X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ is an equivalence of
triangulated categories;

(c) for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, or ${\mathsf{abs}}$, the triangulated functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}$ is an equivalence of triangulated
categories.

###### Proof.

Part (a) is provided by Lemma 5.2.5(b) together with the dual version of
Proposition A.5.6. Parts (b-c) follow from part (a) and Corollary 5.3.2(b-c).
Alternatively, in the case of a semi-separated Noetherian scheme $X$ of finite
Krull dimension, the assertions (b-c) can be obtained directly from Lemma
5.2.1(a). ∎

The following corollary is another restricted version of Theorem 4.6.6; it is
to be compared with Corollary 4.6.10.

######  Corollary 5.4.5.

Let $X$ be a semi-separated Noetherian scheme of finite Krull dimension. Then
for any symbol $\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$,
${\mathsf{abs}}-$, ${\mathsf{co}}$, or ${\mathsf{abs}}$ there is a natural
equivalence of triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$.

###### Proof.

Assuming $\star\neq{\mathsf{co}}$, by Lemma 5.4.1(d) together with the dual
version of Proposition A.5.6 the triangulated functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is an equivalence of categories. In view of Corollary 5.4.2, the same
assertion holds for $\star={\mathsf{co}}$. Hence it remains to recall that the
equivalence of categories from Lemma 4.6.7 identifies
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ with
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ (see the
proof of Corollary 4.6.10(c)). ∎

######  Corollary 5.4.6.

Let $f\colon Y\longrightarrow X$ be a morphism of finite flat dimension
between semi-separated Noetherian schemes. Then for any symbol
$\star={\mathsf{b}}$, $+$, $-$, $\empt$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$,
${\mathsf{co}}$, or ${\mathsf{abs}}$ the equivalences of triangulated
categories
${\mathsf{D}}^{\star}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}^{\star}(Y\allowbreak{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
and
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
from Corollary 5.4.5 transform the right derived functor ${\mathbb{R}}f_{*}$
(65) into the left derived functor ${\mathbb{L}}f_{!}$ (69).

###### Proof.

Can be either deduced from Corollary 4.11.6(c), or proven directly in the
similar way using Lemma 4.11.3(d). ∎

Let $X$ be a locally Noetherian scheme with an open covering ${\mathbf{W}}$.
As in Section 4.11, we denote by
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}$ the
full subcategory of objects of injective dimension $\le d$ in
$X{\operatorname{\mathsf{--qcoh}}}$ and by
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$
the full subcategory of objects of projective dimension $\le d$ in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. For a
Noetherian scheme $X$ of finite Krull dimension, let
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$
denote the full subcategory of objects of projective dimension $\le d$ in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. We set
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{{\operatorname{\mathsf{fpd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\},{{\operatorname{\mathsf{fpd--}}}d}}^{\mathsf{lct}}$
and
$X{\operatorname{\mathsf{--ctrh}}}_{{\operatorname{\mathsf{fpd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\},{{\operatorname{\mathsf{fpd--}}}d}}$.
Clearly, the projective dimension of an object of
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ or
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ does not change when the open
covering ${\mathbf{W}}$ is replaced by its refinement.

The full subcategory
$X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}d}\subset
X{\operatorname{\mathsf{--qcoh}}}$ is closed under extensions, cokernels of
admissible monomoprhisms, and infinite direct sums. The full subcategory
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is closed under
extensions, kernels of admissible epimorphisms, and infinite products (see
Corollary 5.1.5).

######  Corollary 5.4.7.

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

(b) Let $X$ be a locally Noetherian scheme. Then the natural triangulated
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$
are equivalences of categories.

(c) Let $X$ be a Noetherian scheme of finite Krull dimension. Then the natural
triangulated functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}})$
are equivalences of categories.

###### Proof.

Part (a) follows from Corollary 4.11.1(a) and [50, Remark 2.1], while parts
(b-c) follow from Proposition A.5.6 and the same Remark. ∎

A cosheaf of ${\mathcal{O}}_{X}$-modules ${\mathfrak{G}}$ on a scheme $X$ is
said to have _${\mathbf{W}}$ -flat dimension not exceeding $d$_ if the flat
dimension of the ${\mathcal{O}}_{X}(U)$-module ${\mathfrak{G}}[U]$ does not
exceed $d$ for any affine open subscheme $U\subset X$ subordinate to
${\mathbf{W}}$. The flat dimension of a cosheaf of ${\mathcal{O}}_{X}$-modules
is defined as its $\\{X\\}$-flat dimension. ${\mathbf{W}}$-locally
contraherent cosheaves of ${\mathbf{W}}$-flat dimension not exceeding $d$ on a
locally Noetherian scheme $X$ form a full subcategory
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ closed under extensions,
kernels of admissible epimorphisms, and infinite products. We set
$X{\operatorname{\mathsf{--ctrh}}}^{{\operatorname{\mathsf{ffd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{\\{X\\}}^{{\operatorname{\mathsf{ffd--}}}d}$.

The flat dimension of a contraherent cosheaf ${\mathfrak{G}}$ on an affine
Noetherian scheme $U$ is equal to the flat dimension of the
${\mathcal{O}}_{X}(U)$-module ${\mathfrak{G}}[U]$ (see Section 3.7). Over a
semi-separated Noetherian scheme $X$, a ${\mathbf{W}}$-locally contraherent
cosheaf has ${\mathbf{W}}$-flat dimension $\le d$ if and only if it admits a
left resolution of length $\le d$ by ${\mathbf{W}}$-flat
${\mathbf{W}}$-locally contraherent cosheaves (see Corollary 4.4.5(a)).

Hence it follows from Corollary 5.2.2(b) (applied to affine open subschemes
$U\subset X$) that the ${\mathbf{W}}$-flat dimension of a
${\mathbf{W}}$-locally contraherent cosheaf on a locally Noetherian scheme $X$
of finite Krull dimension does not change when the covering ${\mathbf{W}}$ is
replaced by its refinement. According to part (a) of the same Corollary, on a
semi-separated Noetherian scheme of finite Krull dimension the
${\mathbf{W}}$-flat dimension of a ${\mathbf{W}}$-locally contraherent cosheaf
is equal to its colocally flat dimension; so
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}=X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{clfd--}}}d}}$.
By Corollary 5.1.4, the ${\mathbf{W}}$-flat dimension of a locally cotorsion
${\mathbf{W}}$-locally contraherent cosheaf on a locally Noetherian scheme $X$
coincides with its projective dimension in
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ (and also does
not depend on ${\mathbf{W}}$). So one has
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}=X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}$.

######  Lemma 5.4.8.

Let $X$ be a Noetherian scheme of finite Krull dimension $D$. Then a
${\mathbf{W}}$-locally contraherent cosheaf on $X$ has finite projective
dimension in the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ if and only if it has finite
${\mathbf{W}}$-flat dimension. More precisely, the inclusions of full
subcategories
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}\subset
X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}(d+D)}}$
hold in the category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.

###### Proof.

The inclusion
$X{\operatorname{\mathsf{--lcth}}}_{{\mathbf{W}},\,{{\operatorname{\mathsf{fpd--}}}d}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}$
holds due to Corollary 5.2.4. Conversely, by the same Corollary any
${\mathbf{W}}$-locally contraherent cosheaf ${\mathfrak{M}}$ on $X$ has a left
resolution by flat contraherent cosheaves, so the ${\mathbf{W}}$-flat
dimension of ${\mathfrak{M}}$ is equal to its left homological dimension with
respect to $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see Corollary 5.2.2(b)). It
remains to apply the last assertion of Corollary 5.2.6(b). ∎

######  Corollary 5.4.9.

For any Noetherian scheme $X$ of finite Krull dimension and any (finite)
integer $d\ge 0$, the natural triangulated functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})$,
$\mathsf{Hot}^{\pm}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{{\mathsf{abs}}\pm}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})\longrightarrow{\mathsf{D}}^{\pm}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})$,
and
$\mathsf{Hot}^{\mathsf{b}}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d})$
are equivalences of triangulated categories.

###### Proof.

It is clear from Lemma 5.4.8 that the homological dimension of the exact
category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}d}$
is finite, so it remains to apply [50, Remark 2.1] (to obtain the equivalences
between various derived categories of this exact category) and Proposition
A.5.6 (to identify the absolute derived categories with the homotopy
categories of projective objects). Alternatively, one can use [51, Theorem 3.6
and Remark 3.6]. ∎

The following theorem is the main result of this section.

######  Theorem 5.4.10.

(a) For any locally Noetherian scheme $X$, the natural functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of triangulated categories.

(b) For any locally Noetherian scheme $X$, the natural functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}^{\mathsf{lct}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
is an equivalence of triangulated categories.

(c) For any semi-separated Noetherian scheme $X$, the natural functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of triangulated categories.

(d) For any Noetherian scheme $X$ of finite Krull dimension, the natural
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalences of triangulated categories.

###### Proof.

Part (a) is a standard result (see, e. g., [53, Lemma 1.7(b)]) which is a
particular case of [51, Theorem 3.7 and Remark 3.7] and can be also obtained
from the dual version of Proposition A.3.1(b). The key observation is that
there are enough injectives in $X{\operatorname{\mathsf{--qcoh}}}$ and the
full subcategory $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ they form
is closed under infinite direct sums. Similarly, part (b) can be obtained
either from Proposition A.3.1(b), or from the dual version of [51, Theorem 3.7
and Remark 3.7] (see also [51, Section 3.8]). In any case the argument is
based on Theorem 5.1.1(a) and Corollary 5.1.5. Part (c) holds by Proposition
A.3.1(b) together with Lemma 4.3.3, 4.4.1(a), or 4.4.3(a).

Finally, in part (d) the functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X\allowbreak{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$
are equivalences of categories by Corollary 5.4.9, and the functor
${\mathsf{D}}^{\mathsf{ctr}}(X\allowbreak{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of categories by Proposition A.3.1(b) together with
Corollary 5.2.4. A direct proof of the equivalence
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is also possible; it proceeds along the following lines.

