1 Introduction
Over the last few decades, the theory of support varieties has played an increasingly important role in various aspects of representation theory, as this theory has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. The prototype for this has been Quillen’s [Reference Quillen19] description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Carlson [Reference Carlson10] introduced support varieties for modular representations. Their work has inspired the development of analogous theories in various contexts: restricted Lie algebras [Reference Friedlander and Parshall14]; complete intersections in commutative algebra [Reference Avramov2, Reference Avramov and Buchweitz3]; certain finite dimensional algebras [Reference Erdmann, Holloway, Snashall, Solberg and Taillefer12]; and finite group schemes [Reference Friedlander and Pevtsova15, Reference Friedlander and Pevtsova16].
Based on certain localization functors on the category, Benson et al. [Reference Benson, Iyengar and Krause5–Reference Benson, Iyengar and Krause7] developed theories of support and cosupport for objects in any compactly generated triangulated categories admitting set-indexed coproducts. The foundation of their approach is constructions of local cohomology and local homology functors on triangulated categories with respect to a central ring of operators. These works have influenced some of the subsequent work on this topic: Avramov and Iyengar [Reference Avramov and Iyengar4] address the problem of realizing modules over arbitrary associative rings with prescribed cohomological support; Krause [Reference Krause17] studies the classification of thick subcategories of modules over commutative Noetherian rings. Lastly, these works play a pivotal role on a classification theorem for the localizing subcategories of the stable module category of a finite group (see [Reference Benson, Iyengar and Krause8]).
Despite the many ways in which cosupport is dual to the established notion of support, cosupport seems to be more elusive, even in the setting of commutative Noetherian rings. Let R be a graded-commutative Noetherian ring and
$\mathscr {T}$
a compactly generated R-linear triangulated category with small coproducts. This paper focuses on cosupport in
$\mathscr {T}$
.
In Section 4, we characterize cosupport of objects using the boundedness of local cohomology and local homology and seek an efficient way to compute cosupport.
Theorem A. For any object X in
$\mathscr {T}$
, one has the equality
$$ \begin{align*}\mathrm{cosupp}_{R}X=\bigcup_{C\in\mathscr{T}^c,\, \mathfrak{p}\in\mathrm{Spec}R}\mathrm{supp}_R\mathrm{H}^\ast_{C/\hspace {-1.9pt}/\mathfrak{p}}(V^{\mathcal{Z}(\mathfrak{p})}X).\end{align*} $$
In Section 5, we give the comparison of support and cosupport and obtain the following inclusion for cohomologically finite objects.
Theorem B. Let X be a cohomologically finite object in
$\mathscr {T}$
. If
$\mathscr {T}$
is generated by a single compact object or
$\mathrm {dim}(\mathrm {supp}_{R}X)<\infty $
, then there is an inequality
$$ \begin{align*}\mathrm{cosupp}_{R}X\subseteq\mathrm{supp}_{R}X.\end{align*} $$
In Section 6, the big cosupport of objects is studied, and the relation of this notion to cosupport is found.
Our results show that one can get a satisfactory version of the cosupport theories in the setting of triangulated categories, compatible with the known results for the cosupport for modules and complexes.
2 Preliminaries
This section collects some notions of triangulated categories for use throughout this paper. For terminology, we follow [Reference Asadollahi, Salarian and Sazeedeh1, Reference Benson, Iyengar and Krause5–Reference Benson, Iyengar and Krause7].
Compact generation. Let
$\mathscr {T}$
be a triangulated category admitting set-indexed coproducts. An object C in
$\mathscr {T}$
is compact if the functor
$\mathrm {Hom}_{\mathscr {T}}(C,-)$
commutes with all coproducts; we write
$\mathscr {T}^c$
for the full subcategory of compact objects in
$\mathscr {T}$
. The category
$\mathscr {T}$
is compactly generated if it is generated by a set of compact objects. A localizing subcategory of
$\mathscr {T}$
is a full triangulated subcategory that is closed under taking coproducts. We write
$\mathrm {Loc}_{\mathscr {T}}(\mathscr {X})$
for the smallest localizing subcategory containing a given class of objects
$\mathscr {X}$
in
$\mathscr {T}$
, and call it the localizing subcategory generated by
$\mathscr {X}$
. Analogously, a colocalizing subcategory of
$\mathscr {T}$
is a full triangulated subcategory that is closed under taking all products, and
$\mathrm {Coloc}_{\mathscr {T}}(\mathscr {X})$
denotes the colocalizing subcategory of
$\mathscr {T}$
that is cogenerated by
$\mathscr {X}$
. We write
$\mathrm {Thick}_{\mathscr {T}}(\mathscr {X})$
for the smallest thick subcategory containing
$\mathscr {X}$
.
Recall that we write
$\Sigma $
for the suspension on
$\mathscr {T}$
. For objects X and Y in
$\mathscr {T}$
, let
$$ \begin{align*}\mathrm{Hom}^\ast_{\mathscr{T}}(X,Y):=\bigoplus_{i\in\mathbb{Z}}\mathrm{Hom}_{\mathscr{T}}(X,\Sigma^iY)\end{align*} $$
be the graded abelian group of morphisms. Set
$\mathrm {End}^\ast _{\mathscr {T}}(X)=\mathrm {Hom}^\ast _{\mathscr {T}}(X,X)$
; this is a graded ring, and
$\mathrm {Hom}^\ast _{\mathscr {T}}(X,Y)$
is a right
$\mathrm {End}^\ast _{\mathscr {T}}(X)$
and left
$\mathrm {End}^\ast _{\mathscr {T}}(Y)$
-bimodule.
Central ring actions. Let R be a graded-commutative ring, that is, R is
$\mathbb {Z}$
-graded and satisfies
$r\cdot s=(-1)^{|r||s|}s\cdot r$
for any homogeneous elements
$r,s$
in R, where
$|r|$
denotes the degree of r. We say that the triangulated category
$\mathscr {T}$
is R-linear if there is a homomorphism
$R\rightarrow Z^\ast (\mathscr {T})$
of graded rings, where
$Z^\ast (\mathscr {T})$
is the graded center of
$\mathscr {T}$
. This yields for each object X a homomorphism
$\phi _X:R\rightarrow \mathrm {End}^\ast _{\mathscr {T}}(X)$
of graded rings such that for all objects
$X, Y\in \mathscr {T}$
, the R-module structures on
$\mathrm {Hom}^\ast _{\mathscr {T}}(X,Y)$
induced by
$\phi _X$
and
$\phi _Y$
agree, up to the usual sign rule.
We write
$\mathrm {Spec}R$
for the set of homogeneous prime ideals of R. Fix a point
$\mathfrak {p}\in \mathrm {Spec}R$
. We write
$R_{\mathfrak {p}}$
for the homogeneous localization of R with respect to
$\mathfrak {p}$
; it is a graded local ring in the sense of Bruns and Herzog [Reference Bruns and Herzog9, 1.5.13], with maximal ideal
$\mathfrak {p}R_{\mathfrak {p}}$
. Given an R-module M, we let
$M_{\mathfrak {p}}$
denote the homogeneous localization of M at
$\mathfrak {p}$
. The module M is called
$\mathfrak {p}$
-local if the natural map
$M\rightarrow M_{\mathfrak {p}}$
is bijective. The module M is called
$\mathfrak {p}$
-torsion if each element of M is annihilated by a power of
$\mathfrak {p}$
. We also set
$$ \begin{align*}\mathcal{U}(\mathfrak{p})=\{\mathfrak{q}\in \mathrm{Spec}R \mid \mathfrak{q}\subseteq\mathfrak{p}\} \quad\text{and}\quad\mathcal{Z}(\mathfrak{p})=\{\mathfrak{q}\in \mathrm{Spec}R \mid \mathfrak{q}\nsubseteq\mathfrak{p}\}.\end{align*} $$
Given a homogeneous ideal
$\mathfrak {a}$
in R, we set
$$ \begin{align*}\mathcal{V}(\mathfrak{a})=\{\mathfrak{p}\in\textrm{Spec}R \mid \mathfrak{a}\subseteq\mathfrak{p}\}.\end{align*} $$
Let
$\mathcal {U}$
be a subset of
$\mathrm {Spec}R$
. The specialization closure of
$\mathcal {U}$
is the set
$$ \begin{align*}\mathrm{cl}\mathcal{U}=\{\mathfrak{p}\in\textrm{Spec}R \mid \textrm{there\ is}\ \mathfrak{q}\in\mathcal{U}\ \textrm{with}\ \mathfrak{q}\subseteq\mathfrak{p}\}.\end{align*} $$
The subset
$\mathcal {U}$
is specialization closed if
$\mathrm {cl}\mathcal {U}=\mathcal {U}$
. Note that the subsets
$\mathcal {V}(\mathfrak {a})$
and
$\mathcal {Z}(\mathfrak {p})$
are specialization closed.
From now on, R denotes a graded-commutative Noetherian ring and
$\mathscr {T}$
a compactly generated R-linear triangulated category with set-indexed coproducts.
Local cohomology and homology. An exact functor
$L:\mathscr {T}\rightarrow \mathscr {T}$
is called a localization functor if there exists a morphism
$\eta :\mathrm {Id}_{\mathscr {T}}\rightarrow L$
such that the morphism
$L\eta :L\rightarrow L^2$
is invertible and
$L\eta =\eta L$
. Let
$\mathcal {V}$
be a specialization closed subset of
$\mathrm {Spec}R$
and
$L_{\mathcal {V}}$
the associated localization functor. By [Reference Benson, Iyengar and Krause5, Definition 3.2], one then gets an exact functor
$\Gamma _{\mathcal {V}}$
on
$\mathscr {T}$
and for each object X, an exact triangle
$$ \begin{align*}\Gamma_{\mathcal{V}}X\rightarrow X\rightarrow L_{\mathcal{V}}X\rightsquigarrow\!.\end{align*} $$
We call
$\Gamma _{\mathcal {V}}X$
the local cohomology of X supported on
$\mathcal {V}$
, and the essential image of
$\Gamma _{\mathcal {V}}$
is denoted by
$\mathscr {T}_{\mathcal {V}}$
. By [Reference Benson, Iyengar and Krause5, Corollary 6.5], the functors
$L_{\mathcal {V}}$
and
$\Gamma _{\mathcal {V}}$
on
$\mathscr {T}$
preserve coproducts, and hence have right adjoints by Brown representability. Let
$\Lambda ^{\mathcal {V}}$
and
$V^{\mathcal {V}}$
denote right adjoints of
$\Gamma _{\mathcal {V}}$
and
$L_{\mathcal {V}}$
, respectively. They induce a functorial exact triangle
$$ \begin{align*}V^{\mathcal{V}}X\rightarrow X\rightarrow \Lambda^{\mathcal{V}}X\rightsquigarrow,\end{align*} $$
which gives rise to an exact local homology functor
$\Lambda ^{\mathcal {V}}:\mathscr {T}\rightarrow \mathscr {T}$
. The essential image of
$\Lambda ^{\mathcal {V}}$
is denoted by
$\mathscr {T}^{\mathcal {V}}$
.
