# Locally dualisable modular representations
and local regularity

Dave Benson, Srikanth B. Iyengar, Henning Krause
and Julia Pevtsova  Dave Benson
Institute of Mathematics
University of Aberdeen
King’s College
Aberdeen AB24 3UE
Scotland U.K.  Srikanth B. Iyengar
Department of Mathematics
University of Utah
Salt Lake City, UT 84112
U.S.A.  Henning Krause
Fakultät für Mathematik
Universität Bielefeld
33501 Bielefeld
Germany.  Julia Pevtsova
Department of Mathematics
University of Washington
Seattle, WA 98195
U.S.A.

###### Abstract.

This work concerns the stable module category of a finite group over a field
of characteristic dividing the group order. The minimal localising tensor
ideals correspond to the non-maximal homogeneous prime ideals in the
cohomology ring of the group. Given such a prime ideal, a number of
characterisations of the dualisable objects in the corresponding tensor ideal
are given. One characterisation of interest is that they are exactly the
modules whose restriction along a corresponding $\pi$-point are finite
dimensional plus projective. A key insight is the identification of a special
property of the stable module category that controls the cohomological
behaviour of local dualisable objects. This property, introduced in this work
for general triangulated categories and called local regularity, is related to
strong generation. A major part of the paper is devoted to developing this
notion and investigating its ramifications for various special classes of
objects in tensor triangulated categories.

###### Key words and phrases:

dualisable object, finite length object, local regularity, $\pi$-point, stable
module category, tensor triangulated category

###### 2020 Mathematics Subject Classification:

20C20 (primary); 18G80, 20J06 (secondary)

###### Contents

  1. 1 Introduction
  2. 2 Rappels: cohomology and localisation
  3. 3 Tensor structure and dualisable objects
  4. 4 Finite length objects
  5. 5 Brown–Comenetz duality
  6. 6 Strong generation
  7. 7 Local regularity
  8. 8 Finite groups
  9. 9 -points

##  1\. Introduction

Let $G$ be a finite group and $k$ a field of characteristic $p$. In modular
representation theory, traditionally one restricted attention to finitely
generated $kG$-modules. These have a number of desirable properties that make
them easier to work with, such as the Krull–Remak–Schmidt Theorem: every
finitely generated module is a direct sum of indecomposables; the
indecomposables have local endomorphism rings, and the isomorphism types and
multiplicities of the direct summands are uniquely determined. This implies
cancellation: if $M\oplus X\cong N\oplus X$ then $M\cong N$.
Krull–Remak–Schmidt drives the theory of vertices and sources, as well as
various other well known developments in the subject.

In the nineties and subsequent decades, it was found that one has to contend
with some infinitely generated modules, even if one is only interested in
properties of the finitely generated ones. For example, in [5], certain
infinitely generated idempotent modules are used in order to classify the
thick tensor ideals of the stable category of finitely generated modules in
terms of the prime ideal spectrum of the cohomology ring $H^{*}(G,k)$.

An analogy from homotopy theory is as follows. Suppose that all you are really
interested in is finite CW complexes, or even just closed manifolds. You might
start to develop cohomology theories such as ordinary cohomology, $K$-theory,
or cobordism theory. You then find that these are representable, but only by
infinite CW complexes, so you are forced to study these. Cohomology operations
then correspond to maps between these representing objects, and this is what
makes calculation possible.

In modular representation theory, the idempotent modules above are
representing objects in a similar way, and eventually these led to a
classification [8] of the localising tensor ideals of the stable category
$\operatorname{StMod}kG$ of all $kG$-modules. There are local cohomology
functors $\varGamma_{\mathfrak{p}}$ corresponding to the homogeneous prime
ideals of $H^{*}(G,k)$, that pick out the minimal localising tensor ideals
$\varGamma_{\mathfrak{p}}\operatorname{StMod}kG$, and all localising tensor
ideals are determined by the minimal ones that they contain. This sets up a
bijection with the subsets of the homogeneous primes in $H^{*}(G,k)$ that are
different from the maximal ideal of positive degree elements.

The problem with infinitely generated modules is that some rather unsettling
things happen. For example, as soon as the Sylow $p$-subgroups of $G$ are non-
cyclic, there are $kG$-modules $M$ with $M\cong M\oplus M\oplus M$ but
$M\not\cong M\oplus M$. So cancellation is violated, and furthermore, $M$ and
$M\oplus M$ are examples of non-isomorphic modules, each of which is
isomorphic to a direct summand of the other. This and further pathologies are
described in Brenner and Ringel [18]. The idempotent modules appearing above
do not participate in such pathologies, and so it is natural to ask for more
comfortable contexts containing these, and sitting between the finitely
generated modules and all modules.

Our goal in this paper is to describe a thick subcategory of
$\varGamma_{\mathfrak{p}}\operatorname{StMod}kG$ that is large enough to
contain the modules used in various constructions, but small enough that
pathological behaviour is avoided. The simplest description is that this
consists of objects $M$ _of finite length_ in the tensor triangulated category
$\mathcal{T}=\varGamma_{\mathfrak{p}}\operatorname{StMod}kG$. This condition
means that for each compact object $C$ in $\mathcal{T}$ the graded morphisms
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,M)$ form a module of finite length
over the graded endomorphism ring of the tensor unit. In particular, $M$ is
endofinite in the sense of [32]. This subcategory has many descriptions, and
here we give some of these.

######  Theorem 1.1.

Fix $\mathfrak{p}$ in $\operatorname{Proj}H^{*}(G,k)$ and a $\pi$-point
$\alpha\colon K[t]/(t^{p})\to KG$ representing $\mathfrak{p}$. For any object
$M$ in the tensor triangulated category
$\varGamma_{\mathfrak{p}}\operatorname{StMod}kG$, the conditions below are
equivalent.

  1. (1)

$M$ has finite length in $\varGamma_{\mathfrak{p}}\operatorname{StMod}kG$;

  2. (2)

$M$ is dualisable in $\varGamma_{\mathfrak{p}}\operatorname{StMod}kG$;

  3. (3)

$M$ is in the thick subcategory generated by all $\varGamma_{\mathfrak{p}}C$
for $C\in\operatorname{stmod}kG$;

  4. (4)

$\alpha^{*}(K\otimes_{k}M)$ is stably finite dimensional.

There is a one-to-one correspondence between non-maximal homogeneous prime
ideals in $H^{*}(G,k)$ and equivalence classes of $\pi$-points $\alpha$; see
[24]. Condition (4) above means that the restriction of the $KG$-module
$K\otimes_{k}M$ to $K[t]/(t^{p})$ is a direct sum of a finitely generated
module and a projective module.

The compact objects in $\varGamma_{\mathfrak{p}}\operatorname{StMod}kG$ have
finite length, but the converse only holds when the prime $\mathfrak{p}$ is
minimal. In particular, the tensor identity $\varGamma_{\mathfrak{p}}k$ is
dualisable, but compact only when $\mathfrak{p}$ is minimal. So given the
result above we may think of the category of finite length objects as the
tensor triangulated completion of the compact objects.

The equivalence (1)$\Leftrightarrow$(4) is verified in Section 9. It applies
equally to general finite group schemes and not only to finite groups.
Condition (1) is equivalent to a noetherian property of the
$\mathfrak{p}$-completion of $M$. This translation requires a version of
Matlis duality for tensor triangulated categories, developed in Section 5.
Carlson [21] proves that this is equivalent to condition (4) when
$\mathfrak{p}$ is a closed point and $G$ is elementary abelian.

The implications (3)$\Rightarrow$(2)$\Rightarrow$(1) hold in a broader
context; see Sections 5 and 7. They have been established in stable homotopy
theory by Hovey and Strickland [29]. As explained in their work, the
implication (1)$\Rightarrow$(3) does not hold there; see also [14]. The key
property of $\operatorname{StMod}kG$ that is exploited in our proof of this
implication is expressed in the following result.

######  Theorem 1.2.

For each $\mathfrak{p}$ in $\operatorname{Proj}H^{*}(G,k)$, the subcategory

$\operatorname{thick}\\{\varGamma_{\mathfrak{p}}C\mid
C\in\operatorname{stmod}kG\\}$

of $\operatorname{StMod}kG$ has a strong generator.

The notion of a strong generator of a category is due to Bondal and van den
Bergh [17]. For the derived category of a commutative noetherian ring $A$ and
a prime ideal $\mathfrak{p}\subseteq A$, the analogue of the above category is
$\operatorname{thick}\\{\varGamma_{\mathfrak{p}}P\mid P\text{ perfect}\\}$,
and this has a strong generator precisely when the local ring
$A_{\mathfrak{p}}$ is regular. One can thus view the theorem above as stating
that $\operatorname{StMod}kG$ is _locally regular_. Establishing this property
of $\operatorname{StMod}kG$ in its various incarnations and exploring its
ramifications forms a major part of this work. The regularity result itself
appears in Theorem 8.3; see also the discussion following that result. The
proof is quite involved because there are, in general, many more objects in
the thick closure of $\varGamma_{\mathfrak{p}}(\operatorname{stmod}kG)$ than
in its idempotent completion. Examples are given in [22].

We give many other equivalent descriptions of dualisable objects of
$\varGamma_{\mathfrak{p}}\operatorname{StMod}kG$ in Section 8 based on the
ground work in Sections 2-7, leading to many pleasing properties of these
modules. One of them is that the Krull–Remak–Schmidt Theorem holds, so there
are no pathologies of the type mentioned earlier. Another is that
dualisability is preserved by the appropriate versions of induction and
restriction. In particular, this lays the foundation for the theory of
vertices and sources, and makes the Green correspondence work the way one
would desire. There is a version for $\operatorname{StMod}kG$, but it is in
terms of equivalences of quotient categories rather than inducing and
restricting indecomposable objects. The Burry–Carlson–Puig Theorem also works,
as does Green’s Indecomposability Theorem; this fails in
$\operatorname{StMod}kG$ [16].

Here is a brief outline of this paper: Sections 2-3 provide a summary of basic
facts about compactly generated triangulated categories with additional
structure given by a linear action of a ring or a tensor product. Finite
length objects are discussed in Section 4. Brown–Comenetz duality, which may
be viewed as the tensor triangular analogue of Matlis duality from commutative
algebra, is the subject of Section 5. Sections 6-7 are devoted to a notion of
local regularity and various descriptions of local dualisable objects. The
final Sections 8-9 contain our results on the modular representations of
finite groups. As is clear from this description, a major part of this
manuscript develops general machinery that may be of interest beyond modular
representations.

##  2\. Rappels: cohomology and localisation

In this section we recall certain basic notions and constructions in
triangulated categories. Our main interest is in tensor triangulated
categories, but many of the fundamental constructions do not require this
additional structure, and are more transparent when treated in greater
generality. Primary references for the material presented here are [6, 9].

Throughout we fix a graded commutative noetherian ring $R$. We consider only
homogeneous elements and ideals in $R$ and $R$-modules are usually graded.
Morphisms between graded modules are degree preserving, but we suppress shifts
of graded modules when they arise. In this spirit, ‘localisation’ means
homogeneous localisation, and $\operatorname{Spec}R$ denotes the set of
homogeneous prime ideals in $R$. In particular, for
$\mathfrak{p}\in\operatorname{Spec}R$ we write $R_{\mathfrak{p}}$ for the
homogeneous localisation of $R$ at $\mathfrak{p}$; it is a graded local ring
with unique homogeneous maximal ideal $\mathfrak{p}R_{\mathfrak{p}}$.

Fix a compactly generated triangulated category ${\mathcal{T}}$; its full
subcategory consisting of compact objects is denoted
${\mathcal{T}}^{\mathrm{c}}$. Fix as well an $R$-linear action on
${\mathcal{T}}$, in the sense defined below.

### Central actions

For objects $X$ and $Y$ in ${\mathcal{T}}$ set

$\operatorname{Hom}_{\mathcal{T}}^{*}(X,Y)\coloneqq\bigoplus_{i\in\mathbb{Z}}\operatorname{Hom}_{\mathcal{T}}(X,\Sigma^{i}Y)\quad\text{and}\quad\operatorname{End}_{\mathcal{T}}^{*}(X)\coloneqq\operatorname{Hom}_{\mathcal{T}}^{*}(X,X)\,.$

Composition makes $\operatorname{End}_{\mathcal{T}}^{*}(X)$ a graded ring and
$\operatorname{Hom}_{\mathcal{T}}^{*}(X,Y)$ a
left-$\operatorname{End}_{\mathcal{T}}^{*}(Y)$
right-$\operatorname{End}_{\mathcal{T}}^{*}(X)$ module. The category
${\mathcal{T}}$ is said to be _$R$ -linear_ if for each $X$ in ${\mathcal{T}}$
there is a homomorphism of graded rings $\phi_{X}\colon
R\to\operatorname{End}_{\mathcal{T}}^{*}(X)$ such that the induced left and
right actions of $R$ on $\operatorname{Hom}_{\mathcal{T}}^{*}(X,Y)$ are
compatible in the following sense: For any $r\in R$ and
$\alpha\in\operatorname{Hom}^{*}_{\mathcal{T}}(X,Y)$, one has
$\phi_{Y}(r)\alpha=(-1)^{|r||\alpha|}\alpha\phi_{X}(r)$.

The $R$-linear category $\mathcal{T}$ is said to be _noetherian_ if the
$R$-module $\operatorname{Hom}^{*}_{\mathcal{T}}(X,Y)$ is noetherian for all
$X,Y$ in ${\mathcal{T}}^{\mathrm{c}}$.

### Localisation and torsion

Let $V\subseteq\operatorname{Spec}R$ be a _specialisation closed_ subset, so
$\mathfrak{p}\subseteq\mathfrak{q}$ and $\mathfrak{p}\in V$ implies
$\mathfrak{q}\in V$. An $R$-module $M$ is _$V$ -torsion_ if
$M_{\mathfrak{p}}=0$ for all $\mathfrak{p}$ in
$(\operatorname{Spec}R)\setminus V$. Given an ideal $\mathfrak{a}$ in $R$ we
sometimes speak of $\mathfrak{a}$-torsion modules instead of
$V(\mathfrak{a})$-torsion modules.

An object $X$ in ${\mathcal{T}}$ is _$V$ -torsion_ if the $R$-module
$\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)$ is $V$-torsion for all
$C\in{\mathcal{T}}^{\mathrm{c}}$. An object $Y$ in ${\mathcal{T}}$ is _$V$
-local_ if $\operatorname{Hom}_{\mathcal{T}}(X,Y)=0$ for all $V$-torsion
objects $X\in{\mathcal{T}}$. Writing $\mathcal{S}$ for the full subcategory of
$V$-local objects in ${\mathcal{T}}$, one has a recollement of triangulated
categories

(2.1)
${{\mathcal{S}}}$${{\mathcal{T}}}$${{\mathcal{T}}/{\mathcal{S}}}$$\scriptstyle{i}$$\scriptstyle{i_{\lambda}}$$\scriptstyle{i_{\rho}}$$\scriptstyle{p}$$\scriptstyle{p_{\lambda}}$$\scriptstyle{p_{\rho}}$

by [9, Proposition 2.3]. The exact functors

$L_{V}\coloneqq ii_{\lambda}\qquad\text{and}\qquad\varGamma_{V}\coloneqq
p_{\lambda}p$

are called _localisation_ and _local cohomology_ with respect to $V$. They
provide, for each object $X\in{\mathcal{T}}$, a functorial exact triangle

(2.2) $\varGamma_{V}X\longrightarrow X\longrightarrow
L_{V}X\longrightarrow\,.$

An object $X\in\mathcal{T}$ is $V$-torsion if and only if the counit
$\varGamma_{V}X\to X$ is an isomorphism, and $X$ is $V$-local if and only if
the unit $X\to L_{V}X$ is an isomorphism. In particular, the image of
$\varGamma_{V}$ equals the subcategory of $V$-torsion objects in
${\mathcal{T}}$, while the image of $L_{V}$ equals the subcategory of
$V$-local objects in ${\mathcal{T}}$. It is clear from the definition that
$\varGamma_{V}\mathcal{T}$ and $L_{V}\mathcal{T}$ are triangulated
subcategories of $\mathcal{T}$.