One has to use the more advanced features of the results of [51, Sections
3.7–3.8] involving the full generality of the conditions (${*}$)–(${*}{*}$).
Alternatively, one can apply the more general Corollary A.6.2. Specifically,
let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering; then it
follows from Corollary 5.2.4(b) that an infinite product of projective
contraherent cosheaves on $X$ is a direct summand of a direct sum over
$\alpha$ of the direct images of contraherent cosheaves on $U_{\alpha}$
correspoding to infinite products of very flat contraadjusted
${\mathcal{O}}(U_{\alpha})$-modules.

Infinite products of such modules may not be very flat, but they are certainly
flat and contraadjusted. By the last assertion of Corollary 5.2.6(b), one can
conclude that the projective dimensions of infinite products of projective
objects in $X{\operatorname{\mathsf{--ctrh}}}$ do not exceed the Krull
dimension $D$ of the scheme $X$. So the contraherent cosheaf analogue of the
condition (${*}{*}$) holds for $X{\operatorname{\mathsf{--ctrh}}}$, or in
other words, the assumption of Corollary A.6.2 is satisfied by the pair of
exact categories $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}\subset
X{\operatorname{\mathsf{--ctrh}}}$. ∎

The following corollary is to be compared with Corollaries 5.2.8(b) and
5.3.3(b).

######  Corollary 5.4.11.

For any locally Noetherian scheme $X$ with an open covering ${\mathbf{W}}$,
the triangulated functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is an
equivalence of triangulated categories.

###### Proof.

Follows from Theorem 5.4.10(b) applied to the coverings $\\{X\\}$ and
${\mathbf{W}}$ of the scheme $X$. Alternatively, one can apply directly
Proposition A.3.1(b) together with Theorem 5.1.1(a). ∎

###  5.5. Co-contra correspondence over a regular scheme

Let $X$ be a regular semi-separated Noetherian scheme of finite Krull
dimension.

######  Theorem 5.5.1.

(a) The triangulated functor
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ is an equivalence of triangulated
categories.

(b) The triangulated functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
induced by the embedding of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$ is an equivalence of triangulated
categories.

(c) There is a natural equivalence of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
provided by the derived functors
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ and
${\mathcal{O}}_{X}\odot_{X}^{\mathbb{L}}{-}$.

###### Proof.

Part (a) actually holds for any symbol $\star\neq{\mathsf{ctr}}$ in the upper
indices of the derived category signs, and is a particular case of Corollary
4.9.1(a). Indeed, one has
$X{\operatorname{\mathsf{--qcoh}}}=X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{ffd--}}}d}$
provided that $d$ is greater or equal to the Krull dimension of $X$.
Similarly, part (b) actually holds for any symbol $\star\neq{\mathsf{co}}$ in
the upper indices, and is a particular case of Corollary 4.9.1(c). Indeed, one
has
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}=X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{flid--}}}d}$
provided that $d$ is greater or equal to the Krull dimension of $X$.

To prove part (c), notice that all the triangulated functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
and
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalences of categories by Corollary 4.9.5 (since one also has
$X{\operatorname{\mathsf{--qcoh}}}=X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fvfd--}}}d}$
provided that $d$ is greater or equal to the Krull dimension of $X$). So it
remains to apply Theorem 4.6.6. ∎

###  5.6. Co-contra correspondence over a Gorenstein scheme

Let $X$ be a Gorenstein semi-separated Noetherian scheme of finite Krull
dimension. We will use the following formulation of the Gorenstein condition:
for any affine open subscheme $U\subset X$, the classes of
${\mathcal{O}}_{X}(U)$-modules of finite flat dimension, of finite projective
dimension, and of finite injective dimension coincide.

Notice that neither of these dimensions can exceed the Krull dimension $D$ of
the scheme $X$. Accordingly, the class of ${\mathcal{O}}_{X}(U)$-modules
defined by the above finite homological dimension conditions is closed under
both infinite direct sums and infinite products. It is also closed under
extensions and the passages to the cokernels of embeddings and the kernels of
surjections.

Moreover, since the injectivity of a quasi-coherent sheaf on a Noetherian
scheme is a local property, the full subcategories of quasi-coherent sheaves
of finite flat dimension and of finite injective dimension coincide in
$X{\operatorname{\mathsf{--qcoh}}}$. Similarly, the full subcategories of
locally contraherent cosheaves of finite flat dimension and of finite locally
injective dimension coincide in $X{\operatorname{\mathsf{--lcth}}}$. Neither
of these dimensions can exceed $D$.

######  Theorem 5.6.1.

(a) The triangulated functors
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
induced by the embeddings of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ are equivalences of triangulated
categories.

(b) The triangulated functors
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
induced by the embeddings of exact categories
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}\longrightarrow
X{\operatorname{\mathsf{--ctrh}}}$ are equivalences of triangulated
categories.

(c) There is a natural equivalence of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}})$
provided by the derived functors
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ and
${\mathcal{O}}_{X}\odot_{X}^{\mathbb{L}}{-}$.

###### Proof.

Parts (a-b): by Corollary 4.9.1(a,c), the functors
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}^{\star}(X\allowbreak{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$
are equivalences of categories for any symbol $\star\neq{\mathsf{ctr}}$ and
the functors
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
are equivalences of categories for any symbol $\star\neq{\mathsf{co}}$.

To prove that the functor
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of categories, notice that one has
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}\subset
X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}D}=X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$
and the functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}D})$
is an equivalence of categories by Corollary 5.4.7(a), while the composition
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{{\operatorname{\mathsf{fid--}}}D})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of categories by Theorem 5.4.10(a).

Similarly, to prove that the functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equialence of categories, notice that one has
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}D}=X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$
and the functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}D})$
is an equivalence of categories by Corolary 5.4.9, while the composition
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{{\operatorname{\mathsf{ffd--}}}D})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of categories by Theorem 5.4.10(d).

To prove part (c), notice that the functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$
are equivalences of categories by Corollary 5.4.2, while the functors
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
are equivalences of categories by Corollary 4.9.5(b).

Furthermore, consider the intersections
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}}$. As was
explained in Section 4.10, the functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})\allowbreak\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}})$
is an equivalence of triangulated categories for any
$\star\neq{\mathsf{ctr}}$, while the functor
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{flid}})$
is an equivalence of triangulated categories for any $\star\neq{\mathsf{co}}$.

Finally, it is clear from Lemma 4.10.2(a,d) (see also Lemma 4.11.3) that the
equivalence of exact categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\simeq
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}$ of Lemma 4.6.7 identifies
their full exact subcategories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cta}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{ffd}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}\cap
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{flid}}$. So the induced equivalence
of the derived categories ${\mathsf{D}}^{\mathsf{abs}}$ or ${\mathsf{D}}$
provides the desired equivalence of triangulated categories in part (c). ∎

###  5.7. Co-contra correspondence over a scheme with a dualizing complex

Let $X$ be a semi-separated Noetherian scheme with a dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ [29], which
we will view as a finite complex of injective quasi-coherent sheaves on $X$.
The following result complements the covariant Serre–Grothendieck duality
theory as developed in the papers and the thesis [33, 47, 42, 53].

######  Theorem 5.7.1.

There are natural equivalences between the four triangulated categories
${\mathsf{D}}^{{\mathsf{abs}}={\mathsf{co}}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$,
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$,
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$, and
${\mathsf{D}}^{{\mathsf{abs}}={\mathsf{ctr}}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$.
(Here the notation ${\mathsf{abs}}={\mathsf{co}}$ and
${\mathsf{abs}}={\mathsf{ctr}}$ presumes the assertions that the corresponding
derived categories of the second kind coincide for the exact category in
question.) Among these, the equivalences
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$
and
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}})$
do not require a dualizing complex and do not depend on it; all the remaining
equivalences do and do.

###### Proof.

For any quasi-compact semi-separated scheme $X$ with an open covering
${\mathbf{W}}$, one has
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})={\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
by Corollary 4.9.5(b). For any Noetherian scheme $X$ of finite Krull
dimension, one has
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})={\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
by Corollary 5.4.2.

For any semi-separated Noetherian scheme $X$, one has
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq\mathsf{Hot}(X\allowbreak{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$
by Theorem 5.4.10(a) and
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
by Corollary 4.6.8(b). Hence the desired equivalence
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$,
which is provided by the derived functors

${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})\colon{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$

and

${\mathcal{O}}_{X}\odot_{X}^{\mathbb{L}}{-}\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}).$

For any semi-separated Noetherian scheme $X$ of finite Krull dimension, one
has
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
by Corollary 5.4.5,
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
by Theorem 5.4.10(b), and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
by Corollary 5.4.4(b). Alternatively, one can refer to the equivalence
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})$
holding by Corollary 5.4.3,
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$
by Corollary 4.6.10(a), and
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
by Theorem 5.4.10(d). Either way, one gets the same desired equivalence
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$,
which is provided by the derived functors

${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})\colon{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$

and

${\mathcal{O}}_{X}\odot_{X}^{\mathbb{L}}{-}\colon{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu{\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}).$

Now we are going to construct a commutative diagram of equivalences of
triangulated categories

$\dgARROWLENGTH=7em\begin{diagram}$

for any symbol $\star={\mathsf{b}}$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, or
${\mathsf{abs}}$.

The exterior vertical functors are constructed by applying the additive
functors ${\mathcal{O}}_{X}\odot_{X}{-}$ and
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ to the given
complexes termwise. The interior (derived) vertical functors have been defined
in Corollaries 4.6.8(b) and 5.4.5. All the functors invoking the dualizing
complex ${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ are
constructed by applying the respective exact functors of two arguments to
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the
given unbounded complex termwise and totalizing the bicomplexes so obtained.

First of all, one notices that the functors in the interior upper triangle are
right adjoint to the ones in the exterior. This follows from the adjunction
(20) together with the adjunction of the tensor product of quasi-coherent
sheaves and the quasi-coherent internal Hom.

The upper horizontal functors
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{-}$
and
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
are mutually inverse for the reasons explained in [42, Theorem 8.4 and
Proposition 8.9] and [53, Theorem 2.5]. The argument in [53] is based on the
observations that the morphism of finite complexes of flat quasi-coherent
sheaves

${\mathcal{F}}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{F}})$

is a quasi-isomorphism for any sheaf ${\mathcal{F}}\in
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}}$ and the morphism of finite
complexes of injective quasi-coherent sheaves

${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{{\mathcal{O}}_{X}}\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu{\mathcal{J}}$

is a quasi-isomorphism for any sheaf ${\mathcal{J}}\in
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$.