Koszul objects. Let
$r\in R$
be a homogeneous element and X an object in
$\mathscr {T}$
. We denote by
$X/\hspace {-0.11cm}/r$
any object that appears in an exact triangle
$$ \begin{align*}X\xrightarrow{r}\Sigma^{|r|}X\rightarrow X/\hspace{-0.11cm}/r\rightsquigarrow,\end{align*} $$
and call it a Koszul object of r on X. It is well defined up to (nonunique) isomorphism. Given a homogeneous ideal
$\mathfrak {a}$
in R, we write
$X/\hspace {-0.11cm}/\mathfrak {a}$
for any Koszul object obtained by iterating the construction above with respect to some finite sequence of generators for
$\mathfrak {a}$
. This object may depend on the choice of the minimal generating sequence for
$\mathfrak {a}$
.
Cohomologically finite objects. For any objects
$C\in \mathscr {T}^c$
and
$X\in \mathscr {T}$
, set
$$ \begin{align*}\mathrm{H}^\ast_{C}(X)=\mathrm{Hom}^\ast_{\mathscr{T}}(C,X).\end{align*} $$
We say that an object X in
$\mathscr {T}$
is cohomologically finite with respect to C if
$\mathrm {H}^\ast _{C}(X)$
is finitely generated as a graded R-module. The object X is called cohomologically finite if it is cohomologically finite with respect to any
$C\in \mathscr {T}^c$
. We also denote
$$ \begin{align*}\mathrm{inf}_C(X)&=\mathrm{inf}(\mathrm{H}^\ast_C(X))=\mathrm{inf}_C\{n\in\mathbb{Z}\hspace{0.03cm}|\hspace{0.03cm}\mathrm{H}^n_C(X)\neq0\},\\ \mathrm{sup}_C(X)&=\mathrm{sup}(\mathrm{H}^\ast_C(X))=\mathrm{sup}_C\{n\in\mathbb{Z}\hspace{0.03cm}|\hspace{0.03cm}\mathrm{H}^n_C(X)\neq0\}.\end{align*} $$
We assume throughout this paper that R is a graded-commutative Noetherian ring and
$\mathscr {T}$
a compactly generated R-linear triangulated category with set-indexed coproducts.
3 Boundedness of local cohomology and homology
In this section, we provide explicit formulas for computing boundedness of local cohomology and local homology using Koszul objects.
Let
$X_1\xrightarrow {u_1}X_2\xrightarrow {u_2}X_3\xrightarrow {u_3}\cdots $
be a sequence of morphisms in
$\mathscr {T}$
. Its homotopy colimit, denoted by
$\mathrm {hocolim }X_i$
, is defined by an exact triangle
$$ \begin{align*}\bigoplus_{i\geqslant 1}X_i\xrightarrow{\theta}\bigoplus_{i\geqslant 1}X_i\rightarrow\mathrm{hocolim }X_i\rightsquigarrow,\end{align*} $$
where
$\theta $
is the map
$(\mathrm {id}-u_i)$
.
Fix a homogeneous element
$r\in R$
. For each object X in
$\mathscr {T}$
, consider the following commutative diagram:
where each vertical sequence is given by the exact triangle defining
$X/\hspace {-0.11cm}/r^n$
, and the morphisms in the last row are those induced by the commutativity of the upper squares.
Lemma 3.1 [Reference Benson, Iyengar and Krause6, Proposition 2.9]
Let
$r\in R$
be a homogeneous element and X an object in
$\mathscr {T}$
. Then the adjunction morphism
$\Gamma _{\mathcal {V}(r)}X\rightarrow X$
induces an isomorphism
$$ \begin{align*}\mathrm{hocolim }\Sigma^{-1}(X/\hspace{-0.11cm}/r^n)\stackrel{\sim}\longrightarrow\Gamma_{\mathcal{V}(r)}X.\end{align*} $$
The following result provides a formula for computing
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(r)}X)$
, which recovers part of [Reference Foxby and Iyengar13, Theorem 2.1].
Theorem 3.2. Let
$r\in R$
be a homogeneous element and X an object in
$\mathscr {T}$
.
-
(1) If
$|r|\leqslant 0$
, then
$\mathrm {inf}_C(X/\hspace {-0.11cm}/r)+1= \mathrm {inf}_C(\Gamma _{\mathcal {V}(r)}X)$
for any
$C\in \mathscr {T}^c$
. -
(2) If
$|r|>0$
, then
$\mathrm {inf}_C(X/\hspace {-0.11cm}/r)+|r|=\mathrm {inf}_C(\Gamma _{\mathcal {V}(r)}X)$
for any
$C\in \mathscr {T}^c$
.
In particular, if
$\mathfrak {a}$
is a homogeneous ideal of R, then
$$ \begin{align*}\mathrm{H}^{\ast}_C(\Gamma_{\mathcal{V}(\mathfrak{a})}X)\neq0\Longleftrightarrow\mathrm{H}^{\ast}_{C/\hspace {-1.9pt}/\mathfrak{a}}(X)\neq0 \Longleftrightarrow\mathrm{H}^{\ast}_C(X/\hspace{-0.11cm}/\mathfrak{a})\neq0\quad \text{for all}\ C\in\mathscr{T}^c.\end{align*} $$
Proof. Assume that
$|r|\leqslant 0$
. The octahedral axiom yields a commutative diagram of exact triangles in
$\mathscr {T}$
:
The third column triangle induces an exact sequence
$$ \begin{align*}\mathrm{H}^{i+|r|-1}_C(X/\hspace{-0.11cm}/r)\rightarrow\mathrm{H}^i_C(X/\hspace{-0.11cm}/r)\rightarrow\mathrm{H}^i_C(X/\hspace{-0.11cm}/r^2) \rightarrow\mathrm{H}^{i+|r|}_C(X/\hspace{-0.11cm}/r),\end{align*} $$
which implies that
$\mathrm {inf}_C(X/\hspace {-0.11cm}/r^2)\geqslant \mathrm {inf}_C(X/\hspace {-0.11cm}/r)$
. By repeating this process, one has that
$\mathrm {inf}_C(X/\hspace {-0.11cm}/r^n)\geqslant \mathrm {inf}_C(X/\hspace {-0.11cm}/r)$
for any
$n\geqslant 1$
. By Lemma 3.1,
$\mathrm {H}^\ast _C(\Gamma _{\mathcal {V}(r)}X)\cong \mathrm {colim}\mathrm {H}^\ast _C(\Sigma ^{-1}(X/\hspace {-0.11cm}/r^n))$
; it follows that
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(r)}X)\geqslant \mathrm {inf}_C(X/\hspace {-0.11cm}/r)+1$
. However, [Reference Benson, Iyengar and Krause6, Lemma 2.6] gives us an exact triangle
$\Gamma _{\mathcal {V}(r)}X\xrightarrow {r}\Sigma ^{|r|}\Gamma _{\mathcal {V}(r)}X\rightarrow X/\hspace {-0.11cm}/r\rightsquigarrow $
, which induces an exact sequence
$$ \begin{align*}\mathrm{H}^{i+|r|}_C(\Gamma_{\mathcal{V}(r)}X)\rightarrow\mathrm{H}^i_C(X/\hspace{-0.11cm}/r) \rightarrow\mathrm{H}^{i+1}_C(\Gamma_{\mathcal{V}(r)}X)\rightarrow\mathrm{H}^{i+|r|+1}_C(\Gamma_{\mathcal{V}(r)}X).\end{align*} $$
Therefore, one gets that
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(r)}X)\leqslant \mathrm {inf}_C(X/\hspace {-0.11cm}/r)+1$
.
Assume that
$|r|>0$
. Then the exact triangle
$\Gamma _{\mathcal {V}(r)}X\xrightarrow {r}\Sigma ^{|r|}\Gamma _{\mathcal {V}(r)}X\rightarrow X/\hspace {-0.11cm}/r\rightsquigarrow $
implies that
$\mathrm {inf}_C(X/\hspace {-0.11cm}/r)+|r|=\mathrm {inf}_C(\Gamma _{\mathcal {V}(r)}X/\hspace {-0.11cm}/r)+|r|= \mathrm {inf}_C(\Gamma _{\mathcal {V}(r)}X)$
, where the last equality holds by [Reference Asadollahi, Salarian and Sazeedeh1, Lemma 4.8].
This completes the proof of the desired equality.
Corollary 3.3. Let
$r_1,r_2$
be homogeneous elements in R. For any object X in
$\mathscr {T}$
, we have
$$ \begin{align*}\mathrm{inf}_C(\Gamma_{\mathcal{V}(r_1)}X)\leqslant\mathrm{inf}_C(\Gamma_{\mathcal{V}(r_1,r_2)}X)\quad \text{for all}\ C\in\mathscr{T}^c.\end{align*} $$
In particular, for any homogeneous ideals
$\mathfrak {b}\subseteq \mathfrak {a}$
of R and any
$C\in \mathscr {T}^c$
, one has that
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(\mathfrak {b})}X)\leqslant \mathrm {inf}_C(\Gamma _{\mathcal {V}(\mathfrak {a})}X)$
.
