\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}j_{\alpha,\beta}{}_{!}j_{\alpha,\beta}^{!}{\mathfrak{P}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}j_{\alpha}{}_{!}j_{\alpha}^{!}{\mathfrak{P}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu{\mathfrak{P}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

is exact in the exact category of ${\mathbf{W}}$-locally contraherent
cosheaves on $X$. Indeed, the corresponding sequence of cosections over every
affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$ is exact by
Lemma 1.2.6(b). We have constructed a finite left resolution of a
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ by contraherent
cosheaves. When ${\mathfrak{P}}$ is a locally cotorsion or locally injective
${\mathbf{W}}$-locally contraherent cosheaf, the sequence (27) is exact in the
category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ or
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$, respectively.

###  3.4. Coflasque contraherent cosheaves

Let $X$ be a topological space and ${\mathfrak{F}}$ be a cosheaf of abelian
groups on $X$. A cosheaf ${\mathfrak{F}}$ is called _coflasque_ if for any
open subsets $V\subset U\subset X$ the corestriction map
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is injective. Obviously,
the class of coflasque cosheaves of abelian groups is preserved by the
restrictions to open subsets and the direct images with respect to continuous
maps.

######  Lemma 3.4.1.

Let $X=\bigcup_{\alpha}U_{\alpha}$ be an open covering. Then

(a) a cosheaf ${\mathfrak{F}}$ on $X$ is coflasque if and only if its
restriction ${\mathfrak{F}}|_{U_{\alpha}}$ to each open subset $U_{\alpha}$ is
coflasque;

(b) if the cosheaf ${\mathfrak{F}}$ is coflasque, then the Čech complex (22)
has no higher homology groups,
$H_{>0}C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}})=0$.

###### Proof.

One can either check these assertions directly or deduce them from the similar
results for flasque sheaves of abelian groups using the construction of the
sheaf $U\longmapsto\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}[U],I)$ from
the proof of Theorem 2.1.2. Here $I$ is an injective abelian group; clearly,
the sheaf so obtained is flasque for all $I$ if and only if the original
cosheaf ${\mathfrak{F}}$ was coflasque. The sheaf (dual) versions of
assertions (a-b) are well-known [22, Section II.3.1 and Théorème II.5.2.3(a)].
∎

######  Corollary 3.4.2.

Let $X$ be a scheme with an open covering ${\mathbf{W}}$ and ${\mathfrak{F}}$
be a ${\mathbf{W}}$-locally contraherent cosheaf on $X$. Suppose that the
cosheaf ${\mathfrak{F}}$ is coflasque. Then ${\mathfrak{F}}$ is a (globally)
contraherent cosheaf on $X$.

###### Proof.

Follows from Lemmas 3.2.2 and 3.4.1(b). ∎

Assume that the topological space $X$ has a base of the topology consisting of
quasi-compact open subsets. Then one has
${\mathfrak{F}}[Y]\simeq\varinjlim_{U\subset Y}{\mathfrak{F}}[U]$ for any
cosheaf of abelian groups ${\mathfrak{F}}$ on $X$ and any open subset
$Y\subset X$, where the filtered inductive limit is taken over all the quasi-
compact open subsets $U\subset Y$. It follows easily that a cosheaf
${\mathfrak{F}}$ is coflasque if and only if the corestriction map
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is injective for any pair
of embedded quasi-compact open subsets $V\subset U\subset X$.

Moreover, if $X$ is a scheme then it follows from Lemma 3.4.1(a) that a
cosheaf of abelian groups ${\mathfrak{F}}$ on $X$ is coflasque if and only if
the corestriction map ${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is
injective for any affine open subscheme $U\subset X$ and quasi-compact open
subscheme $V\subset U$. It follows that an infinite product of a family of
coflasque contraherent cosheaves on $X$ is coflasque. The following
counterexample shows, however, that coflasqueness of contraherent cosheaves on
schemes cannot be checked on the pairs of embedded affine open subschemes.

######  Example 3.4.3.

Let $X$ be a Noetherian scheme. It is well-known that any injective quasi-
coherent sheaf ${\mathcal{J}}$ on $X$ is a flasque sheaf of abelian groups.
Let ${\mathcal{F}}$ be a quasi-coherent sheaf on $X$ and
${\mathcal{F}}\longrightarrow{\mathcal{J}}$ be an injective morphism. Then one
has $({\mathcal{J}}/{\mathcal{F}})(U)\simeq{\mathcal{J}}(U)/{\mathcal{F}}(U)$
for any affine open subscheme $U\subset X$, so the map
$({\mathcal{J}}/{\mathcal{F}})(U)\longrightarrow({\mathcal{J}}/{\mathcal{F}})(V)$
is surjective for any pair of embedded affine open subschemes $V\subset U$. On
the other hand, if the quotient sheaf ${\mathcal{J}}/{\mathcal{F}}$ were
flasque, one would have $H^{i+1}(X,{\mathcal{F}})\simeq
H^{i}(X,{\mathcal{J}}/{\mathcal{F}})=0$ for $i\ge 1$, which is clearly not the
case in general.

Now let $X$ be a scheme over an affine scheme $\operatorname{Spec}R$. Let
${\mathcal{M}}$ be a quasi-coherent sheaf on $X$ and $J$ be an injective
$R$-module. Then for any quasi-compact quasi-separated open subscheme
$Y\subset X$ one has
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{M}},J)[Y]\simeq\operatorname{Hom}_{R}({\mathcal{M}}(U),J)$.
For a Noetherian scheme $X$, it follows that the cosheaf
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{M}},J)$ is coflasque if and
only if the sheaf ${\mathcal{M}}$ is flasque (cf. Lemma 3.4.6(c) below).

######  Corollary 3.4.4.

Let $X$ be a scheme with an open covering ${\mathbf{W}}$ and
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ be a short exact sequence in
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$ (e. g., a
short exact sequence of ${\mathbf{W}}$-locally contraherent cosheaves on $X$).
Then

(a) the cosheaf ${\mathfrak{Q}}$ is coflasque whenever both the cosheaves
${\mathfrak{P}}$ and ${\mathfrak{R}}$ are;

(b) the cosheaf ${\mathfrak{P}}$ is cosflasque whenever both the cosheaves
${\mathfrak{Q}}$ and ${\mathfrak{R}}$ are;

(c) if the cosheaf ${\mathfrak{R}}$ is coflasque, then the short sequence
$0\longrightarrow{\mathfrak{P}}[Y]\longrightarrow{\mathfrak{Q}}[Y]\longrightarrow{\mathfrak{R}}[Y]\longrightarrow
0$ is exact for any open subscheme $Y\subset X$.

###### Proof.

The assertion actually holds for any short exact sequence in the exact
category of cosheaves of abelian groups on a scheme $X$ with the exact
category structure related to the base of affine open subschemes subordinate
to a covering ${\mathbf{W}}$.

Part (c): let us first consider the case of a semi-separated open subscheme
$Y$. Pick an affine open covering $Y=\bigcup_{\alpha}U_{\alpha}$ subordinate
to ${\mathbf{W}}$. The intersection of any nonempty finite subset of
$U_{\alpha}$ being also an affine open subscheme in $X$, the short sequence of
complexes $0\longrightarrow
C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}}|_{Y})\longrightarrow
C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{Q}}|_{Y})\longrightarrow
C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{R}}|_{Y})\longrightarrow
0$ is exact. Recall that ${\mathfrak{F}}[Y]\simeq
C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}|_{Y})$
for any cosheaf of abelian groups ${\mathfrak{F}}$ on $X$. By Lemma 3.4.1, one
has
$H_{>0}C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{R}}|_{Y})=0$,
and it follows that the short sequence
$0\longrightarrow{\mathfrak{P}}[Y]\longrightarrow{\mathfrak{Q}}[Y]\longrightarrow{\mathfrak{R}}[Y]\longrightarrow
0$ is exact.

Now for any open subscheme $Y\subset X$, pick an open covering
$Y=\bigcup_{\alpha}U_{\alpha}$ of $Y$ by semi-separated open subschemes
$U_{\alpha}$. The intersection of any nonempty finite subset of $U_{\alpha}$
being semi-separated, the short sequence of Čech complexes for the covering
$U_{\alpha}$ and the short exact sequence of cosheaves
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ is exact according to the above, and the same argument concludes the proof.

Parts (a-b): let $U\subset V\subset X$ be any pair of embedded open
subschemes. Assuming that the cosheaf ${\mathfrak{R}}$ is coflasque, according
to part (c) we have short exact sequences of the modules of cosections
$0\longrightarrow{\mathfrak{P}}[V]\longrightarrow{\mathfrak{Q}}[V]\longrightarrow{\mathfrak{R}}[V]\longrightarrow
0$ and
$0\longrightarrow{\mathfrak{P}}[U]\longrightarrow{\mathfrak{Q}}[U]\longrightarrow{\mathfrak{R}}[U]\longrightarrow
0$. If the morphism from the former sequence to the latter is injective on the
rightmost terms, then it is injective on the middle terms if and only if it is
injective on the leftmost terms. ∎

######  Remark 3.4.5.

Let $U$ be an affine scheme. Then a cosheaf of abelian groups ${\mathfrak{F}}$
on $U$ is coflasque if and only if the following three conditions hold:

  1. (i)

for any affine open subscheme $V\subset U$, the corestriction map
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is injective;

  2. (ii)

for any two affine open subschemes $V^{\prime\prime}$,
$V^{\prime\prime}\subset U$, the image of the map
${\mathfrak{F}}[V^{\prime}\cap
V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[U]$ is equal to the
intersection of the images of the maps
${\mathfrak{F}}[V^{\prime}]\longrightarrow{\mathfrak{F}}[U]$ and
${\mathfrak{F}}[V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[U]$;

  3. (iii)

for any finite collection of affine open subschemes $V_{\alpha}\subset U$, the
images of the maps
${\mathfrak{F}}[V_{\alpha}]\longrightarrow{\mathfrak{F}}[U]$ generate a
distributive sublattice in the lattice of all subgroups of the abelian group
${\mathfrak{F}}[U]$.

Indeed, the condition (i) being assumed, in view of the exact sequence
${\mathfrak{F}}[V^{\prime}\cap
V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[V^{\prime}]\oplus
gF[V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[V^{\prime}\cup
V^{\prime\prime}]\longrightarrow 0$ the condition (ii) is equivalent to
injectivity of the corestriction map ${\mathfrak{F}}[V^{\prime}\cup
V^{\prime\prime}]\longrightarrow{\mathfrak{F}}[U]$.

Furthermore, set $W=\bigcup_{\alpha}V_{\alpha}$. The condition (iii) for any
proper subcollection of $V_{\alpha}$ and the conditions (i-ii) being assumed,
the condition (iii) for the collection $\\{V_{\alpha}\\}$ becomes equivalent
to vanishing of the higher homology of the Čech complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\alpha}\\},{\mathfrak{F}}|_{W})$
together with injectivity of the map
${\mathfrak{F}}[W]\longrightarrow{\mathfrak{F}}[U]$. Indeed, the complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\alpha}\\},{\mathfrak{F}}|_{W})\longrightarrow{\mathfrak{F}}[U]$
can be identified with (the abelian group version of) the “cobar complex”
$B^{\text{\smaller\smaller$\scriptstyle\bullet$}}({\mathfrak{F}}[U];\>{\mathfrak{F}}[V_{\alpha}])$
from [49, Proposition 7.2(c*) of Chapter 1] (see also [50, Lemma
11.4.3.1(c*)]).

######  Lemma 3.4.6.

(a) Let ${\mathcal{M}}$ be a flasque quasi-coherent sheaf and ${\mathcal{G}}$
be a flat quasi-coherent sheaf on a locally Noetherian scheme $X$. Then the
quasi-coherent sheaf ${\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{G}}$
on $X$ is flasque.

(b) Let ${\mathcal{M}}$ be a flasque quasi-coherent sheaf and ${\mathfrak{J}}$
be a locally injective contraherent cosheaf on a scheme $X$. Then the
contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ on $X$ is
coflasque.

(c) Let ${\mathcal{M}}$ be a flasque quasi-coherent sheaf and ${\mathcal{J}}$
be an injective quasi-coherent sheaf on an affine Noetherian scheme $U$. Then
the contraherent cosheaf
$\operatorname{\mathfrak{Hom}}_{U}({\mathcal{M}},{\mathcal{J}})$ on $U$ is
coflasque.

(d) Let ${\mathcal{M}}$ be a flasque quasi-coherent sheaf and ${\mathfrak{G}}$
be a flat cosheaf of ${\mathcal{O}}_{U}$-modules on an affine scheme $U$. Then
the quasi-coherent sheaf ${\mathcal{M}}\odot_{U}{\mathfrak{G}}$ on $U$ is
flasque.

###### Proof.

The (co)flasqueness of (co)sheaves being a local property, it suffices to
consider the case of an affine scheme $X=U$ in all assertions (a-d). Then (c)
becomes a restatement of (b) and (d) a restatement of (a). It also follows
that one can extend the assertion (c) to locally injective locally
contraherent cosheaves ${\mathfrak{J}}$ (cf. Section 3.6).

To prove part (a) in the affine case, one notices the isomorphism
$({\mathcal{M}}\otimes_{{\mathcal{O}}_{U}}{\mathcal{G}})(V)\simeq{\mathcal{M}}(V)\otimes_{{\mathcal{O}}(U)}{\mathcal{G}}(U)$
holding for any quasi-compact open subscheme $V$ in an affine scheme $U$,
quasi-coherent sheaf ${\mathcal{M}}$, and a flat quasi-coherent sheaf
${\mathcal{G}}$ on $U$. More generally, one has
$({\mathcal{M}}\otimes_{R}G)(Y)\simeq{\mathcal{M}}(Y)\otimes_{R}G$ for any
quasi-coherent sheaf ${\mathcal{M}}$ with a right $R$-module structure on a
quasi-compact quasi-separated scheme $Y$ and a flat left $R$-module $G$. Part
(b) follows from the similar isomorphism
$\operatorname{\mathfrak{Cohom}}_{U}({\mathcal{M}},{\mathfrak{J}})[V]\simeq\operatorname{Hom}_{{\mathcal{O}}(U)}({\mathcal{M}}(V),{\mathfrak{J}}[U])$
holding for any quasi-coherent sheaf ${\mathcal{M}}$ and a (locally) injective
contraherent cosheaf ${\mathfrak{J}}$ on $U$. More generally,
$\operatorname{\mathfrak{Cohom}}_{R}({\mathcal{M}},J)[Y]\simeq\operatorname{Hom}_{R}({\mathcal{M}}(Y),J)$
for any quasi-compact quasi-separated scheme $Y$ over $\operatorname{Spec}R$,
quasi-coherent sheaf ${\mathcal{M}}$ on $Y$, and an injective $R$-module $J$.
∎

######  Lemma 3.4.7.

(a) Let $X$ be a Noetherian topological space of finite Krull dimension $\le
d+1$, and let
$0\longrightarrow{\mathcal{E}}\longrightarrow{\mathcal{F}}^{0}\longrightarrow\dotsb\longrightarrow{\mathcal{F}}^{d}\longrightarrow{\mathcal{F}}\longrightarrow
0$ be an exact sequence of sheaves of abelian groups on $X$. Suppose that the
sheaves ${\mathcal{F}}^{i}$ are flasque. Then the sheaf ${\mathcal{F}}$ is
flasque.