Let us additionally point out that, according to Lemma 2.5.3(c) and [42, Lemma
8.7], the complex
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a complex of flat cotorsion quasi-coherent sheaves for any complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$. So the functor
$\operatorname{\mathcal{H}\mskip-0.90001mu\text{om}}_{X{\operatorname{\mathrm{-qc}}}}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
actually lands in
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\cap
X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ (as does the functor
${\mathcal{O}}_{X}\odot_{X}{-}$ on the diagram, according to the proof of
Corollary 5.4.5). The interior upper triangle is commutative due to the
natural isomorphism (17). The exterior upper triangle is commutative due to
the natural isomorphism (21).

In order to discuss the equivalence of categories in the lower horizontal
line, we will need the following lemma. It is based on the definitions of the
$\operatorname{\mathfrak{Cohom}}$ functor in Section 3.6 and the contraherent
tensor product functor $\otimes_{X{\operatorname{\mathrm{-ct}}}}$ in Section
3.7.

######  Lemma 5.7.2.

Let ${\mathcal{J}}$ be an injective quasi-coherent sheaf on a semi-separated
Noetherian scheme $X$ with an open covering ${\mathbf{W}}$. Then there are two
well-defined exact functors

$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{J}},{-})\colon
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$

and

${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}\colon
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5muX{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$

between the exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ and
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ of locally injective
${\mathbf{W}}$-locally contraherent cosheaves and colocally flat contraherent
cosheaves on $X$. The functor
${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}$ is left adjoint to
the functor $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{J}},{-})$. Besides,
the functor $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{J}},{-})$ takes
values in the additive subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}\subset
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$, while the functor
${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}$ takes values in the
additive subcategory
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lin}}_{\mathsf{clp}}\subset
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$. For any quasi-
coherent sheaf ${\mathcal{M}}$ and any colocally flat contraherent cosheaf
${\mathfrak{F}}$ on $X$ there is a natural isomorphism

(84)
${\mathcal{M}}\odot_{X}({\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})\simeq({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{J}})\odot_{X}{\mathfrak{F}}$

of quasi-coherent sheaves on $X$.

###### Proof.

Let us show that the locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaf $\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{J}},{\mathfrak{K}})$ is
projective for any locally injective ${\mathbf{W}}$-locally contraherent
cosheaf ${\mathfrak{K}}$ on $X$. Indeed, ${\mathcal{J}}$ is a direct summand
of a finite direct sum of the direct images of injective quasi-coherent
sheaves ${\mathcal{I}}$ from the embeddings of affine open subschemes $j\colon
U\longrightarrow X$ subordinate to ${\mathbf{W}}$. So it suffices to consider
the case of ${\mathcal{J}}=j_{*}{\mathcal{I}}$.

According to (40), there is a natural isomorphism of locally cotorsion
(${\mathbf{W}}$-locally) contraherent cosheaves
$\operatorname{\mathfrak{Cohom}}_{X}(j_{*}{\mathcal{I}},{\mathfrak{K}})\simeq
j_{!}\operatorname{\mathfrak{Cohom}}_{U}({\mathcal{I}},j^{!}{\mathfrak{K}})$
on $X$. The ${\mathcal{O}}(U)$-modules ${\mathcal{I}}(U)$ and
${\mathfrak{K}}[U]$ are injective, so
$\operatorname{Hom}_{{\mathcal{O}}(U)}({\mathcal{I}}(U),{\mathfrak{K}}[U])$ is
a flat cotorsion ${\mathcal{O}}(U)$-module. In other words, the locally
cotorsion contraherent cosheaf
$\operatorname{\mathfrak{Cohom}}_{U}({\mathcal{I}},j^{!}{\mathfrak{K}})$ is
projective on $U$, and therefore its direct image with respect to $j$ is
projective on $X$ (see Lemma 4.4.3(b) or Corollary 4.4.7(b)).

Now let ${\mathfrak{F}}$ be a colocally flat contraherent cosheaf on $X$.
Then, in particular, ${\mathfrak{F}}$ is a flat contraherent cosheaf
(Corollary 4.3.6), so the tensor product
${\mathcal{J}}\otimes_{X}{\mathfrak{F}}$ is a locally injective derived
contrahereable cosheaf on $X$.

Moreover, by Corollary 4.3.4(c), ${\mathfrak{F}}$ is a direct summand of a
finitely iterated extension of the direct images of flat contraherent
cosheaves from affine open subschemes of $X$. It was explained in Section 3.5
that derived contrahereable cosheaves on affine schemes are contrahereable and
the direct images of cosheaves with respect to affine morphisms preserve
contrahereability. Besides, the full subcategory of contrahereable cosheaves
on $X$ is closed under extensions in the exact category of derived
contrahereable cosheaves, and the functor ${\mathcal{J}}\otimes_{X}{-}$ takes
short exact sequences of flat contraherent cosheaves to short exact sequences
of derived contrahereable cosheaves on $X$ (see Section 3.7).

So it follows from the isomorphism (48) that
${\mathcal{J}}\otimes_{X}{\mathfrak{F}}$ is a locally injective contrahereable
cosheaf. Its contraherator
${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}=\operatorname{\mathfrak{C}}({\mathcal{J}}\otimes_{X}{\mathfrak{F}})$
is consequently a locally injective contraherent cosheaf on $X$. Furthermore,
according to Section 3.5 the (global) contraherator construction is an exact
functor commuting with the direct images with respect to affine morphisms.
Hence the contraherent cosheaf
${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}$ is a
direct summand of a finitely iterated extension of the direct images of
(locally) injective contraherent cosheaves from affine open subschemes of $X$,
i. e., ${\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}$
is a colocally projective locally injective contraherent cosheaf.

We have constructed the desired exact functors. A combination of the
adjunction isomorphisms (35) and (30) makes them adjoint to each other.
Finally, for any ${\mathcal{M}}\in X{\operatorname{\mathsf{--qcoh}}}$ and
${\mathfrak{F}}\in X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clf}}$ one has

${\mathcal{M}}\odot_{X}({\mathcal{J}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})={\mathcal{M}}\odot_{X}\operatorname{\mathfrak{C}}({\mathcal{J}}\otimes_{X}{\mathfrak{F}})\simeq{\mathcal{M}}\odot_{X}({\mathcal{J}}\otimes_{X}{\mathfrak{F}})\simeq({\mathcal{M}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{J}})\odot_{X}{\mathfrak{F}}$

according to the isomorphisms (32) and (36). ∎

Now we can return to the proof of Theorem 5.7.1. The functors in the interior
lower triangle are left adjoint to the ones in the exterior, as it follows
from the adjunction (20) and Lemma 5.7.2. Let us show that the lower
horizontal functors are mutually inverse.

According to the proof of Corollary 4.6.8(b), the functor
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})\longrightarrow{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}})$
induced by the embedding
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}\longrightarrow
X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$ is an
equivalence of triangulated categories. Therefore, it suffices to show that
for any cosheaf ${\mathfrak{J}}\in
X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ the morphism
of complexes of contraherent cosheaves

${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}})\mskip
1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip 1.5mu{\mathfrak{J}}$

is a homotopy equivalence (or just a quasi-isomorphism in
$X{\operatorname{\mathsf{--ctrh}}}$), and for any cosheaf ${\mathfrak{P}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ the morphism
of complexes of contraherent cosheaves

${\mathfrak{P}}\mskip 1.5mu\relbar\joinrel\relbar\joinrel\rightarrow\mskip
1.5mu\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{P}})$

is a homotopy equivalence (or just a quasi-isomorphism in
$X{\operatorname{\mathsf{--ctrh}}}$).

According to Corollaries 4.2.8 and 4.4.3, any object of
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}}$ or
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is a direct
summand of a finite direct sum of direct images of objects in the similar
categories on affine open subschemes of $X$. According to (43) and (48)
together with the results of Section 3.5, both functors
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}$
commute with such direct images. So the question reduces to the case of an
affine scheme $U$, for which the distinction between quasi-coherent sheaves
and contraherent cosheaves mostly loses its significance, as both are
identified with (appropriate classes of) ${\mathcal{O}}(U)$-modules. For this
reason, the desired quasi-isomorphisms follow from the similar quasi-
isomorphisms for quasi-coherent sheaves obtained in [53, proof of Theorem 2.5]
(as quoted above).

Alternatively, one can argue in the way similar to the proof in [53].
Essentially, this means using an “inverse image localization” procedure in
place of the “direct image localizaton” above. The argument proceeds as
follows.

Let
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
be a quasi-isomorphism between a finite complex
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of coherent sheaves over $X$ and the complex of injective quasi-coherent
sheaves ${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Then the tensor product
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}}$
is a finite complex of contraherent cosheaves for any flat contraherent
cosheaf ${\mathfrak{F}}$ on $X$. Furthermore, the morphism
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a quasi-isomorphism of finite complexes over the exact category of
coadjusted quasi-coherent cosheaves
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{coa}}$ on $X$, hence the induced
morphism
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}}\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}}$
is a quasi-isomorphism of finite complexes over the exact category of derived
contrahereable cosheaves on $X$. It follows that the morphism
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}}\simeq{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}}$
is a quasi-isomorphism of finite complexes of contraherent cosheaves on $X$
for any flat contraherent cosheaf ${\mathfrak{F}}$.

Let ${\mathfrak{J}}$ be a cosheaf from
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$. In order to
show that the morphism of finite complexes
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}})\longrightarrow{\mathfrak{J}}$
is a quasi-isomorphism over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lin}}$, it suffices
to check that the composition
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}})\longrightarrow{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{J}})\longrightarrow{\mathfrak{J}}$
is a quasi-isomorphism over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$.
The latter assertion can be checked locally, i. e., it simply means that for
any affine open subscheme $U\subset X$ subordinate to ${\mathbf{W}}$ the
morphism
${}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U)\otimes_{{\mathcal{O}}_{X}(U)}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U),{\mathfrak{J}}[U])\longrightarrow{\mathfrak{J}}[U]$
is a quasi-isomorphism of complexes of ${\mathcal{O}}_{X}(U)$-modules. This
can be deduced from the condition that the morphism
${\mathcal{O}}_{X}(U)\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U),{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U))$
is a quasi-isomorphism, as explained in the proof in [53].