Proof. The exact triangle
$X/\hspace {-0.11cm}/r_1\xrightarrow {r_2}\Sigma ^{|r_2|}X/\hspace {-0.11cm}/r_1\rightarrow X/\hspace {-0.11cm}/(r_1,r_2)\rightsquigarrow $
induces an exact sequence
$$ \begin{align} \mathrm{H}^{i+|r_2|}_C(X/\hspace{-0.11cm}/r_1)\rightarrow\mathrm{H}^i_C(X/\hspace{-0.11cm}/(r_1,r_2))\rightarrow\mathrm{H}^{i+1}_C(X/\hspace{-0.11cm}/r_1) \rightarrow\mathrm{H}^{i+|r_2|+1}_C(X/\hspace{-0.11cm}/r_1). \end{align} $$
First, assume that
$|r_1|\leqslant 0$
. If
$|r_2|\leqslant 0$
, then the sequence
implies that
$\mathrm {inf}_C(X/\hspace {-0.11cm}/r_1)\leqslant \mathrm {inf}_C(X/\hspace {-0.11cm}/(r_1,r_2))+1$
. Hence,
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_1)}X)=\mathrm {inf}_C(X/\hspace {-0.11cm}/r_1)+1\leqslant \mathrm {inf}_C(X/\hspace {-0.11cm}/(r_1,r_2))+2=\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_1,r_2)}X)$
by Theorem 3.2(1). If
$|r_2|>0$
, then
$\mathrm {inf}_C(X/\hspace {-0.11cm}/r_1)=\mathrm {inf}_C(X/\hspace {-0.11cm}/(r_1,r_2))+|r_2|$
by [Reference Asadollahi, Salarian and Sazeedeh1, Lemma 4.8]. So
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_1)}X)=\mathrm {inf}_C(X/\hspace {-0.11cm}/r_1)+1=\mathrm {inf}_C(X/\hspace {-0.11cm}/(r_1,r_2))+|r_2|+1 =\mathrm {inf}_C\Gamma _{\mathcal {V}(r_2)}(X/\hspace {-0.11cm}/r_1)+1=\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_2)}X/\hspace {-0.11cm}/r_1)+1=\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_1,r_2)}X)$
by Theorem 3.2. Now assume that
$|r_1|>0$
. If
$|r_2|\leqslant 0$
, then
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_1)}X)=\mathrm {inf}_C(X/\hspace {-0.11cm}/r_1)+|r_1|\leqslant \mathrm {inf}_C(X/\hspace {-0.11cm}/(r_1,r_2))+|r_1|+1 =\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_1)}(X/\hspace {-0.11cm}/r_2))+1=\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_1,r_2)}X)$
by Theorem 3.2. If
$|r_2|>0$
, then
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_1)}X)=\mathrm {inf}_C(X/\hspace {-0.11cm}/r_1)+|r_1|=\mathrm {inf}_C(X/\hspace {-0.11cm}/(r_1,r_2))+|r_1|+|r_2|=\mathrm {inf}_C(\Gamma _{\mathcal {V}(r_1,r_2)}X)$
by Theorem 3.2(2), as claimed.
Since
$\mathscr {T}$
is compactly generated with set-indexed coproducts, it follows from [Reference Neeman18, Proposition 8.4.6] that
$\mathscr {T}$
also admits set-indexed products. Let
$\cdots \xrightarrow {u_4}X_3\xrightarrow {u_3}X_2\xrightarrow {u_2}X_1$
be a sequence of morphisms in
$\mathscr {T}$
. Its homotopy limit, denoted by
$\mathrm {holim }X_i$
, is defined by an exact triangle
$$ \begin{align*}\mathrm{holim }X_i\rightarrow\prod_{i\geqslant 1}X_i\xrightarrow{\theta}\prod_{i\geqslant 1}X_i\rightsquigarrow,\end{align*} $$
where
$\theta $
is the map
$(\mathrm {id}-u_{i+1})$
.
Fix a homogeneous element
$r\in R$
. For each object X in
$\mathscr {T}$
, consider the following commutative diagram:
The next result gives an explicit computation of
$\Lambda ^{\mathcal {V}(r)}X$
.
Lemma 3.4. Let
$r\in R$
be a homogeneous element and X an object in
$\mathscr {T}$
. Then the adjunction morphism
$X\rightarrow \Lambda ^{\mathcal {V}(r)}X$
induces an isomorphism
$$ \begin{align*}\Lambda^{\mathcal{V}(r)}X\stackrel{\sim}\longrightarrow\mathrm{holim }\Sigma^{-|r^n|}(X/\hspace{-0.11cm}/r^n).\end{align*} $$
Proof. By [Reference Benson, Iyengar and Krause7, Lemma 4.12], each
$\Sigma ^{-|r^n|}(X/\hspace {-0.11cm}/r^n)$
is in
$\mathscr {T}^{\mathcal {V}(r)}$
, and hence so is
$\mathrm {holim }\Sigma ^{-|r^n|}(X/\hspace {-0.11cm}/r^n)$
. So
$\mathrm {holim }\Sigma ^{-|r^n|}(X/\hspace {-0.11cm}/r^n)\stackrel {\sim }\rightarrow \Lambda ^{\mathcal {V}(r)}\mathrm {holim }\Sigma ^{-|r^n|}(X/\hspace {-0.11cm}/r^n)$
. Fix
$C\in \mathscr {T}^c$
, by [Reference Benson, Iyengar and Krause6, Proposition 2.9],
$$ \begin{align*}\mathrm{Hom}^\ast_{\mathscr{T}}(C,\Lambda^{\mathcal{V}(r)}\mathrm{holim }\Sigma^{-|r^n|}X) \cong\mathrm{Hom}^\ast_{\mathscr{T}}(\Gamma_{\mathcal{V}(r)}\mathrm{hocolim }\Sigma^{|r^n|}C,X)=0,\end{align*} $$
which implies that
$\Lambda ^{\mathcal {V}(r)}X\stackrel {\sim }\rightarrow \Lambda ^{\mathcal {V}(r)}\mathrm {holim }\Sigma ^{-|r^n|}(X/\hspace {-0.11cm}/r^n)$
. This shows our claim.
By analogy with the proof of Theorem 3.2, one can give the following formula for computing
$\mathrm {sup}_C(\Lambda ^{\mathcal {V}(r)}X)$
, which recovers part of [Reference Foxby and Iyengar13, Theorem 4.1].
Theorem 3.5. Let
$r\in R$
be a homogeneous element and X an object in
$\mathscr {T}$
.
-
(1) If
$|r|\leqslant 0$
, then
$\mathrm {sup}_C(X/\hspace {-0.11cm}/r)+|r|= \mathrm {sup}_C(\Lambda ^{\mathcal {V}(r)}X)$
for any
$C\in \mathscr {T}^c$
. -
(2) If
$|r|>0$
, then
$\mathrm {sup}_C(X/\hspace {-0.11cm}/r)+1= \mathrm {sup}_C(\Lambda ^{\mathcal {V}(r)}X)$
for any
$C\in \mathscr {T}^c$
.
In particular, if
$\mathfrak {a}$
is a homogeneous ideal of R, then
$$ \begin{align*}\mathrm{H}^{\ast}_C(\Lambda^{\mathcal{V}(\mathfrak{a})}X)\neq0\Longleftrightarrow\mathrm{H}^{\ast}_{C/\hspace {-1.9pt}/\mathfrak{a}}(X)\neq0 \Longleftrightarrow\mathrm{H}^{\ast}_C(X/\hspace{-0.11cm}/\mathfrak{a})\neq0.\end{align*} $$
Corollary 3.6. Let
$\mathfrak {a}$
be a homogeneous ideal of R and X an object in
$\mathscr {T}$
. Then the following hold.
-
(1)
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(\mathfrak {a})}X)-\mathrm {inf}_C(X)\geqslant 0$
for any
$C\in \mathscr {T}^c$
. -
(2)
$\mathrm {sup}_C(\Lambda ^{\mathcal {V}(\mathfrak {a})}X)-\mathrm {sup}_C(X)\leqslant 0$
for any
$C\in \mathscr {T}^c$
. -
(3)
$\mathrm {sup}_C(\Gamma _{\mathcal {V}(\mathfrak {a})}X)\geqslant \mathrm {sup}_C(\Lambda ^{\mathcal {V}(\mathfrak {a})}X)$
for any
$C\in \mathscr {T}^c$
. -
(4)
$\mathrm {inf}_C(\Lambda ^{\mathcal {V}(\mathfrak {a})}X)\leqslant \mathrm {inf}_C(\Gamma _{\mathcal {V}(\mathfrak {a})}X)$
for any
$C\in \mathscr {T}^c$
.
In particular,
$\Gamma _{\mathcal {V}(\mathfrak {a})}X\neq 0$
if and only if
$\Lambda ^{\mathcal {V}(\mathfrak {a})}X\neq 0$
.
Proof. We only prove items (1) and (3) since the proofs of items (2) and (4) are similar.
(1) If
$|r|\leqslant 0$
, then
$\mathrm {inf}_C(X/\hspace {-0.11cm}/r)\geqslant \mathrm {inf}_C(X)-1$
. Hence, Theorem 3.2(1) implies that
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(r)}X)-\mathrm {inf}_C(X)\geqslant 0$
. If
$|r|>0$
, then
$\mathrm {inf}_C(X/\hspace {-0.11cm}/r)=\mathrm {inf}_C(X)-|r|$
. Hence, Theorem 3.2(2) implies that
$\mathrm {inf}_C(\Gamma _{\mathcal {V}(r)}X)-\mathrm {inf}_C(X)=0$
, as desired.
(3) Consider the exact triangle
$\Gamma _{\mathcal {V}(r)}X\xrightarrow {r}\Sigma ^{|r|}\Gamma _{\mathcal {V}(r)}X\rightarrow X/\hspace {-0.11cm}/r\rightsquigarrow $
. If
$|r|\leqslant 0$
, then
$\mathrm {sup}_C(X/\hspace {-0.11cm}/r)\leqslant \mathrm {sup}_C(\Gamma _{\mathcal {V}(r)}X)-|r|$
. Hence, Theorem 3.5(1) implies that
$\mathrm {sup}_C(\Gamma _{\mathcal {V}(r)}X)\geqslant \mathrm {sup}_C(\Lambda ^{\mathcal {V}(r)}X)$
. If
$|r|>0$
, then
$\mathrm {sup}_C(X/\hspace {-0.11cm}/r)=\mathrm {sup}_C(\Gamma _{\mathcal {V}(r)}X)-1$
. Hence, Theorem 3.5(2) implies that
$\mathrm {sup}_C(\Gamma _{\mathcal {V}(r)}X)=\mathrm {sup}_C(\Lambda ^{\mathcal {V}(r)}X)$
, as desired.
Remark 3.7. (1) Let R be an
$\mathbb {N}_0$
-graded-commutative Noetherian ring and
$\mathscr {T}=\mathrm {D}(R)$
the derived category of DG-modules over R. Then
$\mathfrak {m}=(\bigoplus _{i\geqslant 1}R_i)\oplus \mathfrak {m}_0$
is the maximal ideal of R, where
$(R_0,\mathfrak {m}_0)$
is a local ring. The number
$\mathrm {inf}(\Gamma _{\mathcal {V}(\mathfrak {m})}X)$
is exactly the depth of the DG-module X introduced by Shaul in [Reference Sather-Wagstaff and Wicklein21].
(2) Let A be a commutative Noetherian ring and
$\mathscr {T}=\mathrm {D}(A)$
the derived category of A-complexes. Let
$\mathfrak {a}$
be an ideal in A and K the Koszul complex on a sequence of n generators for
$\mathfrak {a}$
. Then Theorem 3.2 implies that
$\mathrm {inf}(\Gamma _{\mathcal {V}(\mathfrak {a})}X)=\mathrm {inf}(K\otimes _AX)+n$
; this common value is exactly the
$\mathfrak {a}$
-depth of the complex X introduced by Foxby and Iyengar in [Reference Foxby and Iyengar13]. Additionally, Theorem 3.5 implies that
$-\mathrm {sup}(\Lambda ^{\mathcal {V}(\mathfrak {a})}Y)=-\mathrm {sup}(K\otimes _AY)$
, this common value is exactly the
$\mathfrak {a}$
-width of the complex Y introduced by Foxby and Iyengar in [Reference Foxby and Iyengar13].