######  Lemma 2.3.

The triangulated categories $\varGamma_{V}\mathcal{T}$ and $L_{V}\mathcal{T}$
have coproducts and are compactly generated. We have
${(\varGamma_{V}\mathcal{T})}^{\mathrm{c}}={\mathcal{T}}^{\mathrm{c}}\cap\varGamma_{V}\mathcal{T}$,
while ${(L_{V}\mathcal{T})}^{\mathrm{c}}$ and
$L_{V}({\mathcal{T}}^{\mathrm{c}})$ coincide up to direct summands.

###### Proof.

It follows from [6, Theorem 6.4] that $\varGamma_{V}{\mathcal{T}}$ is
compactly generated by the compact objects from $\mathcal{T}$ that are
$V$-torsion. The claims concerning $L_{V}\mathcal{T}$ are contained in [38,
Theorem 2.1]. ∎

### Completion

The _completion_ with respect to $V$ is the functor

$\varLambda^{V}\coloneqq p_{\rho}p\,,$

and an object $X\in{\mathcal{T}}$ is _$V$ -complete_ if the unit
$X\to\varLambda^{V}X$ is an isomorphism.

The following result describes the interplay between torsion and completion.
This phenomenon was discovered by Greenlees and May [26], and one finds many
similar statements in the literature; see for example [23, Theorem 2.1] or
[28, Theorem 3.3.5], and also [9, Proposition 2.3].

######  Proposition 2.4.

For each $X$ in ${\mathcal{T}}$ the canonical maps $\varGamma_{V}X\to X$ and
$X\to\varLambda^{V}X$ induce isomorphisms

$\varLambda^{V}\varGamma_{V}X\xrightarrow{\
\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\varLambda^{V}X\qquad\text{and}\qquad\varGamma_{V}X\xrightarrow{\
\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\varGamma_{V}\varLambda^{V}X\,.$

Moreover, the functors $(\varGamma_{V},\varLambda^{V})$ form an adjoint pair
and induce mutually quasi-inverse triangle equivalences

${\varGamma_{V}{\mathcal{T}}}$${\varLambda^{V}{\mathcal{T}}\,.}$$\scriptstyle{\varLambda^{V}}$$\scriptstyle{\varGamma_{V}}$

###### Proof.

Since $\varLambda^{V}L_{V}=p_{\rho}pii_{\lambda}=0$, applying $\varLambda^{V}$
to the exact triangle (2.2) yields an isomorphism
$\varLambda^{V}\varGamma_{V}X\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\varLambda^{V}X$.
An analogous argument yields
$\varGamma_{V}X\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\varGamma_{V}\varLambda^{V}X$.

For $X,Y\in{\mathcal{T}}$ we have isomorphisms

$\operatorname{Hom}_{\mathcal{T}}(p_{\lambda}pX,Y)\cong\operatorname{Hom}_{\mathcal{T}/\mathcal{S}}(pX,pY)\cong\operatorname{Hom}_{\mathcal{T}}(X,p_{\rho}pY)$

and hence $(\varGamma_{V},\varLambda^{V})$ form an adjoint pair. The first
part of the lemma then yields the equivalences
$\varGamma_{V}{\mathcal{T}}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\varLambda^{V}{\mathcal{T}}$
and
$\varLambda^{V}{\mathcal{T}}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\varGamma_{V}{\mathcal{T}}$.
∎

### Localisation and completion at a point

Fix a point $\mathfrak{p}\in\operatorname{Spec}R$ and consider the
specialisation closed subsets

$V(\mathfrak{p})\coloneqq\\{\mathfrak{q}\in\operatorname{Spec}R\mid\mathfrak{p}\subseteq\mathfrak{q}\\}\qquad\text{and}\qquad
Z(\mathfrak{p})\coloneqq\\{\mathfrak{q}\in\operatorname{Spec}R\mid\mathfrak{q}\not\subseteq\mathfrak{p}\\}\,.$

For $X\in\mathcal{T}$ the _localisation_ of $X$ at $\mathfrak{p}$ is

$X_{\mathfrak{p}}\coloneqq L_{Z(\mathfrak{p})}X\,.$

The object $X$ is said to be _$\mathfrak{p}$ -local_ if the natural
localisation map is an isomorphism:
$X\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}X_{\mathfrak{p}}$.
Set

${\mathcal{T}}_{\mathfrak{p}}\coloneqq\\{X\in{\mathcal{T}}\mid X\text{ is
$\mathfrak{p}$-local}\\}$

for the full subcategory of $\mathfrak{p}$-local objects in $\mathcal{T}$. For
all $C\in{\mathcal{T}}^{\mathrm{c}}$ the natural morphism $X\to
X_{\mathfrak{p}}$ induces an isomorphism

$\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)_{\mathfrak{p}}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{Hom}_{\mathcal{T}}^{*}(C,X_{\mathfrak{p}})\,.$

Thus an object $X$ is $\mathfrak{p}$-local if and only if
$\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)$ is $\mathfrak{p}$-local for all
$C\in{\mathcal{T}}^{\mathrm{c}}$. For $X,Y$ in ${\mathcal{T}}_{\mathfrak{p}}$
the $R$-module $\operatorname{Hom}_{\mathcal{T}}^{*}(X,Y)$ is
$\mathfrak{p}$-local, so the action of $R$ on ${\mathcal{T}}_{\mathfrak{p}}$
factors through $R_{\mathfrak{p}}$. In particular,
${\mathcal{T}}_{\mathfrak{p}}$ is an $R_{\mathfrak{p}}$-linear category. The
following result is used frequently; the proof is clear.

######  Lemma 2.5.

Let $\mathfrak{p}$ in $\operatorname{Spec}R$. For all
$C\in{\mathcal{T}}^{\mathrm{c}}$ and $X\in\mathcal{T}$ there are isomorphisms
of $R_{\mathfrak{p}}$-modules

$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)_{\mathfrak{p}}\cong\operatorname{Hom}^{*}_{\mathcal{T}}(C,X_{\mathfrak{p}})\cong\operatorname{Hom}^{*}_{\mathcal{T}}(C_{\mathfrak{p}},X_{\mathfrak{p}})\,.$

Thus, if the $R$-linear category $\mathcal{T}$ is noetherian, so is the
$R_{\mathfrak{p}}$-linear category ${\mathcal{T}}_{\mathfrak{p}}$. ∎

We call the functor
$\varGamma_{\mathfrak{p}}\colon{\mathcal{T}}\to\mathcal{T}$ given by
$X\mapsto\varGamma_{V(\mathfrak{p})}X_{\mathfrak{p}}$ the _local cohomology
functor_ with respect to $\mathfrak{p}$. The functor
$\varLambda^{\mathfrak{p}}\colon\mathcal{T}\to\mathcal{T}$ given by
$X\mapsto\varLambda^{V(\mathfrak{p})}ii_{\rho}X$, where the adjoint pair
$(i,i_{\rho})$ is from (2.1) for $V=Z(\mathfrak{p})$, is right adjoint to
$\varGamma_{\mathfrak{p}}$. It is easy to verify from the recollement defining
these functors that the images of $\varGamma_{\mathfrak{p}}$ and
$\varLambda^{\mathfrak{p}}$ are the following subcategories:

$\displaystyle\varGamma_{\mathfrak{p}}{\mathcal{T}}$
$\displaystyle\coloneqq\\{X\in{\mathcal{T}}_{\mathfrak{p}}\mid X\text{ is
$V(\mathfrak{p})$-torsion}\\}$
$\displaystyle\varLambda^{\mathfrak{p}}{\mathcal{T}}$
$\displaystyle\coloneqq\\{X\in{\mathcal{T}}_{\mathfrak{p}}\mid X\text{ is
$V(\mathfrak{p})$-complete}\\}.$

We speak of objects in $\varGamma_{\mathfrak{p}}{\mathcal{T}}$ as being
_$\mathfrak{p}$ -local $\mathfrak{p}$-torsion_ and those in
$\varLambda^{\mathfrak{p}}{\mathcal{T}}$ as being _$\mathfrak{p}$ -local
$\mathfrak{p}$-complete_. Proposition 2.4 implies that
$\varGamma_{\mathfrak{p}}$ and $\varLambda^{\mathfrak{p}}$ restrict to
mutually quasi-inverse triangle equivalences

(2.6)
${\varGamma_{\mathfrak{p}}{\mathcal{T}}}$${\varLambda^{\mathfrak{p}}{\mathcal{T}}\,.}$$\scriptstyle{\varLambda^{\mathfrak{p}}}$$\scriptstyle{\varGamma_{\mathfrak{p}}}$

A caveat is in order here: The equivalence above is not compatible with the
functors from $\mathcal{T}$, in that,
$\varLambda^{\mathfrak{p}}\varGamma_{\mathfrak{p}}\not\cong\varLambda^{\mathfrak{p}}$
and
$\varGamma_{\mathfrak{p}}\varLambda^{\mathfrak{p}}\not\cong\varGamma_{\mathfrak{p}}$
in general. However, restricted to $\mathcal{T}_{\mathfrak{p}}$ they are
isomorphisms, by Proposition 2.4. Moreover, in applications the more natural
completion functor to consider is $\mathfrak{p}$-completion after localisation
at $\mathfrak{p}$, akin to the $\mathfrak{p}$-torsion functor. All these
considerations suggest restricting attention to $\mathfrak{p}$-local objects
and then the following perspective is useful: The inclusion of the full
subcategory

$\mathcal{T}_{<\mathfrak{p}}\coloneqq\\{X\in\mathcal{T}_{\mathfrak{p}}\mid\varGamma_{\mathfrak{p}}X=0\\}=\\{X\in\mathcal{T}_{\mathfrak{p}}\mid\varLambda^{\mathfrak{p}}X=0\\}$

of $\mathfrak{p}$-local objects (co)supported away from $\mathfrak{p}$ yields
a recollement

${{\mathcal{T}}_{<\mathfrak{p}}}$${{\mathcal{T}_{\mathfrak{p}}}}$${{\mathcal{T}_{\mathfrak{p}}}/{\mathcal{T}}_{<\mathfrak{p}}}$$\scriptstyle{q}$$\scriptstyle{q_{\lambda}}$$\scriptstyle{q_{\rho}}$

of triangulated categories. It is a special case of (2.1). Then
$\varGamma_{\mathfrak{p}}\cong q_{\lambda}q$ and
$\varLambda^{\mathfrak{p}}\cong q_{\rho}q$.

### Koszul construction

Given an element $r$ in $R$ and an object $C$ in $\mathcal{T}$, we write
${C}/\\!\\!/{r}$ for a cone of the morphism $C\xrightarrow{r}\Sigma^{|r|}C$.
Given a sequence of elements $\boldsymbol{r}\coloneqq r_{1},\dots,r_{n}$ in
$R$, we write ${C}/\\!\\!/{\boldsymbol{r}}$ for any object obtained by taking
iterated cones; this is the _Koszul construction_ on $C$ with respect to
$\boldsymbol{r}$; see [6, Definition 5.10]. For an ideal $\mathfrak{a}$ of $R$
we write ${C}/\\!\\!/{\mathfrak{a}}$ for any Koszul object
${C}/\\!\\!/{\boldsymbol{r}}$ such that $\boldsymbol{r}$ generates
$\mathfrak{a}$.

######  Lemma 2.7.

Fix $\mathfrak{p}$ in $\operatorname{Spec}R$. There is an equality
${(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}={(\varLambda^{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}$,
and this thick subcategory is generated by objects of the form
${C}/\\!\\!/{\mathfrak{p}}$ as $C$ ranges over the compact objects of
$\mathcal{T}_{\mathfrak{p}}$. Hence for any compact object $C$ in
$\varGamma_{\mathfrak{p}}\mathcal{T}$ or
$\varLambda^{\mathfrak{p}}\mathcal{T}$, there exists an integer $n$ such that

$\mathfrak{p}^{n}\cdot\operatorname{Hom}^{*}_{\mathcal{T}}(C,-)=0=\mathfrak{p}^{n}\cdot\operatorname{Hom}^{*}_{\mathcal{T}}(-,C)\,.$

###### Proof.

The description of the compact objects in
$\varGamma_{\mathfrak{p}}\mathcal{T}$ follows from [6, Theorem 6.4]; see also
[7, Proposition 2.7]. Cosupport theory shows that
$\varLambda^{\mathfrak{p}}({X}/\\!\\!/{\mathfrak{p}})={X}/\\!\\!/{\mathfrak{p}}$
for any object $X$ in $\mathcal{T}_{\mathfrak{p}}$; see [9, §4] and in
particular [9, Lemma 4.12]. This yields the equality
${(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}={(\varLambda^{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}$,
since $\varLambda^{\mathfrak{p}}$ induces an equivalence
$\varGamma_{\mathfrak{p}}\mathcal{T}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\varLambda^{\mathfrak{p}}\mathcal{T}$.
From [6, Lemma 5.11] it follows that
$\mathfrak{p}^{n}\cdot\operatorname{End}^{*}_{\mathcal{T}}(C)=0$ for $n\gg 0$.
This yields the second assertion for compacts in
$\varGamma_{\mathfrak{p}}\mathcal{T}$ and
$\varLambda^{\mathfrak{p}}\mathcal{T}$. ∎

We also record the following observation, for later use. Given a graded
$R$-module $M$, the object ${M}/\\!\\!/{\mathfrak{a}}$ is the Koszul
construction on $M$ with respect to the ideal $\mathfrak{a}$, where $M$ is
viewed as an object in the derived category of graded $R$-modules. In
particular, each $H^{i}({M}/\\!\\!/{\mathfrak{a}})$ is itself a graded
$R$-module.

######  Lemma 2.8.

For objects $X,Y$ in $\mathcal{T}$ and an ideal $\mathfrak{a}\subseteq R$ one
has

$\displaystyle\operatorname{length}_{R}\operatorname{Hom}^{*}_{\mathcal{T}}(X,{Y}/\\!\\!/{\mathfrak{a}})<\infty$
$\displaystyle\iff\operatorname{length}_{R}H^{*}({\operatorname{Hom}^{*}_{\mathcal{T}}(X,Y)}/\\!\\!/{\mathfrak{a}})<\infty$
$\displaystyle\iff\operatorname{length}_{R}\operatorname{Hom}^{*}_{\mathcal{T}}({X}/\\!\\!/{\mathfrak{a}},Y)<\infty\,.$

###### Proof.

We verify the first isomorphism; a similar argument gives the second one. It
suffices to consider the case when $\mathfrak{a}=(r)$. Set
$M=\operatorname{Hom}^{*}_{\mathcal{T}}(X,Y)$ and let $L$ be the submodule of
elements annihilated by $r$. From the construction of the Koszul object
${Y}/\\!\\!/{r}$ one gets an exact sequence of $R$-modules

$0\longrightarrow
M/rM\longrightarrow\operatorname{Hom}^{*}_{\mathcal{T}}(X,{Y}/\\!\\!/{r})\longrightarrow
L\longrightarrow 0\,.$

Thus $\operatorname{Hom}^{*}_{\mathcal{T}}(X,{Y}/\\!\\!/{r})$ has finite
length if and only if both $M/rM$ and $L$ have finite length. For the Koszul
object ${M}/\\!\\!/{r}$ we have

$H^{i}({M}/\\!\\!/{r})=\begin{cases}L&i=-1\,,\\\ M/rM&i=0\,,\\\
0&i\neq-1,0\,.\end{cases}$

The stated equivalences are now obvious. ∎

##  3\. Tensor structure and dualisable objects

In this section we recall some basic facts about compactly generated tensor
triangulated categories. We pay particular attention to the notion of
dualisability; a more detailed discussion can be found in [14].