Let ${\mathfrak{F}}$ be a flat contraherent cosheaf on $X$. Pick a bounded
above complex of very flat quasi-coherent sheaves
${}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X$ together with a quasi-isomorphism
${}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Then the bounded below complex of contraherent cosheaves
$\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}})$
is well-defined. The morphisms of bounded below complexes
$\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}})\longrightarrow\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})$
and
$\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>\allowbreak{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})\longrightarrow\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})$
are quasi-isomorphisms over $X{\operatorname{\mathsf{--ctrh}}}$. Thus in order
to show that the morphism
${\mathfrak{F}}\longrightarrow\operatorname{\mathfrak{Cohom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{\mathfrak{F}})$
is a quasi-isomorphism in $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$,
it suffices to check that the morphism
${\mathfrak{F}}\longrightarrow\operatorname{\mathfrak{Cohom}}_{X}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>\allowbreak{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X}{\mathfrak{F}})$
is a quasi-isomorphism of bounded below complexes over
$X{\operatorname{\mathsf{--ctrh}}}$.

The latter is again a local assertion, meaning simply that the morphism
${\mathfrak{F}}[U]\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U),\>{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U)\otimes_{{\mathcal{O}}_{X}(U)}{\mathfrak{F}}[U])$
is a quasi-isomorphism of complexes of ${\mathcal{O}}_{X}(U)$-modules for any
affine open subscheme $U\subset X$. One proves it by replacing
${}^{\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U)$
by a quasi-isomorphic bounded above complex
${}^{\prime\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U)$
of finitely generated projective ${\mathcal{O}}_{X}(U)$-modules, and reducing
again to the condition that the morphism
${\mathcal{O}}_{X}(U)\longrightarrow\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({}^{\prime\prime\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U),{}^{\prime}{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}(U))$
is a quasi-isomorphism (cf. [53]).

According to the proof of Corollary 4.6.8(b), the functor
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{O}}_{X},{-})$ on the diagram
actually lands in
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{clp}}^{\mathsf{lin}})$
(as does the functor
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\otimes_{X{\operatorname{\mathrm{-ct}}}}{-}$,
according to Lemma 5.7.2). The exterior upper triangle is commutative due to
the natural isomorphism (19). The interior upper triangle is commutative due
to the natural isomorphism (84).

The assertion that the two diagonal functors on the diagram are mutually
inverse follows from the above. It can be also proven directly in the manner
of the former of the above the proofs of the assertion that the two lower
horizontal functors are mutually inverse. One needs to use the natural
isomorphisms (45) and (46) for commutation with the direct images. ∎

###  5.8. Co-contra correspondence over a non-semi-separated scheme

The goal of this section is to obtain partial generalizations of Theorems
4.6.6 and 5.7.1 to the case of a non-semi-separated Noetherian scheme.

######  Theorem 5.8.1.

Let $X$ be a Noetherian scheme of finite Krull dimension. Then for any symbol
$\star={\mathsf{b}}$, $+$, $-$, or $\empt$ there is a natural equivalence of
triangulated categories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})$.

###### Proof.

According to Corollaries 5.3.3 and 5.4.4, one has
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\allowbreak\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$
for any open covering ${\mathbf{W}}$ of the scheme $X$. We will construct an
equivalence of triangulated categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
and then show that it takes the full subcategories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--qcoh}}})\subset{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
into the full subcategories
${\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})\subset{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
and back for all symbols $\star={\mathsf{b}}$, $+$, or $-$.

By Lemma 3.4.7(a), the sheaf ${\mathcal{O}}_{X}$ has a finite right resolution
by flasque quasi-coherent sheaves. We fix such a resolution
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ for the time
being. Given a complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$, we pick a complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ quasi-isomorphic to
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$ (see Theorem 5.4.10(a), cf. Theorem 5.10.2
below) and assign to
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the total
complex of the bicomplex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. Given a complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$, we pick a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ quasi-
isomorphic to ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}$ (see Theorem
5.4.10(b), cf. Theorem 5.10.3(a) below) and assign to
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ the total
complex of the bicomplex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--qcoh}}}$.

Let us first show that the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is acyclic whenever a
complex ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ is. For any scheme point
$x\in X$, let ${\mathfrak{m}}_{x,X}$ denote the maximal ideal of the local
ring ${\mathcal{O}}_{x,X}$. By [29, Proposition II.7.17], any injective quasi-
coherent sheaf ${\mathcal{I}}$ on $X$ can be presented as an infinite direct
sum ${\mathcal{I}}=\bigoplus_{x\in X}\iota_{x}{}_{*}\widetilde{I}_{x}$, where
$\iota_{x}\colon\operatorname{Spec}{\mathcal{O}}_{x,X}\longrightarrow X$ are
the natural morphisms and $\widetilde{I}_{x}$ are the quasi-coherent sheaves
on $\operatorname{Spec}{\mathcal{O}}_{x,X}$ corresponding to infinite direct
sums of copies of the injective envelopes of the ${\mathcal{O}}_{x,X}$-modules
${\mathcal{O}}_{x,X}/{\mathfrak{m}}_{x,X}$. Let
$X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering. Set
$S_{\beta}\subset X$ to be the set-theoretic complement to
$\bigcup_{\alpha<\beta}U_{\alpha}$ in $U_{\beta}$, and consider the direct sum
decomposition ${\mathcal{I}}=\bigoplus_{\alpha=1}^{N}{\mathcal{I}}_{\alpha}$
with ${\mathcal{I}}_{\alpha}=\bigoplus_{z\in
S_{\alpha}}\iota_{z}{}_{*}\widetilde{I}_{z}$.

The associated decreasing filtration
${\mathcal{I}}_{\ge\alpha}=\bigoplus_{\beta\ge\alpha}{\mathcal{I}}_{\beta}$ is
preserved by all morphisms of injective quasi-coherent sheaves ${\mathcal{I}}$
on $X$ (cf. Theorem 5.1.1 and Lemma 5.1.2). We obtain a termwise split
filtration
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\ge\alpha}$ on
the complex ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with the associated quotient complexes
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}$
isomorphic to the direct images
$j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
of complexes of injective quasi-coherent sheaves
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ from
the open embeddings $j_{\alpha}\colon U_{\alpha}\longrightarrow X$. Moreover,
for $\alpha=1$ the complex of quasi-coherent sheaves
${\mathcal{K}}_{1}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
j_{1}^{*}{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
acyclic, since the complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is; and the
complex
$j_{1}{}_{*}{\mathcal{K}}_{1}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is acyclic by Corollary 3.4.9(a) or Lemma 3.4.7(a). It follows by induction
that all the complexes
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ are acyclic over
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}$.

Now one has
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq
j_{\alpha}{}_{!}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
by (45). The complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is quasi-
isomorphic to
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{U_{\alpha}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$,
since ${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a complex over $U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$,
while the complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{U_{\alpha}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$,
since the complex
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
acyclic over $U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}$ (see
Corollary 1.5.7 or Lemma 5.4.1(c)). So the complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$;
by Lemma 3.4.6(c), it is also a complex of coflasque contraherent cosheaves.
By Corollary 3.4.9(c), or alternatively by Lemma 3.4.7(b) together with
Corollary 3.4.8(b), it follows that the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$.

Therefore, so is the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Similarly one proves that the complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is acyclic over $X{\operatorname{\mathsf{--qcoh}}}$ whenever a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ is acyclic
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. One has to use
Theorem 5.1.1 and Lemma 5.1.2 (see the proof of Theorem 5.9.1(c) below), the
isomorphism (47), and Lemma 3.4.6(d).

We have shown that the derived functors
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}^{\mathbb{L}}{-}$
are well defined by the above rules
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
and
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longmapsto{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
It is a standard fact that the adjunction (20) makes such two triangulated
functors adjoint to each other (cf. [50, Lemma 8.3]). Let us check that the
adjunction morphism
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}^{\mathbb{L}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is an isomorphism in ${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ for any
complex ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$. For the reasons explained
above, one can assume
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}=j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
for some complex
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}$. Then
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq
j_{\alpha}{}_{!}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$

Let ${\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
be a complex over
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$
endowed with a quasi-isomorphism
${\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. Then
$j_{\alpha}{}_{!}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$, and the
morphism
$j_{\alpha}{}_{!}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
j_{\alpha}{}_{!}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. So one has
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}^{\mathbb{L}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\simeq{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}j_{\alpha}{}_{!}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
j_{\alpha}{}_{*}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Both ${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and
$j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
being complexes of flasque quasi-coherent sheaves on $U_{\alpha}$, it remains
to show that the natural morphism
$j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is a quasi-isomorphism over $U_{\alpha}{\operatorname{\mathsf{--qcoh}}}$. Now
the morphisms
${\mathcal{O}}_{U_{\alpha}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow
j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
and
${\mathcal{O}}_{U_{\alpha}}\odot_{U_{\alpha}}{\mathfrak{G}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{O}}_{U_{\alpha}}\odot_{U_{\alpha}}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathcal{O}}_{U_{\alpha}}\odot_{U_{\alpha}}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{U_{\alpha}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
are quasi-isomorphisms, and the desired assertion follows.

Similarly one shows that the adjunction morphism
${\mathbb{R}}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{E}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is an isomorphism in
${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}})$ for any
complex ${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. This
finishes the construction of the equivalence of categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$.
To show that it does not depend on the choice of a flasque resolution
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of the sheaf
${\mathcal{O}}_{X}$, consider an acyclic finite complex
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of flasque
quasi-coherent sheaves on $X$. Then for any injective quasi-coherent sheaf
${\mathcal{J}}$ on $X$ the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})$
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is acyclic by
construction. To show that the complex
${\mathcal{L}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}$
is acyclic over $X{\operatorname{\mathsf{--qcoh}}}$ for any cosheaf
${\mathfrak{F}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$, one reduces
the question to the case of an affine scheme $X$ using Theorem 5.1.1(b) and
Lemma 3.4.6(d).