4 Cosupport
Since cosupport is not as well understood as support, we devote this section to some characterizations of cosupport and give the proof of Theorem A.
For each
$\mathfrak {p}$
in
$\mathrm {Spec}R$
, denote the exact functor
$V^{\mathcal {Z}(\mathfrak {p})}\Lambda ^{\mathcal {V}(\mathfrak {p})}\mathscr {T}\rightarrow \mathscr {T}$
by
$\Lambda ^{\mathfrak {p}}$
. The essential image of
$\Lambda ^{\mathfrak {p}}$
is denoted by
$\mathscr {T}^{\{\mathfrak {p}\}}$
; it is a colocalizing subcategory of
$\mathscr {T}$
.
The cosupport of an object X in
$\mathscr {T}$
is a subset of
$\mathrm {Spec}R$
defined as follows:
$$ \begin{align*}\mathrm{cosupp}_RX=\{\mathfrak{p}\in \mathrm{Spec}R\hspace{0.03cm}|\hspace{0.03cm}\Lambda^{\mathfrak{p}}X\neq0\}.\end{align*} $$
For each compact object C in
$\mathscr {T}$
and each injective R-module I, Brown representability yields an object
$T_C(I)$
in
$\mathscr {T}$
and a natural isomorphism:
$$ \begin{align*}\mathrm{Hom}^\ast_{\mathscr{T}}(-,T_C(I))=\mathrm{Hom}^\ast_R(\mathrm{H}^\ast_C(-),I).\end{align*} $$
For
$\mathfrak {p}\in \mathrm {Spec}R$
, let
$I(\mathfrak {p})$
denote the injective envelope of
$R/\mathfrak {p}$
. Then
$I(\mathfrak {p})$
cogenerates the category of
$\mathfrak {p}$
-local R-modules. For any object X in
$\mathscr {T}$
, denote
$$ \begin{align*}X_{\mathfrak{p}}=L_{\mathcal{Z}(\mathfrak{p})}X\quad \text{and}\quad X(\mathfrak{p})=X_{\mathfrak{p}}/\hspace{-0.11cm}/\mathfrak{p}.\end{align*} $$
Next we give an axiomatic description of cosupport via Koszul objects, analogous to the one for support in [Reference Benson, Iyengar and Krause5, Proposition 5.12].
Proposition 4.1. Let X be an object in
$\mathscr {T}$
. For each
$\mathfrak {p}\in \mathrm {Spec}R$
and
$C\in \mathscr {T}^c$
, the following conditions are equivalent:
-
(1)
$\mathrm {Hom}^\ast _{\mathscr {T}}(C,\Lambda ^{\mathfrak {p}}X)\neq 0$
; -
(2)
$\mathrm {Hom}^\ast _{\mathscr {T}}(C/\hspace {-0.11cm}/\mathfrak {p},\Lambda ^{\mathfrak {p}}X)\neq 0$
; -
(3)
$\mathrm {Hom}^\ast _{\mathscr {T}}(C/\hspace {-0.11cm}/\mathfrak {p},V^{\mathcal {Z}(\mathfrak {p})}X)\neq 0$
; -
(4)
$\mathrm {Hom}^\ast _{\mathscr {T}}(C/\hspace {-0.11cm}/\mathfrak {p},\Gamma _{\mathcal {V}(\mathfrak {p})}V^{\mathcal {Z}(\mathfrak {p})}X)\neq 0$
; -
(5)
$\mathrm {Hom}^\ast _{\mathscr {T}}(C(\mathfrak {p}),X)\neq 0$
; -
(6)
$\mathrm {Hom}^\ast _{\mathscr {T}}(\Lambda ^{\mathfrak {p}}X,T_{C}(I(\mathfrak {p})))\neq 0$
; -
(7)
$\mathrm {Hom}^\ast _{\mathscr {T}}(\Lambda ^{\mathfrak {p}}X,T_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p})))\neq 0$
; -
(8)
$\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X,T_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p})))\neq 0$
.
Proof. (1)
$\Leftrightarrow $
(2).
$\mathrm {H}^\ast _C(\Lambda ^{\mathfrak {p}}X)\neq 0$
if and only if
$\mathrm {H}^\ast _C(\Gamma _{\mathcal {V}(\mathfrak {p})}V^{\mathcal {Z}(\mathfrak {p})}X)\neq 0$
by Theorems 3.2 and 3.5 if and only if
$\mathrm {H}^\ast _{C/\hspace {-1.9pt}/\mathfrak {p}}(\Gamma _{\mathcal {V}(\mathfrak {p})}V^{\mathcal {Z}(\mathfrak {p})}X)\neq 0$
since
$\mathrm {H}^\ast _C(\Gamma _{\mathcal {V}(\mathfrak {p})}V^{\mathcal {Z}(\mathfrak {p})}X)$
is
$\mathfrak {p}$
-torsion if and only if
$\mathrm {H}^\ast _{C/\hspace {-1.9pt}/\mathfrak {p}}(\Lambda ^{\mathfrak {p}}X)\neq 0$
by Theorems 3.2 and 3.5 again.
(2)
$\Leftrightarrow $
(3) follows from the isomorphisms
$$ \begin{align*} \mathrm {Hom}^\ast _{\mathscr {T}}(C/\hspace {-0.11cm}/\mathfrak {p}, \Lambda ^{\mathfrak {p}}X)\cong \mathrm {Hom}^\ast _{\mathscr {T}}(\Gamma _{\mathcal {V}(\mathfrak {p})}(C/\hspace {-0.11cm}/\mathfrak {p}), V^{\mathcal {Z}(\mathfrak {p})}X) \cong \mathrm {Hom}^\ast _{\mathscr {T}}(C/\hspace {-0.11cm}/\mathfrak {p},V^{\mathcal {Z}(\mathfrak {p})}X), \end{align*} $$
where the first one holds as
$(\Gamma _{\mathcal {V}(\mathfrak {p})},\Lambda ^{\mathcal {V}(\mathfrak {p})})$
is an adjoint pair, and the second one holds since
$\Gamma _{\mathcal {V}(\mathfrak {p})}(C/\hspace {-0.11cm}/\mathfrak {p})\cong C/\hspace {-0.11cm}/\mathfrak {p}$
.
(3)
$\Leftrightarrow $
(4) follows from the isomorphism
$$ \begin{align*} \mathrm {Hom}^\ast _{\mathscr {T}}(C/\hspace {-0.11cm}/\mathfrak {p},V^{\mathcal {Z}(\mathfrak {p})}X)\cong \mathrm {Hom}^\ast _{\mathscr {T}}(C/\hspace {-0.11cm}/\mathfrak {p}, \Gamma _{\mathcal {V}(\mathfrak {p})}V^{\mathcal {Z}(\mathfrak {p})}X), \end{align*} $$
since
$\Gamma _{\mathcal {V}(\mathfrak {p})}$
is a right adjoint of the inclusion
$\mathscr {T}_{\mathcal {V}(\mathfrak {p})}\rightarrow \mathscr {T}$
.
(3)
$\Leftrightarrow $
(5) follows from the isomorphism
$$ \begin{align*} \mathrm {Hom}^\ast _{\mathscr {T}}(C/\hspace {-0.11cm}/\mathfrak {p},V^{\mathcal {Z}(\mathfrak {p})}X)\cong \mathrm {Hom}^\ast _{\mathscr {T}}(L_{\mathcal {Z}(\mathfrak {p})} (C/\hspace {-0.11cm}/\mathfrak {p}), X)\cong \mathrm {Hom}^\ast _{\mathscr {T}}(C(\mathfrak {p}),X) \end{align*} $$
since
$(L_{\mathcal {Z}(\mathfrak {p})},V^{\mathcal {Z}(\mathfrak {p})})$
is an adjoint pair.
(1)
$\Leftrightarrow $
(6) and (2)
$\Leftrightarrow $
(7) follow from the facts that
$I(\mathfrak {p})$
cogenerates the category of
$\mathfrak {p}$
-local R-modules and the graded R-modules
$\mathrm {H}^\ast _{C}(\Lambda ^{\mathfrak {p}}X)$
and
$\mathrm {H}^\ast _{C/\hspace {-1.9pt}/\mathfrak {p}}(\Lambda ^{\mathfrak {p}}X)$
are
$\mathfrak {p}$
-local.
(7)
$\Leftrightarrow $
(8) follows from the isomorphism
$$ \begin{align*} \mathrm {Hom}^\ast _{\mathscr {T}}(\Lambda ^{\mathfrak {p}}X,T_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p})))\cong \mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X, T_{C/\hspace {-1.9pt}/\mathfrak {p}} (I(\mathfrak {p}))) \end{align*} $$
since
$\Lambda ^{\mathcal {V}(\mathfrak {p})}$
is a left adjoint of the inclusion
$\mathscr {T}^{\mathcal {V}(\mathfrak {p})}\rightarrow \mathscr {T}$
and
$T_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p}))$
is in
$\mathscr {T}^{\mathcal {V}(\mathfrak {p})}$
.
The next result recovers part of [Reference Shaul20, Proposition 4.4].
Proposition 4.2. Let X be an object in
$\mathscr {T}$
and
$\mathfrak {p}\in \mathrm {Spec} R$
. The following conditions are equivalent:
-
(1)
$\mathfrak {p}\in \mathrm {cosupp}_RX$
; -
(2)
$V^{\mathcal {Z}(\mathfrak {p})}(X/\hspace {-0.11cm}/\mathfrak {p})\neq 0$
; -
(3)
$\Gamma _{\mathcal {V}(\mathfrak {p})}V^{\mathcal {Z}(\mathfrak {p})}X\neq 0$
; -
(4)
$ \mathfrak {p}\in \mathrm {cosupp}_R(V^{\mathcal {Z}(\mathfrak {p})}X)$
.
Proof. (1)
$\Leftrightarrow $
(2).
$\Lambda ^{\mathcal {V}(\mathfrak {p})}V^{\mathcal {Z}(\mathfrak {p})}X\neq 0$
if and only if
$\mathrm {H}^\ast _C(\Lambda ^{\mathcal {V}(\mathfrak {p})}V^{\mathcal {Z}(\mathfrak {p})}X)\neq 0$
for some
$C\in \mathscr {T}^c$
if and only if
$\mathrm {H}^\ast _{C}((V^{\mathcal {Z}(\mathfrak {p})}X)/\hspace {-0.11cm}/\mathfrak {p})\neq 0$
for some
$C\in \mathscr {T}^c$
by Theorem 3.5 if and only if
$V^{\mathcal {Z}(\mathfrak {p})}(X/\hspace {-0.11cm}/\mathfrak {p})\neq 0$
since
$(V^{\mathcal {Z}(\mathfrak {p})}X)/\hspace {-0.11cm}/\mathfrak {p}\cong V^{\mathcal {Z}(\mathfrak {p})}(X/\hspace {-0.11cm}/\mathfrak {p})$
.