Fix a compactly generated tensor triangulated category
$({\mathcal{T}},\otimes,\mathds{1})$, with tensor product $\otimes$ and unit
$\mathds{1}$. Throughout we assume the tensor product is symmetric. For now
$\mathds{1}$ need not be compact. Brown representability yields _functions
objects_ $\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,Y)$ satisfying an
adjunction isomorphism

$\operatorname{Hom}_{\mathcal{T}}(X\otimes
Y,Z)\cong\operatorname{Hom}_{\mathcal{T}}(X,\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(Y,Z))\quad\text{for
all $X,Y,Z$ in ${\mathcal{T}}$.}$

The _Spanier–Whitehead dual_ of an object $X$ is

$D^{\scriptscriptstyle{\mathrm{SW}}}X\coloneqq\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,\mathds{1})\,.$

The assignment $X\mapsto D^{\scriptscriptstyle{\mathrm{SW}}}X$ is a
contravariant functor ${\mathcal{T}}\to{\mathcal{T}}$.

### Dualisable objects

An object $X$ in ${\mathcal{T}}$ is said to be _dualisable_ if for all $Y$ in
${\mathcal{T}}$ the natural map

$D^{\scriptscriptstyle{\mathrm{SW}}}X\otimes
Y\longrightarrow\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,Y)\,,$

is an isomorphism.

The adjoint of the composite $X\otimes
D^{\scriptscriptstyle{\mathrm{SW}}}X\xrightarrow{\gamma}D^{\scriptscriptstyle{\mathrm{SW}}}X\otimes
X\xrightarrow{\varepsilon}\mathds{1}$ of the braiding $\gamma$ with the counit
$\varepsilon$ gives the natural double duality map

$\rho\colon X\longrightarrow
D^{\scriptscriptstyle{\mathrm{SW}}}D^{\scriptscriptstyle{\mathrm{SW}}}(X)\,.$

The result below is [14, Proposition 2.2] and collects some useful
observations concerning these notions.

######  Proposition 3.1.

Let $X$ be an object in ${\mathcal{T}}$. The following statements hold.

  1. (1)

The object $X$ is dualisable if and only if there is a map
$\eta\colon\mathds{1}\to X\otimes D^{\scriptscriptstyle{\mathrm{SW}}}X$ making
the following diagram commute

${\mathds{1}}$${X\otimes
D^{\scriptscriptstyle{\mathrm{SW}}}X}$${\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,X)}$${D^{\scriptscriptstyle{\mathrm{SW}}}X\otimes
X}$$\scriptstyle{\eta}$$\scriptstyle{\gamma}$

where the left hand map is the adjoint to the isomorphism $\mathds{1}\otimes
X\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}X$.

  2. (2)

If $X$ is dualisable so is $D^{\scriptscriptstyle{\mathrm{SW}}}X$ and
$\rho\colon X\to
D^{\scriptscriptstyle{\mathrm{SW}}}D^{\scriptscriptstyle{\mathrm{SW}}}X$ is an
isomorphism.

  3. (3)

If $X$ is dualisable and $C\in{\mathcal{T}}$ is compact, then $C\otimes X$ is
compact. ∎

### Central actions

From this point on $({\mathcal{T}},\otimes,\mathds{1})$ is a rigidly compactly
generated tensor triangulated category, where _rigidity_ means that compact
and dualisable objects in $\mathcal{T}$ coincide. Hence $\mathds{1}$ is
compact, ${\mathcal{T}}^{\mathrm{c}}\subseteq\mathcal{T}$ is a tensor ideal
thick subcategory, and Spanier–Whitehead duality induces an auto-equivalence
on it.

Let $R$ be a noetherian graded commutative ring that acts _canonically_ on
$\mathcal{T}$, that is to say, via a map of graded rings
$R\to\operatorname{End}_{\mathcal{T}}^{*}(\mathds{1})$.

We begin by recording some special features of the localisation and completion
functors, recalled in Section 2, in this context. For any specialisation
closed subset $V\subseteq\operatorname{Spec}R$ and object $X\in{\mathcal{T}}$
there are natural isomorphisms

$\varGamma_{V}X\cong\varGamma_{V}\mathds{1}\otimes X\qquad\text{and}\qquad
L_{V}X\cong L_{V}\mathds{1}\otimes X\,.$

By adjunction, these yield for any $Y\in\mathcal{T}$ natural isomorphisms

$\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\varGamma_{V}X,Y)\cong\varLambda^{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,Y)\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,\varLambda^{V}Y)\,.$

See [6, Proposition 8.1] and [9, Proposition 8.3]. In particular, for any
$\mathfrak{p}\in\operatorname{Spec}R$ and objects
$X,Y\in\mathcal{T}_{\mathfrak{p}}$ we have

(3.2)
$\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\varGamma_{\mathfrak{p}}X,Y)\cong\varLambda^{\mathfrak{p}}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,Y)\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,\varLambda^{\mathfrak{p}}Y)\,.$

### Tensor structures of subcategories

We describe the tensor structures of the subcategories $L_{V}\mathcal{T}$ and
$\varGamma_{V}\mathcal{T}$ that are given by a specialisation closed subset
$V\subseteq\operatorname{Spec}R$. In particular we study their dualisable
objects.

The following lemma is about the shadow of Spanier–Whitehead duality in
$L_{V}{\mathcal{T}}$, the subcategory of ${\mathcal{T}}$ consisting of
$V$-local objects.

######  Lemma 3.3.

For each specialisation closed subset $V\subseteq\operatorname{Spec}R$ one has
a commutative diagram of exact functors:

${{\mathcal{T}}^{\mathrm{c}}}$${L_{V}{\mathcal{T}}}$${{\mathcal{T}}^{\mathrm{c}}}$${L_{V}{\mathcal{T}}}$$\scriptstyle{\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,\mathds{1})}$$\scriptstyle{L_{V}}$$\scriptstyle{\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,L_{V}\mathds{1})}$$\scriptstyle{L_{V}}$

###### Proof.

Fix a compact object $C$ in ${\mathcal{T}}$. The following series of
isomorphisms verifies the commutativity of the diagram:

$\displaystyle\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(L_{V}\mathds{1}\otimes
C,L_{V}\mathds{1})$
$\displaystyle\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(L_{V}\mathds{1},\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,L_{V}\mathds{1}))$
$\displaystyle\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(L_{V}\mathds{1},D^{\scriptscriptstyle{\mathrm{SW}}}{C}\otimes
L_{V}\mathds{1})$
$\displaystyle\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(L_{V}\mathds{1},L_{V}D^{\scriptscriptstyle{\mathrm{SW}}}C)$
$\displaystyle\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\mathds{1},L_{V}D^{\scriptscriptstyle{\mathrm{SW}}}C)$
$\displaystyle\cong L_{V}D^{\scriptscriptstyle{\mathrm{SW}}}C\,.$

The second one holds because compact objects are dualisable, by rigidity of
$\mathcal{T}$. ∎

The result below builds on Lemma 2.3.

######  Proposition 3.4.

Let $V\subseteq\operatorname{Spec}R$ be a specialisation closed subset. The
category $L_{V}{\mathcal{T}}$ is compactly generated and inherits a tensor
product from $\mathcal{T}$. The unit is $L_{V}\mathds{1}$ and the function
object is the same as for ${\mathcal{T}}$. The compact objects and dualisable
objects in $L_{V}{\mathcal{T}}$ coincide and are the direct summands of
objects in $L_{V}({\mathcal{T}}^{\mathrm{c}})$.

###### Proof.

The claim about compact generation and the description of compact objects is
from Lemma 2.3. For $X,Y$ in $\mathcal{T}$ the maps $X\to L_{V}X$ and $Y\to
L_{V}Y$ induce isomorphisms

$L_{V}(X\otimes Y)\xrightarrow{\
\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\ }L_{V}X\otimes
Y\xrightarrow{\ \raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}L_{V}X\otimes L_{V}Y\,.$

It follows that when $X,Y$ are $V$-local, so is $X\otimes Y$, and hence that
the tensor product on $\mathcal{T}$ restricts to a tensor product on
$L_{V}\mathcal{T}$. Setting $Y=\mathds{1}$ one gets also that
$L_{V}\mathds{1}$ is the unit in $L_{V}\mathcal{T}$. The isomorphism

$\operatorname{Hom}_{\mathcal{T}}(-,\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,Y))\cong\operatorname{Hom}_{\mathcal{T}}(-\otimes
X,Y)$

shows that $\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,Y)$ is $V$-local
when $Y$ is $V$-local. Thus the function object in $L_{V}{\mathcal{T}}$ is the
same as the one in $\mathcal{T}$.

Since $L_{V}\mathds{1}$ is compact in $L_{V}\mathcal{T}$, dualisable objects
are compact, by Proposition 3.1. As to the converse, it suffices to verify
that objects of the form $L_{V}C$, with $C$ in ${\mathcal{T}}^{\mathrm{c}}$,
are dualisable in $L_{V}{\mathcal{T}}$. For any $Y$ in ${L_{V}\mathcal{T}}$
one has isomorphisms

$\displaystyle\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(L_{V}C,L_{V}\mathds{1})\otimes
Y$ $\displaystyle\cong
L_{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,\mathds{1})\otimes Y$
$\displaystyle\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,\mathds{1})\otimes
Y$ $\displaystyle\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,Y)$
$\displaystyle\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(L_{V}C,Y)$

where the first isomorphism is by the commutativity of the square in Lemma
3.3. Thus $L_{V}C$ is dualisable. ∎

Next we discuss the analogues of the preceding results for
$\varGamma_{V}{\mathcal{T}}$, the subcategory of $\mathcal{T}$ consisting of
$V$-torsion objects.

######  Lemma 3.5.

For each specialisation closed subset $V\subseteq\operatorname{Spec}R$ one has
commutative diagrams of exact functors:

${{\mathcal{T}}^{\mathrm{c}}}$${\varGamma_{V}{\mathcal{T}}}$${{\mathcal{T}}^{\mathrm{c}}}$${\varLambda^{V}{\mathcal{T}}}$$\scriptstyle{\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,\mathds{1})}$$\scriptstyle{\varGamma_{V}}$$\scriptstyle{\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,\varGamma_{V}\mathds{1})}$$\scriptstyle{\varLambda^{V}}$
${{\mathcal{T}}^{\mathrm{c}}}$${\varLambda^{V}{\mathcal{T}}}$${{\mathcal{T}}^{\mathrm{c}}}$${\varGamma_{V}{\mathcal{T}}}$$\scriptstyle{\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,\mathds{1})}$$\scriptstyle{\varLambda^{V}}$$\scriptstyle{\varGamma_{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,\varLambda^{V}\mathds{1})}$$\scriptstyle{\varGamma_{V}}$

###### Proof.

Fix a compact object $C$ in $\mathcal{T}$. The commutativity of the square on
the left is verified below:

$\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\varGamma_{V}C,\varGamma_{V}\mathds{1})\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,\varLambda^{V}\mathds{1})\cong\varLambda^{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,\mathds{1})\,.$

Next we verify the commutativity of the square on the right:

$\displaystyle\varGamma_{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\varLambda^{V}C,\varLambda^{V}\mathds{1})$
$\displaystyle\cong\varGamma_{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,\varLambda^{V}\mathds{1})$
$\displaystyle\cong\varGamma_{V}\varLambda^{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,\mathds{1})$
$\displaystyle\cong\varGamma_{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,\mathds{1})\,.\qed$

It follows from the result below that, typically, $\varGamma_{V}{\mathcal{T}}$
has more dualisable objects than compact ones.

######  Proposition 3.6.

Let $V\subseteq\operatorname{Spec}R$ be a specialisation closed subset. The
category $\varGamma_{V}{\mathcal{T}}$ is compactly generated and inherits a
tensor product from $\mathcal{T}$. The unit is $\varGamma_{V}\mathds{1}$ and
the function object is
$\varGamma_{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,-)$. The compact
objects in $\varGamma_{V}\mathcal{T}$ are

${(\varGamma_{V}\mathcal{T})}^{\mathrm{c}}={\mathcal{T}}^{\mathrm{c}}\cap\varGamma_{V}\mathcal{T}\,,$

and these are dualisable. All objects in
$\varGamma_{V}{({\mathcal{T}}^{\mathrm{c}})}$ are dualisable.

###### Proof.

Compact generation and description of compact objects is from Lemma 2.3. From
the description of $\varGamma_{V}$ as a tensor functor it is clear that the
tensor product on $\mathcal{T}$ restricts to a tensor product on
$\varGamma_{V}{\mathcal{T}}$. The functor $\varGamma_{V}$ provides a right
adjoint for the inclusion $\varGamma_{V}{\mathcal{T}}\to\mathcal{T}$, and
therefore it preserves function objects.

It remains to verify that $\varGamma_{V}C$ is dualisable in
$\varGamma_{V}{\mathcal{T}}$ for any compact object $C$ in $\mathcal{T}$. For
any $Y$ in $\varGamma_{V}{\mathcal{T}}$ one gets

$\displaystyle\varGamma_{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\varGamma_{V}C,\varGamma_{V}\mathds{1})\otimes
Y$
$\displaystyle\cong\varGamma_{V}\varLambda^{V}D^{\scriptscriptstyle{\mathrm{SW}}}{(C)}\otimes
Y$
$\displaystyle\cong\varGamma_{V}D^{\scriptscriptstyle{\mathrm{SW}}}{(C)}\otimes
Y$
$\displaystyle\cong\varGamma_{V}(D^{\scriptscriptstyle{\mathrm{SW}}}{(C)}\otimes
Y)$
$\displaystyle\cong\varGamma_{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(C,Y)$
$\displaystyle\cong\varGamma_{V}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(\varGamma_{V}C,Y)$

where the first isomorphism follows from the commutativity of the second
square in Lemma 3.5. Thus $\varGamma_{V}C$ is dualisable. ∎

Next we focus on $\varGamma_{\mathfrak{p}}{\mathcal{T}}$, the subcategory of
$\mathcal{T}$ consisting of $\mathfrak{p}$-local $\mathfrak{p}$-torsion
objects, for some $\mathfrak{p}$ in $\operatorname{Spec}R$. The
Spanier–Whitehead dual in the category $\varGamma_{\mathfrak{p}}\mathcal{T}$
is given by the functor

$\varGamma_{\mathfrak{p}}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,\mathds{1}_{\mathfrak{p}})=\varGamma_{\mathfrak{p}}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,\varGamma_{\mathfrak{p}}\mathds{1})\,.$

Since
$\varGamma_{\mathfrak{p}}{\mathcal{T}}=\varGamma_{V(\mathfrak{p})}{\mathcal{T}}\cap
L_{Z(\mathfrak{p})}\mathcal{T}$, from Propositions 3.4 and 3.6 we get the
result below.