Finally, it remains to show that the equivalence of categories
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}})$
that we have constructed takes bounded above (resp., below) complexes to
bounded above (resp., below) complexes and vice versa (up to quasi-
isomorphism). If a complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$ is bounded below, it has bounded below
injective resolution
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ and the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is also bounded below.
Now assume that a complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ has bounded above
cohomology.

Arguing as above, consider its decreasing filtration
${\mathcal{J}}_{\ge\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with the associatived quotient complexes
${\mathcal{J}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\simeq
j_{\alpha}{}_{*}{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$.
Using Lemma 3.4.7(a), one shows that the cohomology sheaves of the complexes
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ are also bounded
above. By Corollary 1.5.7, the right homological dimension of
${\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ with
respect to $U_{\alpha}{\operatorname{\mathsf{--qcoh}}}^{\mathsf{cot}}\subset
U_{\alpha}^{\operatorname{\mathsf{--qcoh}}}$ is finite, and it follows that
the complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{X},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is quasi-isomorphic to a bounded above complex over
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$. The complex
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
over $U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ is quasi-
isomorphic to
$\operatorname{\mathfrak{Hom}}_{U_{\alpha}}({\mathcal{O}}_{U_{\alpha}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$.
Finally, the complex
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\allowbreak\simeq
j_{\alpha}{}_{!}\operatorname{\mathfrak{Hom}}_{U_{\alpha}}(j_{\alpha}^{*}{\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{K}}_{\alpha}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is quasi-isomorphic to a bounded above complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ by Lemma 3.4.6(c) and the
other results of Section 3.4.

Similarly one can show that for any complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ quasi-
isomorphic to a bounded below complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$ the complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--qcoh}}}$ has bounded below cohomology sheaves.
∎

Now let $X$ be a Noetherian scheme with a dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ [29,
Chapter 5]. As above, we will consider
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ as a finite
complex of injective quasi-coherent sheaves on $X$. The following partial
version of the covariant Serre–Grothendieck duality holds without the semi-
separatedness assumption on $X$.

######  Theorem 5.8.2.

The choice of a dualizing complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ induces a
natural equivalence of triangulated categories
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--ctrh}}})$.

###### Proof.

According to Corollary 5.4.4(b), one has
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
for any open covering ${\mathbf{W}}$ of the scheme $X$. By Theorem
5.4.10(a-b), one has
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
and
$\mathsf{Hot}(X\allowbreak{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$.
We will show that the functors
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{-}$
induce an equivalence of the homotopy categories
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
for any symbol $\star={\mathsf{b}}$, $+$, $-$, or $\empt$.

Let ${\mathcal{I}}$ be an injective quasi-coherent sheaf on $X$ and $j\colon
U\longrightarrow X$ be the embedding of an affine open subscheme. Then the
results of Section 3.8 provide a natural isomorphism of contraherent cosheaves
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},j_{*}{\mathcal{J}})\simeq
j_{!}\operatorname{\mathfrak{Hom}}_{U}(j^{*}{\mathcal{I}},{\mathcal{J}})$ on
$X$ for any injective quasi-coherent sheaf ${\mathcal{J}}$ on $U$ and a
natural isomorphism of quasi-coherent sheaves
${\mathcal{I}}\odot_{X}j_{!}{\mathfrak{G}}\simeq
j_{*}(j^{*}{\mathcal{I}}\odot_{X}{\mathfrak{G}})$ on $X$ for any flat cosheaf
of ${\mathcal{O}}_{U}$-modules ${\mathfrak{G}}$ on $U$.

Notice that the functor $j_{*}$ takes injective quasi-coherent sheaves to
injective quasi-coherent sheaves and the functor $j_{!}$ takes projective
locally cotorsion contraherent cosheaves to projective locally cotorsion
contraherent cosheaves (Corollary 5.1.6(b)). Furthermore, let
$X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine open covering. It is clear
from the classification theorems (see Theorem 5.1.1(b)) that any injective
quasi-coherent sheaf or projective locally cotorsion contraherent cosheaf on
$X$ is a finite direct sum of the direct images of similar (co)sheaves from
$U_{\alpha}$.

It follows that the functors
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},{-})$ and
${\mathcal{I}}\odot_{X}{-}$ take injective quasi-coherent sheaves to
projective locally cotorsion contraherent cosheaves on $X$ and back. By (20),
these are two adjoint functors between the additive categories
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ and
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.
Substituting
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in place of
${\mathcal{I}}$ and totalizing the finite complexes of complexes of
(co)sheaves, we obtain two adjoint functors
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{-}$
between the homotopy categories
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$ and
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$.

In order to show that these are mutually inverse equivalences, it suffices to
check that the adjunction morphisms
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}})\longrightarrow{\mathcal{J}}$
and
${\mathfrak{P}}\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>\allowbreak{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{P}})$
are quasi-isomorphisms/homotopy equivalences of finite complexes for any
${\mathcal{J}}\in X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ and
${\mathfrak{P}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. Presenting
${\mathcal{J}}$ and ${\mathfrak{P}}$ as finite direct sums of the direct
images of similar (co)sheaves from affine open subschemes of $X$ and taking
again into account the isomorphisms (45), (47) reduces the question to the
case of an affine scheme, where the assertion is already known.

Alternatively, one can work directly in the greater generality of arbitrary
(not necessarily locally cotorsion) and flat contraherent cosheaves. According
to Theorem 5.4.10(d), one has
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$.
Let us show that the functors
$\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{-})$
and
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{-}$
induce an equivalence of triangulated categories
$\mathsf{Hot}^{\star}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})\simeq{\mathsf{D}}^{\star}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$
for any symbol $\star={\mathsf{b}}$, ${\mathsf{abs}}+$, ${\mathsf{abs}}-$, or
${\mathsf{abs}}$.

Given an injective quasi-coherent sheaf ${\mathcal{I}}$ on $X$, let us first
check that the functor
${\mathfrak{F}}\longmapsto{\mathcal{I}}\odot_{X}{\mathfrak{F}}$ takes short
exact sequences of flat contraherent cosheaves to short exact sequences of
quasi-coherent sheaves on $X$. By the adjunction isomorphism (20), for any
injective quasi-coherent sheaf ${\mathcal{J}}$ on $X$ one has
$\operatorname{Hom}_{X}({\mathcal{I}}\odot_{X}{\mathfrak{F}},\>{\mathcal{J}})\simeq\operatorname{Hom}^{X}({\mathfrak{F}},\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},{\mathcal{J}}))$.
The contraherent cosheaf
${\mathfrak{Q}}=\operatorname{\mathfrak{Hom}}_{X}({\mathcal{I}},{\mathcal{J}})$
being locally cotorsion, the functor
${\mathfrak{F}}\longmapsto\operatorname{Hom}^{X}({\mathfrak{F}},{\mathfrak{Q}})$
is exact on $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ by Corollary
5.2.9(a).

Furthermore, by part (b) of the same Corollary any flat contraherent cosheaf
${\mathfrak{F}}$ on $X$ is a direct summand of a finitely iterated extension
of the direct images $j_{!}{\mathfrak{G}}$ of flat contraherent cosheaves
${\mathfrak{G}}$ on affine open subschemes $U\subset X$. Using the isomorphism
(47), we conclude that the quasi-coherent sheaf
${\mathcal{I}}\odot_{X}{\mathfrak{F}}$ is injective. It follows, in
particular, that the complex of quasi-coherent sheaves
${\mathcal{I}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
is contractible for any acyclic complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the exact
category $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. Therefore, the
same applies to the complex
${\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
over $X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$.

Finally, to prove that the map
${\mathfrak{F}}\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}})$
is a quasi-isomorphism for any flat contraherent cosheaf ${\mathfrak{F}}$, it
suffices again to consider the case ${\mathfrak{F}}=j_{!}{\mathfrak{G}}$, when
the assertion follows from the isomorphisms (45), (47). Hence the morphism
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow\operatorname{\mathfrak{Hom}}_{X}({\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}},\>{\mathcal{D}}_{X}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\odot_{X}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
has a cone absolutely acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ for any compex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. ∎

###  5.9. Compact generators

Let ${\mathsf{D}}$ be a triangulated category where arbitrary infinite direct
sums exist. We recall that object $C\in{\mathsf{D}}$ is called _compact_ if
the functor $\operatorname{Hom}_{\mathsf{D}}(C,{-})$ takes infinite direct
sums in ${\mathsf{D}}$ to infinite direct sums of abelian groups [45]. A set
of compact objects ${\mathsf{C}}\subset{\mathsf{D}}$ is said to _generate_
${\mathsf{D}}$ if any object $X\in{\mathsf{D}}$ such that
$\operatorname{Hom}_{\mathsf{D}}(C,X[*])=0$ for all $C\in{\mathsf{C}}$
vanishes in ${\mathsf{D}}$.

Equivalently, this means that any full triangulated subcategory of
${\mathsf{D}}$ containing ${\mathsf{C}}$ and closed under infinite direct sums
coincides with ${\mathsf{D}}$. If ${\mathsf{C}}$ is a set of compact
generators for ${\mathsf{D}}$, then an object of ${\mathsf{D}}$ is compact if
and only if it belongs to the minimal thick subcategory of ${\mathsf{D}}$
containing ${\mathsf{C}}$.

Let $X{\operatorname{\mathsf{--coh}}}$ denote the abelian category of coherent
sheaves on a Noetherian scheme $X$.

######  Theorem 5.9.1.

(a) For any scheme $X$, the coderived category
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ admits
arbitrary infinite direct sums, while the contraderived categories
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
admit infinite products.

(b) For any Noetherian scheme $X$, the coderived category
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$ is compactly
generated. The triangulated functor
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--coh}}})\longrightarrow{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$
induced by the embedding of abelian categories
$X{\operatorname{\mathsf{--coh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$ is fully faithful, and its image is the
full subcategory of compact objects in
${\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}})$.

(c) For any Noetherian scheme $X$ of finite Krull dimension, the contraderived
categories
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
admit arbitrary infinite direct sums and are compactly generated.