(b) Let $X$ be a scheme with an open covering ${\mathbf{W}}$ and
$0\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{F}}_{d}\longrightarrow\dotsb\longrightarrow{\mathfrak{F}}_{0}\longrightarrow{\mathfrak{E}}\longrightarrow
0$ be an exact sequence in
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$. Suppose that
the cosheaves ${\mathfrak{F}}_{i}$ are coflasque and the underlying
topological space of the scheme $X$ is Noetherian of finite Krull dimension
$\le d+1$. Then the cosheaf ${\mathfrak{F}}$ is coflasque.

###### Proof.

Part (a) is a version of Grothendieck’s vanishing theorem [24, Théorème
3.6.5]; it can be either proven directly along the lines of Grothendieck’s
proof, or, making a slightly stronger assumption that the dimension of $X$
does not exceed $d$, deduced from the assersion of Grothendieck’s theorem.
Indeed, let ${\mathcal{G}}$ denote the image of the morphism of sheaves
${\mathcal{F}}^{d-1}\longrightarrow{\mathcal{F}}^{d}$; by Grothendieck’s
theorem, one has $H^{1}(V,{\mathcal{G}}|_{V})\simeq
H^{d+1}(V,{\mathcal{E}}|_{V})=0$ for any open subset $V\subset X$. Hence the
map ${\mathcal{F}}^{d}(V)\longrightarrow{\mathcal{F}}(V)$ is surjective, and
it follows that the map ${\mathcal{F}}(X)\longrightarrow{\mathcal{F}}(V)$ is
surjective, too.

Part (b): given an injective abelian group $I$, the sequence
$0\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{E}}[U],I)\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}_{0}[U],I)\longrightarrow\dotsb\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}_{d}[U],I)\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}[U],I)\longrightarrow
0$ is exact for any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$. Hence the construction of the sheaf
$V\longmapsto\operatorname{Hom}_{\mathbb{Z}}({-}[V],I)$, where $V\subset X$
are arbitrary open subschemes, transforms our sequence of cosheaves into an
exact sequence of sheaves of ${\mathcal{O}}_{X}$-modules
$0\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{E}},I)\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}_{0},I)\longrightarrow\dotsb\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}_{d},I)\longrightarrow\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}},I)\longrightarrow
0$. Now the sheaves $\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}}^{i},I)$
are flasque, and by part (a) it follows that the sheaf
$\operatorname{Hom}_{\mathbb{Z}}({\mathfrak{F}},I)$ is. Therefore, the cosheaf
${\mathfrak{F}}$ is coflasque. (Cf. Section A.5.) ∎

We have shown, in particular, that coflasque contraherent cosheaves on a
scheme $X$ form a full subcategory of $X{\operatorname{\mathsf{--ctrh}}}$
closed under extensions, kernels of admissible epimorphisms, and infinite
products. Hence this full subcategory acquires an induced exact category
structure, which we will denote by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$.

Similarly, coflasque locally cotorsion contraherent cosheaves on $X$ form a
full subcategory of $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ closed
under extensions, kernels of admissible epimorphisms, and infinite products.
We denote the induced exact category structure on this full subcategory by
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$.

######  Corollary 3.4.8.

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

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

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

###### Proof.

Since the contraadjustness/contraherence conditions, local cotorsion, and
exactness of short sequences in $X{\operatorname{\mathsf{--ctrh}}}$ or
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ only depend on the
restrictions to affine open subschemes, while the coflasqueness is preserved
by restrictions to any open subschemes, we can assume that the scheme $X$ is
affine. Then the scheme $Y$ is quasi-compact and quasi-separated.

Let $Y=\bigcup_{\alpha}V_{\alpha}$ be a finite affine open covering and
${\mathfrak{F}}$ be a coflasque contraherent cosheaf on $Y$. Then the Čech
complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\alpha}\\},{\mathfrak{F}})$
is a finite left resolution of the ${\mathcal{O}}(X)$-module
${\mathfrak{F}}[Y]$. Let us first consider the case when $Y$ is semi-
separated, so the intersection of any nonempty subset of $V_{\alpha}$ is
affine.

By Lemma 1.2.2(a), our resolution consists of contraadjusted
${\mathcal{O}}(X)$-modules. When ${\mathfrak{F}}$ is a locally cotorsion
contraherent cosheaf, by Lemma 1.3.4(a) this resolution even consists of
cotorsion ${\mathcal{O}}(X)$-modules. It follows that the
${\mathcal{O}}(X)$-module $(f_{!}{\mathfrak{F}})[X]={\mathfrak{F}}[Y]$ is
contraadjusted in the former case and cotorsion in the latter one.

Now let $U\subset X$ be an affine open subscheme. Then one has

${\mathfrak{F}}[f^{-1}(U)\cap
V]\simeq\operatorname{Hom}_{{\mathcal{O}}_{Y}(V)}({\mathcal{O}}_{Y}(f^{-1}(U)\cap
V),\>{\mathfrak{F}}[V])\simeq\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}_{X}(U),{\mathfrak{F}}[V])$

for any affine open subscheme $V\subset Y$; so the complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{f^{-1}(U)\cap
V_{\alpha}\\},\>{\mathfrak{F}}|_{f^{-}1(U)})$ can be obtained by applying the
functor $\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}_{X}(U),{-})$ to
the complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\alpha}\\},{\mathfrak{F}})$.
It follows that
$(f_{!}{\mathfrak{F}})[U]\simeq{\mathfrak{F}}[f^{-1}(U)]\simeq\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}_{X}(U),{\mathfrak{F}}[Y])$
and the contraherence condition holds for $f_{!}{\mathfrak{F}}$.

Finally, let us turn to the general case. According to the above, for any
semi-separated quasi-compact open subscheme $V\subset Y$ the
${\mathcal{O}}(X)$-module ${\mathfrak{F}}[V]$ is contraadjusted (and even
cotorsion if ${\mathfrak{F}}$ is locally cotorsion), and for any affine open
subscheme $U\subset X$ one has ${\mathfrak{F}}[f^{-1}(U)\cap
V]\simeq\operatorname{Hom}_{{\mathcal{O}}(X)}({\mathcal{O}}_{X}(U),{\mathfrak{F}}[V])$.
Given that the intersection of any nonempty subset of $V_{\alpha}$ is
separated (being quasi-affine) and quasi-compact, the same argument as above
goes through. ∎

######  Corollary 3.4.9.

Let $f\colon Y\longrightarrow X$ be a quasi-compact quasi-separated morphism
of schemes and ${\mathbf{T}}$ be an open covering of $Y$. Then

(a) for any complex
${\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of flasque
quasi-coherent sheaves on $Y$ that is acyclic as a complex over
$Y{\operatorname{\mathsf{--qcoh}}}$, the complex
$f_{*}{\mathcal{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of flasque
quasi-coherent sheaves on $X$ is acyclic as a complex over
$X{\operatorname{\mathsf{--qcoh}}}$;

(b) for any complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of coflasque
contraherent cosheaves on $Y$ that is acyclic as a complex over
$Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}$, the complex
$f_{!}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
coflasque contraherent cosheaves on $X$ is acyclic as a complex over
$X{\operatorname{\mathsf{--ctrh}}}$;

(c) for any complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of coflasque
locally cotorsion contraherent cosheaves on $Y$ that is acyclic as a complex
over $Y{\operatorname{\mathsf{--lcth}}}_{\mathbf{T}}^{\mathsf{lct}}$, the
complex $f_{!}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of coflasque locally cotorsion contraherent cosheaves on $X$ is acyclic as a
complex over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$.

###### Proof.

Let us prove part (c), parts (a-b) being analogous. In view of the results of
Section 3.2, it suffices to consider the case of an affine scheme $X$ and a
quasi-compact quasi-separated scheme $Y$. We have to show that the complex of
cotorsion ${\mathcal{O}}(X)$-modules
$\Delta(Y,{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$ is
acyclic over ${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$.
Considering the case of a semi-separated scheme $Y$ first and the general case
second, one can assume that $Y$ has a finite open covering
$Y=\bigcup_{\alpha=1}^{N}U_{\alpha}$ by quasi-compact quasi-separated schemes
$U_{\alpha}$ subordinate to ${\mathbf{T}}$ such that for any intersection $V$
of a nonempty subset of $U_{\alpha}$ the complex
$\Delta(V,{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$ is
acyclic over ${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$.

Consider the Čech bicomplex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
of the complex of cosheaves
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on the scheme
$Y$ with the open covering $U_{\alpha}$. There is a natural morphism of
bicomplexes
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\Delta(X,{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$;
and for every degree $i$ the complex
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}^{i})\longrightarrow\Delta(X,{\mathfrak{F}}^{i})$
is a finite (and uniformly bounded) acyclic complex of cotorsion
${\mathcal{O}}(X)$-modules. It follows that the induced morphism of total
complexes is a quasi-isomorphism (in fact, a morphism with an absolutely
acyclic cone) of complexes over
${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. On the other
hand, for every $k$ the complex
$C_{k}(\\{U_{\alpha}\\},{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is by assumption acyclic over over
${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$. It follows
that the total complex of
$C_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
and hence also the complex
$\Delta(Y,{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$, is
acyclic over ${\mathcal{O}}(X){\operatorname{\mathsf{--mod}}}^{\mathsf{cot}}$.
∎

###  3.5. Contrahereable cosheaves and the contraherator

A cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on a scheme $X$ is
said to be _derived contrahereable_ if

  1. (i∘)

for any affine open subscheme $U\subset X$ and its finite affine open covering
$U=\bigcup_{\alpha=1}^{N}U_{\alpha}$ the homological Čech sequence (cf. (22))

(28) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathfrak{P}}[U_{1}\cap\dotsb\cap U_{N}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\
\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}{\mathfrak{P}}[U_{\alpha}\cap U_{\beta}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}{\mathfrak{P}}[U_{\alpha}]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathfrak{P}}[U]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu0$

is exact; and

  2. (ii)

for any affine open subscheme $U\subset X$, the ${\mathcal{O}}_{X}(U)$-module
${\mathfrak{P}}[U]$ is contraadjusted.

We will call (i∘) the _exactness condition_ and (ii) the _contraadjustness
condition_.

Notice that the present contraadjustness condition (ii) is an equivalent
restatement of the contraadjustness condition (ii) of Section 2.2, while the
exactness condition (i∘) is weaker than the contraherence condition (i) of
Section 2.2 (provided that the condition (ii) is assumed). The condition (i∘)
is also weaker than the coflasqueness condition on a cosheaf of
${\mathcal{O}}_{X}$-modules discussed in Section 3.4.

By Remark 2.1.4, the exactness condition (i∘) can be thought of as a
strengthening of the cosheaf property (6) of a covariant functor with an
${\mathcal{O}}_{X}$-module structure on the category of affine open subschemes
of $X$. Any such functor satisfying (i∘) can be extended to a cosheaf of
${\mathcal{O}}_{X}$-modules in a unique way.

Let ${\mathbf{W}}$ be an open covering of a scheme $X$. A cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on $X$ is called _${\mathbf{W}}$
-locally derived contrahereable_ if its restrictions ${\mathfrak{P}}|_{W}$ to
all the open subschemes $W\in{\mathbf{W}}$ are derived contrahereable on $W$.
In other words, this means that the conditions (i∘) and (ii) must hold for all
the affine open subschemes $U\subset X$ subordinate to ${\mathbf{W}}$. A
cosheaf of ${\mathcal{O}}_{X}$-modules is called locally derived
contrahereable if it is ${\mathbf{W}}$-locally derived contrahereable for some
open covering ${\mathbf{W}}$. Any contraherent cosheaf is derived
contrahereable, and any ${\mathbf{W}}$-locally contraherent cosheaf is
${\mathbf{W}}$-locally derived contrahereable. Conversely, according to Lemma
3.2.2, if a ${\mathbf{W}}$-locally contraherent cosheaf is derived
contrahereable, then it is contraherent.

A ${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{P}}$ on
$X$ is called _locally cotorsion_ (respectively, _locally injective_) if the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{P}}[U]$ is cotorsion (resp.,
injective) for any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$. A locally derived contrahereable cosheaf ${\mathfrak{P}}$ is
locally cotorsion (resp., locally injective) if and only if the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{P}}[U]$ is cotorsion (resp.,
injective) for every affine open subscheme $U\subset X$ such that the cosheaf
${\mathfrak{P}}|_{U}$ is derived contrahereable.

In the exact category of cosheaves of ${\mathcal{O}}_{X}$-modules
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$ defined in
Section 3.1, the ${\mathbf{W}}$-locally derived contrahereable cosheaves form
an exact subcategory closed under extensions, infinite products, and cokernels
of admissible monomorphisms. Hence there is the induced exact category
structure on the category of ${\mathbf{W}}$-locally derived contrahereable
cosheaves on $X$.

Let $U$ be an affine scheme and ${\mathfrak{Q}}$ be a derived contrahereable
cosheaf on $U$. The _contraherator_
$\operatorname{\mathfrak{C}}{\mathfrak{Q}}$ of the cosheaf ${\mathfrak{Q}}$ is
defined in this simplest case as the contraherent cosheaf on $U$ corresponding
to the contraadjusted ${\mathcal{O}}(U)$-module ${\mathfrak{Q}}[U]$, that is
$\operatorname{\mathfrak{C}}{\mathfrak{Q}}=\widecheck{{\mathfrak{Q}}[U]}$.
There is a natural morphism of derived contrahereable cosheaves
${\mathfrak{Q}}\longrightarrow\operatorname{\mathfrak{C}}{\mathfrak{Q}}$ on
$U$ (see Lemma 2.2.4). For any affine open subscheme $V\subset U$ there is a
natural morphism of contraherent cosheaves
$\operatorname{\mathfrak{C}}({\mathfrak{Q}}|_{V})\longrightarrow(\operatorname{\mathfrak{C}}{\mathfrak{Q}})|_{V}$
on $V$. Our next goal is to extend this construction to an appropriate class
of cosheaves of ${\mathcal{O}}_{X}$-modules on quasi-compact semi-separated
schemes $X$ (cf. [61, Appendix B]).

Let $X$ be such a scheme with an open covering ${\mathbf{W}}$ and a finite
affine open covering $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ subordinate to
${\mathbf{W}}$. The _contraherator complex_
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
of a ${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{P}}$ on
$X$ is a finite Čech complex of contraherent cosheaves on $X$ of the form

(29) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muj_{1,\dotsc,N}{}_{!}\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{U_{1}\cap\dotsb\cap
U_{N}})\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\dotsb\\\ \textstyle\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}j_{\alpha,\beta}{}_{!}\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{U\alpha\cap
U_{\beta}})\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}j_{\alpha}{}_{!}\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{U_{\alpha}}),$

where $j_{\alpha_{1},\dotsc,\alpha_{k}}$ is the open embedding
$U_{\alpha_{1}}\cap\dotsb\cap U_{\alpha_{k}}\longrightarrow X$ and the
notation $\operatorname{\mathfrak{C}}({\mathfrak{Q}})$ was explained above.
The differentials in this complex are constructed using the adjunction of the
direct and inverse images of contraherent cosheaves and the above morphisms
$\operatorname{\mathfrak{C}}({\mathfrak{Q}}|_{V})\longrightarrow(\operatorname{\mathfrak{C}}{\mathfrak{Q}})|_{V}$.