######  Corollary 3.7.

For each $\mathfrak{p}$ in $\operatorname{Spec}R$ the category
$\varGamma_{\mathfrak{p}}{\mathcal{T}}$ is compactly generated and inherits a
tensor product from $\mathcal{T}$. The unit is
$\varGamma_{\mathfrak{p}}\mathds{1}$ and the function object is
$\varGamma_{\mathfrak{p}}\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,-)$.
The compact objects in $\varGamma_{\mathfrak{p}}\mathcal{T}$ are

${(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}={(\mathcal{T}_{\mathfrak{p}})}^{\mathrm{c}}\cap\varGamma_{\mathfrak{p}}\mathcal{T}\,.$

All objects in $\varGamma_{\mathfrak{p}}({\mathcal{T}}^{\mathrm{c}})$ are
dualisable. ∎

Here is the analogous result for $\varLambda^{\mathfrak{p}}\mathcal{T}$.

######  Corollary 3.8.

For each $\mathfrak{p}$ in $\operatorname{Spec}R$ the category
$\varLambda^{\mathfrak{p}}\mathcal{T}$ is compactly generated and tensor
triangulated, with product $\varLambda^{\mathfrak{p}}(-\otimes-)$. The unit is
$\varLambda^{\mathfrak{p}}(\mathds{1}_{\mathfrak{p}})$ and the function object
is $\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(-,-)$, the function object
on $\mathcal{T}$. The compact objects in
$\varLambda^{\mathfrak{p}}\mathcal{T}$ are

${(\varLambda^{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}={(\mathcal{T}_{\mathfrak{p}})}^{\mathrm{c}}\cap\varGamma_{\mathfrak{p}}\mathcal{T}={(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}\,.$

All objects in
$\varLambda^{\mathfrak{p}}({\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}})$ are
dualisable. ∎

##  4\. Finite length objects

Let $R$ be a graded commutative noetherian ring and $\mathcal{T}$ a compactly
generated $R$-linear triangulated category; it need not be tensor
triangulated. In what follows we consider various types of finiteness
conditions for objects in $\mathcal{T}$ and its various subcategories. Here
are the relevant concepts.

######  Definition 4.1.

An object $X$ in $\mathcal{T}$ is said to be

  1. (1)

_noetherian_ over $R$ if the $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is noetherian for all
$C\in{\mathcal{T}}^{\mathrm{c}}$,

  2. (2)

_artinian_ over $R$ if the $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is artinian for all
$C\in{\mathcal{T}}^{\mathrm{c}}$,

  3. (3)

_of finite length_ over $R$ if $\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is
of finite length over $R$ for all $C\in{\mathcal{T}}^{\mathrm{c}}$.

For tensor triangulated categories there is a more intrinsic notion of a
finite length object.

######  Remark 4.2.

Let $\mathcal{T}=(\mathcal{T},\otimes,\mathds{1})$ be tensor triangulated and
suppose it admits an $R$-action via a map of graded rings $R\to
S=\operatorname{End}_{\mathcal{T}}^{*}(\mathds{1})$. Then we say that an
object $X$ in $\mathcal{T}$ is _of finite length_ if
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is of finite length over $S$ for
all $C\in{\mathcal{T}}^{\mathrm{c}}$. This condition coincides with the one
over $R$ when, for example, $S$ is finite over $R$ or when $R$ is local and
$S$ is its completion at the maximal ideal.

######  Remark 4.3.

The above finiteness conditions on graded morphisms imply the analogous
conditions for the degree zero morphisms, as a module over $R^{0}$, because
for any objects $X,Y$ in $\mathcal{T}$ and
$M=\operatorname{Hom}^{*}_{\mathcal{T}}(X,Y)$, the poset of submodules of the
$R^{0}$-module $M^{0}=\operatorname{Hom}_{\mathcal{T}}(X,Y)$ embeds into the
poset of submodules of the graded $R$-module $M$. The map sends $U\subseteq
M^{0}$ to $RU$, and its inverse sends $V\subseteq M$ to $V^{0}$.

We proceed with a characterisation of the finite length objects in
$\varGamma_{\mathfrak{p}}\mathcal{T}$.

######  Proposition 4.4.

For $\mathfrak{p}\in\operatorname{Spec}R$ and $X\in\mathcal{T}_{\mathfrak{p}}$
the conditions below are equivalent.

  1. (1)

$X$ is in $\varGamma_{\mathfrak{p}}\mathcal{T}$ and of finite length over
$R_{\mathfrak{p}}$;

  2. (2)

$X$ is artinian over $R_{\mathfrak{p}}$;

  3. (3)

$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is artinian over
$R_{\mathfrak{p}}$ for each compact object $C$ in ${\mathcal{T}}$.

###### Proof.

By Lemmas 2.3 and 2.5 we can pass to $\mathcal{T}_{\mathfrak{p}}$ and
$R_{\mathfrak{p}}$ and suppose $R$ is local with maximal ideal $\mathfrak{p}$.
In particular, (2) and (3) are equivalent.

(1)$\Rightarrow$(2): Let $C$ be a compact object in $\mathcal{T}$ and fix a
sequence of elements $\boldsymbol{r}\coloneqq r_{1},\ldots,r_{n}$ which
generates the ideal $\mathfrak{p}$. Then ${C}/\\!\\!/{\boldsymbol{r}}$ is
$\mathfrak{p}$-torsion and hence in
${(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}$. The hypothesis yields
that the $R$-module
$\operatorname{Hom}_{\mathcal{T}}^{*}({C}/\\!\\!/{\boldsymbol{r}},X)$ has
finite length, and in particular it is artinian. Since
${C}/\\!\\!/{\boldsymbol{r}}$ is obtained from $C$ by an iterated Koszul
construction, it suffices to prove that for any $r\in\mathfrak{p}$ and compact
object $C$ in $\mathcal{T}$, if the $R$-module
$\operatorname{Hom}_{\mathcal{T}}^{*}({C}/\\!\\!/{r},X)$ is artinian, then so
is $\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)$.

By definition of ${C}/\\!\\!/{r}$ one has an exact triangle
$C\xrightarrow{r}\Sigma^{|r|}C\to{C}/\\!\\!/{r}\to$ and that induces an exact
sequence of $R$-modules

$\operatorname{Hom}_{\mathcal{T}}^{*}({C}/\\!\\!/{r},X)\longrightarrow\operatorname{Hom}_{\mathcal{T}}^{*-|r|}(C,X)\xrightarrow{\
r\ }\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)\,.$

The module $\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)$ is
$\mathfrak{p}$-torsion, since $X$ is in $\varGamma_{\mathfrak{p}}\mathcal{T}$.
As $\operatorname{Hom}_{\mathcal{T}}^{*}({C}/\\!\\!/{r},X)$ is artinian, Lemma
4.5 below yields that $\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)$ is also
artinian.

(2)$\Rightarrow$(1): Artinian $R$-modules are $\mathfrak{p}$-torsion, so the
hypothesis implies $X$ is in $\varGamma_{\mathfrak{p}}\mathcal{T}$. Any
compact object $C$ in $\varGamma_{\mathfrak{p}}\mathcal{T}$ is also compact in
$\mathcal{T}$, so the hypothesis implies that the $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is artinian; it is also
annihilated by a power of $\mathfrak{p}$, by Lemma 2.7. Thus it has finite
length. ∎

The following observation was used in the argument above.

######  Lemma 4.5.

Let $R$ be a local ring with maximal ideal $\mathfrak{m}$ and $M$ an
$\mathfrak{m}$-torsion $R$-module. If for an $r\in\mathfrak{m}$ the $R$-module
$\operatorname{Ker}(M\xrightarrow{r}M)$ is artinian, then so is $M$.

###### Proof.

Since $M$ is $\mathfrak{m}$-torsion its injective hull is a direct sum of
copies of twists of $I(\mathfrak{m})$, the number of copies being equal to the
length of $\operatorname{Soc}M$, the socle of $M$. The module
$I(\mathfrak{m})$ is artinian by Matlis duality, so it suffices to verify that
$\operatorname{Soc}M$ is finitely generated.

Set $L\coloneqq\operatorname{Ker}(M\xrightarrow{r}M)$ and consider the exact
sequence of $R$-modules

$0\longrightarrow L\longrightarrow M\xrightarrow{\ r\ }M\,.$

Applying $\operatorname{Hom}^{*}_{R}(R/\mathfrak{m},-)$ to it yields the exact
sequence

$0\longrightarrow\operatorname{Soc}L\longrightarrow\operatorname{Soc}M\xrightarrow{\
r\ }\operatorname{Soc}M$

and the right hand map is zero. Thus $\operatorname{Soc}M$ is isomorphic to
$\operatorname{Soc}L$ and therefore finitely generated, because $L$ is
artinian. ∎

### Minimal primes

Let $\operatorname{supp}_{R}\mathcal{T}$ denote the _support_ of
$\mathcal{T}$, the set of points $\mathfrak{p}$ in $\operatorname{Spec}R$ such
that $\varGamma_{\mathfrak{p}}X\neq 0$ for some $X\in\mathcal{T}$. The result
below augments Proposition 4.4.

######  Lemma 4.6.

Suppose $\mathcal{T}$ is noetherian and fix $\mathfrak{p}$ in
$\operatorname{supp}_{R}\mathcal{T}$. Then $\mathfrak{p}$ is minimal in
$\operatorname{supp}_{R}{\mathcal{T}}$ if and only if in
$\mathcal{T}_{\mathfrak{p}}$ artinian and finite length objects over
$R_{\mathfrak{p}}$ coincide.

###### Proof.

First observe that $\mathfrak{p}\in\operatorname{supp}_{R}\mathcal{T}$ is
minimal if and only if
$\varGamma_{\mathfrak{p}}\mathcal{T}=\mathcal{T}_{\mathfrak{p}}$. From this it
follows with Proposition 4.4 that artinian and finite length coincide for the
$R_{\mathfrak{p}}$-module $\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)$ when
$\mathfrak{p}$ is minimal.

As to the converse, we use the object $T_{\mathfrak{p}}C$ in
$\varGamma_{\mathfrak{p}}\mathcal{T}$ which is defined below via (5.2). In
particular, we have an isomorphism of $R$-modules

$\operatorname{Hom}^{*}_{R}(\operatorname{End}^{*}_{\mathcal{T}}(C),I(\mathfrak{p}))\cong\operatorname{Hom}^{*}_{\mathcal{T}}(C,T_{\mathfrak{p}}C)\,.$

Since $\operatorname{End}^{*}_{\mathcal{T}}(C)$ is noetherian,
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,T_{\mathfrak{p}}C)$ is artinian.
Moreover one of them has finite length if and only if the other does, by
Matlis duality. This means that any compact object $C$ in
$\mathcal{T}_{\mathfrak{p}}$ belongs to $\varGamma_{\mathfrak{p}}\mathcal{T}$.
Thus $\varGamma_{\mathfrak{p}}\mathcal{T}=\mathcal{T}_{\mathfrak{p}}$, and
therefore $\mathfrak{p}$ is minimal by our first observation. ∎

### Pure-injectivity

Artinian objects enjoy further finiteness properties. To explain this we
recall the notion of purity for compactly generated triangulated categories;
see [30, §1]. A triangle $X\to Y\to Z\to$ is _pure-exact_ if for each compact
object $C$ the induced sequence

$0\to\operatorname{Hom}_{\mathcal{T}}(C,X)\to\operatorname{Hom}_{\mathcal{T}}(C,Y)\to\operatorname{Hom}_{\mathcal{T}}(C,Z)\to
0$

is exact. In that case $X\to Y$ is a _pure monomorphism_. One calls an object
$X$ _pure-injective_ if each pure monomorphism $X\to Y$ splits, and $X$ is
_$\Sigma$ -pure-injective_ if every coproduct of copies of $X$ is pure-
injective.

######  Proposition 4.7.

Let $\mathcal{T}$ be a compactly generated $R$-linear triangulated category
and $X\in\mathcal{T}$ artinian over $R$. Then $X$ is $\Sigma$-pure-injective
and decomposes into a coproduct of indecomposable objects with local
endomorphism rings.

###### Proof.

Let $C\in\mathcal{T}$ be compact. A subgroup of
$\operatorname{Hom}_{\mathcal{T}}(C,X)$ is said to be _of finite definition_
if it is the image of a map
$\operatorname{Hom}_{\mathcal{T}}(C^{\prime},X)\to\operatorname{Hom}_{\mathcal{T}}(C,X)$
that is induced by a morphism $C\to C^{\prime}$ between compact objects. In
[32, Theorem 4.10] it is proved that when the descending chain condition on
subgroups of finite definition holds for all compact $C$ in $\mathcal{T}$, the
object $X$ is $\Sigma$-pure-injective and admits a decomposition into
indecomposable objects with local endomorphism rings. From this the assertion
about artinian objects follows, using Remark 4.3. ∎

######  Corollary 4.8.

Suppose that $\mathcal{T}$ admits a compact generator. Then the category of
objects in $\mathcal{T}$ that are artinian over $R$ is a Krull–Schmidt
category.

###### Proof.

Fix a compact generator $C$. We have already seen that any artinian object has
a decomposition $X=\coprod_{i\in I}X_{i}$ into indecomposables. We need to
show that $I$ is finite, but this is clear from the isomorphism
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)\cong\coprod_{i\in
I}\operatorname{Hom}^{*}_{\mathcal{T}}(C,X_{i})$ since this module is
artinian. ∎

##  5\. Brown–Comenetz duality

In this section we explain how classical Matlis duality from commutative
algebra, between artinian modules and noetherian modules over complete local
rings lifts to the world of tensor triangulated categories, leading to
Brown–Comenetz duality. The motivation for this is our interest in the
category of finite length objects in $\varGamma_{\mathfrak{p}}\mathcal{T}$.
Three other categories arise naturally in their study: artinian objects in
$\varGamma_{\mathfrak{p}}\mathcal{T}$, noetherian objects in
$\varLambda^{\mathfrak{p}}\mathcal{T}$, and a certain category of Matlis lifts
of compacts objects, introduced below. These categories are related via
Brown–Comenetz duality; see for instance Corollary 5.9.

To begin with we record basic results on Matlis duality. Since this is a local
construction, throughout this section we fix a graded commutative noetherian
ring $R$ and $\mathfrak{p}$ in $\operatorname{Spec}R$. We write
$I(\mathfrak{p})$ for the injective hull of $R/\mathfrak{p}$ and, for any
$R$-module $M$, write $M{}^{\wedge}_{\mathfrak{p}}$ for the completion of
$M_{\mathfrak{p}}$ with respect to $\mathfrak{p}R_{\mathfrak{p}}$. We remind
the reader that these constructions take place in the world of graded rings
and modules.

### Matlis duality

On the category of $R$-modules _Matlis duality at $\mathfrak{p}$_ is the
functor

$M\longmapsto
D_{\mathfrak{p}}M\coloneqq\operatorname{Hom}_{R}^{*}(M,I(\mathfrak{p}))\,.$

There is a canonical biduality map $M\to D_{\mathfrak{p}}^{2}M$; it is a
monomorphism and we say $M$ is _Matlis reflexive_ when it is an isomorphism.
For $M=R$ the map induces an isomorphism
$R{}^{\wedge}_{\mathfrak{p}}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{End}_{R}^{*}(I(\mathfrak{p}))$;
this is part of Matlis duality which is given below. Therefore the Matlis dual
$D_{\mathfrak{p}}M$ admits a canonical $R{}^{\wedge}_{\mathfrak{p}}$-action.