###### Proof.

Part (a) holds, because the abelian category
$X{\operatorname{\mathsf{--qcoh}}}$ admits arbitrary infinite direct sums and
the full subcategory of coacyclic complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})$ is closed under infinite
direct sums (see [46, Proposition 1.2.1 and Lemma 3.2.10]). Analogously, the
exact categories
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ admit arbitrary infinite
products and the full subcategories of contraacyclic complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and $\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ are closed
under infinite products.

In the assumption of part (b), the abelian category
$X{\operatorname{\mathsf{--qcoh}}}$ is a locally Noetherian Grothendieck
category, so the assertions hold by Theorem 5.4.10(a) and [37, Proposition
2.3] (see also Lemma A.1.2). A more generally applicable assertion/argument
can be found in [53, Proposition 1.5(d)] and/or [51, Section 3.11].

Part (c): notice first of all that all the categories
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalent to each other by Corollaries 5.3.3 and 5.4.4(b). Furthermore,
if the scheme $X$ admits a dualizing complex, the assertion of part (c)
follows from Theorem 5.8.2 and part (b). The following more complicated
argument allows to prove the desired assertion in the stated generality.

By Theorem 5.4.10(b), the triangulated category in question is equivalent to
the homotopy category
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$.
Let us first consider the case when the scheme $X$ is semi-separated. Then
Corollary 5.4.5 identifies our triangulated category with
${\mathsf{D}}^{\mathsf{abs}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}^{\mathsf{co}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$.
It follows immediately that this triangulated category admits arbitrary
infinite direct sums.

In the case of an affine Noetherian scheme $U$ of finite Krull dimension,
another application of Proposition A.5.6 allows to identify
${\mathsf{D}}^{\mathsf{co}}(U{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
with the homotopy category of complexes of projective
${\mathcal{O}}(U)$-modules, which is compactly generated by [34, Theorem 2.4].
More generally, the category
${\mathsf{D}}(U{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ is equivalent
to the homotopy category of projective ${\mathcal{O}}(U)$-modules for any
affine scheme $U$ by [47, Section 8] and is compactly generated for any affine
Noetherian scheme $U$ by [47, Proposition 7.14] (see also [48]). Finally, for
any semi-separated Noetherian scheme $X$ the triangulated category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ is compactly
generated by [42, Theorem 4.10].

Now let us turn to the general case. First we have to show that the category
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
admits arbitrary infinite direct sums. Let
$X=\bigcup_{\alpha=1}^{N}U_{\alpha}$ be a finite affine open covering, and let
$S_{\beta}\subset X$ denote the set-theoretic complement to
$\bigcup_{\alpha<\beta}U_{\alpha}$ in $U_{\beta}$. Let $j_{\alpha}\colon
U_{\alpha}\longrightarrow X$ denote the open embedding morphisms; then the
direct image functor
$j_{\alpha}{}_{!}\colon\mathsf{Hot}(U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
is left adjoint to the inverse image functor
$j_{\alpha}^{!}\colon\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\longrightarrow\mathsf{Hot}(U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
(see Corollaries 5.1.3(a) and 5.1.6(b), and the adjunction (26)). Hence the
functor $j_{\alpha}{}_{!}$ preserves infinite direct sums.

As explained in the proof of Theorem 5.1.1, any projective locally cotorsion
contraherent cosheaf ${\mathfrak{F}}$ on $X$ decomposes into a direct sum
${\mathfrak{F}}=\bigoplus_{\alpha=1}^{N}{\mathfrak{F}}_{\alpha}$, where each
direct summand ${\mathfrak{F}}_{\alpha}$ is an infinite product over the
points $z\in S_{\alpha}$ of the direct images of contraherent cosheaves on
$\operatorname{Spec}{\mathcal{O}}_{z,X}$ corresponding to free contramodules
over $\widehat{\mathcal{O}}_{z,X}$. According to Lemma 5.1.2, the associated
increasing filtration
${\mathfrak{F}}_{\le\alpha}=\bigoplus_{\beta\le\alpha}{\mathfrak{F}}_{\beta}$
on ${\mathfrak{F}}$ is preserved by all morphisms of cosheaves
${\mathfrak{F}}\in
X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$.

Given a family
${}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
complexes over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$, we now see
that every complex
${}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
endowed with a finite termwise split filtration
${}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\le\alpha}$
such that the family of associated quotient complexes
${}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}$
can be obtained by applying the direct image functor $j_{\alpha}{}_{!}$ to a
family of complexes over
$U_{\alpha}{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$. It
follows that the object
$\bigoplus_{i}{}^{(i)}{\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}_{\alpha}$
exists in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$,
and it remains to apply the following lemma (which is slightly stronger than
[46, Proposition 1.2.1]).

######  Lemma 5.9.2.

Let $A_{i}\longrightarrow B_{i}\longrightarrow C_{i}\longrightarrow A_{i}[1]$
be a family of distinguished triangles in a triangulated category
${\mathsf{D}}$. Suppose that the infinite direct sums $\bigoplus_{i}A_{i}$ and
$\bigoplus_{i}B_{i}$ exist in ${\mathsf{D}}$. Then a cone $C$ of the natural
morphism $\bigoplus_{i}A_{i}\longrightarrow\bigoplus_{i}B_{i}$ is the infinite
direct sum of the family of objects $C_{i}$ in ${\mathsf{D}}$.

###### Proof.

Set $A=\bigoplus_{i}A_{i}$ and $B=\bigoplus_{i}B_{i}$. By one of the
triangulated category axioms, there exist morphisms of distinguished triangles
$(A_{i}\to B_{i}\to C_{i}\to A_{i}[1])\longrightarrow(A\to B\to C\to A[1])$
whose components $A_{i}\longrightarrow A$ and $B_{i}\longrightarrow B$ are the
natural embeddings. For any object $E\in{\mathsf{D}}$, apply the functor
$\operatorname{Hom}_{\mathsf{D}}({-},E)$ to this family of morphisms of
triangles and pass to the infinite product (of abelian groups) over $i$. The
resulting morphism from the long exact sequence
$\dotsb\longrightarrow\operatorname{Hom}_{\mathsf{D}}(A[1],E)\longrightarrow\operatorname{Hom}_{\mathsf{D}}(C,E)\longrightarrow\operatorname{Hom}_{\mathsf{D}}(B,E)\longrightarrow\operatorname{Hom}_{\mathsf{D}}(A,E)\longrightarrow\dotsb$
to the long exact sequence
$\dotsb\longrightarrow\prod_{i}\operatorname{Hom}_{\mathsf{D}}(A_{i}[1],E)\longrightarrow\prod_{i}\operatorname{Hom}_{\mathsf{D}}(C_{i},E)\longrightarrow\prod_{i}\operatorname{Hom}_{\mathsf{D}}(B_{i},E)\longrightarrow\prod_{i}\operatorname{Hom}_{\mathsf{D}}(A_{i},E)\longrightarrow\dotsb$
is an isomorphism at the two thirds of all the terms, and consequently an
isomorphism at the remaining terms, too. ∎

Denote temporarily the homotopy category
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
by ${\mathsf{D}}(X)$. To show that the category ${\mathsf{D}}(X)$ is compactly
generated, we will use the result of [56, Theorem 5.15]. Let $Y\subset X$ be
an open subscheme such that the category ${\mathsf{D}}(Y)$ is compactly
generated (e. g., we already know this to hold when $Y$ is semi-separated).
Let $j\colon Y\longrightarrow X$ denote the open embedding morphism.

The composition $j^{!}j_{!}$ of the direct image and inverse image functors
$j_{!}\colon{\mathsf{D}}(Y)\longrightarrow{\mathsf{D}}(X)$ and
$j^{!}\colon{\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(Y)$ is isomorphic to
the identity endofunctor of ${\mathsf{D}}(Y)$, so the functor $j_{!}$ is fully
faithful and the functor $j^{!}$ is a Verdier localization functor. Applying
again Lemma 5.1.2, we conclude that the kernel of $j^{!}$ is the homotopy
category of projective locally cotorsion contraherent cosheaves on $X$ with
vanishing restrictions to $Y$. Denote this homotopy category by
${\mathsf{D}}(Z,X)$, where $Z=X\setminus Y$, and its identity embedding
functor by $i_{!}\colon{\mathsf{D}}(Z,X)\longrightarrow{\mathsf{D}}(X)$.

The functor $j_{!}$ is known to preserve infinite products, and the
triangulated category ${\mathsf{D}}(Y)$ is assumed to be compactly generated;
so it follows that there exists a triangulated functor
$j^{*}\colon{\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(Y)$ left adjoint to
$j_{!}$ (see [46, Remark 6.4.5 and Theorem 8.6.1] and [37, Proposition
3.3(2)]). The existence of the functor $j_{!}$ left adjoint to $j^{!}$ implies
existence of a functor
$i^{*}\colon{\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(Z,X)$ left adjoint to
$i_{!}$; and the existence of the functor $j^{*}$ left adjoint to $j_{!}$
implies existence of a functor
$i_{+}\colon{\mathsf{D}}(Z,X)\longrightarrow{\mathsf{D}}(X)$ left adjoint to
$i^{*}$.

The functors $j^{*}$ and $i_{+}$ have double right adjoints (i. e., the right
adjoints and the right adjoints to the right adjoints), hence they not only
preserve infinite direct sums, but also take compact objects to compact
objects. Furthermore, for any open subscheme $W\subset X$ with the embedding
morphism $h\colon W\longrightarrow X$ one has the base change isomorphism
$h^{!}j_{!}\simeq j^{\prime}_{!}h^{\prime}{}^{!}$, where $j^{\prime}$ and
$h^{\prime}$ denote the open embeddings $W\cap Y\longrightarrow W$ and $W\cap
Y\longrightarrow Y$. If the triangulated category ${\mathsf{D}}(W\cap Y)$ is
compactly generated, one can pass to the left adjoint functors, obtaining an
isomorphism of triangulated functors $j^{*}h_{!}\simeq h^{\prime}_{!}\mskip
1.5muj^{\prime}{}^{*}$.