(1)
$\Leftrightarrow $
(3) follows from Corollary 3.6.
(1)
$\Leftrightarrow $
(4) follows from
$V^{\mathcal {Z}(\mathfrak {p})}V^{\mathcal {Z}(\mathfrak {p})}X\cong V^{\mathcal {Z}(\mathfrak {p})}X$
.
Analogous to [Reference Benson, Iyengar and Krause5, Theorem 5.2], we give the following computation of cosupport.
Theorem 4.3. For each object X in
$\mathscr {T}$
, one has the equality
$$ \begin{align*}\mathrm{cosupp}_{R}X &=\bigcup_{C\in\mathscr{T}^c,\ \mathfrak{p}\in\mathrm{Spec}R}\mathrm{cosupp}_R\mathrm{Hom}^\ast_{\mathscr{T}}(V^{\mathcal{Z}(\mathfrak{p})}X,T_{C/\hspace{-1.9pt}/\mathfrak{p}}(I(\mathfrak{p})))\\&=\bigcup_{C\in\mathscr{T}^c,\ \mathfrak{p}\in\mathrm{Spec}R}\mathrm{supp}_R\mathrm{H}^\ast_{C/\hspace{-1.9pt}/\mathfrak{p}}(V^{\mathcal{Z}(\mathfrak{p})}X), \end{align*} $$
where
$\mathrm {cosupp}_R\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X,T_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p})))$
(respectively
$\mathrm {supp}_R\mathrm {H}^\ast _{C/\hspace {-1.9pt}/\mathfrak {p}}(V^{\mathcal {Z}(\mathfrak {p})}X))$
is the cosupport (respectively support) of the graded R-module
$\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X,T_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p})))$
(respectively
$\mathrm {H}^\ast _{C/\hspace {-1.9pt}/\mathfrak {p}}(V^{\mathcal {Z}(\mathfrak {p})}X))$
.
Proof. By [Reference Benson, Iyengar and Krause7, Theorem 4.5 and Proposition 4.4],
$X=0$
if and only if
$\mathrm {H}^\ast _{C/\hspace {-1.9pt}/\mathfrak {p}}(V^{\mathcal {Z}(\mathfrak {p})}X)\cong \mathrm {Hom}^\ast _{\mathscr {T}}(C(\mathfrak {p}),X)=0$
for all
$C\in \mathscr {T}^c$
and
$\mathfrak {p}\in \mathrm {Spec}R$
if and only if we have
$\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X, T_{C/\hspace {-1.9pt}/\mathfrak {p}} (I(\mathfrak {p})))=0$
for all
$C\in \mathscr {T}^c$
and
$\mathfrak {p}\in \mathrm {Spec}R$
. So assume that
$X\neq 0$
. Let
$\mathfrak {p}$
be a point in
$\mathrm {Spec}R$
such that
$\Lambda ^{\mathfrak {p}}X\neq 0$
. Then there is
$C\in \mathscr {T}^c$
such that
$\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X,T_{C/\hspace {-1.9pt}/\mathfrak {p}} (I(\mathfrak {p})))\neq 0$
by Proposition 4.1. Hence, [Reference Shaul20, Proposition 4.10] implies that
$$ \begin{align*} \mathrm {cosupp}_R\mathrm {Hom}^\ast _{\mathscr {T}} (V^{\mathcal {Z}(\mathfrak {p})}X, T_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p})))= \mathrm {supp}_R\mathrm {H}^\ast _{C/\hspace {-1.9pt}/\mathfrak {p}}(V^{\mathcal {Z}(\mathfrak {p})}X)\cap \mathrm {cosupp}_RT_{C/\hspace {-1.9pt}/\mathfrak {p}} (I(\mathfrak {p}))=\{\mathfrak {p}\}. \end{align*} $$
This justifies the inclusion
$$ \begin{align*} \mathrm {cosupp}_{R}X\subseteq \bigcup _{C\in \mathscr {T}^c,\ \mathfrak {p}\in \mathrm {Spec}R}\mathrm {cosupp}_R\mathrm {Hom}^\ast _{\mathscr {T}} (V^{\mathcal {Z}(\mathfrak {p})}X,T_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p}))). \end{align*} $$
Let now
$\mathfrak {p}\in \mathrm {Spec}R$
. Then there is an object C in
$\mathscr {T}^c$
so that
$\mathrm {cosupp}_R\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X, T_{C/\hspace {-1.9pt}/\mathfrak {p}} (I(\mathfrak {p})))= \{\mathfrak {p}\}$
, and so
$\mathrm {Hom}^\ast _{\mathscr {T}}(C(\mathfrak {p}),X)\neq 0$
. Consequently,
$\mathfrak {p}\in \mathrm {cosupp}_RX$
by [Reference Benson, Iyengar and Krause7, Proposition 4.4]. This shows the first equality. The second equality follows from the isomorphism
$$ \begin{align*}\mathrm{Hom}^\ast_{\mathscr{T}}(V^{\mathcal{Z}(\mathfrak{p})}X,T_{C/\hspace {-1.9pt}/\mathfrak{p}}(I(\mathfrak{p}))) \cong\mathrm{Hom}^\ast_R(\mathrm{H}^\ast_{C/\hspace {-1.9pt}/\mathfrak{p}}(V^{\mathcal{Z}(\mathfrak{p})}X), I(\mathfrak{p}))\end{align*} $$
and the module
$\mathrm {H}^\ast _{C/\hspace {-1.9pt}/\mathfrak {p}}(V^{\mathcal {Z}(\mathfrak {p})}X)$
is
$\mathfrak {p}$
-local. This completes the proof.
Corollary 4.4. For any specialization closed subset
$\mathcal {V}$
of
$\mathrm {Spec}R$
, one has that
$$ \begin{align*}\begin{aligned}\mathscr{T}^{\mathcal{V}}\! &=\{X\in\mathscr{T}\hspace{0.03cm}|\hspace{0.03cm}\mathrm{cosupp}_RX\subseteq\mathcal{V}\}\\&=\{X\in\mathscr{T}\hspace{0.03cm}|\hspace{0.03cm}\mathrm{cosupp}_R\mathrm{Hom}^\ast_{\mathscr{T}}(V^{\mathcal{Z}(\mathfrak{p})}X,T_{C}(I(\mathfrak{p}))){\kern-2pt}\subseteq{\kern-2pt}\mathcal{V}, \text{ for all } C{\kern-2pt}\in{\kern-2pt}\mathscr{T}^c, \text{ for all } \mathfrak{p}\in\mathcal{V}\}.\end{aligned}\end{align*} $$
Proof. By Theorem 4.3,
$\mathrm {cosupp}_RX\subseteq \mathcal {V}$
if and only if
$\mathrm {supp}_R\mathrm {H}^\ast _{C/\hspace {-1.9pt}/\mathfrak {p}}(V^{\mathcal {Z}(\mathfrak {p})}X)\subseteq \mathcal {V}$
for all
$C\in \mathscr {T}^c$
and
$\mathfrak {p}\in \mathrm {Spec}R$
. Hence, [Reference Benson, Iyengar and Krause5, Theorem 5.13] implies that
$\mathrm {supp}_R\mathrm {H}^\ast _{C}(V^{\mathcal {Z}(\mathfrak {p})}X)\subseteq \mathcal {V}$
for all
$C\in \mathscr {T}^c$
and
$\mathfrak {p}\in \mathrm {Spec}R$
. However,
$\mathrm {cosupp}_R\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X,T_{C}(I(\mathfrak {p})))=\mathrm {supp}_R\mathrm {H}^\ast _{C}\ (V^{\mathcal {Z}(\mathfrak {p})}X)$
for all
$C\in \mathscr {T}^c$
and
$\mathfrak {p}\in \mathrm {Spec}R$
, so the proof is complete.
Corollary 4.5. For each specialization closed subset
$\mathcal {V}\subseteq \mathrm {Spec}R$
and each object X in
$\mathscr {T}$
, one has that
$$ \begin{align*}\mathrm{cosupp}_{R}(\Gamma_{\mathcal{V}}X)\cap\mathcal{V}=\mathrm{cosupp}_{R}X\cap\mathcal{V}.\end{align*} $$
In particular,
$\Gamma _{\mathcal {V}}X=0$
if and only if
$\mathrm {cosupp}_{R}X\cap \mathcal {V}=\emptyset $
.
Proof. By Theorem 4.3, we have the following equalities:
$$ \begin{align*}\mathrm{cosupp}_{R}(\Gamma_{\mathcal{V}}X)\cap\mathcal{V} &=\bigcup_{C\in\mathscr{G},\mathfrak{p}\in\mathcal{V}}\mathrm{cosupp}_R\mathrm{Hom}^\ast_{\mathscr{T}}(V^{\mathcal{Z}(\mathfrak{p})}\Gamma_{\mathcal{V}}X,T_{C/\hspace {-1.9pt}/\mathfrak{p}}(I(\mathfrak{p})))\\[-1.5pt]&=\bigcup_{C\in\mathscr{G},\mathfrak{p}\in\mathcal{V}}\mathrm{cosupp}_R\mathrm{Hom}^\ast_R(\mathrm{Hom}^\ast_{\mathscr{T}}(C(\mathfrak{p}),\Gamma_{\mathcal{V}}X),I(\mathfrak{p}))\\[-1.5pt]&=\bigcup_{C\in\mathscr{G},\mathfrak{p}\in\mathcal{V}}\mathrm{cosupp}_R\mathrm{Hom}^\ast_R(\mathrm{Hom}^\ast_{\mathscr{T}}(C/\hspace{-0.11cm}/\mathfrak{p},V^{\mathcal{Z}(\mathfrak{p})}X),I(\mathfrak{p}))\\[-2pt]&=\mathrm{cosupp}_{R}X\cap\mathcal{V}.\end{align*} $$
We obtain the equality we seek.
If
$\Gamma _{\mathcal {V}}X=0$
, then
$\mathrm {cosupp}_{R}X\cap \mathcal {V}=\emptyset $
. Assume that
$\mathrm {cosupp}_{R}X\cap \mathcal {V}=\emptyset $
. Then
$\mathrm {cosupp}_{R}X\subseteq \mathrm {Spec}R\backslash \mathcal {V}$
, and so
$X\cong L_{\mathcal {V}}X$
by [Reference Benson, Iyengar and Krause7, Corollary 4.9]. Consequently,
$\Gamma _{\mathcal {V}}X\cong \Gamma _{\mathcal {V}}L_{\mathcal {V}}X=0$
.