For any pair of $R$-modules $M,N$ there are natural isomorphisms of
$R{}^{\wedge}_{\mathfrak{p}}$-modules

$\operatorname{Hom}^{*}_{R}(M,D_{\mathfrak{p}}N)\cong\operatorname{Hom}^{*}_{R}(M\otimes_{R}N,I(\mathfrak{p}))\cong\operatorname{Hom}^{*}_{R}(N,D_{\mathfrak{p}}M)\,.$

Let $\operatorname{art}R$ and $\operatorname{noeth}R$ denote the categories of
$R$-modules that are artinian and noetherian, respectively. The result below
is due to Matlis [34, §4].

######  Proposition 5.1.

Let $R$ be a graded commutative noetherian ring and
$\mathfrak{p}\in\operatorname{Spec}R$. Then the following functors are
mutually quasi-inverse contravariant equivalences:

${\operatorname{art}R_{\mathfrak{p}}=\operatorname{art}R{}^{\wedge}_{\mathfrak{p}}}$${\operatorname{noeth}R{}^{\wedge}_{\mathfrak{p}}}$$\scriptstyle{\operatorname{Hom}^{*}_{R{}^{\wedge}_{\mathfrak{p}}}(-,I(\mathfrak{p}))=\operatorname{Hom}^{*}_{R}(-,I(\mathfrak{p}))}$$\scriptstyle{\operatorname{Hom}^{*}_{R{}^{\wedge}_{\mathfrak{p}}}(-,I(\mathfrak{p}))}$

Moreover, for a finitely generated $R$-module $M$ the canonical map $M\to
D_{\mathfrak{p}}^{2}M$ induces an isomorphism
$M{}^{\wedge}_{\mathfrak{p}}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}D_{\mathfrak{p}}^{2}M$.

These constructions and results carry over to the world of triangulated
categories.

### Matlis lifts

Let $\mathcal{T}$ be a compactly generated $R$-linear triangulated category;
it need not be tensor triangulated. For each object $C$ in
${\mathcal{T}}^{\mathrm{c}}$ Brown representability yields an object
$T_{\mathfrak{p}}C$ in ${\mathcal{T}}$ such that

(5.2)
$\operatorname{Hom}_{R}(\operatorname{Hom}^{*}_{\mathcal{T}}(C,-),I(\mathfrak{p}))\cong\operatorname{Hom}_{\mathcal{T}}(-,T_{\mathfrak{p}}C)\,.$

One can think of $T_{\mathfrak{p}}C$ as a _Matlis lift_ of $I(\mathfrak{p})$
to $\mathcal{T}$, with respect to $C$. The assignment $C\mapsto
T_{\mathfrak{p}}C$ yields a functor
${\mathcal{T}}^{\mathrm{c}}\to{\mathcal{T}}$, and it follows from Lemma 2.5
that $T_{\mathfrak{p}}$ admits a canonical factorisation

${\mathcal{T}}^{\mathrm{c}}\twoheadrightarrow{\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}}\to\mathcal{T}_{\mathfrak{p}}\rightarrowtail\mathcal{T}\,.$

Matlis duality implies that when $\mathcal{T}$ is noetherian,
$T_{\mathfrak{p}}C$ is an artinian object in the $R_{\mathfrak{p}}$-linear
triangulated category $\mathcal{T}_{\mathfrak{p}}$ and belongs therefore to
$\varGamma_{\mathfrak{p}}\mathcal{T}$; see also Lemma 6.2. For tensor
triangulated categories, one can lift Matlis duality to the level of objects,
and not just on their cohomology. This is explained below.

### Local Brown–Comenetz duality

Let $\mathcal{T}$ be a rigidly compactly generated tensor triangulated
category. We fix a graded commutative noetherian ring $R$ and a canonical
$R$-action on $\mathcal{T}$.

For an object $X\in\mathcal{T}$ Brown representability yields an object
$D_{\mathfrak{p}}{X}$ in ${\mathcal{T}}$ such that

$\operatorname{Hom}_{R}(\operatorname{Hom}^{*}_{\mathcal{T}}(\mathds{1},X\otimes-),I(\mathfrak{p}))\cong\operatorname{Hom}_{\mathcal{T}}(-,D_{\mathfrak{p}}X).$

This yields a contravariant functor
$D_{\mathfrak{p}}\colon{\mathcal{T}}\to{\mathcal{T}}$ called _local
Brown–Comenetz duality_. The same notation $D_{\mathfrak{p}}$ is used for
modules and for objects in a triangulated category, but there is no danger of
confusion.

In the following we collect some basic properties which will be used
throughout, sometimes without any further reference.

######  Lemma 5.3.

For $C\in{\mathcal{T}}^{\mathrm{c}}$ and $X\in\mathcal{T}$ there are natural
isomorphism of $R$-modules

$\operatorname{Hom}_{\mathcal{T}}^{*}(C,D_{\mathfrak{p}}X)\cong
D_{\mathfrak{p}}\operatorname{Hom}^{*}_{\mathcal{T}}(D^{\scriptscriptstyle{\mathrm{SW}}}C,X)\,.$

###### Proof.

The isomorphism is immediate from the defining isomorphism of
$D_{\mathfrak{p}}X$, using the dualisablity of $C$. ∎

For any pair of objects $X,Y$ in $\mathcal{T}$ there is a natural isomorphism
of $R$-modules

$\operatorname{Hom}^{*}_{\mathcal{T}}(X,D_{\mathfrak{p}}Y)\cong\operatorname{Hom}^{*}_{\mathcal{T}}(Y,D_{\mathfrak{p}}X)$

which means that $D_{\mathfrak{p}}$ yields an adjoint pair of functors
$\mathcal{T}^{\mathrm{op}}\rightleftarrows\mathcal{T}$. Also, it is easy to
verify that there is a natural isomorphism

$D_{\mathfrak{p}}X\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,D_{\mathfrak{p}}\mathds{1})\,.$

Therefore $D_{\mathfrak{p}}$ restricts to a functor
$\varGamma_{\mathfrak{p}}\mathcal{T}\to\varLambda^{\mathfrak{p}}\mathcal{T}$,
thanks to (3.2). For any compact (equivalently, dualisable) object $C$, one
has that

$D_{\mathfrak{p}}C\cong D^{\scriptscriptstyle{\mathrm{SW}}}C\otimes
D_{\mathfrak{p}}\mathds{1}\,.$

The following observation connects Matlis lifts with Brown–Comenetz duality.

######  Lemma 5.4.

For any compact object $C$ in $\mathcal{T}$ there are natural isomorphisms

$T_{\mathfrak{p}}C\cong C\otimes T_{\mathfrak{p}}\mathds{1}\cong
D_{\mathfrak{p}}(D^{\scriptscriptstyle{\mathrm{SW}}}C)\,.$

###### Proof.

Since $T_{\mathfrak{p}}\mathds{1}\cong D_{\mathfrak{p}}\mathds{1}$, from
definitions, one gets the isomorphism on the right. The one on the left holds
because for each $X\in\mathcal{T}$ there are natural isomorphisms:

$\displaystyle\operatorname{Hom}_{\mathcal{T}}(X,T_{\mathfrak{p}}C)$
$\displaystyle\cong\operatorname{Hom}_{R}(\operatorname{Hom}^{*}_{\mathcal{T}}(C,X),I(\mathfrak{p}))$
$\displaystyle\cong\operatorname{Hom}_{R}(\operatorname{Hom}^{*}_{\mathcal{T}}(\mathds{1},D^{\scriptscriptstyle{\mathrm{SW}}}C\otimes
X),I(\mathfrak{p}))$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{T}}(D^{\scriptscriptstyle{\mathrm{SW}}}C\otimes
X,T_{\mathfrak{p}}\mathds{1})$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{T}}(X,C\otimes
T_{\mathfrak{p}}\mathds{1})\,.$

The second and last isomorphisms hold because $C$ is dualisable. ∎

### Completed linear actions

We wish to establish a Matlis duality theorem for $\mathcal{T}$ and need to
assume that $\mathcal{T}$ is noetherian over $R$. Recall that this means
compact objects are noetherian: the $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,D)$ is finitely generated for all
$C,D\in{\mathcal{T}}^{\mathrm{c}}$. The first step is to show that the
$\mathfrak{p}$-torsion category and the $\mathfrak{p}$-complete category admit
$R{}^{\wedge}_{\mathfrak{p}}$-linear actions that extend the $R$-linear
actions.

######  Lemma 5.5.

Let $\mathcal{T}$ be noetherian. For a compact object
$X\in\mathcal{T}_{\mathfrak{p}}$ the canonical morphism $X\to
D_{\mathfrak{p}}^{2}X$ induces an isomorphism
$\varLambda^{\mathfrak{p}}X\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}D_{\mathfrak{p}}^{2}X$.

###### Proof.

We may assume that $R$ is local with maximal ideal $\mathfrak{p}$. Let
$C\in{\mathcal{T}}^{\mathrm{c}}$. Since $\mathcal{T}$ is noetherian over $R$,
the $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(D^{\scriptscriptstyle{\mathrm{SW}}}C,X)$
is finitely generated. Hence, the $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,D_{\mathfrak{p}}X)\cong
D_{\mathfrak{p}}\operatorname{Hom}^{*}_{\mathcal{T}}(D^{\scriptscriptstyle{\mathrm{SW}}}C,X)$
is artinian. In particular $D_{\mathfrak{p}}X$ is $\mathfrak{p}$-torsion, and
hence
$D_{\mathfrak{p}}^{2}X\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(D_{\mathfrak{p}}X,D_{\mathfrak{p}}\mathds{1})$
is $\mathfrak{p}$-complete, by (3.2).

It follows that the canonical morphism $X\to D_{\mathfrak{p}}^{2}X$ factors
through $X\to\varLambda^{\mathfrak{p}}X$. For $\varLambda^{\mathfrak{p}}X\to
D_{\mathfrak{p}}^{2}X$ to be an isomorphism, it suffices to show that the
induced map

$\operatorname{Hom}^{*}_{\mathcal{T}}({C}/\\!\\!/{\mathfrak{p}},\varLambda^{\mathfrak{p}}X)\longrightarrow\operatorname{Hom}^{*}_{\mathcal{T}}({C}/\\!\\!/{\mathfrak{p}},D_{\mathfrak{p}}^{2}X)$

is a bijection; see Lemma 2.7. For the same reason the induced map

$\operatorname{Hom}^{*}_{\mathcal{T}}({C}/\\!\\!/{\mathfrak{p}},X)\longrightarrow\operatorname{Hom}^{*}_{\mathcal{T}}({C}/\\!\\!/{\mathfrak{p}},\varLambda^{\mathfrak{p}}X)$

is a bijection. But
$\operatorname{Hom}^{*}_{\mathcal{T}}({C}/\\!\\!/{\mathfrak{p}},X)$ is of
finite length over $R$, so it is Matlis reflexive. This yields the first
isomorphism below

$\operatorname{Hom}^{*}_{\mathcal{T}}({C}/\\!\\!/{\mathfrak{p}},X)\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}D_{\mathfrak{p}}^{2}\operatorname{Hom}^{*}_{\mathcal{T}}({C}/\\!\\!/{\mathfrak{p}},X)\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{Hom}^{*}_{\mathcal{T}}({C}/\\!\\!/{\mathfrak{p}},D_{\mathfrak{p}}^{2}X)$

and the second is from Lemma 5.3. ∎

######  Lemma 5.6.

The tensor triangulated categories $\varGamma_{\mathfrak{p}}\mathcal{T}$ and
$\varLambda^{\mathfrak{p}}\mathcal{T}$ have canonical
$R{}^{\wedge}_{\mathfrak{p}}$-linear actions, extending the $R$-action
inherited from $\mathcal{T}$.

###### Proof.

For each $X\in\mathcal{T}$ there is a sequence of homomorphisms

$R\to R_{\mathfrak{p}}\to
R{}^{\wedge}_{\mathfrak{p}}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{End}_{R}^{*}(I(\mathfrak{p}))\to\operatorname{End}^{*}_{\mathcal{T}}(D_{\mathfrak{p}}\mathds{1})\to\operatorname{End}^{*}_{\mathcal{T}}(D_{\mathfrak{p}}X)\,,$

where the isomorphism is from Matlis duality. This yields a factorisation of
the canonical homomorphism
$R\to\operatorname{End}^{*}_{\mathcal{T}}(D_{\mathfrak{p}}X)$. We have
$\varLambda^{\mathfrak{p}}\mathds{1}_{\mathfrak{p}}\cong
D_{\mathfrak{p}}^{2}\mathds{1}_{\mathfrak{p}}$ by Lemma 5.5, and this gives
the factorisation of
$R\to\operatorname{End}^{*}_{\mathcal{T}}(\varLambda^{\mathfrak{p}}\mathds{1}_{\mathfrak{p}})$
by taking $X=D_{\mathfrak{p}}\mathds{1}_{\mathfrak{p}}$. This establishes the
canonical $R{}^{\wedge}_{\mathfrak{p}}$-linear action for
$\varLambda^{\mathfrak{p}}\mathcal{T}$, since
$\varLambda^{\mathfrak{p}}\mathds{1}_{\mathfrak{p}}$ is the tensor identity.
The equivalence
$\varLambda^{\mathfrak{p}}\mathcal{T}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\varGamma_{\mathfrak{p}}\mathcal{T}$
from (2.6) preserves the tensor identity, since
$\varGamma_{\mathfrak{p}}(\varLambda^{\mathfrak{p}}\mathds{1}_{\mathfrak{p}})\cong\varGamma_{\mathfrak{p}}\mathds{1}$,
and this yields the assertion for $\varGamma_{\mathfrak{p}}\mathcal{T}$. ∎

### Duality over the completion

In the following we restrict to $R$-modules and objects in $\mathcal{T}$ that
admit an $R{}^{\wedge}_{\mathfrak{p}}$-action. Then we consider the following
modification of the Matlis duality:

$M\longmapsto
D{}^{\wedge}_{\mathfrak{p}}M\coloneqq\operatorname{Hom}_{R{}^{\wedge}_{\mathfrak{p}}}^{*}(M,I(\mathfrak{p}))$

Note that there is a natural morphism $D{}^{\wedge}_{\mathfrak{p}}M\to
D_{\mathfrak{p}}M$. This is an isomorphism when $M$ is $\mathfrak{p}$-torsion,
because the class of $R{}^{\wedge}_{\mathfrak{p}}$-modules $M$ such that this
is an isomorphism includes the finite length modules and is closed under
colimits, so it includes all $\mathfrak{p}$-torsion modules. There is also a
canonical monomorphism $M\to(D{}^{\wedge}_{\mathfrak{p}})^{2}M$, which is an
isomorphism when $M$ is artinian or noetherian.