Let $X=\bigcup_{\alpha}U_{\alpha}$ be a finite affine (or, more generally,
semi-separated) open covering, $Z_{\alpha}=X\setminus U_{\alpha}$ be the
corresponding closed complements, and
$i_{\alpha}{}_{+}\colon{\mathsf{D}}(Z_{\alpha},X)\longrightarrow{\mathsf{D}}(X)$
be the related fully faithful triangulated functors. It follows from the above
that the images of the functors $i_{\alpha}{}_{+}$ form a collection of
Bousfield subcategories in ${\mathsf{D}}(X)$ pairwise intersecting properly in
the sense of [56, Lemma 5.7(2)]. Furthermore, the category ${\mathsf{D}}(X)$
being generated by the images of the functors $j_{\alpha}{}_{!}$, the
intersection of the kernels of the functors
$j_{\alpha}^{*}\colon{\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(U_{\alpha})$
is zero. These coincide with the images of the functors $i_{\alpha}{}_{+}$.
Thus the triangulated subcategories
$i_{\alpha}{}_{+}{\mathsf{D}}(Z_{\alpha},X)\subset{\mathsf{D}}(X)$ form a
_cocovering_ (in the sense of [56]).

It remains to check that intersections of the images of
$i_{\beta}{}_{+}{\mathsf{D}}(Z_{\beta},X)$ under the localization morphism
${\mathsf{D}}(X)\longrightarrow{\mathsf{D}}(U_{\alpha})$ are compactly
generated in ${\mathsf{D}}(U_{\alpha})$. Let $Y$ be a semi-separated
Noetherian scheme of finite Krull dimension and $V\subset Y$ be an open
subscheme with the closed complement $Z=Y\setminus V$. We will show that the
image of the fully faithful triangulated functor
$i_{+}\colon{\mathsf{D}}(Z,Y)\longrightarrow{\mathsf{D}}(Y)$ is compactly
generated in ${\mathsf{D}}(Y)$; this is clearly sufficient.

The result of Corollary 5.4.5 identifies
${\mathsf{D}}(Y)=\mathsf{Hot}(Y{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
with ${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$ and
${\mathsf{D}}(V)=\mathsf{Hot}(V{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
with ${\mathsf{D}}(V{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$.
According to Corollary 5.4.6, this identification transforms the functor
$j_{!}\colon{\mathsf{D}}(V)\longrightarrow{\mathsf{D}}(Y)$ into the derived
functor
${\mathbb{R}}j_{*}\colon{\mathsf{D}}(V{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
constructed in (65). The latter functor is right adjoint to the functor
$j^{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(V{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$,
which is therefore identified with the functor
$j^{*}\colon{\mathsf{D}}(Y)\longrightarrow{\mathsf{D}}(V)$.

Finally, we refer to [42, Proposition 4.5 and Theorem 4.10] for the assertion
that the kernel of the functor
$j^{*}\colon{\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\longrightarrow{\mathsf{D}}(V{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$
is compactly generated in
${\mathsf{D}}(Y{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})$. ∎

######  Theorem 5.9.3.

(a) For any scheme $X$, the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ admits infinite direct sums,
while the derived categories
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and ${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ admit
infinite products.

(b) For any quasi-compact semi-separated scheme $X$, the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ is compactly generated. The
full triangulated subcategory of perfect complexes in
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{vfl}})\subset{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{fl}})\subset{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}})\subset{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
is the full subcategory of compact objects in
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$.

(c) For any Noetherian scheme $X$, the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$ is compactly generated. The
full triangulated subcategory of perfect complexes in
${\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--coh}}})\subset{\mathsf{D}}^{\mathsf{b}}(X{\operatorname{\mathsf{--qcoh}}})\subset{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
is the full subcategory of compact objects in
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$.

(d) For any quasi-compact semi-separated scheme $X$, the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ admits infinite
direct sums and is compactly generated.

(e) For any Noetherian scheme $X$ of finite Krull dimension, the derived
categories
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and ${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ admit
infinite direct sums and are compactly generated.

###### Proof.

The proof of part (a) is similar to that of Theorem 5.9.1(a): the assersions
hold, since the class of acyclic complexes over
$X{\operatorname{\mathsf{--qcoh}}}$ is closed under infinite direct sums, and
the classes of acyclic complexes over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ and
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ are closed under infinite
products.

Parts (b) and (c) are particular cases of [56, Theorem 6.8], according to
which the derived category ${\mathsf{D}}(X)$ of complexes of sheaves of
${\mathcal{O}}_{X}$-modules with quasi-coherent cohomology sheaves is
compactly generated for any quasi-compact quasi-separated scheme $X$. Here one
needs to know that the natural functor
${\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})\longrightarrow{\mathsf{D}}(X)$
is an equivalence of categories when $X$ is either quasi-compact and semi-
separated, or else Noetherian (cf. [61, Appendix B]). In the semi-separated
case, this was proven in [8, Sections 5–6]. The proof in the Noetherian case
is similar.

Alternatively, one can prove parts (b) and (c) directly in the way analogous
to the argument in [56]. In either approach, one needs to know that the
functor ${\mathbb{R}}j_{*}$ of derived direct image of complexes over
$Y{\operatorname{\mathsf{--qcoh}}}$ with respect to an open embedding $j\colon
Y\longrightarrow X$ of schemes from the class under consideration is well-
behaved. E. g., it needs to be local in the base, or form a commutative square
with the derived functor of direct image of complexes of
${\mathcal{O}}_{Y}$-modules, etc. (cf. [41, Theorems 31 and 42]).

In the semi-separated case, one can establish such properties using
contraadjusted resolutions and (the proof of) Corollary 4.1.13(a) (see the
construction of the functor ${\mathbb{R}}f_{*}$ in Section 4.8 above). In the
Noetherian case, one needs to use flasque resolutions and Corollary 3.4.9(a)
(see the construction of the functor ${\mathbb{R}}f_{*}$ in Section 5.12
below).

Part (d) follows from part (b) together with Theorem 4.6.6. Part (e) follows
from part (c) together with Theorem 5.8.1 and Corollary 5.4.4(b). ∎

###  5.10. Homotopy projective complexes

Let $X$ be a scheme. A complex
${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of quasi-
coherent sheaves on $X$ is called _homotopy injective_ if the complex of
abelian groups
$\operatorname{Hom}_{X}({\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any acyclic complex of quasi-coherent sheaves
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$. The
full subcategory of homotopy injective complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})$ is denoted by
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})^{\mathsf{inj}}$ and the full
subcategory of complexes of injective quasi-coherent sheaves that are also
homotopy injective is denoted by
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})^{\mathsf{inj}}\subset\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$.

Similarly, a complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaves is called _homotopy
projective_ if the complex of abelian groups
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any acyclic complex
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the exact
category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. The
full subcategory of homotopy projective complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
is denoted by
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})_{\mathsf{prj}}$.
We will see below in this section that the property of a complex of locally
cotorsion ${\mathbf{W}}$-locally contraherent cosheaves on a Noetherian scheme
$X$ of finite Krull dimension to be homotopy projective does not change when
the covering ${\mathbf{W}}$ is replaced by its refinement.

Finally, a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
${\mathbf{W}}$-locally contraherent cosheaves is called _homotopy projective_
if the complex of abelian groups
$\operatorname{Hom}^{X}({\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any acyclic complex
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over the exact
category $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$. The full
subcategory of homotopy projective complexes in
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ is denoted by
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})_{\mathsf{prj}}$.
Let us issue a _warning_ that our terminology is misleading: a homotopy
projective complex of locally cotorsion ${\mathbf{W}}$-locally contraherent
cosheaves need not be homotopy projective as a complex of
${\mathbf{W}}$-locally contraherent cosheaves. It will be shown below that the
property of a complex of ${\mathbf{W}}$-locally contraherent cosheaves on a
Noetherian scheme $X$ of finite Krull dimension to be homotopy projective does
not change when the covering ${\mathbf{W}}$ is replaced by its refinement.

######  Lemma 5.10.1.

(a) Let $X$ be a locally Noetherian scheme of finite Krull dimension with an
open covering ${\mathbf{W}}$. Then a complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ belongs to
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})_{\mathsf{prj}}$
if and only if the complex
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ acyclic with
respect to $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$.

(b) Let $X$ be a Noetherian scheme of finite Krull dimension with an open
covering ${\mathbf{W}}$. Then a complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ belongs to
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})_{\mathsf{prj}}$
if and only if the complex
$\operatorname{Hom}^{X}({\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$ acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$.

###### Proof.

We will prove part (a), part (b) being similar. The “only if” assertion holds
by the definition. To check the “if”, consider a complex
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. By (the proof
of) Theorem 5.4.10(b), there exists a complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ together
with a morphism of complexes of locally contraherent cosheaves
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
with a cone contraacyclic with respect to
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. Moreover, the
complex $\operatorname{Hom}^{X}$ from any complex of projective locally
cotorsion contraherent cosheaves to a contraacyclic complex over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$ is acyclic.
Hence the morphism
$\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})\longrightarrow\operatorname{Hom}^{X}({\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is a quasi-isomorphism. Finally, if the complex
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is acyclic
over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$, then so
is the complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$, and by Lemma
5.3.1(b) it follows that the complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is also
acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{cfq}}$. ∎

According to Lemma 5.10.1, the property of a complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$
(respectively, over $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}}$) to
belong to
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})_{\mathsf{prj}}$
(resp.,
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})_{\mathsf{prj}}$)
does not depend on the covering ${\mathbf{W}}$ (in the assumptions of the
respective part of the lemma). We will denote the full subcategory in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})$
(resp., $\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$)
consisting of the homotopy projective complexes by
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})_{\mathsf{prj}}$
(resp.,
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}$).
It is a standard fact that bounded above complexes of projectives are homotopy
projective,
$\mathsf{Hot}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})_{\mathsf{prj}}$
and
$\mathsf{Hot}^{-}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}$.

The next result is essentially well-known.

######  Theorem 5.10.2.

Let $X$ be a locally Noetherian scheme. Then the natural functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})^{\mathsf{inj}}\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})^{\mathsf{inj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
are equivalences of triangulated categories.

###### Proof.

It is clear that both functors are fully faithful. The functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}})^{\mathsf{inj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--qcoh}}})$
is an equivalence of categories by [1, Theorem 5.4]; this is applicable to any
Grothendieck abelian category in place of $X{\operatorname{\mathsf{--qcoh}}}$
(for an even more general statement, see [38, Theorem 6]).