######  Lemma 3.5.1.

Let ${\mathfrak{P}}$ be a ${\mathbf{W}}$-locally derived contrahereable
cosheaf on a quasi-compact semi-separated scheme $X$. Then the object of the
bounded derived category
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}})$ of the exact
category of contraherent cosheaves on $X$ represented by the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
does not depend on the choice of a finite affine open covering
$\\{U_{\alpha}\\}$ of $X$ subordinate to the covering ${\mathbf{W}}$.

###### Proof.

Let us adjoin another affine open subscheme $V\subset X$, subordinate to
${\mathbf{W}}$, to the covering $\\{U_{\alpha}\\}$. Then the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
embeds into the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V,U_{\alpha}\\},{\mathfrak{P}})$
by a termwise split morphism of complexes with the cokernel isomorphic to the
complex
$k_{!}\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V\cap
U_{\alpha}\\},{\mathfrak{P}}|_{V})\longrightarrow
k_{!}\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{V})$.

The complex of contraherent cosheaves
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V\cap
U_{\alpha}\\},{\mathfrak{P}}|_{V})\longrightarrow\operatorname{\mathfrak{C}}({\mathfrak{P}}|_{V})$
on $V$ corresponds to the acyclic complex of contraadjusted
${\mathcal{O}}(V)$-modules (28) for the covering of an affine open subscheme
$V\subset X$ by the affine open subschemes $V\cap U_{\alpha}$. Hence the
cokernel of the admissible monomorphism of complexes
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})\longrightarrow\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V,U_{\alpha}\\},{\mathfrak{P}})$
is an acyclic complex of contraherent cosheaves on $X$.

Now, given two affine open coverings
$X=\bigcup_{\alpha}U_{\alpha}=\bigcup_{\beta}V_{\beta}$ subordinate to
${\mathbf{W}}$, one compares both complexes
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
and
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{V_{\beta}\\},{\mathfrak{P}})$
with the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha},V_{\beta}\\},{\mathfrak{P}})$
corresponding to the union of the two coverings $\\{U_{\alpha},V_{\beta}\\}$.
∎

A ${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{P}}$ on a
quasi-compact semi-separated scheme $X$ is called _${\mathbf{W}}$ -locally
contrahereable_ if the complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
for some particular (or equivalently, for any) finite affine open covering
$X=\bigcup_{\alpha}U_{\alpha}$ is quasi-isomorphic in
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ to
a ${\mathbf{W}}$-locally contraherent cosheaf on $X$ (viewed as a complex
concentrated in homological degree $0$). The ${\mathbf{W}}$-locally
contraherent cosheaf that appears here is called the (_${\mathbf{W}}$ -local_)
_contraherator_ of ${\mathfrak{P}}$ and denoted by
$\operatorname{\mathfrak{C}}{\mathfrak{P}}$. A derived contrahereable cosheaf
${\mathfrak{P}}$ on $X$ is called _contrahereable_ if it is locally
contrahereable with respect to the covering $\\{X\\}$.

Any derived contrahereable cosheaf ${\mathfrak{Q}}$ on an affine scheme $U$ is
contrahereable, because the contraherator complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U\\},{\mathfrak{Q}})$
is concentrated in homological degree $0$. The contraherator cosheaf
$\operatorname{\mathfrak{C}}{\mathfrak{Q}}$ constructed in this way coincides
with the one defined above specifically in the affine scheme case; so our
notation and terminology is consistent. Any ${\mathbf{W}}$-locally
contraherent cosheaf ${\mathfrak{P}}$ on a quasi-compact semi-separated scheme
$X$ is ${\mathbf{W}}$-locally contrahereable; the corresponding contraherator
complex
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{\mathfrak{P}})$
is the contraherent Čech resolution (27) of a ${\mathbf{W}}$-locally
contraherent cosheaf
${\mathfrak{P}}=\operatorname{\mathfrak{C}}{\mathfrak{P}}$.

The contraherator complex construction
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{-})$
is an exact functor from the exact category of ${\mathbf{W}}$-locally
contrahereable cosheaves to the exact category of finite complexes of
contraherent cosheaves on $X$, or to the bounded derived category
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}})$. The full
subcategory of ${\mathbf{W}}$-locally contrahereable cosheaves in the exact
category of ${\mathbf{W}}$-locally derived contrahereable cosheaves is closed
under extensions and infinite products. Hence it acquires the induced exact
category structure. The contraherator $\operatorname{\mathfrak{C}}$ is an
exact functor from the exact category of ${\mathbf{W}}$-locally contrahereable
cosheaves to that of ${\mathbf{W}}$-locally contraherent ones.

All the above applies to locally cotorsion and locally injective
${\mathbf{W}}$-locally derived contrahereable cosheaves as well. These form
full exact subcategories closed under extensions, infinite products, and
cokernels of admissible monomorphisms in the exact category of
${\mathbf{W}}$-locally derived contrahereable cosheaves. The contraherator
complex construction
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{-})$
takes locally cotorsion (resp., locally injective) ${\mathbf{W}}$-locally
derived contrahereable cosheaves to finite complexes of locally cotorsion
(resp., locally injective) contraherent cosheaves on $X$.

A locally cotorsion (resp., locally injective) ${\mathbf{W}}$-locally derived
contrarehreable cosheaf is called ${\mathbf{W}}$-locally contrahereable if it
is ${\mathbf{W}}$-locally contrahereable as a ${\mathbf{W}}$-locally derived
contrahereable cosheaf. The contraherator functor
$\operatorname{\mathfrak{C}}$ takes locally cotorsion (resp., locally
injective) ${\mathbf{W}}$-locally contrahereable cosheaves to locally
cotorsion (resp., locally injective) ${\mathbf{W}}$-locally contraherent
cosheaves.

Let $f\colon Y\longrightarrow X$ be a $({\mathbf{W}},{\mathbf{T}})$-affine
morphism of schemes. Then the direct image functor $f_{!}$ takes
${\mathbf{T}}$-locally derived contrahereable cosheaves on $Y$ to
${\mathbf{W}}$-locally derived contrahereable cosheaves on $X$. For a
$({\mathbf{W}},{\mathbf{T}})$-affine morphism $f$ of quasi-compact semi-
separated schemes, the functor $f_{!}$ also commutes with the contraherator
complex construction, as there is a natural isomorphism of complexes of
contraherent cosheaves

$f_{!}\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{f^{-1}(U_{\alpha})\\},{\mathfrak{P}})\simeq\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},f_{!}{\mathfrak{P}})$

for any finite affine open covering $U_{\alpha}$ of $X$ subordinate to
${\mathbf{W}}$. It follows that the functor $f_{!}$ takes
${\mathbf{T}}$-locally contrahereable cosheaves to ${\mathbf{W}}$-locally
contrahereable cosheaves and commutes with the functor
$\operatorname{\mathfrak{C}}$.

For any ${\mathbf{W}}$-locally contrahereable cosheaf ${\mathfrak{P}}$ and
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ on a quasi-
compact semi-separated scheme $X$, there is a natural isomorphism of the
groups of morphisms

(30)
$\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathfrak{P}},{\mathfrak{Q}})\simeq\operatorname{Hom}^{X}(\operatorname{\mathfrak{C}}{\mathfrak{P}},{\mathfrak{Q}}).$

In other words, the functor $\operatorname{\mathfrak{C}}$ is left adjoint to
the identity embedding functor of the category of ${\mathbf{W}}$-locally
contrahereable cosheaves into the category of ${\mathbf{W}}$-locally
contraherent ones. Indeed, applying the contraherator complex construction
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},{-})$
to a morphism ${\mathfrak{P}}\longrightarrow{\mathfrak{Q}}$ and passing to the
zero homology, we obtain the corresponding morphism
$\operatorname{\mathfrak{C}}{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}$.
Conversely, any cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ is the
cokernel of the rightmost arrow of the complex

(31) $\textstyle 0\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muj_{1,\dotsc,N}{}_{!}({\mathfrak{P}}|_{U_{1}\cap\dotsb\cap U_{N}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\dotsb\\\
\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha<\beta}j_{\alpha,\beta}{}_{!}({\mathfrak{P}}|_{U\alpha\cap
U_{\beta}})\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\bigoplus_{\alpha}j_{\alpha}{}_{!}({\mathfrak{P}}|_{U_{\alpha}})$

in the additive category of cosheaves of ${\mathcal{O}}_{X}$-modules. A
cosheaf ${\mathfrak{P}}$ satisfying the exactness condition (i∘) for affine
open subschemes $U\subset X$ subordinate to ${\mathbf{W}}$ is also quasi-
isomorphic to the whole complex (31) in the exact category
${\mathcal{O}}_{X}{\operatorname{\mathsf{--cosh}}}_{\mathbf{W}}$. Passing to
the zero homology of the natural morphism between the complexes (31) and (29),
we produce the desired adjunction morphism
${\mathfrak{P}}\longrightarrow\operatorname{\mathfrak{C}}{\mathfrak{P}}$.

For any ${\mathbf{W}}$-locally contrahereable cosheaf ${\mathfrak{P}}$ and any
quasi-coherent sheaf ${\mathcal{M}}$ on $X$ the natural morphism of cosheaves
of ${\mathcal{O}}_{X}$-modules
${\mathfrak{P}}\longrightarrow\operatorname{\mathfrak{C}}{\mathfrak{P}}$
induces an isomorphism of the contratensor products

(32)
${\mathcal{M}}\odot_{X}{\mathfrak{P}}\simeq{\mathcal{M}}\odot_{X}\operatorname{\mathfrak{C}}({\mathfrak{P}}).$

Indeed, for any injective quasi-coherent sheaf ${\mathcal{J}}$ on $X$ one has

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

in view of the isomorphism (20).

###  3.6. $\operatorname{\mathfrak{Cohom}}$ into a locally derived
contrahereable cosheaf

We start with discussing the $\operatorname{\mathfrak{Cohom}}$ from a quasi-
coherent sheaf to a locally contraherent cosheaf.

Let ${\mathbf{W}}$ be an open covering of a scheme $X$. Let ${\mathcal{F}}$ be
a very flat quasi-coherent sheaf on $X$, and let ${\mathfrak{P}}$ be a
${\mathbf{W}}$-locally contraherent cosheaf on $X$. The ${\mathbf{W}}$-locally
contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ on the
scheme $X$ is defined by the rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$. The
contraadjustness and ${\mathbf{W}}$-local contraherence conditions can be
verified in the same way as it was done in Section 2.4.

Similarly, if ${\mathcal{F}}$ is a flat quasi-coherent sheaf and
${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf on $X$, then the locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ is
defined by the same rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$.

Finally, if ${\mathcal{M}}$ is a quasi-coherent sheaf and ${\mathfrak{J}}$ is
a locally injective ${\mathbf{W}}$-locally contraherent cosheaf on $X$, then
the locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ is defined
by the same rule as above. If ${\mathcal{F}}$ is a flat quasi-coherent sheaf
and ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$, then the ${\mathbf{W}}$-locally contraherent
cosheaf $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{J}})$ is
locally injective.

For any two very flat quasi-coherent sheaves ${\mathcal{F}}$ and
${\mathcal{G}}$ on a scheme $X$ and any ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{P}}$ on $X$ there is a natural isomorphism of
${\mathbf{W}}$-locally contraherent cosheaves

(33)
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{G}},\>{\mathfrak{P}})\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{G}},\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})).$

Similarly, for any two flat quasi-coherent sheaves ${\mathcal{F}}$ and
${\mathcal{G}}$ and a locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{P}}$ on $X$ there is a natural isomorphism (33) of locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaves. Finally, for any flat
quasi-coherent sheaf ${\mathcal{F}}$, quasi-coherent sheaf ${\mathcal{M}}$,
and locally injective ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{J}}$ on $X$ there are natural isomorphisms of locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaves

(34)
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}},\>{\mathfrak{J}})\\\
\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{J}}))\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})).$

More generally, let ${\mathcal{F}}$ be a very flat quasi-coherent sheaf on
$X$, and let ${\mathfrak{P}}$ be a ${\mathbf{W}}$-locally derived
contrahereable cosheaf on $X$. The ${\mathbf{W}}$-locally derived
contrahereable cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ on the
scheme $X$ is defined by the rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$. For
any pair of embedded affine open subschemes $V\subset U$ the restriction and
corestriction morphisms ${\mathcal{F}}(U)\longrightarrow{\mathcal{F}}(V)$ and
${\mathfrak{P}}[V]\longrightarrow{\mathfrak{P}}[U]$ induce a morphism of
${\mathcal{O}}_{X}(U)$-modules

$\operatorname{Hom}_{{\mathcal{O}}_{X}(V)}({\mathcal{F}}(V),{\mathfrak{P}}[V])\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{\mathfrak{P}}[U]),$

so our rule defines a covariant functor with ${\mathcal{O}}_{X}$-module
structure on the category of affine open subschemes $U\subset X$ subordinate
to ${\mathbf{W}}$. The contraherence condition clearly holds; and to check the
exactness condition (i∘) of Section 3.5 for this covariant functor, it
suffices to apply the functor
$\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{F}}(U),{-})$ to the exact
sequence of contraadjusted ${\mathcal{O}}_{X}(U)$-modules (28) for the cosheaf
${\mathfrak{P}}$.

Similarly, if ${\mathcal{F}}$ is a flat quasi-coherent sheaf and
${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally derived
contrahereable cosheaf on $X$, then the locally cotorsion
${\mathbf{W}}$-locally derived contrahereable cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})$ is defined
by the same rule
$U\longmapsto\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}\allowbreak({\mathcal{F}}(U),{\mathfrak{P}}[U])$
for any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$.
Finally, if ${\mathcal{M}}$ is a quasi-coherent sheaf and ${\mathfrak{J}}$ is
a locally injective ${\mathbf{W}}$-locally derived contrahereable cosheaf on
$X$, then the locally cotorsion ${\mathbf{W}}$-locally derived contrahereable
cosheaf $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})$ is
defined by the very same rule. If ${\mathcal{F}}$ is a flat quasi-coherent
sheaf and ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
derived contrahereable cosheaf on $X$, then the ${\mathbf{W}}$-locally derived
cotrahereable cosheaf
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{J}})$ is locally
injective.

The associativity isomorphisms (33–34) hold for the
$\operatorname{\mathfrak{Cohom}}$ into ${\mathbf{W}}$-locally derived
contrahereable cosheaves under the assumptions similar to the ones made above
in the locally contraherent case.

###  3.7. Contraherent tensor product

Let ${\mathfrak{P}}$ be a cosheaf of ${\mathcal{O}}_{X}$-modules on a scheme
$X$ and ${\mathcal{M}}$ be a quasi-coherent cosheaf on $X$. Define a covariant
functor with an ${\mathcal{O}}_{X}$-module structure on the category of affine
open subschemes of $X$ by the rule
$U\longmapsto{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U]$.
To a pair of embedded affine open subschemes $V\subset U\subset X$ this
functor assigns the ${\mathcal{O}}_{X}(U)$-module homomorphism

${\mathcal{M}}(V)\otimes_{{\mathcal{O}}_{X}(V)}{\mathfrak{P}}[V]\simeq{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[V]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U].$

Obviously, this functor satisfies the condition (6) of Theorem 2.1.2 (since
the restriction of the cosheaf ${\mathfrak{P}}$ to affine open subschemes of
$X$ does). Hence the functor
$U\longmapsto{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{P}}[U]$
extends uniquely to a cosheaf of ${\mathcal{O}}_{X}$-modules on $X$, which we
will denote by ${\mathcal{M}}\otimes_{X}{\mathfrak{P}}$.