Analogously, for any object $X\in\mathcal{T}$ such that the canonical
$R$-action on $\operatorname{End}^{*}_{\mathcal{T}}(X)$ factors through $R\to
R{}^{\wedge}_{\mathfrak{p}}$, Brown representability yields an object
$D{}^{\wedge}_{\mathfrak{p}}X$ in ${\mathcal{T}}$ with a natural isomorphism

$\operatorname{Hom}_{R{}^{\wedge}_{\mathfrak{p}}}(\operatorname{Hom}^{*}_{\mathcal{T}}(\mathds{1},X\otimes-),I(\mathfrak{p}))\cong\operatorname{Hom}_{\mathcal{T}}(-,D{}^{\wedge}_{\mathfrak{p}}X).$

The asssignment $X\mapsto D{}^{\wedge}_{\mathfrak{p}}X$ yields a pair of
contravariant functors $\varGamma_{\mathfrak{p}}\mathcal{T}\to\mathcal{T}$ and
$\varLambda^{\mathfrak{p}}\mathcal{T}\to\mathcal{T}$, equipped with natural
transformations $D{}^{\wedge}_{\mathfrak{p}}\to D_{\mathfrak{p}}$.

######  Lemma 5.7.

For $X\in\varGamma_{\mathfrak{p}}\mathcal{T}$ the morphism
$D{}^{\wedge}_{\mathfrak{p}}X\to D_{\mathfrak{p}}X$ is an isomorphism.

###### Proof.

It suffices to show that the induced map
$\operatorname{Hom}_{\mathcal{T}}^{*}(C,D{}^{\wedge}_{\mathfrak{p}}X)\to\operatorname{Hom}_{\mathcal{T}}^{*}(C,D_{\mathfrak{p}}X)$
is an isomorphism for all compact $C\in\mathcal{T}$. But this follows from
Lemma 5.3 and its analogue for $D{}^{\wedge}_{\mathfrak{p}}$, since
$\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)$ is $\mathfrak{p}$-torsion. ∎

We formulate an analogue of Matlis duality for modules. To this end, consider
the following subcategories of $\mathcal{T}$:

$\displaystyle\mathcal{A}_{\mathfrak{p}}$
$\displaystyle\coloneqq\\{X\in\varGamma_{\mathfrak{p}}\mathcal{T}\mid\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)\in\operatorname{art}R{}^{\wedge}_{\mathfrak{p}}\text{
for all }C\in{\mathcal{T}}^{\mathrm{c}}\\}$
$\displaystyle\mathcal{N}_{\mathfrak{p}}$
$\displaystyle\coloneqq\\{X\in\varLambda^{\mathfrak{p}}\mathcal{T}\mid\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)\in\operatorname{noeth}R{}^{\wedge}_{\mathfrak{p}}\text{
for all }C\in{\mathcal{T}}^{\mathrm{c}}\\}$

######  Theorem 5.8.

Let $\mathcal{T}$ be a rigidly compactly generated tensor triangulated
category with a canonical $R$-action such that $\mathcal{T}$ is noetherian
over $R$. Fix $\mathfrak{p}$ in $\operatorname{Spec}R$. Then there are
mutually quasi-inverse contravariant equivalences:

${\mathcal{A}_{\mathfrak{p}}}$${\mathcal{N}_{\mathfrak{p}}}$$\scriptstyle{D{}^{\wedge}_{\mathfrak{p}}=D_{\mathfrak{p}}}$$\scriptstyle{D{}^{\wedge}_{\mathfrak{p}}}$

Moreover

${(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}={(\varLambda^{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}\subseteq\mathcal{A}_{\mathfrak{p}}\cap\mathcal{N}_{\mathfrak{p}}\,,$

and there are equalities

$(D{}^{\wedge}_{\mathfrak{p}})^{-1}(\mathcal{A}_{\mathfrak{p}})\cap\mathcal{T}_{\mathfrak{p}}=\mathcal{N}_{\mathfrak{p}}\qquad\text{and}\qquad(D{}^{\wedge}_{\mathfrak{p}})^{-1}(\mathcal{N}_{\mathfrak{p}})\cap\mathcal{T}_{\mathfrak{p}}=\mathcal{A}_{\mathfrak{p}}\,.$

###### Proof.

We apply the analogue of Lemma 5.3 for $D{}^{\wedge}_{\mathfrak{p}}$ and use
the Matlis duality from Proposition 5.1. In particular, for objects with an
$R{}^{\wedge}_{\mathfrak{p}}$-action, the assignment $X\mapsto
D{}^{\wedge}_{\mathfrak{p}}X$ sends artinian to noetherian objects, and vice
versa. Any artinian $R_{\mathfrak{p}}$-module is $\mathfrak{p}$-torsion, so
$D{}^{\wedge}_{\mathfrak{p}}X\in\mathcal{A}_{\mathfrak{p}}$ for
$X\in\mathcal{N}_{\mathfrak{p}}$. On the other hand, for
$X\in\varGamma_{\mathfrak{p}}\mathcal{T}$ there are isomorphisms

$D{}^{\wedge}_{\mathfrak{p}}X\cong
D_{\mathfrak{p}}X\cong\operatorname{\mathcal{H}\\!\\!\;\mathit{om}}(X,D_{\mathfrak{p}}\mathds{1})\,,$

and this lies in $\varLambda^{\mathfrak{p}}\mathcal{T}$. Thus
$D{}^{\wedge}_{\mathfrak{p}}X\in\mathcal{N}_{\mathfrak{p}}$ for
$X\in\mathcal{A}_{\mathfrak{p}}$. For a compact $C\in\mathcal{T}$ there is a
natural isomorphism

$\operatorname{Hom}_{\mathcal{T}}^{*}(C,(D{}^{\wedge}_{\mathfrak{p}})^{2}X)\cong(D{}^{\wedge}_{\mathfrak{p}})^{2}\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)\,.$

Thus the canonical morphism $X\to(D{}^{\wedge}_{\mathfrak{p}})^{2}X$ is an
isomorphism when $X$ is in $\mathcal{A}_{\mathfrak{p}}$ or in
$\mathcal{N}_{\mathfrak{p}}$. This gives the desired pair of equivalences.

For the statement about the categories of compacts, we recall from Lemma 2.7
that
${(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}={(\varLambda^{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}$,
and
${(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}\subseteq\mathcal{A}_{\mathfrak{p}}\cap\mathcal{N}_{\mathfrak{p}}$
holds since $\mathcal{T}$ is noetherian. For the final pair of equalities, let
$X\in\mathcal{T}_{\mathfrak{p}}$ such that $D{}^{\wedge}_{\mathfrak{p}}X$ is
in $\mathcal{A}_{\mathfrak{p}}$. Then $(D{}^{\wedge}_{\mathfrak{p}})^{2}X$ is
in $\mathcal{N}_{\mathfrak{p}}$. The map
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)\to\operatorname{Hom}^{*}_{\mathcal{T}}(C,(D{}^{\wedge}_{\mathfrak{p}})^{2}X)$
is a monomorphism for all compact $C$, again by Lemma 5.3. Thus $X$ is in
$\mathcal{N}_{\mathfrak{p}}$. The other inclusion
$(D{}^{\wedge}_{\mathfrak{p}})^{-1}(\mathcal{A}_{\mathfrak{p}})\cap\mathcal{T}_{\mathfrak{p}}\supseteq\mathcal{N}_{\mathfrak{p}}$
is clear. The proof of the equality for $\mathcal{A}_{\mathfrak{p}}$ is
analogous. ∎

The following consequence is pertinent to the discussion in the next section.

######  Corollary 5.9.

For $\mathfrak{p}$ in $\operatorname{Spec}R$, there are equivalences of
triangulated categories

${\operatorname{thick}(T_{\mathfrak{p}}{({\mathcal{T}}^{\mathrm{c}})})^{\mathrm{op}}=\operatorname{thick}(D_{\mathfrak{p}}{({\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}})})^{\mathrm{op}}}$${\operatorname{thick}(\varLambda^{\mathfrak{p}}{({\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}})})}$${\operatorname{thick}(\varGamma_{\mathfrak{p}}{({\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}})})\,.}$$\scriptstyle{D{}^{\wedge}_{\mathfrak{p}}}$$\scriptstyle{D{}^{\wedge}_{\mathfrak{p}}}$$\scriptstyle{\varGamma_{\mathfrak{p}}}$$\scriptstyle{\varLambda^{\mathfrak{p}}}$

###### Proof.

The equality on the left follows from Lemma 5.4. The next equivalence uses
Lemma 5.5 and is a restriction of the equivalence from Theorem 5.8. The
equivalence on the right is obtained from the Dwyer–Greenlees correspondence
(2.6). ∎

### Noetherian objects

The result below, which is an analogue of Proposition 4.4, characterises
objects in the local category $\varLambda^{\mathfrak{p}}\mathcal{T}$ with
noetherian cohomology.

######  Proposition 5.10.

Fix $\mathfrak{p}\in\operatorname{Spec}R$. An object
$X\in\varLambda^{\mathfrak{p}}\mathcal{T}$ is in $\mathcal{N}_{\mathfrak{p}}$
if and only if $X$ has finite length over $R_{\mathfrak{p}}$.

###### Proof.

The module $\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is
$\mathfrak{p}$-torsion by Lemma 2.7, and therefore the forward implication is
clear.

Suppose the $R_{\mathfrak{p}}$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ has finite length for each compact
object $C$ in $\varLambda^{\mathfrak{p}}{\mathcal{T}}$. Then Lemma 5.3 implies
that $\operatorname{Hom}^{*}_{\mathcal{T}}(C,D_{\mathfrak{p}}X)$ has finite
length over $R_{\mathfrak{p}}$ for each compact object $C$ in
$\varGamma_{\mathfrak{p}}\mathcal{T}$. Thus Proposition 4.4 yields that the
object $\varGamma_{\mathfrak{p}}D_{\mathfrak{p}}X$ is artinian over
$R_{\mathfrak{p}}$. Then its Brown–Comenetz dual is in
$\mathcal{N}_{\mathfrak{p}}$, by Theorem 5.8.

The natural map $\varGamma_{\mathfrak{p}}D_{\mathfrak{p}}X\to
D_{\mathfrak{p}}X$ induces the map on the right below:

$X\longrightarrow D_{\mathfrak{p}}^{2}X\longrightarrow
D_{\mathfrak{p}}{\varGamma_{\mathfrak{p}}D_{\mathfrak{p}}X}\,.$

The one of the left is the biduality map. So it suffices to verify that the
composite $X\to D_{\mathfrak{p}}{\varGamma_{\mathfrak{p}}D_{\mathfrak{p}}X}$
is an isomorphism. Since source and target are in
$\varLambda^{\mathfrak{p}}\mathcal{T}$, it suffices to verify that the induced
map

$\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)\longrightarrow\operatorname{Hom}_{\mathcal{T}}^{*}(C,D_{\mathfrak{p}}{\varGamma_{\mathfrak{p}}D_{\mathfrak{p}}X})$

is an isomorphism for each compact object $C$ in
$\varLambda^{\mathfrak{p}}\mathcal{T}$. It remains to observe that one has
isomorphisms

$\displaystyle\operatorname{Hom}_{\mathcal{T}}^{*}(C,D_{\mathfrak{p}}{\varGamma_{\mathfrak{p}}D_{\mathfrak{p}}X})$
$\displaystyle\cong
D_{\mathfrak{p}}\operatorname{Hom}_{\mathcal{T}}^{*}(D^{\scriptscriptstyle{\mathrm{SW}}}C,\varGamma_{\mathfrak{p}}D_{\mathfrak{p}}X)$
$\displaystyle\cong
D_{\mathfrak{p}}\operatorname{Hom}_{\mathcal{T}}^{*}(D^{\scriptscriptstyle{\mathrm{SW}}}C,D_{\mathfrak{p}}X)$
$\displaystyle\cong
D_{\mathfrak{p}}^{2}\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)\,.$

The first and the third isomorphism follow from Lemma 5.3. The second
isomorphism uses that $D^{\scriptscriptstyle{\mathrm{SW}}}C$ is in
$\varGamma_{\mathfrak{p}}\mathcal{T}$, and the last one holds because the
$R_{\mathfrak{p}}$-module $\operatorname{Hom}_{\mathcal{T}}^{*}(C,X)$ has
finite length. ∎

### Dualisable objects

Our goal is a cohomological description of dualisable objects in a local
stratum of a tensor triangulated category, and here is a first step.

######  Proposition 5.11.

Let $\mathcal{T}$ be a rigidly compactly generated tensor triangulated
category with a canonical $R$-action such that $\mathcal{T}$ is noetherian
over $R$. Fix $\mathfrak{p}$ in $\operatorname{Spec}R$. The following
statements hold.

  1. (1)

The dualisable objects of $\varGamma_{\mathfrak{p}}{\mathcal{T}}$ are in
$\mathcal{A}_{\mathfrak{p}}$, and the dualisable objects of
$\varLambda^{\mathfrak{p}}{\mathcal{T}}$ are in $\mathcal{N}_{\mathfrak{p}}$.

  2. (2)

There are inclusions
$\varGamma_{\mathfrak{p}}({\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}})\subseteq\mathcal{A}_{\mathfrak{p}}$
and
$\varLambda^{\mathfrak{p}}({\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}})\subseteq\mathcal{N}_{\mathfrak{p}}$.

###### Proof.

(1) Let $\mathcal{S}$ denote either $\varGamma_{\mathfrak{p}}\mathcal{T}$ or
$\varLambda^{\mathfrak{p}}\mathcal{T}$ and view it as an
$R_{\mathfrak{p}}$-linear category. Fix a dualisable object $X\in\mathcal{S}$.
Given Propositions 4.4 and 5.10 it suffices to verify that the
$R_{\mathfrak{p}}$-module $\operatorname{Hom}^{*}_{\mathcal{S}}(C,X)$ has
finite length for each compact object $C$. Consider the isomorphism of
$R_{\mathfrak{p}}$-modules

$\operatorname{Hom}_{\mathcal{S}}^{*}(C,X)\cong\operatorname{Hom}^{*}_{\mathcal{S}}(\mathds{1},D^{\scriptscriptstyle{\mathrm{SW}}}C\otimes
X)\,,$

where the unit and tensor product are the ones in $\mathcal{S}$. The object
$Y\coloneqq D^{\scriptscriptstyle{\mathrm{SW}}}C\otimes X$ is compact in
$\mathcal{S}$, by Proposition 3.1. Lemma 2.7 gives

${(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}={\mathcal{S}}^{\mathrm{c}}={(\varLambda^{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}\,.$

This yields an isomorphism
$\operatorname{Hom}_{\mathcal{S}}^{*}(\mathds{1},Y)\cong\operatorname{Hom}_{\mathcal{T}_{\mathfrak{p}}}^{*}(\mathds{1}_{\mathfrak{p}},Y)$
of $R_{\mathfrak{p}}$-modules, where $\mathds{1}_{\mathfrak{p}}$ denote the
unit in $\mathcal{T}_{\mathfrak{p}}$, and
$\operatorname{Hom}_{\mathcal{T}_{\mathfrak{p}}}^{*}(\mathds{1}_{\mathfrak{p}},Y)$
has finite length over $R_{\mathfrak{p}}$.

(2) We apply part (1). Any object in
$\varGamma_{\mathfrak{p}}({{\mathcal{T}}}^{\mathrm{c}}_{\mathfrak{p}})$ is
dualisable and therefore in $\mathcal{A}_{\mathfrak{p}}$. In the same vein,
any object in
$\varLambda^{\mathfrak{p}}({{\mathcal{T}}_{\mathfrak{p}}}^{\mathrm{c}})$ is
dualisable and therefore in $\mathcal{N}_{\mathfrak{p}}$. ∎

The diagram below illustrates the findings in this section. Corollary 7.9
provides a context where the bottom-most inclusion is an equality.