Proposition 4.6. For any
$\mathfrak {p}\in \mathrm {Spec}R$
and any object X in
$\mathscr {T}$
, one has an exact triangle
$$ \begin{align*}X'\rightarrow X\rightarrow X"\rightsquigarrow,\end{align*} $$
where
$X'\in {^\bot }(\mathscr {T}^{\{\mathfrak {p}\}})$
and
$X"\in \mathscr {T}^{\{\mathfrak {p}\}}$
.
Proof. Let X be an object in
$\mathscr {T}$
. The exact triangle
$\Gamma _{\mathcal {Z}(\mathfrak {p})}X\rightarrow X\rightarrow L_{\mathcal {Z}(\mathfrak {p})}X\rightsquigarrow $
induces an exact triangle
$\Lambda ^{\mathcal {V}(\mathfrak {p})}\Gamma _{\mathcal {Z}(\mathfrak {p})}X\rightarrow \Lambda ^{\mathcal {V}(\mathfrak {p})}X\rightarrow \Lambda ^{\mathcal {V}(\mathfrak {p})}L_{\mathcal {Z}(\mathfrak {p})}X\rightsquigarrow $
. By the octahedral axiom, one has a commutative diagram of triangles in
$\mathscr {T}$
:
Set
$\Lambda ^{\mathcal {V}(\mathfrak {p})}L_{\mathcal {Z}(\mathfrak {p})}X=X"$
. Since
$\Lambda ^{\mathfrak {p}}X"\cong X"$
by [Reference Benson, Iyengar and Krause7, Proposition 2.3(3)], it follows that
$X"\in \mathscr {T}^{\{\mathfrak {p}\}}$
by [Reference Benson, Iyengar and Krause7, page 176]. However, one has the following isomorphisms:
$$ \begin{align*}\begin{aligned}\mathrm{Hom}^\ast_{\mathscr{T}}(\Lambda^{\mathcal{V}(\mathfrak{p})}L_{\mathcal{Z}(\mathfrak{p})}X,T_{C/\hspace{-1.9pt}/\mathfrak{p}}(I(\mathfrak{p}))) &\cong\mathrm{Hom}^\ast_{\mathscr{T}}(L_{\mathcal{Z}(\mathfrak{p})}X,T_{C/\hspace{-1.9pt}/\mathfrak{p}}(I(\mathfrak{p})))\\&\cong\mathrm{Hom}^\ast_{\mathscr{T}}(X,T_{C/\hspace{-1.9pt}/\mathfrak{p}}(I(\mathfrak{p})))\quad \text{for all}\ C\in\mathscr{T}^c.\end{aligned}\end{align*} $$
Hence,
$\mathrm {Hom}^\ast _{\mathscr {T}}(X',T_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p})))=0$
for all
$C\in \mathscr {T}^c$
, and therefore
$Y\in {^\bot }(\mathscr {T}^{\{\mathfrak {p}\}})$
by [Reference Benson, Iyengar and Krause7, Proposition 5.4], as claimed.
5 Comparison of support and cosupport
The task of this section is to compare the support and cosupport for cohomologically finite objects, and give the proof of Theorem B. For any cohomologically finite object X, we show that
$\mathrm {supp}_RX=\mathrm {Supp}_RX$
, and obtain some applications of this equality.
Consider the exact functor
$\Gamma _{\mathfrak {p}}:\mathscr {T}\rightarrow \mathscr {T}$
defined by
$\Gamma _{\mathfrak {p}}X=\Gamma _{\mathcal {V}(\mathfrak {p})}L_{\mathcal {Z}(\mathfrak {p})}X$
. The essential image of
$\Gamma _{\mathfrak {p}}$
is denoted by
$\mathscr {T}_{\{\mathfrak {p}\}}$
, it is a localizing subcategory of
$\mathscr {T}$
.
The support of an object X in
$\mathscr {T}$
is a subset of
$\mathrm {Spec}R$
defined as follows:
$$ \begin{align*}\mathrm{supp}_RX=\{\mathfrak{p}\in \mathrm{Spec}R\hspace{0.03cm}|\hspace{0.03cm}\Gamma_{\mathfrak{p}}X\neq0\}.\end{align*} $$
For any graded R-module M, set
$\mathrm {Supp}_RM=\{\mathfrak {p}\in \mathrm {Spec}R\hspace {0.03cm}|\hspace {0.03cm}M_{\mathfrak {p}}\neq 0\}$
. This subset is sometimes referred to as the ‘big’ support of M to distinguish it from its ‘homological’ support,
$\mathrm {supp}_RM$
. For any object X in
$\mathscr {T}$
, [Reference Asadollahi, Salarian and Sazeedeh1, Theorem 3.3] proved that
$$ \begin{align*}\mathrm{Supp}_RX=\{\mathfrak{p}\in\mathrm{Spec}R\hspace{0.03cm}|\hspace{0.03cm}X_{\mathfrak{p}}\neq0\}=\bigcup_{C\in\mathscr{T}^c}\mathrm{Supp}_R\mathrm{H}^\ast_C(X).\end{align*} $$
Lemma 5.1. For each object X of
$\mathscr {T}$
, there is an inclusion of sets
$\mathrm {supp}_RX\subseteq \mathrm {Supp}_RX$
; equality holds if X is cohomologically finite.
Proof. The containment
$\mathrm {supp}_RX\subseteq \mathrm {Supp}_RX$
is clear by definition. If X is cohomologically finite, then
$\mathrm {Supp}_RX=\bigcup _{C\in \mathscr {T}^c}\mathrm {Supp}_R\mathrm {H}^\ast _{C}(X)=\bigcup _{C\in \mathscr {T}^c}\mathrm {supp}_R\mathrm {H}^\ast _{C}(X)\subseteq \mathrm {supp}_RX$
by [Reference Benson, Iyengar and Krause5, Theorem 5.5], as claimed.
Let
$\mathcal {U}$
be a subset of
$\mathrm {Spec}R$
. We denote by
$\mathrm \min \hspace {0.05cm}\mathcal {U}$
the set of minimal elements with respect to inclusion in
$\mathcal {U}$
. The dimension of a subset
$\mathcal {U}$
of
$\mathrm {Spec}R$
, denoted by
$\mathrm {dim}\hspace {0.05cm}\mathcal {U}$
, is the supremum of all integers n such that there exists a chain
$\mathfrak {p}_0\subsetneq \mathfrak {p}_1\subsetneq \cdots \subsetneq \mathfrak {p}_n$
in
$\mathcal {U}$
. The set
$\mathcal {U}$
is called discrete if
$\mathrm {dim}\hspace {0.05cm}\mathcal {U}=0$
.
Our goal of the following results is to study when the inclusion
$\mathrm {cosupp}_{R}X\subseteq \mathrm {supp}_{R}X$
holds.
Theorem 5.2. Let X be a cohomologically finite object in
$\mathscr {T}$
. If
$\mathrm \min (\mathrm {supp}_{R}X)$
is finite, then there is an inequality
$$ \begin{align*}\mathrm{cosupp}_{R}X\subseteq\mathrm{supp}_{R}X.\end{align*} $$
Proof. If
$X=0$
, then
$\mathrm {cosupp}_{R}X=\emptyset =\mathrm {supp}_{R}X$
, and we are done. Next, assume that
$X\neq 0$
. One has that
$\mathrm {supp}_{R}X=\mathrm {Supp}_{R}X$
. Let
$\mathrm \min (\mathrm {supp}_{R}X)=\{\mathfrak {p}_1,\ldots ,\mathfrak {p}_s\}$
. It follows from [Reference Benson, Iyengar and Krause8, Lemma 3.9] that
$X\in \mathrm {Thick}_{\mathscr {T}}(\{X/\hspace {-0.11cm}/\mathfrak {p}_i\hspace {0.03cm}|\hspace {0.03cm}i=1,\ldots ,s\})$
. Therefore,
$\mathrm {cosupp}_{R}X\subseteq \mathcal {V}(\mathfrak {p}_1)\cup \cdots \cup \mathcal {V}(\mathfrak {p}_s)=\mathrm {supp}_{R}X$
, as claimed.
Theorem 5.3. Let X be an object in
$\mathscr {T}$
with
$\mathrm {supp}_{R}X=\mathrm {Supp}_{R}X$
. If
$\mathrm {dim}(\mathrm {supp}_{R}X)<\infty $
, then there is an inclusion
$$ \begin{align*}\mathrm{cosupp}_{R}X\subseteq\mathrm{supp}_{R}X.\end{align*} $$
Proof. Assume that
$X\neq 0$
and
$\mathrm {supp}_{R}X=\mathrm {Supp}_{R}X=\mathcal {V}$
. Set
$\mathrm {dim}(\mathrm {supp}_{R}X)=n$
. If
$n=0$
, then
$\mathrm {cosupp}_{R}X\subseteq \mathcal {V}$
by [Reference Benson, Iyengar and Krause7, Theorem 4.13]. For
$n>0$
, set
$\mathcal {V}'=\mathcal {V}\backslash \mathrm \min \hspace {0.05cm}\mathcal {V}$
. Then
$\mathcal {V}'$
is specialization closed. Since
$\mathrm {dim}\hspace {0.05cm}\mathcal {V}'=n-1$
, the induction hypothesis yields that
$\mathrm {cosupp}_{R}\Gamma _{\mathcal {V}'}X\subseteq \mathcal {V}$
. However,
$\mathrm {supp}_R(L_{\mathcal {V}'}X)=\mathrm \min \hspace {0.05cm}\mathcal {V}$
is discrete and hence
$\mathrm {cosupp}_R(L_{\mathcal {V}'}X)=\mathrm \min \hspace {0.05cm}\mathcal {V}$
. Consequently,
$\mathrm {cosupp}_{R}X\subseteq \mathcal {V}$
, as claimed.
Remark 5.4. (1) If
$\mathscr {T}$
is generated by finitely many compact objects
$\{C_1,\ldots ,C_n\}$
, then for any cohomologically finite object X in
$\mathscr {T},$
$$ \begin{align*}\mathrm{supp}_{R}X &=\bigcup_{i=1}^n\mathrm{supp}_{R}\mathrm{H}^\ast_{C_i}(X)\\ &=\bigcup_{i=1}^n\mathcal{V}(\mathrm{ann}_R\mathrm{H}^\ast_{C_i}(X))\\ &=\mathcal{V}\bigg(\bigcap_{i=1}^n\mathrm{ann}_R\mathrm{H}^\ast_{C_i}(X)\bigg). \end{align*} $$
Thus, the set
$\mathrm \min (\mathrm {supp}_{R}X)$
is finite. Set
$\mathfrak {a}=\bigcap _{i=1}^n\mathrm {ann}_R\mathrm {H}^\ast _{C_i}(X)$
. One also has that
$$ \begin{align*}\mathrm{Thick}_{\mathscr{T}}(X)=\mathrm{Thick}_{\mathscr{T}}(X/\hspace{-0.11cm}/\mathfrak{a}).\end{align*} $$
(2) The inclusion in the preceding theorems can be strict. For example, let A be a commutative Noetherian ring with
$\mathrm {dim}A\geqslant 1$
and
$\mathscr {T}=\mathrm {D}(A)$
. If
$(A,\mathfrak {m})$
is a complete local ring, then
$\mathrm {cosupp}_AA=\{\mathfrak {m}\}\subsetneq \mathrm {Spec}A=\mathrm {supp}_AA$
.
