--adj}}}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}})$
provides the desired functor ${\mathbb{R}}f^{!}$ (61). When both schemes $X$
and $Y$ are quasi-compact and semi-separated, one can use the equivalences of
categories from Corollary 4.6.3(a) in order to obtain the right derived
functor

(62)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}),$

which is right adjoint to the functor ${\mathbb{L}}f_{!}$ (51) from Section
4.8.

For a morphism $f\colon Y\longrightarrow X$ of flat dimension $\le D$ into a
quasi-compact semi-separated scheme $X$, any open coverings ${\mathbf{W}}$ and
${\mathbf{T}}$ of the schemes $X$ and $Y$ for which the morphism $f$ is
$({\mathbf{W}},{\mathbf{T}})$-coaffine, and any symbol
$\star\neq{\mathsf{co}}$, one can similarly construct the right derived
functor

(63)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathbf{W}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathbf{T}}^{\mathsf{lct}}).$

More precisely, a locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{P}}$ on $X$ is said to be _adjusted to $f$_ if for any
affine open subschemes $U\subset X$ and $V\subset Y$ such that $U$ is
subordinate to ${\mathbf{W}}$ and $f(V)\subset U$, and for any flat
${\mathcal{O}}_{Y}(V)$-module $G$, one has
$\operatorname{Ext}_{{\mathcal{O}}_{X}(U)}(G,{\mathfrak{P}}[U])=0$. This
condition does not change when restricted to open subschemes $V\subset Y$
subordinate to ${\mathbf{T}}$, nor is it changed by any refinement of the
covering ${\mathbf{W}}$ of the scheme $X$.

As above, locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves on
$X$ adjusted to $f$ form a full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ closed under extensions,
cokernels of admissible monomorphisms, and infinite products. It follows from
Lemma 1.5.2(a) that any locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf on $X$ has a finite right resolution of length $\le D$ by objects of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}}$.
Hence the natural functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
is an equivalence of triangulated categories.

Now Lemma 1.5.5(b) allows to construct an exact functor

$f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muY{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}},$

and the induced functor
$f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\mathsf{lct}},\,{f{\operatorname{\mathsf{--adj}}}}})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}})$
provides the desired derived functor (63). When both schemes $X$ and $Y$ are
quasi-compact and semi-separated, one can use Corollary 4.6.4(a) in order to
obtain the right derived functor

(64)
${\mathbb{R}}f^{!}\colon{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}),$

which is right adjoint to the functor ${\mathbb{L}}f_{!}$ (52).

Let $X$ be a quasi-compact semi-separated scheme with an open covering
${\mathbf{W}}$.

######  Lemma 4.10.1.

(a) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering,
then the right homological dimension of any quasi-coherent sheaf of flat
dimension $\le d$ on $X$ with respect to the full exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ does not
exceed $N$.

(b) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering,
then the right homological dimension of any quasi-coherent sheaf of very flat
dimension $\le d$ on $X$ with respect to the full exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ does
not exceed $N$.

(c) If $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ is a finite affine open covering
subordinate to ${\mathbf{W}}$, then the left homological dimension of any
${\mathbf{W}}$-locally contraherent cosheaf of locally injective dimension
$\le d$ on $X$ with respect to the full exact subcategory
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
does not exceed $N-1$.

###### Proof.

Part (a): in view of Lemma 4.6.1(a) and the dual version of Corollary A.5.3,
it suffices to show that there exists an injective morphism from any given
quasi-coherent sheaf belonging to
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ into a
quasi-coherent sheaf belonging to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ with the
cokernel belonging to
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$. This
follows from Corollary 4.1.4(b) or 4.1.11(b). The proof of part (b) is
similar.

The proof of part (c) is analogous up to duality, and based on Lemma 4.6.1(b)
and Corollary 4.2.5(b) (alternatively, the argument from the proof of Lemma
4.9.3 is sufficient in this case). ∎

######  Lemma 4.10.2.

Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering. Then

(a) a quasi-coherent sheaf on $X$ belongs to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of quasi-coherent sheaves from
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$;

(b) a quasi-coherent sheaf on $X$ belongs to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of quasi-coherent sheaves from
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$;

(c) a quasi-coherent sheaf on $X$ belongs to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of quasi-coherent sheaves from
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$;

(d) a contraherent cosheaf on $X$ belongs to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$ if and
only if it is a direct summand of a finitely iterated extension of the direct
images of contraherent cosheaves from
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$.

###### Proof.

One has to repeat the arguments in Sections 4.1 and 4.2 working with,
respectively, quasi-coherent sheaves of (very) flat dimension $\le d$ only or
locally contraherent cosheaves of locally injective dimension $\le d$ only
throughout (cf. Lemma 4.1.6). ∎

Let $f\colon Y\longrightarrow X$ be a morphism of finite flat dimension $\le
D$ between quasi-compact semi-separated schemes $X$ and $Y$.

######  Corollary 4.10.3.

(a) The exact functor $f_{*}\colon
Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ takes objects of
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$ to
objects of $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}(d+D)}$.