For any quasi-coherent sheaf ${\mathcal{M}}$, cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$, and locally injective
${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{J}}$ on a
scheme $X$ there is a natural isomorphism of abelian groups

(35)
$\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathcal{M}}\otimes_{X}{\mathfrak{P}},\>{\mathfrak{J}})\simeq\operatorname{Hom}^{{\mathcal{O}}_{X}}({\mathfrak{P}},\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})).$

The analogous adjunction isomorphism holds in the other cases mentioned in
Section 3.6 when the functor $\operatorname{\mathfrak{Cohom}}$ from a quasi-
coherent sheaf to a locally derived contrahereable cosheaf is defined. In
other words, the functors
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{-})$ and
${\mathcal{M}}\otimes_{X}{-}$ between subcategories of the category of
cosheaves of ${\mathcal{O}}_{X}$-modules are adjoint “wherever the former
functor is defined”.

For a locally free sheaf of finite rank ${\mathcal{E}}$ and a
${\mathbf{W}}$-locally contraherent (resp., ${\mathbf{W}}$-locally derived
contrahereable) cosheaf ${\mathfrak{P}}$ on a scheme $X$, the cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathcal{E}}\otimes_{X}{\mathfrak{P}}$ is
${\mathbf{W}}$-locally contraherent (resp., ${\mathbf{W}}$-locally derived
contrahereable). There is a natural isomorphism of ${\mathbf{W}}$-locally
contraherent (resp., ${\mathbf{W}}$-locally derived contrahereable) cosheaves
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{{\mathcal{O}}_{X}}({\mathcal{E}},{\mathcal{O}}_{X})\otimes_{X}{\mathfrak{P}}\simeq\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{E}},{\mathfrak{P}})$
on $X$.

The isomorphism
$j_{*}j^{*}({\mathcal{K}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{M}})\simeq
j_{*}j^{*}{\mathcal{K}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{M}}(U)$ (15)
for quasi-coherent sheaves ${\mathcal{K}}$ and ${\mathcal{M}}$ and the
embedding of an affine open subscheme $j\colon U\longrightarrow X$ allows to
construct a natural isomorphism of quasi-coherent sheaves

(36)
$({\mathcal{K}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{M}})\odot_{X}{\mathfrak{P}}\simeq{\mathcal{K}}\odot_{X}({\mathcal{M}}\otimes_{X}{\mathfrak{P}})$

for any quasi-coherent sheaves ${\mathcal{K}}$ and ${\mathcal{M}}$ and a
cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on a semi-separated
scheme $X$.

Let ${\mathbf{W}}$ be an open covering of a scheme $X$. We will call a cosheaf
of ${\mathcal{O}}_{X}$-modules ${\mathfrak{F}}$  _${\mathbf{W}}$ -flat_ if the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{F}}[U]$ is flat for every affine
open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$. A cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathfrak{F}}$ is said to be _flat_ if it is
$\\{X\\}$-flat. Clearly, the direct image of a ${\mathbf{T}}$-flat cosheaf of
${\mathcal{O}}_{Y}$-modules with respect to a flat
$({\mathbf{W}},{\mathbf{T}})$-affine morphism of schemes $f\colon
Y\longrightarrow X$ is ${\mathbf{W}}$-flat.

One can easily see that whenever a cosheaf ${\mathfrak{F}}$ is
${\mathbf{W}}$-flat and satisfies the “exactness condition” (i∘) of Section
3.5 for finite affine open coverings of affine open subschemes $U\subset X$
subordinate to ${\mathbf{W}}$, the cosheaf
${\mathcal{M}}\otimes_{X}{\mathfrak{F}}$ also satisfies the condition (i∘) for
such open affines $U\subset X$. Similarly, whenever a quasi-coherent sheaf
${\mathcal{F}}$ is flat and a cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathfrak{P}}$ satisfies the condition (i∘), so does the cosheaf
${\mathcal{F}}\otimes_{X}{\mathfrak{P}}$.

The full subcategory of ${\mathbf{W}}$-flat ${\mathbf{W}}$-locally
contraherent cosheaves is closed under extensions and kernels of admissible
epimorphisms in the exact category of ${\mathbf{W}}$-locally contraherent
cosheaves $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ on $X$. Hence it
acquires the induced exact category structure, which we will denote by
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{fl}}$. The category
$X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{\mathsf{fl}}$ will be denoted by
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. Similarly, there is the
exact category structure on the category of ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally derived contrahereable cosheaves on $X$ induced from
the exact category of ${\mathbf{W}}$-locally derived contrahereable cosheaves.

A quasi-coherent sheaf ${\mathcal{K}}$ on a scheme $X$ is called _coadjusted_
if the ${\mathcal{O}}_{X}(U)$-module ${\mathcal{K}}(U)$ is coadjusted (see
Section 1.6) for every affine open subscheme $U\subset X$. By Lemma 1.6.9, the
coadjustness of a quasi-coherent sheaf is a local property. By the definition,
if a cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{P}}$ on $X$ satisfies
the contraherence condition (ii) of Section 2.2 or 3.5 and a quasi-coherent
sheaf ${\mathcal{K}}$ on $X$ is coadjusted, then the cosheaf of
${\mathcal{O}}_{X}$-modules ${\mathcal{K}}\otimes_{X}{\mathfrak{P}}$ also
satisfies the condition (ii).

The full subcategory of coadjusted quasi-coherent sheaves is closed under
extensions and the passage to quotient objects in the abelian category of
quasi-coherent sheaves $X{\operatorname{\mathsf{--qcoh}}}$ on $X$. Hence it
acquires the induced exact category structure, which we will denote by
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{coa}}$.

Let ${\mathcal{K}}$ be a coadjusted quasi-coherent sheaf and ${\mathfrak{F}}$
be a ${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent (or more
generally, ${\mathbf{W}}$-locally derived contrahereable) cosheaf on a scheme
$X$. Then the tensor product ${\mathcal{K}}\otimes_{X}{\mathfrak{F}}$
satisfies both conditions (i∘) and (ii) for affine open subschemes $U\subset
X$ subordinate to ${\mathbf{W}}$, i. e., it is ${\mathbf{W}}$-locally derived
contrahereable. (Of course, the cosheaf
${\mathcal{K}}\otimes_{X}{\mathfrak{F}}$ is _not_ in general locally
contraherent, even if the cosheaf ${\mathfrak{F}}$ was ${\mathbf{W}}$-locally
contraherent.)

Assuming the scheme $X$ is quasi-compact and semi-separated, the contraherator
complex construction now allows to assign to this cosheaf of
${\mathcal{O}}_{X}$-modules a complex of contraherent cosheaves
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},\>{\mathcal{K}}\otimes_{X}{\mathfrak{F}})$
on $X$. To a short exact sequence of coadjusted quasi-coherent sheaves or
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent (or
${\mathbf{W}}$-locally derived contrahereable) cosheaves on $X$, the functor
$\operatorname{\mathfrak{C}}_{\text{\smaller\smaller$\scriptstyle\bullet$}}(\\{U_{\alpha}\\},\>{-}\otimes_{X}{-})$
assigns a short exact sequence of complexes of contraherent cosheaves.

By Lemma 3.5.1, the corresponding object of the bounded derived category
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}})$ does not depend
on the choice of a finite affine open covering $\\{U_{\alpha}\\}$. We will
denote it by
${\mathcal{K}}\otimes_{X{\operatorname{\mathrm{-ct}}}}^{\mathbb{L}}{\mathfrak{F}}$
and call the _derived contraherent tensor product_ of a coadjusted quasi-
coherent sheaf ${\mathcal{K}}$ and a ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on a quasi-
compact semi-separated scheme $X$.

When the derived category object
${\mathcal{K}}\otimes_{X{\operatorname{\mathrm{-ct}}}}^{\mathbb{L}}{\mathfrak{F}}$,
viewed as an object of the derived category
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
via the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, turns out to be isomorphic to
an object of the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$, we say that the (underived)
_contraherent tensor product_ of ${\mathcal{K}}$ and ${\mathfrak{F}}$ is
defined, and denote the corresponding object by
${\mathcal{K}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}\mskip
1.5mu\in X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. In other words, for
a coadjusted quasi-coherent sheaf ${\mathcal{K}}$ and a ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on a quasi-
compact semi-separated scheme $X$ one sets
${\mathcal{K}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}=\operatorname{\mathfrak{C}}({\mathcal{K}}\otimes_{X}{\mathfrak{F}})$
whenever the right-hand side is defined (where $\operatorname{\mathfrak{C}}$
denotes the ${\mathbf{W}}$-local contraherator).

Now assume that the scheme $X$ is locally Noetherian. Then, by Corollary
1.6.5(a), a contraherent cosheaf ${\mathfrak{F}}$ on an affine open subscheme
$U\subset X$ is flat if and only if the contraadjusted
${\mathcal{O}}(U)$-module ${\mathfrak{F}}[U]$ is flat. Besides, the full
subcategory of ${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent
cosheaves in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is closed under
infinite products. In addition, any coherent sheaf on $X$ is coadjusted, as is
any injective quasi-coherent sheaf and any quasi-coherent quotient sheaf of an
injective one.

For any injective quasi-coherent sheaf ${\mathcal{J}}$ and any
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally derived contrahereable cosheaf
${\mathfrak{F}}$ on $X$, the tensor product
${\mathcal{J}}\otimes_{X}{\mathfrak{F}}$ is a locally injective
${\mathbf{W}}$-locally derived contrahereable cosheaf on $X$. For any flat
quasi-coherent sheaf ${\mathcal{F}}$ and any locally injective
${\mathbf{W}}$-locally derived contrahereable cosheaf ${\mathfrak{J}}$ on $X$,
the tensor product ${\mathcal{F}}\otimes_{X}{\mathfrak{J}}$ is a locally
injective ${\mathbf{W}}$-locally derived contrahereable cosheaf. These
assertions hold since the tensor product of a flat module and an injective
module over a Noetherian ring is injective.

Let ${\mathcal{M}}$ be a coherent sheaf and ${\mathfrak{F}}$ be a
${\mathbf{W}}$-flat ${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then
the cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathcal{M}}\otimes_{X}{\mathfrak{F}}$ is ${\mathbf{W}}$-locally
contraherent. Indeed, by Corollary 1.6.3(a), the ${\mathcal{O}}_{X}(U)$-module
${\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}(U)$ is
contraadjusted for any affine open subscheme $U\subset X$. For a pair of
embedded affine open subschemes $V\subset U\subset X$ subordinate to the
covering ${\mathbf{W}}$, one has

${\mathcal{M}}(V)\otimes_{{\mathcal{O}}_{X}(V)}{\mathfrak{F}}[V]\simeq{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[V]\\\
\simeq{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),{\mathfrak{F}}[U])\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{X}(V),\>{\mathcal{M}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[U])$

according to Corollary 1.6.3(c). If the scheme $X$ is semi-separated and
Noetherian, the contraherent tensor product
${\mathcal{M}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}$ is
defined and isomorphic to the tensor product
${\mathcal{M}}\otimes_{X}{\mathfrak{F}}$.

Similarly, it follows from Corollary 1.6.4 that for any coherent sheaf
${\mathcal{M}}$ and ${\mathbf{W}}$-flat locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on $X$ the tensor
product ${\mathcal{M}}\otimes_{X}{\mathfrak{F}}$ is a locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on $X$. (See Sections 5.1–5.2 and
Lemma 5.7.2 for further discussion.)

###  3.8. Compatibility of direct and inverse images with the tensor
operations

Let ${\mathbf{W}}$ be an open covering of a scheme $X$ and ${\mathbf{T}}$ be
an open covering of a scheme $Y$. Let $f\colon Y\longrightarrow X$ be a
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism.

Let ${\mathcal{F}}$ be a flat quasi-coherent sheaf and ${\mathfrak{J}}$ be a
locally injective ${\mathbf{W}}$-locally contraherent cosheaf on the scheme
$X$. Then there is a natural isomorphism of locally injective
${\mathbf{T}}$-locally contraherent cosheaves

(37)
$f^{!}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{J}})\simeq\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{F}},f^{!}{\mathfrak{J}})$

on the scheme $Y$.

Assume additionally that $f$ is a flat morphism. Let ${\mathcal{M}}$ be a
quasi-coherent sheaf and ${\mathfrak{J}}$ be a locally injective
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is a natural
isomorphism of locally cotorsion ${\mathbf{T}}$-locally contraherent cosheaves

(38)
$f^{!}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},{\mathfrak{J}})\simeq\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{M}},f^{!}{\mathfrak{J}})$

on $Y$. Analogously, if ${\mathcal{F}}$ is a flat quasi-coherent sheaf and
${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf on $X$, then there is a natural isomorphism of locally cotorsion
${\mathbf{T}}$-locally contraherent cosheaves

(39)
$f^{!}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},{\mathfrak{P}})\simeq\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{F}},f^{!}{\mathfrak{P}})$

on the scheme $Y$.

Assume that, moreover, $f$ is a very flat morphism. Let ${\mathcal{F}}$ be a
very flat quasi-coherent sheaf and ${\mathfrak{P}}$ be a
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is the natural
isomorphism (39) of ${\mathbf{T}}$-locally contraherent cosheaves on $Y$.

Let $f\colon Y\longrightarrow X$ be a $({\mathbf{W}},{\mathbf{T}})$-affine
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism. Let ${\mathcal{N}}$ be a
quasi-coherent cosheaf on $Y$ and ${\mathfrak{J}}$ be a locally injective
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is a natural
isomorphism of locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves

(40)
$\operatorname{\mathfrak{Cohom}}_{X}(f_{*}{\mathcal{N}},{\mathfrak{J}})\simeq
f_{!}\operatorname{\mathfrak{Cohom}}_{Y}({\mathcal{N}},f^{!}{\mathfrak{J}})$

on the scheme $X$. This is one version of the projection formula for the
$\operatorname{\mathfrak{Cohom}}$ from a quasi-coherent sheaf to a
contraherent cosheaf.

Assume additionally that $f$ is a flat morphism. Let ${\mathcal{G}}$ be a flat
quasi-coherent sheaf on $Y$ and ${\mathfrak{P}}$ be a locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is a natural
isomorphism of locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves

(41)
$\operatorname{\mathfrak{Cohom}}_{X}(f_{*}{\mathcal{G}},{\mathfrak{P}})\simeq
f_{!}\operatorname{\mathfrak{Cohom}}_{Y}({\mathcal{G}},f^{!}{\mathfrak{P}})$

on the scheme $X$.

Assume that, moreover, $f$ is a very flat morphism. Let ${\mathcal{G}}$ be a
very flat quasi-coherent sheaf on $Y$ and ${\mathfrak{P}}$ be a
${\mathbf{W}}$-locally contraherent cosheaf on $X$. Then there is the natural
isomorphism (41) of ${\mathbf{W}}$-locally contraherent cosheaves on $X$.