${\mathcal{T}_{\mathfrak{p}}}$${{\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}}}$${\varGamma_{\mathfrak{p}}\mathcal{T}}$${\varLambda^{\mathfrak{p}}\mathcal{T}}$${\mathcal{A}_{\mathfrak{p}}}$${\mathcal{N}_{\mathfrak{p}}}$${\mathcal{A}_{\mathfrak{p}}\cap\mathcal{N}_{\mathfrak{p}}}$${{(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}={(\varLambda^{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}}$$\scriptstyle{\varGamma_{\mathfrak{p}}}$$\scriptstyle{\varLambda^{\mathfrak{p}}}$$\scriptstyle{D{}^{\wedge}_{\mathfrak{p}}}$
Figure 1. Subcategories of $\mathcal{T}_{\mathfrak{p}}$

##  6\. Strong generation

We introduce a notion of local regularity for compactly generated triangulated
categories which builds on the notion of strong generation. A key consequence
of this property is a strong finiteness condition for the class of artinian
objects, analogous to finite global dimension in the setting of abelian
categories. We begin by recalling the notion of a strong generator from [17,
41].

### Strong generation

Fix a triangulated category $\mathcal{T}$ with suspension
$\Sigma\colon\mathcal{T}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\mathcal{T}$.
For an object $X$ in $\mathcal{T}$ and $n\geq 0$ set

$\operatorname{thick}^{n}(X)\coloneqq\begin{cases}\\{0\\}&n=0,\\\
\operatorname{add}\\{\Sigma^{i}X\mid i\in\mathbb{Z}\\}&n=1,\\\
\operatorname{add}\\{\operatorname{cone}\phi\mid\phi\in\operatorname{Hom}_{\mathcal{T}}(\operatorname{thick}^{1}(X),\operatorname{thick}^{n-1}(X))&n>1,\end{cases}$

where $\operatorname{add}\mathcal{X}$ denotes the smallest full subcategory
containing $\mathcal{X}$ that is closed under finite direct sums and direct
summands. An object $X$ is a _strong generator_ for $\mathcal{T}$ if
$\mathcal{T}=\operatorname{thick}^{n}(X)$ for some integer $n\geq 0$. The
category $\mathcal{T}$ is _strongly generated_ if it admits a strong
generator; in this case any object $X$ satisfying
$\operatorname{thick}(X)=\mathcal{T}$ is a strong generator.

######  Example 6.1.

Let $\mathcal{A}$ be an abelian category with an injective cogenerator $I$
such that every object embeds into a finite direct sum of copies of $I$.
Viewing $I$ as a complex concentrated in degree zero, the equality
$\mathbf{D}^{\mathrm{b}}(\mathcal{A})=\operatorname{thick}(I)$ holds if and
only if each object in $\mathcal{A}$ has finite injective dimension. Moreover,
$\operatorname{thick}(I)$ is strongly generated if and only if $\mathcal{A}$
has finite global dimension. This is a consequence of the ghost lemma, cf.
[31, Lemma 2.4].

There is also a dual statement when $\mathcal{A}$ has a projective generator.
It implies, for example, that the category of perfect complexes over a local
noetherian ring admits a strong generator if and only if the ring is regular;
see [41, Proposition 7.25].

### Artinian objects

Let $\mathcal{T}$ be a compactly generated $R$-linear triangulated category.
The subcategory of objects that are artinian over $R$ is denoted
$\operatorname{art}_{R}\mathcal{T}$.

######  Lemma 6.2.

When the $R$-linear category $\mathcal{T}$ is noetherian one has inclusions

$\displaystyle\operatorname{thick}\\{\varGamma_{\mathfrak{m}}C\mid
C\in{\mathcal{T}}^{\mathrm{c}},\,\mathfrak{m}\in\operatorname{Spec}R\text{
maximal}\\}\subseteq\operatorname{art}_{R}\mathcal{T}$
$\displaystyle\operatorname{thick}\\{T_{\mathfrak{m}}C\mid
C\in{\mathcal{T}}^{\mathrm{c}},\,\mathfrak{m}\in\operatorname{Spec}R\text{
maximal}\\}\subseteq\operatorname{art}_{R}\mathcal{T}\,.$

###### Proof.

Since $\operatorname{art}_{R}\mathcal{T}$ is a thick subcategory of
$\mathcal{T}$, it suffices to verify that it contains
$\varGamma_{\mathfrak{m}}C$ for $\mathfrak{m}$ maximal and $C$ compact. Using
Proposition 4.4 it suffices to show that the $R_{\mathfrak{m}}$-module
$\operatorname{Hom}^{*}_{\mathcal{T}_{\mathfrak{m}}}(X,\varGamma_{\mathfrak{m}}C)$
has finite length for all
$X\in{(\varGamma_{\mathfrak{m}}\mathcal{T})}^{\mathrm{c}}$. We have
$\operatorname{Hom}^{*}_{\mathcal{T}_{\mathfrak{m}}}(X,\varGamma_{\mathfrak{m}}C)\cong\operatorname{Hom}^{*}_{\mathcal{T}_{\mathfrak{m}}}(X,C_{\mathfrak{m}})$
for such $X$. This module is noetherian and $\mathfrak{m}$-torsion, so of
finite length. For the other inclusion, one uses the isomorphism defining
$T_{\mathfrak{m}}C$. Matlis duality yields that
$\operatorname{Hom}_{R}^{*}(-,I(\mathfrak{m}))$ takes finitely generated
modules to artinian modules, since $I(\mathfrak{m})$ is artinian. ∎

######  Definition 6.3.

We say that $\mathcal{T}$ is _locally regular over $R$_ if it is noetherian
and has a compact generator $G$ such that
$\operatorname{thick}(T_{\mathfrak{p}}G)$ is strongly generated for each
$\mathfrak{p}\in\operatorname{Spec}R$.

Local regularity has useful consequences for the class of artinian objects. In
particular, $\operatorname{thick}(T_{\mathfrak{p}}G)$ does not depend on the
choice of the compact generator $G$.

######  Proposition 6.4.

Let $R$ be a graded local commutative noetherian ring with maximal ideal
$\mathfrak{m}$, and $\mathcal{T}$ a compactly generated $R$-linear
triangulated category. Assume $\mathcal{T}$ is noetherian and has a compact
generator $G$. If $\operatorname{thick}(T_{\mathfrak{m}}G)$ is strongly
generated, then
$\operatorname{thick}(T_{\mathfrak{m}}G)=\operatorname{art}_{R}\mathcal{T}$.
If in addition $\operatorname{thick}(\varGamma_{\mathfrak{m}}G)$ is strongly
generated, then

$\operatorname{thick}(\varGamma_{\mathfrak{m}}G)=\operatorname{art}_{R}\mathcal{T}=\operatorname{thick}(T_{\mathfrak{m}}G)\,.$

###### Proof.

We write $R{}^{\wedge}_{\mathfrak{m}}$ for the completion of $R$ at
$\mathfrak{m}$ and note that $\operatorname{End}^{*}_{R}(I(\mathfrak{m}))\cong
R{}^{\wedge}_{\mathfrak{m}}$. Set
$\mathcal{I}=\operatorname{thick}(T_{\mathfrak{m}}G)$ and fix
$X\in\operatorname{art}_{R}\mathcal{T}$. Then
$\operatorname{Hom}^{*}_{\mathcal{T}}(X,Y)$ is naturally a finitely generated
graded $R{}^{\wedge}_{\mathfrak{m}}$-module for all $Y\in\mathcal{I}$. This
yields a cohomological functor

$\operatorname{Hom}^{*}_{\mathcal{T}}(X,-)|_{\mathcal{I}}\colon\mathcal{I}\longrightarrow\operatorname{noeth}R{}^{\wedge}_{\mathfrak{m}}\,.$

Since $\mathcal{I}$ is strongly generated, a variation of Brown
representability due to Rouquier [41] and extended by Letz to the graded
setting in [33, Theorem 2.7], implies that this functor is graded
representable. Let $X^{\prime}$ denote the representing object. By Yoneda’s
lemma, an isomorphism

$\operatorname{Hom}^{*}_{\mathcal{I}}(X^{\prime},-)\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{Hom}^{*}_{\mathcal{T}}(X,-)|_{\mathcal{I}}$

induces a morphism $X\to X^{\prime}$, which is an isomorphism since
$T_{\mathfrak{m}}G$ is a cogenerator for
$\varGamma_{\mathfrak{m}}\mathcal{T}$. Thus $X$ belongs to $\mathcal{I}$.

The proof for $\mathcal{J}=\operatorname{thick}(\varGamma_{\mathfrak{m}}G)$ is
analogous, using cohomological functors
$\mathcal{J}^{\mathrm{op}}\to\operatorname{noeth}R{}^{\wedge}_{\mathfrak{m}}$.
Note that $\operatorname{Hom}^{*}_{\mathcal{T}}(X,Y)$ is naturally a finitely
generated graded $R{}^{\wedge}_{\mathfrak{m}}$-module for all
$X,Y\in\mathcal{J}$ since $\mathcal{J}\subseteq\mathcal{I}$. ∎

### Hom-symmetry

The following symmetry statement is needed later. It assumes a weak Gorenstein
property that holds when $\mathcal{T}$ is a locally regular tensor
triangulated category; see Lemma 7.2 and Proposition 6.4

######  Lemma 6.5.

Let $\mathfrak{p}\in\operatorname{Spec}R$ and assume

$\operatorname{thick}\\{\varGamma_{\mathfrak{p}}C\mid
C\in{\mathcal{T}}^{\mathrm{c}}\\}=\operatorname{thick}\\{T_{\mathfrak{p}}C\mid
C\in{\mathcal{T}}^{\mathrm{c}}\\}\,.$

Then the following holds for each $X\in\varGamma_{\mathfrak{p}}\mathcal{T}$.

  1. (1)

The $R_{\mathfrak{p}}$-module $\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ has
finite length for each $C\in{\mathcal{T}}^{\mathrm{c}}$ if and only if
$\operatorname{Hom}^{*}_{\mathcal{T}}(X,C)$ has finite length for each
$C\in{\mathcal{T}}^{\mathrm{c}}$.

  2. (2)

The $R_{\mathfrak{p}}$-module $\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ has
finite length for each
$C\in{(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}$ if and only if
$\operatorname{Hom}^{*}_{\mathcal{T}}(X,C)$ has finite length for each
$C\in{(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}$.

###### Proof.

(1) The $R_{\mathfrak{p}}$-module $\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$
has finite length if and only if

$D_{\mathfrak{p}}\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)\cong\operatorname{Hom}^{*}_{\mathcal{T}}(X,T_{\mathfrak{p}}C)$

has finite length. The assumption on $\mathcal{T}$ implies that it is
equivalent to the finiteness of the length of

$\operatorname{Hom}^{*}_{\mathcal{T}}(X,\varGamma_{\mathfrak{p}}C)\cong\operatorname{Hom}^{*}_{\mathcal{T}}(X,C)\,.$

(2) One has
${(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}=\operatorname{thick}\\{{C_{\mathfrak{p}}}/\\!\\!/{\mathfrak{p}}\mid
C\in{\mathcal{T}}^{\mathrm{c}}\\}$, by Lemma 2.7. It then follows from Lemma
2.8 that the $R_{\mathfrak{p}}$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ has finite length for each
$C\in{(\varGamma_{\mathfrak{p}}\mathcal{T})}^{\mathrm{c}}$ if and only if
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,{X}/\\!\\!/{\mathfrak{p}})$ has finite
length for each $C\in{\mathcal{T}}^{\mathrm{c}}$. The same argument applies to
$\operatorname{Hom}^{*}_{\mathcal{T}}(X,C)$. So we have reduced to the
statement of part (1). ∎

### Descent

Let $f\colon\mathcal{T}\to\mathcal{U}$ be an exact functor between $R$-linear
triangulated categories. We say that $f$ is _$R$ -linear_ if for each object
$X$ in $\mathcal{T}$ the canonical map

$\operatorname{End}^{*}_{\mathcal{T}}(X)\longrightarrow\operatorname{End}^{*}_{\mathcal{U}}(fX)$

is $R$-linear. An equivalent condition is that these are maps of $R$-algebras.

In the following we study adjoint pairs $(f,g)$ of functors between compactly
generated triangulated categories. We make frequent use of the fact that $f$
preserves compacts if and only if $g$ preserves coproducts.

######  Proposition 6.6.

Let $f\colon\mathcal{T}\rightleftarrows\mathcal{U}\nobreak\mskip 6.0mu plus
1.0mu\mathpunct{}\nonscript\mkern-3.0mu{:}\mskip 2.0mug$ be an adjoint pair of
$R$-linear exact functors between compactly generated $R$-linear triangulated
categories such that both functors preserve products and coproducts. Suppose
there exists an integer $d$ such that for each $X\in\mathcal{T}$ one has
$X\in\operatorname{thick}^{d}(gfX)$. If $\mathcal{U}$ is locally regular over
$R$, then so is $\mathcal{T}$.

A sufficient condition for $(f,g)$ to preserve products and coproducts is the
existence of an equivalence
$u\colon\mathcal{U}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\mathcal{U}$
such that $(g,uf)$ is an adjoint pair, because then the pair $(gu,f)$ is
adjoint.

###### Proof.

First observe that $f$ admits a left adjoint
$e\colon\mathcal{U}\to\mathcal{T}$ that preserves compacts. This follows from
Brown representability since $f$ preserves products and coproducts. Also, $f$
preserves compacts since $g$ preserves coproducts.

Assume $\mathcal{U}$ is noetherian over $R$. By adjunction, the $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,gfD)$ is noetherian for each pair of
compact objects $C,D$ in $\mathcal{T}$. Since $D$ is in
$\operatorname{thick}(gfD)$ we conclude that that
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,D)$ is noetherian. Thus $\mathcal{T}$
is noetherian over $R$.

Suppose that $\mathcal{U}$ is locally regular with compact generator $G$. Fix
$\mathfrak{p}\in\operatorname{Spec}R$. There exists an integer $n$ such that

$\operatorname{thick}^{n}(T_{\mathfrak{p}}G)=\operatorname{thick}(T_{\mathfrak{p}}G)=\operatorname{art}_{R_{\mathfrak{p}}}\mathcal{U}_{\mathfrak{p}}\,,$

where the first equality is from the definition of local regularity and the
second one is by Propostion 6.4. We claim

  1. (1)

$eG$ is a compact generator for $\mathcal{T}$,

  2. (2)

$g(T_{\mathfrak{p}}G)\cong T_{\mathfrak{p}}(eG)$, and

  3. (3)

$\operatorname{thick}^{d+n}(T_{\mathfrak{p}}(eG))=\operatorname{thick}(T_{\mathfrak{p}}(eG))$,

which would prove that $\mathcal{T}$ is locally regular over $R$, as desired.

Indeed, (1) holds because $e$ preserves compactness and for each
$X\in\mathcal{T}$ one has an isomorphism of $R$-modules

$\operatorname{Hom}^{*}_{\mathcal{U}}(G,fX)\cong\operatorname{Hom}_{\mathcal{T}}(eG,X)\,.$

This isomorphism also yields isomorphisms

$\displaystyle\operatorname{Hom}^{*}_{\mathcal{T}}(X,g(T_{\mathfrak{p}}G))$
$\displaystyle\cong\operatorname{Hom}^{*}_{\mathcal{U}}(fX,T_{\mathfrak{p}}G)$
$\displaystyle\cong\operatorname{Hom}_{R}(\operatorname{Hom}^{*}_{\mathcal{U}}(G,fX),I(\mathfrak{p}))$
$\displaystyle\cong\operatorname{Hom}_{R}(\operatorname{Hom}^{*}_{\mathcal{T}}(eG,X),I(\mathfrak{p}))\,.$

Therefore $g(T_{\mathfrak{p}}G)\cong T_{\mathfrak{p}}(eG)$ holds, which
justifies (2).