To prove that any homotopy injective complex of quasi-coherent sheaves on a
locally Noetherian scheme is homotopy equivalent to a homotopy injective
complex of injective quasi-coherent sheaves, one can use (the proof of)
Theorem 5.4.10(a). From any complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}$ there exists a closed morphism into a
complex ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}}$ with a coacyclic cone
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. If the complex
${\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ was homotopy
injective, the morphism
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}[1]$
is homotopic to zero, hence the complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a direct
summand of ${\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ in
$\mathsf{Hot}(X{\operatorname{\mathsf{--qcoh}}}^{\mathsf{inj}})$. Any morphism
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}\longrightarrow{\mathcal{J}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$
being also homotopic to zero, it follows that the complex
${\mathcal{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
contractible. ∎

######  Theorem 5.10.3.

Let $X$ be a Noetherian scheme of finite Krull dimension with an open covering
${\mathbf{W}}$. Then

(a) the natural functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}})_{\mathsf{prj}}\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})_{\mathsf{prj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}^{\mathsf{lct}}_{\mathbf{W}})$
are equivalences of triangulated categories;

(b) the natural functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}\longrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})_{\mathsf{prj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
are equivalences of triangulated categories.

###### Proof.

We will prove part (b), part (a) being similar. Since both functors are
clearly fully faithful, it suffices to show that the composition
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
is an equivalence of categories. This is equivalent to saying that the
localization functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
has a left adjoint whose image is essentially contained in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})$.

The functor in question factorizes into the composition of two localization
functors
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$.
The functor
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\allowbreak\longrightarrow{\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
has a left adjoint functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\simeq\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})\hookrightarrow\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
provided by Theorem 5.4.10(d); so it remains to show that the functor
${\mathsf{D}}^{\mathsf{ctr}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})\longrightarrow{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
has a left adjoint.

Since the latter functor preserves infinite products, the assertion follows
from Theorem 5.9.1(c) and [37, Proposition 3.3(2)]. Here one also needs to
know that the derived category
${\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})\simeq{\mathsf{D}}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$
“exists” (i. e., morphisms between any given two objects form a set rather
than a class). This is established by noticing that the classes of quasi-
isomorphisms are locally small in
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}})$
and $\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})$ (see [62,
Section 10.3.6 and Proposition 10.4.4]).

Similarly one can prove Theorem 5.10.2 for a Noetherian scheme $X$ using
Theorems 5.4.10(a) and 5.9.1(b); the only difference is that this time one
needs a right adjoint functor, so [37, Proposition 3.3(1)] has to be applied.
(Cf. [51, Section 5.5]). ∎

A complex ${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ of
${\mathbf{W}}$-locally contraherent cosheaves on a scheme $X$ is called
_homotopy flat_ if the complex of abelian groups
$\operatorname{Hom}^{X}({\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex of locally cotorsion ${\mathbf{W}}$-locally
contraherent cosheaves
${\mathfrak{M}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ acyclic over
the exact category
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$. On a locally
Noetherian scheme $X$ of finite Krull dimension, the latter condition is
equivalent to the acyclicity over
$X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}$ (see Corollary 1.5.7). Notice
that the property of a complex of ${\mathbf{W}}$-locally contraherent
cosheaves to be homotopy flat may possibly change when the covering
${\mathbf{W}}$ is replaced by its refinement.

######  Lemma 5.10.4.

Let $X$ be a Noetherian scheme of finite Krull dimension with an open covering
${\mathbf{W}}$. Then a complex of flat contraherent cosheaves
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ on $X$ is
homotopy flat if and only if the complex
$\operatorname{Hom}^{X}({\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}},{\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}})$
is acyclic for any complex
${\mathfrak{E}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ acyclic with
respect to $X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{cfq}}$.

###### Proof.

Similar to that of Lemma 5.10.1. The only difference is that one has to use
Corollary 5.2.9(a) in order to show that the complex $\operatorname{Hom}^{X}$
from any complex of flat contraherent cosheaves on $X$ to a contraacyclic
complex over $X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}}^{\mathsf{lct}}$
is acyclic. ∎

According to Lemma 5.10.4, the property of a complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ to belong to
$\mathsf{Hot}(X{\operatorname{\mathsf{--lcth}}}_{\mathbf{W}})^{\mathsf{fl}}$
does not depend on the covering ${\mathbf{W}}$ (on a Noetherian scheme $X$ of
finite Krull dimension). We denote the full subcategory in
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})$ consisting of
the homotopy flat complexes by
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})^{\mathsf{fl}}$.
One can easily check that bounded above complexes of flat contraherent
cosheaves are homotopy flat,
$\mathsf{Hot}^{-}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})^{\mathsf{fl}}$.

######  Theorem 5.10.5.

Let $X$ be a Noetherian scheme of finite Krull dimension. Then the quotient
category of the homotopy category of homotopy flat complexes of flat
contraherent cosheaves
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})^{\mathsf{fl}}$
on $X$ by its thick subcategory of acyclic complexes over the exact category
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ is equivalent to the derived
category ${\mathsf{D}}(X{\operatorname{\mathsf{--ctrh}}})$.

###### Proof.

By Corollary 5.2.6(b) and [50, Remark 2.1], any acyclic complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ is absolutely acyclic; and
one can see from Corollary 5.2.9(a) that any absolutely acyclic complex over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ is homotopy flat. According
to (the proof of) Theorem 5.10.3(b), there is a quasi-isomorphism into any
complex over $X{\operatorname{\mathsf{--ctrh}}}$ from a complex belonging to
$\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}_{\mathsf{prj}})_{\mathsf{prj}}\subset\mathsf{Hot}(X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}})^{\mathsf{fl}}$.
In view of [51, Lemma 1.6], it remains to show that any homotopy flat complex
of flat contraherent cosheaves that is acyclic over
$X{\operatorname{\mathsf{--ctrh}}}$ is also acyclic over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$.

According again to Corollary 5.2.6(b) and the dual version of the proof of
Proposition A.5.6, any complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$ admits a morphism into a
complex ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ over
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}_{\mathsf{prj}}$ with a cone
absolutely acyclic with respect to
$X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. If the complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ was homotopy
flat, it follows that the complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is homotopy
flat, too. This means that
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is a homotopy
projective complex of locally cotorsion contraherent cosheaves on $X$. If the
complex ${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ was
also acyclic over $X{\operatorname{\mathsf{--ctrh}}}$, so is the complex
${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$. It follows
that ${\mathfrak{P}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is
acyclic over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{lct}}$, and therefore
contractible. We have proven that the complex
${\mathfrak{F}}^{\text{\smaller\smaller$\scriptstyle\bullet$}}$ is absolutely
acyclic over $X{\operatorname{\mathsf{--ctrh}}}^{\mathsf{fl}}$. ∎

###  5.11. Special inverse image of contraherent cosheaves

Recall that an affine morphism of schemes $f\colon Y\longrightarrow X$ is
called _finite_ if for any affine open subscheme $U\subset X$ the ring
${\mathcal{O}}_{Y}(f^{-1}(U))$ is a finitely generated module over the ring
${\mathcal{O}}_{X}(U)$. One can easily see that this condition on a morphism
$f$ is local in $X$.

Let $f\colon Y\longrightarrow X$ be a finite morphism of locally Noetherian
schemes. Given a quasi-coherent sheaf ${\mathcal{M}}$ on $X$, one defines the
quasi-coherent sheaf $f^{!}{\mathcal{M}}$ on $Y$ by the rule

$(f^{!}{\mathcal{M}})(V)={\mathcal{O}}_{Y}(V)\otimes_{{\mathcal{O}}_{Y}(f^{-1}(U))}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{Y}(f^{-1}(U)),\>{\mathcal{M}}(U))$

for any affine open subschemes $V\subset Y$ and $U\subset X$ such that
$f(V)\subset U$ [29, Section III.6]. The construction is well-defined, since
for any pair of embedded affine open subschemes $U^{\prime}\subset U\subset X$
one has

$\operatorname{Hom}_{{\mathcal{O}}_{X}(U^{\prime})}({\mathcal{O}}_{Y}(f^{-1}(U^{\prime}),\>{\mathcal{M}}(U^{\prime}))\\\
\simeq\operatorname{Hom}_{{\mathcal{O}}_{X}(U^{\prime})}({\mathcal{O}}_{X}(U^{\prime})\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{O}}_{Y}(f^{-1}(U)),\>{\mathcal{O}}_{X}(U^{\prime})\otimes_{{\mathcal{O}}_{X}(U)}{\mathcal{M}}(U))\\\
\simeq{\mathcal{O}}_{X}(U^{\prime})\otimes_{{\mathcal{O}}_{X}(U)}\operatorname{Hom}_{{\mathcal{O}}_{X}(U)}({\mathcal{O}}_{Y}(f^{-1}(U)),\>{\mathcal{M}}(U)).$

Indeed, one has $\operatorname{Hom}_{R}(L,\>F\otimes_{R}M)\simeq
F\otimes_{R}\operatorname{Hom}_{R}(L,M)$ for any module $M$, finitely
presented module $L$, and flat module $F$ over a commutative ring $R$. (See
Section 3.3 for a treatment of the non-semi-separatedness issue.)

The functor $f^{!}\colon X{\operatorname{\mathsf{--qcoh}}}\longrightarrow
Y{\operatorname{\mathsf{--qcoh}}}$ is right adjoint to the exact functor
$f_{*}\colon Y{\operatorname{\mathsf{--qcoh}}}\longrightarrow
X{\operatorname{\mathsf{--qcoh}}}$. Indeed, it suffices to define a morphism
of quasi-coherent sheaves on $Y$ on the modules of sections over the affine
open subschemes $f^{-1}(U)\subset Y$. So given quasi-coherent sheaves
${\mathcal{M}}$ on $X$ and ${\mathcal{N}}$ on $Y$, both groups of morphisms
$\operatorname{Hom}_{X}(f_{*}{\mathcal{N}},{\mathcal{M}})$ and
$\operatorname{Hom}_{Y}({\mathcal{N}},f^{!}{\mathcal{M}})$ are identified with
the group of all compatible collections of morphisms of
${\mathcal{O}}_{X}(U)$-modules