Let $f\colon Y\longrightarrow X$ be a $({\mathbf{W}},{\mathbf{T}})$-affine
morphism. Let ${\mathcal{F}}$ be a very flat quasi-coherent sheaf on $X$ and
${\mathfrak{Q}}$ be a ${\mathbf{T}}$-locally contraherent cosheaf on $Y$. Then
there is a natural isomorphism of ${\mathbf{W}}$-locally contraherent
cosheaves

(42)
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{F}},f_{!}{\mathfrak{Q}})\simeq
f_{!}\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{F}},{\mathfrak{Q}})$

on the scheme $X$. The similar isomorphism of locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaves on $X$ holds for any flat quasi-
coherent sheaf ${\mathcal{F}}$ on $X$ and locally cotorsion
${\mathbf{T}}$-locally contraherent cosheaf ${\mathfrak{Q}}$ on $Y$. This is
another version of the projection formula for
$\operatorname{\mathfrak{Cohom}}$.

Assume additionally that $f$ is a flat morphism. Let ${\mathcal{M}}$ be a
quasi-coherent sheaf on $X$ and ${\mathfrak{I}}$ be a locally injective
${\mathbf{T}}$-locally contraherent cosheaf on $Y$. Then there is a natural
isomorphism of locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves

(43)
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{M}},f_{!}{\mathfrak{I}})\simeq
f_{!}\operatorname{\mathfrak{Cohom}}_{Y}(f^{*}{\mathcal{M}},{\mathfrak{I}})$

on the scheme $X$.

Let $f\colon Y\longrightarrow X$ be either an affine morphism of semi-
separated schemes, or a morphism of quasi-compact semi-separated schemes. Let
${\mathcal{F}}$ be a very flat quasi-coherent sheaf on $X$ and ${\mathcal{Q}}$
be a contraadjusted quasi-coherent sheaf on $Y$. Then there is a natural
isomorphism of contraherent cosheaves

(44)
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},f_{*}{\mathcal{Q}})\simeq
f_{!}\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{F}},{\mathcal{Q}})$

on $X$. Here the quasi-coherent sheaf $f_{*}{\mathcal{Q}}$ on $X$ is
contraadjusted according to Section 2.5 above (if $f$ is affine) or Corollary
4.1.13 below (if $X$ and $Y$ are quasi-compact). The right-hand side is, by
construction, a contraherent cosheaf if the morphism $f$ is affine, and a
cosheaf of ${\mathcal{O}}_{X}$-modules otherwise (see Section 2.3). Both sides
are, in fact, contraherent in the general case, because the isomorphism holds
and the left-hand side is. This is a version of the projection formula for
$\operatorname{\mathfrak{Hom}}$.

Indeed, let $j\colon U\longrightarrow X$ be an embedding of an affine open
subscheme; set $V=U\times_{X}Y$. Let $j^{\prime}\colon V\longrightarrow Y$ and
$f^{\prime}\colon V\longrightarrow U$ be the natural morphisms. Then one has

$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{F}},f_{*}{\mathcal{Q}})[U]\simeq\operatorname{Hom}_{X}(j_{*}j^{*}{\mathcal{F}},f_{*}{\mathcal{Q}})\simeq\operatorname{Hom}_{Y}(f^{*}\\!\mskip
1.5muj_{*}j^{*}{\mathcal{F}},{\mathcal{Q}})\\\
\simeq\operatorname{Hom}_{Y}(j^{\prime}_{*}f^{\prime}{}^{*}\\!\mskip
1.5muj^{*}{\mathcal{F}},{\mathcal{Q}})\simeq\operatorname{Hom}_{Y}(j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{F}},{\mathcal{Q}})\simeq\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{F}},{\mathcal{Q}})[V].$

Here we are using the fact that the direct images of quasi-coherent sheaves
with respect to affine morphisms of schemes commute with the inverse images in
the base change situations. Notice that, when the morphism $f$ is not affine,
neither is the scheme $V$; however, the scheme $V$ is quasi-compact and the
open embedding morphism $j^{\prime}\colon V\longrightarrow Y$ is affine, so
Lemma 2.5.2(a) applies. The similar isomorphism of locally cotorsion
contraherent cosheaves on $X$ holds for any flat quasi-coherent sheaf
${\mathcal{F}}$ on $X$ and any cotorsion quasi-coherent sheaf ${\mathcal{Q}}$
on $Y$.

Now let $f$ be a flat quasi-compact morphism of semi-separated schemes. Let
${\mathcal{M}}$ be a quasi-coherent sheaf on $X$ and ${\mathcal{I}}$ be an
injective quasi-coherent sheaf on $Y$. Then there is a natural isomorphism of
locally cotorsion contraherent cosheaves

(45)
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{M}},f_{*}{\mathcal{I}})\simeq
f_{!}\operatorname{\mathfrak{Hom}}_{Y}(f^{*}{\mathcal{M}},{\mathcal{I}})$

on the scheme $Y$. The proof is similar to the above. For any flat quasi-
compact morphism of quasi-separated schemes $f\colon Y\longrightarrow X$, a
quasi-coherent sheaf ${\mathcal{M}}$ on $X$, and an injective quasi-coherent
sheaf ${\mathcal{I}}$ on $Y$, there is a natural morphism of cosheaves of
${\mathcal{O}}_{X}$-modules from the right-hand side to the left-hand side of
(45); this morphism is an isomorphism whenever the morphism $f$ is also
affine.

Finally, let $f\colon Y\longrightarrow X$ be a quasi-compact open embedding of
quasi-separated schemes. Then the isomorphism (45) holds for any flasque
quasi-coherent sheaf ${\mathcal{M}}$ on $X$ and injective quasi-coherent sheaf
${\mathcal{I}}$ on $Y$. In fact, the direct images of quasi-coherent sheaves
with respect to quasi-compact quasi-separated morphisms commute with the
inverse images with respect to flat morphisms of schemes in the base change
situations, while the restrictions to open subschemes also preserve the
flasqueness; so one can apply Lemma 2.5.2(d).

Let $f\colon Y\longrightarrow X$ be an affine morphism of semi-separated
schemes. Let ${\mathcal{M}}$ be a quasi-coherent sheaf on $X$ and
${\mathfrak{Q}}$ be a cosheaf of ${\mathcal{O}}_{Y}$-modules. Then there is a
natural isomorphism of quasi-coherent sheaves

(46) ${\mathcal{M}}\odot_{X}f_{!}{\mathfrak{Q}}\simeq
f_{*}(f^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{Q}})$

on the scheme $X$. This is a version of the projection formula for the
contratensor product of quasi-coherent sheaves and cosheaves of
${\mathcal{O}}_{X}$-modules.

Indeed, in the notation above, for any affine open subscheme $U\subset X$ we
have

$j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}(f_{!}{\mathfrak{Q}})[U]\simeq
j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{Q}}[V]\\\
\simeq(j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{Y}(V))\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\simeq
j_{*}(j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{Y}(V))\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\\\
\simeq j_{*}f^{\prime}_{*}f^{\prime}{}^{*}\\!\mskip
1.5muj^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\simeq
f_{*}j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\\\
\simeq f_{*}(j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]).$

As it was explained Section 2.6, the contratensor product
$f^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{Q}}$ can be computed as the inductive
limit over the diagram ${\mathbf{D}}$ formed by the affine open subschemes
$V\subset Y$ of the form $V=U\times_{X}Y$, where $U$ are affine open
subschemes in $X$. It remains to use the fact that the direct image of quasi-
coherent sheaves with respect to an affine morphism is an exact functor. The
same isomorphism (46) holds for a flat affine morphism $f$ of quasi-separated
schemes.

Furthermore, there is a natural morphism from the left-hand side to the right-
hand side of (46) for any quasi-compact morphism of quasi-separated schemes
$f\colon Y\longrightarrow X$. It is constructed as the composition

$\textstyle{\mathcal{M}}\odot_{X}f_{!}{\mathfrak{Q}}\mskip 1.5mu=\mskip
1.5mu\varinjlim_{U}j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{Q}}[f^{-1}(U)]\mskip
1.5mu\simeq\mskip
1.5mu\varinjlim_{U}j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}(\varinjlim_{V\subset
f^{-1}(U)}{\mathfrak{Q}}[V])\\\ \textstyle\simeq\mskip
1.5mu\varinjlim_{U}\varinjlim_{f(V)\subset
U}j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{Q}}[V]\mskip
1.5mu\simeq\mskip 1.5mu\varinjlim_{f(V)\subset
U}j_{*}j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{Q}}[V]\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu\\\
\textstyle\varinjlim_{f(V)\subset
U}j_{*}(j^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{Y}(V))\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\mskip
1.5mu\simeq\mskip 1.5mu\varinjlim_{f(V)\subset
U}f_{*}j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\\\
\textstyle\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muf_{*}\varinjlim_{f(V)\subset U}j^{\prime}_{*}j^{\prime}{}^{*}\\!\mskip
1.5muf^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{Q}}[V]\mskip
1.5mu\simeq\mskip 1.5muf_{*}(f^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{Q}}).$

Here the inductive limit is taken firstly over affine open subschemes
$U\subset X$, then over affine open subschemes $V\subset Y$ such that
$f(V)\subset U$, and eventually over pairs of affine open subschemes $U\subset
X$ and $V\subset Y$ such that $f(V)\subset U$. The open embeddings
$U\longrightarrow X$ and $V\longrightarrow Y$ are denoted by $j$ and
$j^{\prime}$, while the morphism $V\longrightarrow U$ is denoted by
$f^{\prime}$. The final isomorphism holds, since the contratensor product
$f^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{Q}}$ can be computed over the diagram
${\mathbf{D}}$ formed by all the pairs of affine open subschemes $(U,V)$ such
that $f(V)\subset U$.

In particular, the isomorphism

(47) ${\mathcal{M}}\odot_{X}h_{!}{\mathfrak{F}}\simeq
h_{*}(h^{*}{\mathcal{M}}\odot_{Y}{\mathfrak{F}})$

holds for any open embedding $h\colon Y\longrightarrow X$ of an affine scheme
$Y$ into a quasi-separated scheme $X$, any quasi-coherent sheaf
${\mathcal{M}}$ on $X$, and any flat cosheaf of ${\mathcal{O}}_{X}$-modules
${\mathfrak{F}}$ on $Y$. Indeed, in this case one has

$\textstyle\varinjlim_{h(V)\subset
U}h_{*}j^{\prime}_{*}j^{\prime}{}^{*}h^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{F}}[V]\simeq\varinjlim_{V}h_{*}j^{\prime}_{*}j^{\prime}{}^{*}h^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}_{Y}(V)}{\mathfrak{F}}[V]\\\
\simeq
h_{*}h^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}(Y)}{\mathfrak{F}}[Y]\simeq
h_{*}(h^{*}{\mathcal{M}}\otimes_{{\mathcal{O}}(Y)}{\mathfrak{F}}[Y]),$

where the $\varinjlim_{V}$ is taken over all affine open subschemes $V\subset
Y$, which is clearly equivalent to considering $V=Y$ only.

Let $f\colon Y\longrightarrow X$ be an affine morphism of schemes. Then for
any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ and any cosheaf of
${\mathcal{O}}_{Y}$-modules ${\mathfrak{Q}}$ there is a natural isomorphism of
cosheaves of ${\mathcal{O}}_{X}$-modules

(48)
$f_{!}(f^{*}{\mathcal{M}}\otimes_{Y}{\mathfrak{Q}})\simeq{\mathcal{M}}\otimes_{X}f_{!}{\mathfrak{Q}}.$

This is a version of the projection formula for the tensor product of quasi-
coherent sheaves and cosheaves of ${\mathcal{O}}_{X}$-modules.

##  4\. Quasi-compact Semi-separated Schemes

###  4.1. Contraadjusted and cotorsion quasi-coherent sheaves

Recall that the definition of a very flat quasi-coherent sheaf was given in
Section 1.7 and the definition of a contraadjusted quasi-coherent sheaf in
Section 2.5 (cf. Remark 2.5.4).

In particular, a quasi-coherent sheaf ${\mathcal{P}}$ over an affine scheme
$U$ is very flat (respectively, contraadjusted) if and only if the
${\mathcal{O}}(U)$-module ${\mathcal{P}}(U)$ is very flat (respectively,
contraadjusted). The class of very flat quasi-coherent sheaves is preserved by
inverse images with respect to arbitrary morphisms of schemes and direct
images with respect to very flat affine morphisms (which includes affine open
embeddings). The class of contraadjusted quasi-coherent sheaves is preserved
by direct images with respect to affine morphisms of schemes.

The class of very flat quasi-coherent sheaves on any scheme $X$ is closed
under the passage to the kernel of a surjective morphism. Both the full
subcategories of very flat and contraadjusted quasi-coherent sheaves are
closed under extensions in the abelian category of quasi-coherent sheaves.
Hence they acquire the induced exact category structures, which we denote by
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$ and
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$, respectively.

Let us introduce one bit of categorical terminology. Given an exact category
${\mathsf{E}}$ and a class of objects ${\mathsf{C}}\subset{\mathsf{E}}$, we
say that an object $X\in{\mathsf{E}}$ is a _finitely iterated extension_ of
objects from ${\mathsf{C}}$ if there exists a nonnegative integer $N$ and a
sequence of admissible monomorphisms $0=X_{0}\longrightarrow
X_{1}\longrightarrow\dotsb\longrightarrow X_{N-1}\longrightarrow X_{N}=X$ in
${\mathsf{E}}$ such that the cokernels of all the morphisms
$X_{i-1}\longrightarrow X_{i}$ belong to ${\mathsf{C}}$ (cf. Section 1.1).

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

######  Lemma 4.1.1.

Any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ can be included in a short
exact sequence
$0\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{M}}\longrightarrow
0$, where ${\mathcal{F}}$ is a very flat quasi-coherent sheaf and
${\mathcal{P}}$ is a finitely iterated extension of the direct images of
contraadjusted quasi-coherent sheaves from affine open subschemes in $X$.

###### Proof.

The proof is based on the construction from [53, Section A.1] and Theorem
1.1.1(b). We argue by a kind of induction in the number of affine open
subschemes covering $X$. Assume that for some open subscheme $h\colon
V\longrightarrow X$ there is a short exact sequence
$0\longrightarrow{\mathcal{Q}}\longrightarrow{\mathcal{K}}\longrightarrow{\mathcal{M}}\longrightarrow
0$ of quasi-coherent sheaves on $X$ such that the restriction
$h^{*}{\mathcal{K}}$ of the sheaf ${\mathcal{K}}$ to the open subscheme $V$ is
very flat, while the sheaf ${\mathcal{Q}}$ is a finitely iterated extension of
the direct images of contraadjusted quasi-coherent sheaves from affine open
subschemes in $X$. Let $j\colon U\longrightarrow X$ be an affine open
subscheme; we will construct a short exact sequence
$0\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{M}}\longrightarrow
0$ having the same properties with respect to the open subscheme $U\cup
V\subset X$.

Pick an short exact sequence
$0\longrightarrow{\mathcal{R}}\longrightarrow{\mathcal{G}}\longrightarrow
j^{*}{\mathcal{K}}\longrightarrow 0$ of quasi-coherent sheaves on the affine
scheme $U$ such that the sheaf ${\mathcal{G}}$ is very flat and the sheaf
${\mathcal{R}}$ is contraadjusted. Consider its direct image $0\longrightarrow
j_{*}{\mathcal{R}}\longrightarrow j_{*}{\mathcal{G}}\longrightarrow
j_{*}j^{*}{\mathcal{K}}\longrightarrow 0$ with respect to the affine open
embedding $j$, and take its pull-back with respect to the adjunction morphism
${\mathcal{K}}\longrightarrow j_{*}j^{*}{\mathcal{K}}$. Let ${\mathcal{F}}$
denote the middle term of the resulting short exact sequence of quasi-coherent
sheaves on $X$.