(3) Let A be a commutative Noetherian ring and
$\mathscr {T}=\mathrm {D}(A)$
. The cohomologically finite objects in
$\mathrm {D}(A)$
are cohomologically bounded complexes X such that each cohomology module
$\mathrm {H}^i(X)$
is finitely generated. In this case, Theorem 5.2 is exactly [Reference Shaul20, Theorem 6.7].
(4) Assume that
$(A,\mathfrak {m})$
is a commutative local Noetherian ring and not Artinian and set
$I=I(\mathfrak {m})$
. Then
$\mathrm {supp}_AI=\{\mathfrak {m}\}\subsetneq \mathrm {Spec}A=\mathrm {cosupp}_AI$
. Therefore, the assumption that X is cohomologically finite in
$\mathscr {T}$
in Theorem 5.2 is essential.
(5) Let A be a commutative Noetherian ring and
$\mathscr {T}=\mathrm {D}(A)$
. Let X be a complex in
$\mathrm {D}(A)$
with each
$\mathrm {H}^i(X)$
of finite length. Then each
$\mathfrak {p}\in \mathrm {Supp}_A\mathrm {H}^i(X)$
is a maximal ideal of A by [Reference Enochs and Jenda11, page 60, Exercise 5]. Thus
$\mathrm {supp}_AX=\mathrm {Supp}_AX$
is discrete.
(6) If the ring R has finite Krull dimension, then
$\mathrm {dim}(\mathrm {supp}_{R}X)<\infty $
for any X in
$\mathscr {T}$
.
Following [Reference Benson, Iyengar and Krause6], we say that
$\mathscr {T}$
is stratified by R if the following conditions hold.
-
(S1) The local-global principle holds for localizing subcategories of
$\mathscr {T}$
, that is, for each object X in
$\mathscr {T}$
, there is an equality
$\mathrm {Loc}_{\mathscr {T}}(X)=\mathrm {Loc}_{\mathscr {T}}(\{\Gamma _{\mathfrak {p}}X\hspace {0.03cm}|\hspace {0.03cm}\mathfrak {p}\in \mathrm {Spec}R\})$
. -
(S2) For each
$\mathfrak {p}\in \mathrm {Spec}R$
, the localizing subcategory
$\mathscr {T}_{\{\mathfrak {p}\}}$
has no proper nonzero localizing subcategories.
The following result was proved by Benson, Iyengar, and Krause when
$\mathscr {T}$
is Noetherian and stratified by R (see [Reference Benson, Iyengar and Krause5, Corollary 5.3]).
Proposition 5.5. If
$\mathscr {T}$
is stratified by R, then for each
$C\in \mathscr {T}^c$
and each cohomologically finite object X, there is an equality
$$ \begin{align*}\mathrm{supp}_R\mathrm{Hom}^\ast_{\mathscr{T}}(C,X)=\mathrm{supp}_RC\cap\mathrm{supp}_RX.\end{align*} $$
In particular,
$\mathrm {Hom}^\ast _{\mathscr {T}}(C,X)=0$
if and only if
$\mathrm {supp}_RC\cap \mathrm {supp}_RX=\emptyset $
.
Proof. Note that
$\mathrm {supp}_R\mathrm {Hom}^\ast _{\mathscr {T}}(C,X)=\mathrm {Supp}_R\mathrm {Hom}^\ast _{\mathscr {T}}(C,X)$
. Let
$\mathfrak {p}\in \mathrm {Spec}R$
. Suppose
$\mathrm {Hom}^\ast _{\mathscr {T}}(C,X)_{\mathfrak {p}}\neq 0$
. Then
$\mathrm {Hom}^\ast _{\mathscr {T}}(C,X_{\mathfrak {p}})\neq 0$
, and so
$\mathrm {Hom}^\ast _{\mathscr {T}}(C(\mathfrak {p}),X_{\mathfrak {p}})\cong \mathrm {Hom}^\ast _{\mathscr {T}}(C/\hspace {-0.11cm}/\mathfrak {p},X_{\mathfrak {p}})\neq 0$
by [Reference Benson, Iyengar and Krause5, Lemma 5.11(3)]. Thus,
$C(\mathfrak {p})\neq 0$
and
$X_{\mathfrak {p}}\neq 0$
, so
$\mathfrak {p}\in \mathrm {supp}_RC\cap \mathrm {supp}_RX$
. Now suppose
$\mathrm {Hom}^\ast _{\mathscr {T}}(C,X)_{\mathfrak {p}}=0$
. Then,
$\mathrm {Hom}^\ast _{\mathscr {T}}(C_{\mathfrak {p}},X_{\mathfrak {p}})=0$
. Note that
$\Gamma _{\mathfrak {p}}C$
is in
$\mathrm {Loc}_{\mathscr {T}}(C_{\mathfrak {p}})$
, so one gets
$\mathrm {Hom}^\ast _{\mathscr {T}}(\Gamma _{\mathfrak {p}}C,\Gamma _{\mathfrak {p}}X)\cong \mathrm {Hom}^\ast _{\mathscr {T}}(\Gamma _{\mathfrak {p}}C,X_{\mathfrak {p}})=0$
. Thus, one of
$\Gamma _{\mathfrak {p}}C$
or
$\Gamma _{\mathfrak {p}}X$
is zero since
$\mathscr {T}$
is stratified by R. This shows the equality we seek.
Corollary 5.6. If
$\mathscr {T}$
is stratified by R, then for each
$C\in \mathscr {T}^c$
and each cohomologically finite object X, there is an equality
$$ \begin{align*}\mathrm{cosupp}_R\mathrm{Hom}^\ast_{\mathscr{T}}(X,T_{C}(I(\mathfrak{p})))=\mathrm{supp}_RX\cap\mathrm{cosupp}_RT_{C}(I(\mathfrak{p})).\end{align*} $$
Proof. It follows from the definition of the object
$T_{C}(I(\mathfrak {p}))$
and Proposition 5.5 that
$$ \begin{align*}\begin{aligned}\mathrm{cosupp}_R\mathrm{Hom}^\ast_{\mathscr{T}}(X,T_{C}(I(\mathfrak{p}))) &=\mathrm{supp}_R\mathrm{Hom}^\ast_{\mathscr{T}}(C,X)\cap\mathrm{cosupp}_RI(\mathfrak{p})\\ &=\mathrm{supp}_RX\cap\mathrm{supp}_RC\cap\mathrm{cosupp}_RI(\mathfrak{p}).\end{aligned}\end{align*} $$
Hence, [Reference Benson, Iyengar and Krause7, Proposition 5.4] yields the desired equality.
Following [Reference Benson, Iyengar and Krause6], we say that
$\mathscr {T}$
is costratified by R if the following conditions hold.
(C1) The local-global principle holds for colocalizing subcategories of
$\mathscr {T}$
, that is, for each object X in
$\mathscr {T}$
, there is an equality
$\mathrm {Coloc}_{\mathscr {T}}(X)=\mathrm {Coloc}_{\mathscr {T}}(\{\Lambda ^{\mathfrak {p}}X\hspace {0.03cm}|\hspace {0.03cm}\mathfrak {p}\in \mathrm {Spec}R\})$
.
(C2) For each
$\mathfrak {p}\in \mathrm {Spec}R$
, the colocalizing subcategory
$\mathscr {T}^{\{\mathfrak {p}\}}$
contains no proper nonzero colocalizing subcategories.
Lemma 5.7. Suppose
$\mathscr {G}$
is a set of compact generators for
$\mathscr {T}$
. If
$\mathscr {T}$
is costratified by R, then for each specialization closed subset
$\mathcal {V}\subseteq \mathrm {Spec}R$
, there are equalities
$$ \begin{align*}\mathscr{T}^{\mathcal{V}}=\mathrm{Coloc}_{\mathscr{T}}(C(\mathfrak{p})\hspace{0.03cm}|\hspace{0.03cm}C\in\mathscr{G}, \mathfrak{p}\in\mathcal{V})=\mathrm{Coloc}_{\mathscr{T}}(T_{C/\hspace{-1.9pt}/\mathfrak{p}}(I(\mathfrak{p}))\hspace{0.03cm}|\hspace{0.03cm}C\in\mathscr{G}, \mathfrak{p}\in\mathcal{V}).\end{align*} $$
Proof. Let
$C\in \mathscr {G}$
. It follows from [Reference Benson, Iyengar and Krause7, Corollary 4.9] that
$\mathrm {cosupp}_RC(\mathfrak {p})\subseteq \mathcal {U}(\mathfrak {p})$
. Also,
$\mathrm {cosupp}_RC(\mathfrak {p})=\mathrm {cosupp}_RC_{\mathfrak {p}}/\hspace {-0.11cm}/\mathfrak {p}\subseteq \mathcal {V}(\mathfrak {p})$
by [Reference Benson, Iyengar and Krause7, Lemma 4.12]. However,
$\mathfrak {p}\in \mathrm {supp}_RC$
if and only if
$C(\mathfrak {p})\neq 0$
if and only if
$\mathfrak {p}\in \mathrm {cosupp}_RC(\mathfrak {p})$
, so
$\mathrm {cosupp}_RC(\mathfrak {p})=\mathrm {supp}_RC\cap \{\mathfrak {p}\}$
. Also,
$\mathrm {cosupp}_RT_{C/\hspace {-1.9pt}/\mathfrak {p}}(I(\mathfrak {p}))=\mathrm {supp}_RC\cap \{\mathfrak {p}\}$
by [Reference Benson, Iyengar and Krause7, Theorem 5.4]. Therefore, one has the desired equalities by [Reference Benson, Iyengar and Krause6, Remark 5.7].
The next result obtains a description of the cosupport of the right orthogonal of the category
$\mathscr {T}^{\mathcal {V}}$
.
Proposition 5.8. Suppose that
$\mathscr {G}$
is a set of compact generators for
$\mathscr {T}$
. If
$\mathscr {T}$
is costratified by R, then for each specialization closed subset
$\mathcal {V}\subseteq \mathrm {Spec}R$
,
$$ \begin{align*}(\mathscr{T}^{\mathcal{V}})^\bot=\{X\in\mathscr{T}\hspace{0.03cm}|\hspace{0.03cm}\mathrm{cosupp}_RX\subseteq\mathrm{Spec}R\backslash\mathcal{V}\}.\end{align*} $$
Proof. By Proposition 4.1, one has that
$$ \begin{align*}\mathrm{cosupp}_RX\subseteq\mathrm{Spec}R\backslash\mathcal{V} \Longleftrightarrow\mathrm{Hom}^\ast_{\mathscr{T}}(C(\mathfrak{p}),X)=0\quad \text{for all } C\in\mathscr{G}\text{ for all } \mathfrak{p}\in\mathcal{V}.\end{align*} $$
Hence, Lemma 5.7 shows our claim.