(b) If the morphism $f$ has very flat dimension not exceeding $D$, then the
exact functor $f_{*}\colon
Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ takes objects of
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$ to
objects of $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}(d+D)}$.

(c) The exact functor $f_{!}\colon
Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ takes objects of
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}$ to
objects of $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}(d+D)}$.

###### Proof.

Part (a) follows from Lemma 4.10.2(a) together with the fact that the direct
image with respect to an affine morphism of flat dimension $\le D$ takes
quasi-coherent sheaves of flat dimension $\le d$ to quasi-coherent sheaves of
flat dimension $\le d+D$. The latter is provided by Lemma 1.5.2(a). The proof
of part (b) is similar and based on Lemmas 4.10.2(c) and 1.5.3(b).

Finally, part (c) follows from Lemma 4.10.2(d) together with the fact that the
direct image with respect to a $({\mathbf{W}},{\mathbf{T}})$-affine morphism
of flat dimension $\le D$ takes ${\mathbf{T}}$-locally contraherent cosheaves
of locally injective dimension $\le d$ to ${\mathbf{W}}$-locally contraherent
cosheaves of locally injective dimension $\le d+D$. The latter is provided by
Lemma 1.5.2(b). ∎

According to Lemma 4.10.1(a) and the dual version of Proposition A.5.6, for
any symbol $\star\neq{\mathsf{ctr}}$ the natural functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})$
is an equivalence of triangulated categories (as is the similar functor for
sheaves over $X$). So one can construct the right derived functor

${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}(d+D)})$

as the functor on the derived categories induced by the exact functor
$f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
Y{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}(d+D)}$ from
Corollary 4.10.3(a). Passing to the inductive limits as $d\to\infty$, we
obtain the right derived functor

(65)
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}),$

which is right adjoint to the functor ${\mathbb{L}}f^{*}$ (57). For a morphism
$f$ of finite very flat dimension, the right derived functor

(66)
${\mathbb{R}}f_{*}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fvfd}})$

right adjoint to the functor ${\mathbb{L}}f^{*}$ (58) is constructed in the
similar way.

Analogously, according to Lemma 4.10.1(c) and Proposition A.5.6, for any
symbol $\star\neq{\mathsf{co}}$ the natural functor
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d})\longrightarrow{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d})$
is an equivalence of triangulated categories (as is the similar functor for
cosheaves over $X$). Thus one can construct the left derived functor

${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}(d+D)})$

as the functor on the derived categories induced by the exact functor
$f_{!}\colon Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
Y{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}d}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{flid--}}}(d+D)}$
from Corollary 4.10.3(c). Passing to the inductive limits as $d\to\infty$, we
obtain the left derived functor

(67)
${\mathbb{L}}f_{!}\colon{\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}),$

which is left adjoint to the functor ${\mathbb{R}}f^{!}$ (59).

###  4.11. Finite injective and projective dimension

For any scheme $X$, we denote the full subcategory of objects of injective
dimension $\le d$ in the abelian category $X{\operatorname{\mathsf{--qcoh}}}$
by $X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}$. For
a quasi-compact semi-separated scheme $X$, the full subcategory of objects of
projective dimension $\le d$ in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is denoted by
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$
and the full subcategory of objects of projective dimension $\le d$ in the
exact category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$
by
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$.
Set
$X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{fpd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\},\,{{\operatorname{\mathsf{fpd--}}}d}}$
and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}d}=X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{\\{X\\},\,{{\operatorname{\mathsf{fpd--}}}d}}$.

Furthermore, the _colocally flat dimension_ of a ${\mathbf{W}}$-locally
contraherent cosheaf on a quasi-compact semi-separated scheme $X$ is defined
as its left homological dimension with respect to the exact subcategory
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see Section A.5). The
colocally flat dimension is well-defined by Corollaries 4.3.2(b) and 4.3.4(b).
A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on $X$ has
colocally flat dimension $\le d$ if and only if
$\operatorname{Ext}^{>d}({\mathfrak{F}},{\mathfrak{P}})=0$ for any locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$.
The full subcategory of objects of colocally flat dimension $\le d$ in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is denoted by
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{clfd--}}}d}}$.
We set
$X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{clfd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\},\,{{\operatorname{\mathsf{clfd--}}}d}}$.

Since the subcategories of projective objects in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ do not depend
on the covering ${\mathbf{W}}$, and the full subcategories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}\subset
X{\operatorname{\mathsf{--lcth}}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}\subset
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ are closed under kernels of
admissible epimorphisms, the projective dimension of an object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ or
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ does not
change when the open covering ${\mathbf{W}}$ is replaced by its refinement.
Similarly, the colocally flat dimension of a ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ does not depend on the covering ${\mathbf{W}}$.

One can easily see that the full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}$ is closed under extensions and cokernels of
admissible monomorphisms, while the full subcategories
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$,
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, and
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{clfd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ are closed under extensions
and kernels of admissible epimorphisms.

######  Corollary 4.11.1.

(a) For any scheme $X$, the natural triangulated functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d})$
are equivalences of categories.

(b) For any quasi-compact semi-separated scheme $X$, the natural triangulated
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,