By Lemma 1.2.6(a), it suffices to show that the restrictions of
${\mathcal{F}}$ to $U$ and $V$ are very flat in order to conclude that the
restriction to $U\cup V$ is. We have $j^{*}{\mathcal{F}}\simeq{\mathcal{G}}$,
which is very flat by the construction. On the other hand, the sheaf
$j^{*}{\mathcal{K}}$ is very flat over $V\cap U$, hence so is the sheaf
${\mathcal{R}}$, as the kernel of a surjective map
${\mathcal{G}}\longrightarrow j^{*}{\mathcal{K}}$. The embedding $U\cap
V\longrightarrow V$ is a very flat affine morphism, so the sheaf
$j_{*}{\mathcal{R}}$ is very flat over $V$. Now it is clear from the short
exact sequence $0\longrightarrow
j_{*}{\mathcal{R}}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{K}}\longrightarrow
0$ that the sheaf ${\mathcal{F}}$ is very flat over $V$.

Finally, the kernel ${\mathcal{P}}$ of the composition of surjective morphisms
${\mathcal{F}}\longrightarrow{\mathcal{K}}\longrightarrow{\mathcal{M}}$ is an
extension of the sheaves ${\mathcal{Q}}$ and $j_{*}{\mathcal{R}}$, the latter
of which is the direct image of a contraadjustested quasi-coherent sheaf from
an affine open subscheme of $X$, and the former is a finitely iterated
extension of such. ∎

######  Corollary 4.1.2.

(a) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is contraadjusted if and
only if the functor $\operatorname{Hom}_{X}({-},{\mathcal{P}})$ takes short
exact sequences of very flat quasi-coherent sheaves on $X$ to short exact
sequences of abelian groups.

(b) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is contraadjusted if and
only if $\operatorname{Ext}_{X}^{>0}({\mathcal{F}},{\mathcal{P}})=0$ for any
very flat quasi-coherent sheaf ${\mathcal{F}}$ on $X$.

(c) The class of contraadjusted quasi-coherent sheaves on $X$ is closed with
respect to the passage to the cokernels of injective morphisms.

###### Proof.

While the condition in part (a) is _a priori_ weaker and the condition in part
(b) is _a priori_ stronger than our definition of a contraherent cosheaf
${\mathcal{P}}$ by the condition
$\operatorname{Ext}_{X}^{1}({\mathcal{F}},{\mathcal{P}})=0$ for any very flat
${\mathcal{F}}$, all the three conditions are easily seen to be equivalent
provided that every quasi-coherent sheaf on $X$ is the quotient sheaf of a
very flat one. That much we know from Lemma 4.1.1. The condition in (b)
clearly has the property (c). ∎

######  Lemma 4.1.3.

Any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ can be included in a short
exact sequence
$0\longrightarrow{\mathcal{M}}\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow
0$, where ${\mathcal{F}}$ is a very flat quasi-coherent sheaf and
${\mathcal{P}}$ is a finitely iterated extension of the direct images of
contraadjusted quasi-coherent sheaves from affine open subschemes in $X$.

###### Proof.

Any quasi-coherent sheaf on a quasi-compact quasi-separated scheme can be
embedded into a finite direct sum of direct images of injective quasi-coherent
sheaves from affine open subschemes constituting a finite covering. So an
embedding ${\mathcal{M}}\longrightarrow{\mathcal{J}}$ of a sheaf
${\mathcal{M}}$ into a sheaf ${\mathcal{J}}$ with the desired (an even
stronger) properties exists, and it remains to make sure that the quotient
sheaf has the desired properties.

One does this using Lemma 4.1.1 and (the dual version of) the procedure used
in the second half of the proof of Theorem 10 in [17] (see the proof of Lemma
1.1.3). Present the quotient sheaf ${\mathcal{J}}/{\mathcal{M}}$ as the
quotient sheaf of a very flat sheaf ${\mathcal{F}}$ by a subsheaf
${\mathcal{Q}}$ representable as a finitely iterated extension of the desired
kind. Set ${\mathcal{P}}$ to be the fibered product of ${\mathcal{J}}$ and
${\mathcal{F}}$ over ${\mathcal{J}}/{\mathcal{M}}$; then ${\mathcal{P}}$ is an
extension of ${\mathcal{J}}$ and ${\mathcal{Q}}$, and there is a natural
injective morphism ${\mathcal{M}}\longrightarrow{\mathcal{P}}$ with the
cokernel ${\mathcal{F}}$. ∎

######  Corollary 4.1.4.

(a) Any quasi-coherent sheaf on $X$ admits a surjective map onto it from a
very flat quasi-coherent sheaf such that the kernel is contraadjusted.

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

(c) A quasi-coherent sheaf on $X$ is contraadjusted if and only if it is a
direct summand of a finitely iterated extension of the direct images of
contraadjusted quasi-coherent sheaves from affine open subschemes of $X$.

###### Proof.

Parts (a) and (b) follow from Lemmas 4.1.1 and 4.1.3, respectively. The proof
of part (c) uses (the dual version of) the argument from the proof of
Corollary 1.1.4. Given a contraadjusted quasi-coherent sheaf ${\mathcal{P}}$,
use Lemma 4.1.3 to embed it into a finitely iterated extension ${\mathcal{Q}}$
of the desired kind in such a way that the cokernel ${\mathcal{F}}$ is a very
flat quasi-coherent sheaf. Since
$\operatorname{Ext}^{1}_{X}({\mathcal{F}},{\mathcal{P}})=0$ by the definition,
we can conclude that ${\mathcal{P}}$ is a direct summand of ${\mathcal{Q}}$. ∎

######  Lemma 4.1.5.

A quasi-coherent sheaf on $X$ is very flat and contraadjusted if and only if
it is a direct summand of a finite direct sum of the direct images of very
flat contraadjusted quasi-coherent sheaves from affine open subschemes of $X$.

###### Proof.

The “if” assertion is clear. To prove “only if”, notice that the very flat
contraadjusted quasi-coherent sheaves are the injective objects of the exact
category of very flat quasi-coherent sheaves (cf. Section 1.4). So it remains
to show that there are enough injectives of the kind described in the
formulation of Lemma in the exact category
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}}$.

Indeed, let ${\mathcal{F}}$ be a very flat quasi-coherent sheaf on $X$ and
$X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering. Denote by
$j_{\alpha}$ the identity open embeddings $U_{\alpha}\longrightarrow X$. For
each $\alpha$, pick an injective morphism
$j_{\alpha}^{*}{\mathcal{F}}\longrightarrow{\mathcal{G}}_{\alpha}$ from a very
flat quasi-coherent sheaf $j_{\alpha}^{*}{\mathcal{F}}$ to a very flat
contraadjusted quasi-coherent sheaf ${\mathcal{G}}_{\alpha}$ on $U_{\alpha}$
such that the cokernel
${\mathcal{G}}_{\alpha}/j_{\alpha}^{*}{\mathcal{F}}_{\alpha}$ is a very flat.
Then $\bigoplus_{\alpha}j_{\alpha}{}_{*}{\mathcal{G}}_{\alpha}$ is a very flat
contraadjusted quasi-coherent sheaf on $X$ and the cokernel of the natural
morphism
${\mathcal{F}}\longrightarrow\bigoplus_{\alpha}j_{\alpha}{}_{*}{\mathcal{G}}_{\alpha}$
is very flat (since its restriction to each $U_{\alpha}$ is). ∎

######  Lemma 4.1.6.

A quasi-coherent sheaf on $X$ is flat and contraadjusted if and only if it is
a direct summand of a finitely iterated extension of the direct images of flat
contraadjusted quasi-coherent sheaves from affine open subschemes of $X$.

###### Proof.

Given a flat quasi-coherent sheaf ${\mathcal{E}}$, we apply the constructions
of Lemmas 4.1.1 and 4.1.3 in order to obtain a short exact sequence of quasi-
coherent sheaves
$0\longrightarrow{\mathcal{E}}\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow
0$ with a very flat quasi-coherent sheaf ${\mathcal{F}}$. One can verify step
by step that the whole construction is performed entirely inside the exact
category of flat quasi-coherent sheaves on $X$, so the quasi-coherent sheaf
${\mathcal{P}}$ it produces is a finitely iterated extension of the direct
images of flat contraadjusted quasi-coherent sheaves from affine open
subschemes of $X$. Now if the sheaf ${\mathcal{E}}$ was also contraadjusted,
then the short exact sequence splits by Corollary 4.1.2(b), providing the
desired result. ∎

The following corollary provides equivalent definitions of contraadjusted and
very flat quasi-coherent sheaves on a quasi-compact semi-separated scheme
resembling the corresponding definitions for modules over a ring in Section
1.1.

######  Corollary 4.1.7.

(a) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is contraadjusted if and
only if
$\operatorname{Ext}_{X}^{>0}(j_{*}j^{*}{\mathcal{O}}_{X},{\mathcal{P}})=0$ for
any affine open embedding of schemes $j\colon Y\longrightarrow X$.

(b) A quasi-coherent sheaf ${\mathcal{F}}$ on $X$ is very flat if and only if
$\operatorname{Ext}_{X}^{1}({\mathcal{F}},{\mathcal{P}})=0$ for any
contraadjusted quasi-coherent sheaf ${\mathcal{P}}$ on $X$.

###### Proof.

Part (a): the “only if” assertion follows from Corollary 4.1.2(b). To prove
“if”, notice that any very flat sheaf ${\mathcal{F}}$ on $X$ has a finite
right Čech resolution by finite direct sums of sheaves of the form
$j_{*}j^{*}{\mathcal{F}}$, where $j\colon U\longrightarrow X$ are embeddings
of affine open subschemes. Hence the condition
$\operatorname{Ext}_{X}^{>0}(j_{*}j^{*}{\mathcal{F}},{\mathcal{P}})=0$ for all
such $j$ implies $\operatorname{Ext}_{X}^{>0}({\mathcal{F}},{\mathcal{P}})=0$.

Furthermore, a very flat quasi-coherent sheaf $j^{*}{\mathcal{F}}$ on $U$ is a
direct summand of a transfinitely iterated extension of the direct images of
the structure sheaves of principal affine open subschemes $V\subset U$ (by
Corollary 1.1.4). Since the direct images with respect to affine morphisms
preserve transfinitely iterated extensions, it remains to use the quasi-
coherent sheaf version of the result that
$\operatorname{Ext}^{1}$-orthogonality is preserved by transfinitely iterated
extensions in the first argument [17, Lemma 1].

Part (b): “only if” holds by the definition of contraadjusted sheaves, and
“if” can be deduced from Corollary 4.1.4(a) by an argument similar to the
proof of Corollary 1.1.4 (and dual to that of Corollary 4.1.4(c)). ∎

Now we proceed to formulate the analogues of the above assertions for
cotorsion quasi-coherent sheaves. The definition of these was given in Section
2.5. The class of cotorsion quasi-coherent sheaves is closed under extensions
in the abelian category of quasi-coherent sheaves on a scheme and under the
direct images with respect to affine morphisms of schemes. We denote the
induced exact category structure on the category of of cotorsion quasi-
coherent sheaves on a scheme $X$ by
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$.

As above, in the sequel $X$ denotes a quasi-compact semi-separated scheme.

######  Lemma 4.1.8.

Any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ can be included in a short
exact sequence
$0\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow{\mathcal{M}}\longrightarrow
0$, where ${\mathcal{F}}$ is a flat quasi-coherent sheaf and ${\mathcal{P}}$
is a finitely iterated extension of the direct images of cotorsion quasi-
coherent sheaves from affine open subschemes in $X$.

###### Proof.

Similar to that of Lemma 4.1.1, except that Theorem 1.3.1(b) is being used in
place of Theorem 1.1.1(b). ∎

######  Corollary 4.1.9.

(a) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is cotorsion if and only if
the functor $\operatorname{Hom}_{X}({-},{\mathcal{P}})$ takes short exact
sequences of flat quasi-coherent sheaves on $X$ to short exact sequences of
abelian groups.

(b) A quasi-coherent sheaf ${\mathcal{P}}$ on $X$ is cotorsion if and only if
$\operatorname{Ext}_{X}^{>0}({\mathcal{F}},{\mathcal{P}})=0$ for any flat
quasi-coherent sheaf ${\mathcal{F}}$ on $X$.

(c) The class of cotorsion quasi-coherent sheaves on $X$ is closed with
respect to the passage to the cokernels of injective morphisms.

###### Proof.

Similar to that of Corollary 4.1.2. ∎

######  Lemma 4.1.10.

Any quasi-coherent sheaf ${\mathcal{M}}$ on $X$ can be included in a short
exact sequence
$0\longrightarrow{\mathcal{M}}\longrightarrow{\mathcal{P}}\longrightarrow{\mathcal{F}}\longrightarrow
0$, where ${\mathcal{F}}$ is a flat quasi-coherent sheaf and ${\mathcal{P}}$
is a finitely iterated extension of the direct images of cotorsion quasi-
coherent sheaves from affine open subschemes in $X$.

###### Proof.

Similar to that of Lemma 4.1.3. ∎

######  Corollary 4.1.11.

(a) Any quasi-coherent sheaf on $X$ admits a surjective map onto it from a
flat quasi-coherent sheaf such that the kernel is cotorsion.

(b) Any quasi-coherent sheaf on $X$ can be embedded into a cotorsion quasi-
coherent sheaf in such a way that the cokernel is flat.

(c) A quasi-coherent sheaf on $X$ is cotorsion if and only if it is a direct
summand of a finitely iterated extension of the direct images of cotorsion
quasi-coherent sheaves from affine open subschemes of $X$.

###### Proof.

Similar to that of Corollary 4.1.4. ∎

######  Lemma 4.1.12.

A quasi-coherent sheaf on $X$ is flat and cotorsion if and only if it is a
direct summand of a finite direct sum of the direct images of flat cotorsion
quasi-coherent sheaves from affine open subschemes of $X$.

###### Proof.

Similar to that of Lemma 4.1.5. ∎

The following result shows that contraadjusted (and in particular, cotorsion)
quasi-coherent sheaves are adjusted to direct images with respect to nonaffine
morphisms of quasi-compact semi-separated schemes (cf. Corollary 4.5.3 below).

######  Corollary 4.1.13.

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

(a) the functor $f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ takes the full exact subcategory
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
Y{\operatorname{\mathsf{--qcoh}}}$ into the full exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\subset
X{\operatorname{\mathsf{--qcoh}}}$, and induces an exact functor between these
exact categories;

(b) the functor $f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ takes the full exact subcategory
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\subset
Y{\operatorname{\mathsf{--qcoh}}}$ into the full exact subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\subset
X{\operatorname{\mathsf{--qcoh}}}$, and induces an exact functor between these
exact categories.