6 Big cosupport
In this section, we introduce the notion of big cosupport for an object and develop systematically a theory of big cosupport to make it a viable tool.
definition 6.1. Let X be an object of
$\mathscr {T}$
. We define the big cosupport of X, denoted by
$\mathrm {Cosupp}_RX$
, to be the set
$$ \begin{align*}\mathrm{Cosupp}_RX=\{\mathfrak{p}\in\mathrm{Spec}R\hspace{0.03cm}|\hspace{0.03cm}V^{\mathcal{Z}(\mathfrak{p})}X\neq0\}.\end{align*} $$
Remark 6.2. (1) Our notion of ‘big cosupport’ is an extension of the ‘large co-support’ of complexes introduced by Sather-Wagstaff and Wicklein in [Reference Shaul20, Definition 4.1].
(2) For any object X in
$\mathscr {T}$
, one has that
$\mathrm {Cosupp}_{R}X=\mathrm {Cosupp}_{R}\Sigma X$
and
$\mathrm {cosupp}_RX\subseteq \mathrm {Cosupp}_RX$
.
(3) For any exact triangle
$X\rightarrow Y\rightarrow Z\rightsquigarrow $
in
$\mathscr {T}$
, we have
$$ \begin{align*}\mathrm{Cosupp}_{R}Y\subseteq\mathrm{Cosupp}_{R}X\cup\mathrm{Cosupp}_{R}Z.\end{align*} $$
(4) For any object X in
$\mathscr {T}$
, the subset
$\mathrm {Cosupp}_RX$
is specialization closed.
(5)
$\mathrm {Cosupp}_R(\Lambda ^{\mathcal {V}}X)=\mathrm {Cosupp}_RX\cap \mathcal {V}$
by [Reference Benson, Iyengar and Krause7, (4.2)], and so
$$ \begin{align*} \mathrm {Cosupp}_RX=\mathrm {Cosupp}_R (\Lambda ^{\mathcal {V}}X)\cup \mathrm {Cosupp}_R(V^{\mathcal {V}}X) \end{align*} $$
by the exact triangle
$V^{\mathcal {V}}X\rightarrow X\rightarrow \Lambda ^{\mathcal {V}}X\rightsquigarrow $
, where
$\mathcal {V}$
is a specialization closed subset of
$\mathrm {Spec}R$
.
The following result provides a computation of big cosupport.
Theorem 6.3. For each object X in
$\mathscr {T}$
, one has that
$$ \begin{align*}\begin{aligned}\mathrm{Cosupp}_{R}X &=\bigcup_{C\in\mathscr{T}^c,\ \mathfrak{p}\in\mathrm{Spec}R}\mathrm{Cosupp}_R\mathrm{Hom}^\ast_{\mathscr{T}}(V^{\mathcal{Z}(\mathfrak{p})}X,T_{C}(I(\mathfrak{p})))\\ &=\bigcup_{C\in\mathscr{T}^c,\ \mathfrak{p}\in\mathrm{Spec}R}\mathrm{Supp}_R\mathrm{H}^\ast_{C}(V^{\mathcal{Z}(\mathfrak{p})}X). \end{aligned}\end{align*} $$
Proof. Let
$\mathfrak {p}\in \mathrm {Spec}R$
be such that
$V^{\mathcal {Z}(\mathfrak {p})}X\neq 0$
. By Theorem 4.3, there is a compact object
$C\in \mathscr {T}^c$
such that
$\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X,T_{C}(I(\mathfrak {p})))\neq 0$
. However,
$$ \begin{align*}\mathrm{Hom}^\ast_R(R_{\mathfrak{p}},\mathrm{Hom}^\ast_{\mathscr{T}}(V^{\mathcal{Z}(\mathfrak{p})}X,T_{C}(I(\mathfrak{p})))) \cong\mathrm{Hom}^\ast_{\mathscr{T}}(V^{\mathcal{Z}(\mathfrak{p})}X,T_{C}(I(\mathfrak{p}))),\end{align*} $$
so
$\mathfrak {p}\in \mathrm {Cosupp}_R\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X,T_{C}(I(\mathfrak {p})))$
. Conversely, let
$\mathfrak {q}$
be a point in
$\mathrm {Spec}R$
such that
$\mathrm {Hom}^\ast _R(R_{\mathfrak {q}},\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {p})}X,T_{C}(I(\mathfrak {p}))))\neq 0$
for some
$C\in \mathscr {T}^c$
and
$\mathfrak {p}\in \mathrm {Spec}R$
. Assume to the contrary that
$V^{\mathcal {Z}(\mathfrak {q})}X=0$
. So
$\mathrm {cosupp}_{R}X\subseteq \mathcal {Z}(\mathfrak {q})$
by [Reference Benson, Iyengar and Krause7, Corollary 4.8]. Therefore,
$\mathrm {cosupp}_R\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {u})}X,T_{D}(I(\mathfrak {u})))\subseteq \mathcal {Z}(\mathfrak {q})$
for any
$D\in \mathscr {T}^c$
and
$\mathfrak {u}\in \mathrm {Spec}R$
by Corollary 4.4, that is to say,
$\mathrm {Hom}_R(R_{\mathfrak {q}},\mathrm {Hom}^\ast _{\mathscr {T}}(V^{\mathcal {Z}(\mathfrak {u})}X,T_{D}(I(\mathfrak {u}))))=0$
for all
$D\in \mathscr {T}^c$
and
$\mathfrak {u}\in \mathrm {Spec}R$
, which contradicts our assumption. The second equality follows from the isomorphism
$$ \begin{align*}\mathrm{Hom}^\ast_{\mathscr{T}}(V^{\mathcal{Z}(\mathfrak{p})}X,T_{C}(I(\mathfrak{p}))) \cong\mathrm{Hom}^\ast_R(\mathrm{H}^\ast_{C}(V^{\mathcal{Z}(\mathfrak{p})}X), I(\mathfrak{p}))\end{align*} $$
and the module
$\mathrm {H}^\ast _{C}(V^{\mathcal {Z}(\mathfrak {p})}X)$
is
$\mathfrak {p}$
-local. This completes the proof.
Proposition 6.4. Let X be an object of
$\mathscr {T}$
. The sets
$\mathrm {cosupp}_RX$
and
$\mathrm {Cosupp}_RX$
have the same minimal elements with respect to containment, that is,
$\mathrm \min (\mathrm {cosupp}_RX)=\mathrm \min (\mathrm {Cosupp}_RX)$
.
Proof. For the containment
$\mathrm \min (\mathrm {cosupp}_RX)\supseteq \mathrm \min (\mathrm {Cosupp}_RX)$
, fix
$\mathfrak {p}\in \mathrm \min (\mathrm {Cosupp}_R (X))$
. Then
$\mathrm {cosupp}_R(V^{\mathcal {Z}(\mathfrak {p})}X)\neq \emptyset $
, this is to say, there is
$\mathfrak {q}\in \mathrm {Spec}R$
such that
$\Lambda ^{\mathfrak {q}}(V^{\mathcal {Z}(\mathfrak {p})}X)\neq 0$
. However,
$\mathfrak {q}\subseteq \mathfrak {p}$
by [Reference Benson, Iyengar and Krause7, Corollary 4.9], so
$\Lambda ^{\mathfrak {q}}X\cong \Lambda ^{\mathfrak {q}}(V^{\mathcal {Z}(\mathfrak {p})}X)\neq 0$
, which implies that
$\mathfrak {q}\in \mathrm {cosupp}_RX\subseteq \mathrm {Cosupp}_RX$
. Thus, the minimality of
$\mathfrak {p}$
in
$\mathrm {Cosupp}_RX$
implies that
$\mathfrak {p}=\mathfrak {q}\in \mathrm {cosupp}_RX$
. From the containment
$\mathrm {cosupp}_RX\subseteq \mathrm {Cosupp}_RX$
, the fact that
$\mathfrak {p}$
is minimal in
$\mathrm {Cosupp}_RX$
implies that it is also minimal in
$\mathrm {cosupp}_RX$
.
For the reverse containment, let
$\mathfrak {p}\in \mathrm \min (\mathrm {cosupp}_RX)\subseteq \mathrm {Cosupp}_RX$
. Suppose that
$\mathfrak {p}$
is not minimal in
$\mathrm {Cosupp}_RX$
, so there is a prime
$\mathfrak {q}\in \mathrm {Cosupp}_RX$
such that
$\mathfrak {q}\subsetneq \mathfrak {p}$
, that is to say,
$V^{\mathcal {Z}(\mathfrak {q})}X\neq 0$
, so there is a prime
$\mathfrak {u}\in \mathrm {Spec}R$
such that
$\mathfrak {u}\subseteq \mathfrak {q}$
and
$\mathfrak {u}\in \mathrm {cosupp}_{R}(V^{\mathcal {Z}(\mathfrak {q})}X)$
. As in the previous paragraph, this implies that
$\mathfrak {u}\in \mathrm {cosupp}_RX$
, so the minimality of
$\mathfrak {p}$
implies that
$\mathfrak {p}=\mathfrak {u}\subseteq \mathfrak {q}\subsetneq \mathfrak {p}$
, which contradicts the assumption.
Corollary 6.5. Let X be an object of
$\mathscr {T}$
and
$\mathfrak {a}$
a homogeneous ideal of R. One has that
$\mathrm {Cosupp}_RX\subseteq \mathcal {V}(\mathfrak {a})$
if and only if
$\mathrm {cosupp}_RX\subseteq \mathcal {V}(\mathfrak {a})$
.
Proof. The forward implication is by the containment
$\mathrm {cosupp}_RX\subseteq \mathrm {Cosupp}_RX$
. For the converse, assume that
$\mathrm {cosupp}_RX\subseteq \mathcal {V}(\mathfrak {a})$
, and let
$\mathfrak {p}\in \mathrm {Cosupp}_RX$
. It follows that
$\mathfrak {p}$
is contained in a minimal element
$\mathfrak {q}$
of
$\mathrm {Cosupp}_RX$
, which is in
$\mathrm {cosupp}_RX$
by Proposition 6.4. In other words, we have
$\mathfrak {a}\subseteq \mathfrak {q}\subseteq \mathfrak {p}$
, so
$\mathfrak {p}\in \mathcal {V}(\mathfrak {a})$
.
Acknowledgements
This paper began during a visit to the University of Utah. I am very grateful to Professor Srikanth Iyengar for many helpful discussions and reading multiple versions of this paper. The author expresses his sincere thanks to the anonymous referees for their careful reading and helpful suggestions.