###### Proof.

For any affine morphism $g\colon V\longrightarrow Y$ into a quasi-compact
semi-separated scheme $Y$, the inverse image functor $g^{*}$ takes quasi-
coherent sheaves that can be represented as finitely iterated extensions of
the direct images of quasi-coherent sheaves from affine open subschemes in $Y$
to quasi-coherent sheaves of the similar type on $V$. This follows easily from
the fact that direct images of quasi-coherent sheaves with respect to affine
morphisms of schemes commute with inverse images in the base change
situations. In particular, it follows from Corollary 4.1.4(c) that the functor
$g^{*}$ takes contraadjusted quasi-coherent sheaves on $Y$ to quasi-coherent
sheaves that are direct summands of finitely iterated extensions of the direct
images of quasi-coherent sheaves from affine open subschemes $W\subset V$.

The quasi-coherent sheaves on $V$ that can be represented as such iterated
extensions form a full exact subcategory in the abelian category of quasi-
coherent sheaves. The functor of global sections $\Gamma(V,{-})$ is exact on
this exact category. Indeed, there is a natural isomorphism of the Ext groups
$\operatorname{Ext}_{V}^{*}({\mathcal{F}},h_{*}{\mathcal{G}})\simeq\operatorname{Ext}_{W}^{*}(h^{*}{\mathcal{F}},{\mathcal{G}})$
for any quasi-coherent sheaves ${\mathcal{F}}$ on $V$ and ${\mathcal{G}}$ on
$W$, and a flat affine morphism $h\colon W\longrightarrow V$. Applying this
isomorphism in the case when $h$ is the embedding of an affine open subscheme
and ${\mathcal{F}}={\mathcal{O}}_{V}$, one concludes that
$\operatorname{Ext}_{V}^{>0}({\mathcal{O}}_{V},{\mathcal{G}})=0$ for all
quasi-coherent sheaves ${\mathcal{G}}$ from the exact category in question.

Specializing to the case of the open subschemes $V=U\times_{X}Y\subset Y$,
where $U$ are affine open subschemes in $X$, we deduce the assertion that the
functor $f_{*}\colon
Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ is exact. It remains to recall that the
direct images of contraadjusted quasi-coherent sheaves with respect to affine
morphisms of schemes are contraadjusted in order to show that $f_{*}$ takes
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$ to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}$. Since the direct images of
cotorsion quasi-coherent sheaves with respect to affine morphisms of schemes
are cotorsion, it similarly follows that $f_{*}$ takes
$Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ to
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$. ∎

###  4.2. Colocally projective contraherent cosheaves

Let $X$ be a scheme and ${\mathbf{W}}$ be its open covering. A
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$ is called
_colocally projective_ if for any short exact sequence
$0\longrightarrow{\mathfrak{I}}\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{K}}\longrightarrow
0$ of locally injective ${\mathbf{W}}$-locally contraherent cosheaves on $X$
the short sequence of abelian groups
$0\longrightarrow\operatorname{Hom}^{X}({\mathfrak{P}},{\mathfrak{I}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{P}},{\mathfrak{J}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{P}},{\mathfrak{K}})\longrightarrow
0$ is exact.

Obviously, the class of colocally projective ${\mathbf{W}}$-locally
contraherent cosheaves on $X$ is closed under direct summands. It follows from
the adjunction isomorphism (23) of Section 3.3 that the functor of direct
image of ${\mathbf{T}}$-locally contraherent cosheaves $f_{!}$ with respect to
any $({\mathbf{W}},{\mathbf{T}})$-affine
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism of schemes $f\colon
Y\longrightarrow X$ takes colocally projective ${\mathbf{T}}$-locally
contraherent cosheaves on $Y$ to colocally projective ${\mathbf{W}}$-locally
contraherent cosheaves on $X$. It is also clear that _any_ contraherent
cosheaf on an affine scheme $U$ with the covering $\\{U\\}$ is colocally
projective.

######  Lemma 4.2.1.

On any scheme $X$ with an open covering ${\mathbf{W}}$, any coflasque
contraherent cosheaf is colocally projective.

###### Proof.

We will prove a somewhat stronger assertion: any short exact sequence
$0\longrightarrow{\mathfrak{I}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$ in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ with ${\mathfrak{F}}\in
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$ and ${\mathfrak{I}}\in
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ splits. It will
follow easily that the functor $\operatorname{Hom}^{X}({\mathfrak{F}},{-})$
takes any short exact sequence in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ to a short
exact sequence of abelian groups.

We proceed by applying Zorn’s lemma to the partially ordered set of sections
$\phi_{Y}\colon{\mathfrak{F}}|_{Y}\longrightarrow{\mathfrak{Q}}|_{Y}$ of the
morphism of cosheaves ${\mathfrak{Q}}\longrightarrow{\mathfrak{F}}$ defined
over open subsets $Y\subset X$. Since it suffices to define a morphism of
cosheaves on the modules of cosections over affine open subschemes, which are
quasi-compact, a compatible system of sections $\phi_{Y_{i}}$ defined over a
linearly ordered family of open subsets $Y_{i}\subset X$ extends uniquely to a
section over the union $\bigcup_{i}Y_{i}$.

Now let $\phi_{Y}$ be a section over $Y$ and $U\subset X$ be an affine open
subscheme subordinate to ${\mathbf{W}}$. Set $V=Y\cap U$; by assumption, the
${\mathcal{O}}(U)$-module homomorphism
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{F}}[U]$ is injective and the
${\mathcal{O}}(U)$-module ${\mathfrak{I}}[U]$ is injective. The short exact
sequence of ${\mathcal{O}}(U)$-modules
$0\longrightarrow{\mathfrak{I}}[U]\longrightarrow{\mathfrak{Q}}[U]\longrightarrow{\mathfrak{F}}[U]\longrightarrow
0$ splits, and the difference between two such splittings
${\mathfrak{F}}[U]\birarrow{\mathfrak{Q}}[U]$ is an arbitrary
${\mathcal{O}}(U)$-module morphism
${\mathfrak{F}}[U]\longrightarrow{\mathfrak{I}}[U]$. The composition
${\mathfrak{F}}[V]\longrightarrow{\mathfrak{Q}}[U]$ of the morphism
$\phi_{Y}[V]\colon{\mathfrak{F}}[V]\longrightarrow{\mathfrak{Q}}[V]$ and the
corestriction morphism ${\mathfrak{Q}}[V]\longrightarrow{\mathfrak{Q}}[U]$ can
therefore be extended to an ${\mathcal{O}}(U)$-linear section
${\mathfrak{F}}[U]\longrightarrow{\mathfrak{Q}}[U]$ of the surjection
${\mathfrak{Q}}[U]\longrightarrow{\mathfrak{F}}[U]$.

We have constructed a morphism of contraherent cosheaves
$\phi_{U}\colon{\mathfrak{F}}|_{U}\longrightarrow{\mathfrak{Q}}|_{U}$ whose
restriction to $V$ coincides with the restriction of the morphism
$\phi_{Y}\colon{\mathfrak{F}}|_{Y}\longrightarrow{\mathfrak{Q}}|_{Y}$. Set
$Z=Y\cup U$; the pair of morphisms of cosheaves $\phi_{Y}$ and $\phi_{U}$
extends uniquely to a morphism of cosheaves
$\phi_{Z}\colon{\mathfrak{F}}|_{Z}\longrightarrow{\mathfrak{Q}}|_{Z}$. Since
the morphisms $\phi_{Y}$ and $\phi_{U}$ were some sections of of the
surjection ${\mathfrak{Q}}\longrightarrow{\mathfrak{F}}$ over $Y$ and $U$, the
morphism $\phi_{Z}$ is a section of this surjection over $Z$. ∎

Generally speaking, according to the above definition the colocal projectivity
property of a locally contraherent cosheaf ${\mathfrak{P}}$ on a scheme $X$
may depend not only on the cosheaf ${\mathfrak{P}}$ itself, but also on the
covering ${\mathbf{W}}$. No such dependence occurs on quasi-compact semi-
separated schemes. Indeed, we will see below in this section that on such a
scheme any colocally projective ${\mathbf{W}}$-locally contraherent cosheaf is
(globally) contraherent. Moreover, the class of colocally projective
${\mathbf{W}}$-locally contraherent cosheaves coincides with the class of
colocally projective contraherent cosheaves and does not depend on the
covering ${\mathbf{W}}$.

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

######  Lemma 4.2.2.

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

(a) any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on $X$
can be included in an exact triple
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{P}}\longrightarrow
0$, where ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{P}}$ is a finitely iterated
extension of the direct images of contraherent cosheaves from the affine open
subschemes $U_{\alpha}\subset X$;

(b) any locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{M}}$ on $X$ can be included in an exact triple
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{P}}\longrightarrow
0$, where ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{P}}$ is a finitely iterated
extension of the direct images of locally cotorsion contraherent cosheaves
from the affine open subschemes $U_{\alpha}\subset X$.

###### Proof.

The argument is a dual version of the proofs of Lemmas 4.1.1 and 4.1.8. Let us
prove part (a); the proof of part (b) is completely analogous.

Arguing by induction in $1\le\beta\le N$, we consider the open subscheme
$V=\bigcup_{\alpha<\beta}U_{\alpha}$ with the induced covering
${\mathbf{W}}|_{V}=\\{V\cap W\mid W\in{\mathbf{W}}\\}$ and the identity
embedding $h\colon V\longrightarrow X$. Assume that we have constructed an
exact triple
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{Q}}\longrightarrow
0$ of ${\mathbf{W}}$-locally contraherent cosheaves on $X$ such that the
restriction $h^{!}{\mathfrak{K}}$ of the ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{K}}$ to the open subscheme $V\subset X$ is locally
injective, while the cosheaf ${\mathfrak{Q}}$ on $X$ is a finitely iterated
extension of the direct images of contraherent cosheaves from the affine open
subschemes $U_{\alpha}\subset X$, $\alpha<\beta$. When $\beta=1$, it suffices
to take ${\mathfrak{K}}={\mathfrak{M}}$ and ${\mathfrak{Q}}=0$ for the
induction base. Set $U=U_{\beta}$ and denote by $j\colon U\longrightarrow X$
the identity open embedding.

Let $0\longrightarrow
j^{!}{\mathfrak{K}}\longrightarrow{\mathfrak{I}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ be an exact triple of contraherent cosheaves on the affine scheme $U$ such
that the contraherent cosheaf ${\mathfrak{I}}$ is (locally) injective.
Consider its direct image $0\longrightarrow
j_{!}j^{!}{\mathfrak{K}}\longrightarrow j_{!}{\mathfrak{I}}\longrightarrow
j_{!}{\mathfrak{R}}\longrightarrow 0$ with respect to the affine open
embedding $j$, and take its push-forward with respect to the adjunction
morphism $j_{!}j^{!}{\mathfrak{K}}\longrightarrow{\mathfrak{K}}$. Let us show
that in the resulting exact triple
$0\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{J}}\longrightarrow
j_{!}{\mathfrak{R}}\longrightarrow 0$ the ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{J}}$ is locally injective in the restriction to $U\cup V$.
By Lemma 1.3.6(b), it suffices to show that the restrictions of
${\mathfrak{J}}$ to $U$ and $V$ are locally injective.

Indeed, in the restriction to $U$ we have $j^{!}j_{!}j^{!}{\mathfrak{K}}\simeq
j^{!}{\mathfrak{K}}$, hence $j^{!}{\mathfrak{J}}\simeq
j^{!}j_{!}{\mathfrak{I}}\simeq{\mathfrak{I}}$ is a (locally) injective
contraherent cosheaf. On the other hand, if $j^{\prime}\colon U\cap
V\longrightarrow V$ and $h^{\prime}\colon U\cap V\longrightarrow U$ denote the
embeddings of $U\cap V$, then $h^{!}j_{!}{\mathfrak{R}}\simeq
j^{\prime}_{!}h^{\prime}{}^{!}{\mathfrak{R}}$ (as explained in the end of
Section 3.3). Notice that the contraherent cosheaf
$h^{\prime}{}^{!}j^{!}{\mathfrak{K}}\simeq
j^{\prime}{}^{!}h^{!}{\mathfrak{K}}$ is locally injective, hence the
contraherent cosheaf $h^{\prime}{}^{!}{\mathfrak{R}}$ is locally injective as
the cokernel of the admissible monomorphism of locally injective contraherent
cosheaves $h^{\prime}{}^{!}j^{!}{\mathfrak{K}}\longrightarrow
h^{\prime}{}^{!}{\mathfrak{I}}$. Since the local injectivity of
${\mathbf{T}}$-locally contraherent cosheaves is preserved by the direct
images with respect to flat $({\mathbf{W}},{\mathbf{T}})$-affine morphisms,
the contraherent cosheaf $j^{\prime}_{!}h^{\prime!}{\mathfrak{R}}$ is locally
injective, too. Now in the exact triple $0\longrightarrow
h^{!}{\mathfrak{K}}\longrightarrow h^{!}{\mathfrak{J}}\longrightarrow
h^{!}j_{!}{\mathfrak{R}}\longrightarrow 0$ of ${\mathbf{W}}|_{V}$-locally
contraherent cosheaves on $V$ the middle term is locally injective, because so
are the other two terms.

Finally, the composition of admissible monomorphisms of ${\mathbf{W}}$-locally
contraherent cosheaves
${\mathfrak{M}}\longrightarrow{\mathfrak{K}}\longrightarrow{\mathfrak{J}}$ on
$X$ is an admissible monomorphism with the cokernel isomorphic to an extension
of the contraherent cosheaves $j_{!}{\mathfrak{R}}$ and ${\mathfrak{Q}}$,
hence also a finitely iterated extension of the direct images of contraherent
cosheaves from the affine open subschemes $U_{\alpha}\subset X$,
$\alpha\le\beta$. The induction step is finished, and the whole lemma is
proven. ∎

We denote by $\operatorname{Ext}^{X,*}({-},{-})$ the $\operatorname{Ext}$
groups in the exact category of ${\mathbf{W}}$-locally contraherent cosheaves
on $X$. Notice that these do not depend on the covering ${\mathbf{W}}$ and
coincide with the $\operatorname{Ext}$ groups in the whole category of locally
contraherent cosheaves $X{\operatorname{\mathsf{--lcth}}}$. Indeed, the full
exact subcategory $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is closed
under extensions and the passage to kernels of admissible epimorphisms in
$X{\operatorname{\mathsf{--lcth}}}$ (see Section 3.2), and for any object
${\mathfrak{P}}\in X{\operatorname{\mathsf{--lcth}}}$ there exists an
admissible epimorphism onto ${\mathfrak{P}}$ from an object of
$X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see the resolution (27) in
Section 3.3).

For the same reasons (up to duality), the $\operatorname{Ext}$ groups computed
in the exact subcategories of locally cotorsion and locally injective
${\mathbf{W}}$-locally contraherent cosheaves
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ agree with
those in $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (and also in
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$). Indeed, the full exact
subcategories $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$
and $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ are closed
under extensions and the passage to cokernels of admissible monomorphisms in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see Section 3.1), and we
have just constructed in Lemma 4.2.2 an admissible monomorphism from any
${\mathbf{W}}$-locally contraherent cosheaf to a locally injective one. We
refer to Sections A.2–A.3 for further details.

######  Corollary 4.2.3.

(a) A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$ is
colocally projective if and only if
$\operatorname{Ext}^{X,1}({\mathfrak{P}},{\mathfrak{J}})=0$ and if and only if
$\operatorname{Ext}^{X,>0}({\mathfrak{P}},{\mathfrak{J}})=0$ for all locally
injective ${\mathbf{W}}$-locally contraherent cosheaves ${\mathfrak{J}}$ on
$X$.

(b) The class of colocally projective ${\mathbf{W}}$-locally contraherent
cosheaves on $X$ is closed under extensions and the passage to kernels of
admissible epimorphisms in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.

###### Proof.

Part (a) follows from the existence of an admissible monomorphism from any
${\mathbf{W}}$-locally contraherent cosheaf on $X$ into a locally injective
${\mathbf{W}}$-locally contraherent cosheaf (a weak form of Lemma 4.2.2(a)).
Part (b) follows from part (a). ∎

######  Lemma 4.2.4.

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

(a) any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on $X$
can be included in an exact triple
$0\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$, where ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{P}}$ is a finitely iterated
extension of the direct images of contraherent cosheaves from the affine open
subschemes $U_{\alpha}\subset X$;

(b) any locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{M}}$ on $X$ can be included in an exact triple
$0\longrightarrow{\mathfrak{J}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$, where ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{P}}$ is a finitely iterated
extension of the direct images of locally cotorsion contraherent cosheaves
from the affine open subschemes $U_{\alpha}\subset X$.

###### Proof.

There is an admissible epimorphism
$\bigoplus_{\alpha}j_{\alpha}{}_{!}j_{\alpha}^{!}{\mathfrak{M}}\longrightarrow{\mathfrak{M}}$
(see (27) for the notation and explanation) onto any ${\mathbf{W}}$-locally
contraherent cosheaf ${\mathfrak{M}}$ from a finite direct sum of the direct
images of contraherent cosheaves from the affine open subschemes $U_{\alpha}$.
When ${\mathfrak{M}}$ is a locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf, this is an admissible epimorphism in the category of
locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves, and
$j_{\alpha}^{!}{\mathfrak{M}}$ are locally cotorsion contraherent cosheaves on
$U_{\alpha}$.

Given that, the desired exact triples in Lemma can be obtained from those of
Lemma 4.2.2 by the construction from the second half of the proof of Theorem
10 in [17] (see the proof of Lemma 1.1.3; cf. the proofs of Lemmas 4.1.3 and
4.1.10). ∎

######  Corollary 4.2.5.

(a) For any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$ there exists an admissible monomorphism from ${\mathfrak{M}}$ into a
locally injective ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{J}}$
on $X$ such that the cokernel ${\mathfrak{P}}$ is a colocally projective
${\mathbf{W}}$-locally contraherent cosheaf.

(b) For any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$ there exists an admissible epimorphism onto ${\mathfrak{M}}$ from a
colocally projective ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{P}}$ on $X$ such that the kernel ${\mathfrak{J}}$ is a locally
injective ${\mathbf{W}}$-locally contraherent cosheaf.

(c) 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 colocally projective if and only if it is (a contraherent
cosheaf and) a direct summand of a finitely iterated extension of the direct
images of contraherent cosheaves from the affine open subschemes
$U_{\alpha}\subset X$.

###### Proof.

The “if” assertion in part (c) follows from Corollary 4.2.3(b) together with
our preliminary remars in the beginning of this section. This having been
shown, part (a) follows from Lemma 4.2.2(a) and part (b) from Lemma 4.2.4(a).

The “only if” assertion in (c) follows from Corollary 4.2.3(a) and Lemma
4.2.4(a) by the argument from the proof of Corollary 1.1.4 (cf. Corollaries
4.1.4(c) and 4.1.11(c)). Notice that the functors of direct image with respect
to the open embeddings $U_{\alpha}\longrightarrow X$ take contraherent
cosheaves to contraherent cosheaves, and the full subcategory of contraherent
cosheaves $X{\operatorname{\mathsf{--ctrh}}}\subset
X{\operatorname{\mathsf{--lcth}}}$ is closed under extensions. ∎

By a colocally projective locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf we will mean a ${\mathbf{W}}$-locally contraherent
cosheaf that is simultaneously colocally projective and locally cotorsion.

######  Corollary 4.2.6.

(a) For any locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{M}}$ on $X$ there exists an admissible monomorphism from
${\mathfrak{M}}$ into a locally injective ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{J}}$ on $X$ such that the cokernel ${\mathfrak{P}}$ is a
colocally projective locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf.

(b) For any locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{M}}$ on $X$ there exists an admissible epimorphism onto
${\mathfrak{M}}$ from a colocally projective locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$ on $X$ such that
the kernel ${\mathfrak{J}}$ is a locally injective ${\mathbf{W}}$-locally
contraherent cosheaf.

(c) 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 colocally projective if and only if it is (a
contraherent cosheaf and) a direct summand of a finitely iterated extension of
the direct images of locally cotorsion contraherent cosheaves from the affine
open subschemes $U_{\alpha}\subset X$.

###### Proof.

Same as Corollary 4.2.5, except that parts (b) of Lemmas 4.2.2 and 4.2.4 need
to be used. Parts (a-b) can be also easily deduced from Corollary 4.2.5(a-b).
∎

######  Corollary 4.2.7.

The full subcategory of colocally projective ${\mathbf{W}}$-locally
contraherent cosheaves in the exact category of all locally contraherent
cosheaves on $X$ does not depend on the choice of the open covering
${\mathbf{W}}$.

###### Proof.

Given two open coverings ${\mathbf{W}}^{\prime}$ and
${\mathbf{W}}^{\prime\prime}$ of the scheme $X$, pick a finite affine open
covering $X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ subordinate to both
${\mathbf{W}}^{\prime}$ and ${\mathbf{W}}^{\prime\prime}$, and apply Corollary
4.2.5(c). ∎

As a full subcategory closed under extensions and kernels of admissible
epimorphisms in $X{\operatorname{\mathsf{--ctrh}}}$, the category of colocally
projective contraherent cosheaves on $X$ acquires the induced exact category
structure. We denote this exact category by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$. The (similarly
constructed) exact category of colocally projective locally cotorsion
contraherent cosheaves on $X$ is denoted by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lct}}$.

The full subcategory of contraherent sheaves that are simultaneously colocally
projective and locally injective will be denoted by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$. Clearly,
any extension of two objects from this subcategory is trivial in
$X{\operatorname{\mathsf{--ctrh}}}$, so the category of colocally projective
locally injective contraherent cosheaves is naturally viewed as an additive
category endowed with the trivial exact category structure.

It follows from Corollary 4.2.5(a-b) that the additive category
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ is
simultaneously the category of projective objects in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ and the
category of injective objects in
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$, and that it has enough of
both such projectives and injectives.

######  Corollary 4.2.8.

Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering. Then a
contraherent cosheaf on $X$ is colocally projective and locally injective if
and only if it is isomorphic to a direct summand of a finite direct sum of the
direct images of (locally) injective contraherent cosheaves from the open
embeddings $U_{\alpha}\longrightarrow X$.

###### Proof.

For any locally injective ${\mathbf{W}}$-locally contraherent cosheaf
${\mathfrak{J}}$ on $X$, the map
$\bigoplus_{\alpha}j_{\alpha}{}_{!}j_{\alpha}^{!}{\mathfrak{J}}\longrightarrow{\mathfrak{J}}$
is an admissible epimorphism in the category of locally injective
${\mathbf{W}}$-locally contraherent cosheaves. Now if ${\mathfrak{J}}$ is also
colocally projective, then the extension splits. ∎

######  Corollary 4.2.9.

The three full subcategories of colocally projective cosheaves
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$,
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}}$, and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}$ are closed
with respect to infinite products in the category
$X{\operatorname{\mathsf{--ctrh}}}$.

###### Proof.

The assertions are easily deduced from the descriptions of the full
subcategories of colocally projective cosheaves given in Corollaries 4.2.5(c),
4.2.6(c), and 4.2.8 together with the fact that the functor of direct image of
contraherent cosheaves with respect to an affine morphism of schemes preserves
infinite products. ∎

######  Corollary 4.2.10.

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

(a) the functor of inverse image of locally injective locally contraherent
cosheaves $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lin}}$ takes the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}$ into
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}$;

(b) assuming that the morphism $f$ is also flat, the functor of inverse image
of locally cotorsion locally contraherent cosheaves $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ takes the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}}$ into
$Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{clp}}$;

(c) assuming that the morphism $f$ is also very flat, the functor of inverse
image of locally contraherent cosheaves $f^{!}\colon
X{\operatorname{\mathsf{--lcth}}}\longrightarrow
Y{\operatorname{\mathsf{--lcth}}}$ takes the full subcategory
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ into
$Y{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$.

###### Proof.

Parts (a-c) follow from Corollaries 4.2.5(c), 4.2.6(c), and 4.2.8,
respectively, together with the base change results from the second half of
Section 3.3. ∎

###  4.3. Colocally flat contraherent cosheaves

Let $X$ be a scheme and ${\mathbf{W}}$ be its open covering. A
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on $X$ is called
_colocally flat_ if for any short exact sequence
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{Q}}\longrightarrow{\mathfrak{R}}\longrightarrow
0$ of locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaves on $X$
the short sequence of abelian groups
$0\longrightarrow\operatorname{Hom}^{X}({\mathfrak{F}},{\mathfrak{P}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{F}},{\mathfrak{Q}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{F}},{\mathfrak{R}})\longrightarrow
0$ is exact.

Let us issue a _warning_ that our terminology is misleading: the colocal
flatness is, by the definition, a stronger condition that the colocal
projectivity. It follows from the adjunction isomorphism (24) that the functor
of direct image of ${\mathbf{T}}$-locally contraherent cosheaves $f_{!}$ with
respect a flat $({\mathbf{W}},{\mathbf{T}})$-affine
$({\mathbf{W}},{\mathbf{T}})$-coaffine morphism of schemes $f\colon
Y\longrightarrow X$ takes colocally flat ${\mathbf{T}}$-locally contraherent
cosheaves on $Y$ to colocally flat ${\mathbf{W}}$-locally contraherent
cosheaves on $X$. Clearly, a contraherent cosheaf ${\mathfrak{F}}$ on an
affine scheme $U$ with the covering $\\{U\\}$ is colocally flat whenever the
contraadjusted ${\mathcal{O}}(U)$-module ${\mathfrak{F}}[U]$ is flat; the
converse assertion can be deduced from Theorem 1.3.1(b) (cf. Corollary
4.3.4(c) below).

Let $X$ be a quasi-compact semi-separated scheme. It follows from the results
of Section 4.2 that any colocally flat ${\mathbf{W}}$-locally contraherent
cosheaf on $X$ is contraherent. We will see below in this section that the
class of colocally flat ${\mathbf{W}}$-locally contraherent cosheaves on $X$
coincides with the class of colocally flat contraherent cosheaves and does not
depend on the covering ${\mathbf{W}}$.

######  Lemma 4.3.1.

Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then any ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{M}}$ on $X$ can be included in an exact triple
$0\longrightarrow{\mathfrak{M}}\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow
0$, where ${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{F}}$ is a finitely iterated
extension of the direct images of contraherent cosheaves on $U_{\alpha}$
corresponding to flat contraadjusted ${\mathcal{O}}(U_{\alpha})$-modules.

###### Proof.

Similar to the proof of Lemma 4.2.2, except that Theorem 1.3.1(a) needs to be
used to resolve a contraherent cosheaf on an affine open subscheme $U\subset
X$ (cf. the proof of Lemma 4.1.1). Besides, one has to use Lemma 1.3.6(a) and
the fact that the class of locally cotorsion contraherent cosheaves is
preserved by direct images with respect to affine morphisms. ∎

######  Corollary 4.3.2.

(a) A ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on $X$ is
colocally flat if and only if
$\operatorname{Ext}^{X,1}({\mathfrak{F}},{\mathfrak{P}})=0$ and if and only if
$\operatorname{Ext}^{X,>0}({\mathfrak{F}},{\mathfrak{P}})=0$ for all locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaves ${\mathfrak{P}}$ on
$X$.

(b) The class of colocally flat ${\mathbf{W}}$-locally contraherent cosheaves
on $X$ is closed under extensions and the passage to kernels of admissible
epimorphisms in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.

###### Proof.

Similar to the proof of Corollary 4.2.3. ∎

######  Lemma 4.3.3.

Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering
subordinate to ${\mathbf{W}}$. Then any ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{M}}$ on $X$ can be included in an exact triple
$0\longrightarrow{\mathfrak{P}}\longrightarrow{\mathfrak{F}}\longrightarrow{\mathfrak{M}}\longrightarrow
0$, where ${\mathfrak{P}}$ is a locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaf on $X$ and ${\mathfrak{F}}$ is a finitely iterated
extension of the direct images of contraherent cosheaves on $U_{\alpha}$
corresponding to flat contraadjusted ${\mathcal{O}}(U_{\alpha})$-modules.

###### Proof.

The proof is similar to that of Lemma 4.2.4 and based on Lemma 4.3.1. The key
is to show that there is an admissible epimorphism in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ onto any
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ from a finitely
iterated extension (in fact, even a finite direct sum) of the direct images of
contraherent cosheaves on $U_{\alpha}$ corresponding to flat contraadjusted
${\mathcal{O}}(U_{\alpha})$-modules.

Here it suffices to pick admissible epimorphisms from such contraherent
cosheaves ${\mathfrak{F}}_{\alpha}$ on $U_{\alpha}$ onto the restrictions
$j_{\alpha}^{!}{\mathfrak{M}}$ of ${\mathfrak{M}}$ to $U_{\alpha}$ and
consider the corresponding morphism
$\bigoplus_{\alpha}j_{\alpha}{}_{!}{\mathfrak{F}}_{\alpha}\longrightarrow{\mathfrak{M}}$
of ${\mathbf{W}}$-locally contraherent cosheaves on $X$. To check that this is
an admissible epimorphism, one can, e. g., notice that it is so in the
restriction to each $U_{\alpha}$ and recall that being an admissible
epimorphism of ${\mathbf{W}}$-locally contraherent cosheaves is a local
property (see Section 3.2). ∎

######  Corollary 4.3.4.

(a) For any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$ there exists an admissible monomorphism from ${\mathfrak{M}}$ into a
locally cotorsion ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{P}}$
on $X$ such that the cokernel ${\mathfrak{F}}$ is a colocally flat
${\mathbf{W}}$-locally contraherent cosheaf.

(b) For any ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on
$X$ there exists an admissible epimorphism onto ${\mathfrak{M}}$ from a
colocally flat ${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{F}}$ on
$X$ such that the kernel ${\mathfrak{P}}$ is a locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf.

(c) 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 colocally flat if and only if it is a direct summand of a
finitely iterated extension of the direct images of contraherent cosheaves on
$U_{\alpha}$ corresponding to flat contraadjusted
${\mathcal{O}}(U_{\alpha})$-modules.

###### Proof.

Similar to the proof of Corollary 4.2.5 and based on Lemmas 4.3.1, 4.3.3 and
Corollary 4.3.2. ∎

######  Corollary 4.3.5.

The full subcategory of colocally flat ${\mathbf{W}}$-locally contraherent
cosheaves in the exact category $X{\operatorname{\mathsf{--lcth}}}$ does not
depend on the open covering ${\mathbf{W}}$.

###### Proof.

Similar to the proof of Corollary 4.2.7 and based on Corollary 4.3.4(c). ∎

As a full subcategory closed under extensions and kernels of admissible
epimorphisms in $X{\operatorname{\mathsf{--ctrh}}}$, the category of colocally
flat contraherent cosheaves on $X$ acquires the induced exact category
structure, which we denote by
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$.