(3) Given Lemma 6.2 it suffices to prove that each artinian object $X$ in
$\mathcal{T}_{\mathfrak{p}}$ is in
$\operatorname{thick}^{d+n}(T_{\mathfrak{p}}(eG))$. It is easy to verify that
$fX$ is in $\operatorname{art}_{R_{\mathfrak{p}}}\mathcal{U}_{\mathfrak{p}}$,
and hence in $\operatorname{thick}^{n}(T_{\mathfrak{p}}G)$. Thus we get that

$X\in\operatorname{thick}^{d}(gfX)\subseteq\operatorname{thick}^{d+n}(g(T_{\mathfrak{p}}G))=\operatorname{thick}^{d+n}(T_{\mathfrak{p}}(eG))\,,$

where the equality is by (2). This justifies (3). ∎

### Change of rings

Let $\varphi\colon R\to S$ be a finite morphism of graded commutative
noetherian rings. The following lemma shows that local regularity is preserved
when the linear action on a triangulated category is changed via $\varphi$.

######  Lemma 6.7.

Let $\mathcal{T}$ be a compactly generated $S$-linear triangulated category.
Then one has
$\operatorname{art}_{R}\mathcal{T}=\operatorname{art}_{S}\mathcal{T}$, and for
each $\mathfrak{p}\in\operatorname{Spec}R$ and compact object $C$ in
$\mathcal{T}$ there is an isomorphism

$T_{\mathfrak{p}}C\cong\bigoplus_{\begin{subarray}{c}\mathfrak{q}\in\operatorname{Spec}S\\\
\varphi^{-1}(\mathfrak{q})=\mathfrak{p}\end{subarray}}T_{\mathfrak{q}}C\,.$

Moreover $\mathcal{T}$ is locally regular over $R$ if and only if the same
holds over $S$.

###### Proof.

The first assertion is clear, because an $S$-module is artinian if and only if
it is artinian when viewed as an $R$-module via restriction of scalars along
$\varphi$. One has an isomorphism of $S$-modules

$\operatorname{Hom}_{R}(S,I(\mathfrak{p}))\cong\bigoplus_{\begin{subarray}{c}\mathfrak{q}\in\operatorname{Spec}S\\\
\varphi^{-1}(\mathfrak{q})=\mathfrak{p}\end{subarray}}I(\mathfrak{q})\,.$

See, for instance, [40, Theorem 1.1]. Combining it with the adjunction
isomorphism

$\operatorname{Hom}_{R}(\operatorname{Hom}^{*}_{\mathcal{T}}(C,-),I(\mathfrak{p}))\cong\operatorname{Hom}_{S}(\operatorname{Hom}^{*}_{\mathcal{T}}(C,-),\operatorname{Hom}_{R}(S,I(\mathfrak{p})))$

and the defining isomorphism of $T_{\mathfrak{p}}C$, yields the isomorphism
concerning $T_{\mathfrak{p}}C$.

Let $\mathfrak{p}\in\operatorname{Spec}R$. We localise at $\mathfrak{p}$ and
may assume $R$ is local with maximal ideal $\mathfrak{p}$. Consider the
induced map $\operatorname{Spec}S\to\operatorname{Spec}R$. Since $R\to S$ is
finite, the fibre over $\mathfrak{p}$ consists of a finite collection of
maximal ideals, say $\mathfrak{q}_{1},\dots,\mathfrak{q}_{r}$. It follows that
any object $X$ in $\mathcal{T}$ admits a functorial decomposition

$\varGamma_{\mathfrak{p}}X=\varGamma_{\mathfrak{q}_{1}}X\oplus\cdots\oplus\varGamma_{\mathfrak{q}_{r}}X\,.$

In particular, any exact triangle supported at $\mathfrak{p}$ decomposes into
a direct sum of exact triangles supported at the $\mathfrak{q}_{i}$. When
applied to $T_{\mathfrak{p}}C$ for a compact object $C$, one obtains the above
decomposition $T_{\mathfrak{p}}C\cong T_{\mathfrak{q}_{1}}C\oplus\cdots\oplus
T_{\mathfrak{q}_{r}}C$. Here, one uses that $T_{\mathfrak{p}}C$ is in
$\varGamma_{\mathfrak{p}}\mathcal{T}$, so
$\varGamma_{\mathfrak{p}}T_{\mathfrak{p}}=T_{\mathfrak{p}}$.

Now suppose $\mathcal{T}$ is locally regular over $S$, with compact generator
$G$. This yields integers $n_{i}$ such that

$\operatorname{thick}^{n_{i}}(T_{\mathfrak{q}_{i}}G)=\operatorname{thick}(T_{\mathfrak{q}_{i}}G)\qquad(i=1,\ldots,r)\,.$

Using the above decomposition
$\varGamma_{\mathfrak{p}}=\bigoplus_{i}\varGamma_{\mathfrak{q}_{i}}$, each
object in $\operatorname{thick}(T_{\mathfrak{p}}G)$ decomposes into objects
from the categories $\operatorname{thick}(T_{\mathfrak{q}_{i}}G)$. Hence
$\operatorname{thick}^{n}(T_{\mathfrak{p}}G)=\operatorname{thick}(T_{\mathfrak{p}}G)$
for $n=\max(n_{1},\ldots,n_{r})$, and $\mathcal{T}$ is regular over $R$.

For the converse suppose that $\mathcal{T}$ is locally regular over $R$, with
compact generator $G$, and fix $\mathfrak{q}\in\operatorname{Spec}S$. Set
$\mathfrak{p}=\varphi^{-1}(\mathfrak{q})$. As above, we can localise at
$\mathfrak{p}$ and assume $R$ is local with maximal ideal $\mathfrak{p}$, and
that $\mathfrak{q}$ lies over $\mathfrak{p}$. It is easy to verify that

$\operatorname{art}_{S_{\mathfrak{q}}}\mathcal{T}_{\mathfrak{q}}\subseteq\operatorname{art}_{S}\mathcal{T}=\operatorname{art}_{R}\mathcal{T}\,.$

It follows from Proposition 6.4 that each artinian object $X$ in the
$S_{\mathfrak{q}}$-linear category $\mathcal{T}_{\mathfrak{q}}$ belongs to
$\operatorname{thick}^{n}(T_{\mathfrak{p}}G)$ for some integer $n$. The
decomposition of $T_{\mathfrak{p}}G$ noted above implies that
$\varGamma_{\mathfrak{q}}T_{\mathfrak{p}}G=T_{\mathfrak{q}}G$, so applying the
functor $\varGamma_{\mathfrak{q}}$ yields that $X$ belongs to
$\operatorname{thick}^{n}(T_{\mathfrak{q}}G)$. Thus $\mathcal{T}$ is locally
regular over $S$. ∎

### Regular endomorphism rings

Let $\mathcal{T}$ be a compactly generated triangulated category and $G$ a
compact generator. Set $A=\operatorname{End}^{*}_{\mathcal{T}}(G)$ and
consider the cohomological functor

$H\colon\mathcal{T}\longrightarrow\operatorname{Mod}A,\quad\text{where}\quad
X\mapsto\operatorname{Hom}^{*}_{\mathcal{T}}(G,X).$

For an injective $A$-module $E$ let $T(E)$ denote the object in $\mathcal{T}$
such that

$\operatorname{Hom}_{A}(H(X),E)\cong\operatorname{Hom}_{\mathcal{T}}(X,T(E)).$

The existence and uniqueness of $T(E)$ follows from Brown representability.

The result below uses Adams resolutions and is well known; see for instance
[36].

######  Lemma 6.8.

Let $X\in\mathcal{T}$ be an object such that the $A$-module $H(X)$ admits an
injective resolution $\eta\colon H(X)\to I$ where $I^{s}=0$ for $s>n$. Then
$X$ belongs to the triangulated subcategory of $\mathcal{T}$ generated by
$T(I^{0}),\ldots,T(I^{n})$.

###### Proof.

The argument is an induction on $n$. The morphism $\eta\colon H(X)\to I^{0}$
corresponds to a morphism $\widetilde{\eta}\colon X\to T(I^{0})$. If $n=0$,
then $\eta$ is an isomorphism. It follows that
$\operatorname{Hom}_{\mathcal{T}}^{*}(G,\widetilde{\eta})$ is an isomorphism,
and hence so is $\widetilde{\eta}$, because $G$ generates $\mathcal{T}$.

Assume $n\geq 1$ and let $X^{\prime}$ denote the cone of the morphism
$\widetilde{\eta}$. This yields an exact sequence $0\to H(X)\to I^{0}\to
H(X^{\prime})\to 0$. The induction hypothesis implies that $X^{\prime}$
belongs to the triangulated subcategory generated by
$T(I^{1}),\ldots,T(I^{n})$. Thus $X$ belongs to the triangulated subcategory
generated by $T(I^{0}),\ldots,T(I^{n})$. ∎

######  Theorem 6.9.

Let $\mathcal{T}$ be a compactly generated $R$-linear category that is
noetherian. If there exists a compact generator $G$ for $\mathcal{T}$ such
that $\operatorname{End}^{*}_{\mathcal{T}}(G)$ has finite global dimension,
then $\mathcal{T}$ is locally regular over $R$.

###### Proof.

Set $A=\operatorname{End}^{*}_{\mathcal{T}}(G)$ and fix
$\mathfrak{p}\in\operatorname{Spec}R$. Then $G_{\mathfrak{p}}$ is a compact
generator for $\mathcal{T}_{\mathfrak{p}}$, and one has an isomorphism of
rings

$\operatorname{End}^{*}_{\mathcal{T}_{\mathfrak{p}}}(G_{\mathfrak{p}})\cong\operatorname{End}^{*}_{\mathcal{T}}(G)_{\mathfrak{p}}=A\otimes_{R}R_{\mathfrak{p}}\,.$

Since $A\otimes_{R}R_{\mathfrak{p}}$ is also of finite global dimension, we
can replace $R$ and $\mathcal{T}$ by their localisations at $\mathfrak{p}$ and
assume $R$ is local, with maximal ideal $\mathfrak{p}$. The $A$-module
$J=\operatorname{Hom}_{R}(A,I(\mathfrak{p}))$ is injective, and it is
straightforward to verify that $T_{\mathfrak{p}}G=T(J)$. The desired result is
thus that $\operatorname{thick}(T(J))$ has a strong generator. Since
$\mathcal{T}$ is noetherian, arguing as in the proof of Lemma 6.2 one deduces
that $T(J)$, and hence any object in $\operatorname{thick}(T(J))$, is
artinian.

Let $X$ be any artinian object in $\mathcal{T}$ and set
$M=\operatorname{Hom}^{*}_{\mathcal{T}}(G,X)$. Then $M$ is artinian as an
$R$-module, so it embeds into a finite direct sum of copies of
$I(\mathfrak{p})$. Thus as an $A$-module $M$ embeds into a finite direct sum
of copies of $J$. The cokernel of such an embedding is again artinian over
$R$, and we obtain therefore an injective resolution of $M$ over $A$ involving
in each degree finite direct sums of copies of $J$ and their direct summands.
Thus, if the global dimension of $A$ is bounded by $n$, then $X$ belongs to
$\operatorname{thick}^{n+1}(T(J))$, as desired; see Lemma 6.8 and its proof. ∎

##  7\. Local regularity

In this section we consider a rigidly compactly generated tensor triangulated
category $\mathcal{T}=(\mathcal{T},\otimes,\mathds{1})$ and introduce in this
setting a notion of local regularity. Then we are able to describe for each
point its dualisable objects.

######  Definition 7.1.

Set $R=\operatorname{End}^{*}_{\mathcal{T}}(\mathds{1})$. We say $\mathcal{T}$
is _noetherian_ to mean that $\mathcal{T}$ is noetherian with respect to the
canonical $R$-action, so the graded $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,D)$ is noetherian for all compact
objects $C,D$ in $\mathcal{T}$. In particular, the graded ring $R$, which is
always commutative, is noetherian.

We say that $\mathcal{T}$ is _locally regular_ if it is noetherian and admits
a compact generator $G$ such that
$\operatorname{thick}(\varGamma_{\mathfrak{p}}G)$ is strongly generated for
each $\mathfrak{p}\in\operatorname{Spec}R$.

For any pair $G,G^{\prime}$ of compact generators, one has
$\operatorname{thick}(\varGamma_{\mathfrak{p}}G)=\operatorname{thick}(\varGamma_{\mathfrak{p}}G^{\prime})$
for each $\mathfrak{p}\in\operatorname{Spec}R$. Thus the definition of local
regularity does not depend on the choice of a generator.

The result below reconciles this notion of local regularity with the one
introduced in Section 6 for triangulated categories with linear actions. For
tensor triangulated categories, the definition above is more intrinsic. In
fact, the linear action of $\operatorname{End}^{*}_{\mathcal{T}}(\mathds{1})$
plays a major role in our arguments. If this ring is too small, then the
definition ought to be formulated in terms of prime ideals of the tensor
triangulated category ${\mathcal{T}}^{\mathrm{c}}$. An example to keep in mind
is the derived category of a non-affine scheme. Another such example from
representation theory is the derived category of a Schur algebra, with
subcategory of compact objects equivalent to the homotopy category of Young
permutation modules.

######  Lemma 7.2.

Let $\mathcal{T}$ be a rigidly compactly generated tensor triangulated
category. Assume $\mathcal{T}$ is noetherian and has a compact generator $G$.
Then the local regularity of $\mathcal{T}$ is equivalent to strong generation
of any one, equivalently, all of the triangulated categories:
$\operatorname{thick}(T_{\mathfrak{p}}G)$,
$\operatorname{thick}(D_{\mathfrak{p}}G)$ and
$\operatorname{thick}(\varLambda^{\mathfrak{p}}G_{\mathfrak{p}})$, for each
$\mathfrak{p}\in\operatorname{Spec}R$.

###### Proof.

This follows from Corollary 5.9. ∎

### Examples

The definition of local regularity is motivated by the following example. For
a noetherian ring $A$ we write $\mathbf{D}(\operatorname{Mod}A)$ for its full
derived category; it is a rigidly compactly generated tensor triangulated
category.

######  Proposition 7.3.

Let $A$ a commutative noetherian ring. Then the ring $A$ is regular if and
only if the tensor triangulated category $\mathbf{D}(\operatorname{Mod}A)$ is
locally regular.

###### Proof.

Local regularity is a local condition. For each
$\mathfrak{p}\in\operatorname{Spec}A$ we have

$\mathbf{D}(\operatorname{Mod}A)_{\mathfrak{p}}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\mathbf{D}(\operatorname{Mod}A_{\mathfrak{p}})\,.$

Thus we may assume $A$ is local with maximal ideal $\mathfrak{p}$; set
$k=A/\mathfrak{p}$.

The triangulated category $\mathbf{D}(\operatorname{Mod}A)$ has a compact
generator, namely, $A$, and is noetherian as an $A$-linear category. It is
convenient to check local regularity via the strong generation of
$\operatorname{thick}(T_{\mathfrak{p}}A)$; see Lemma 7.2. The $A$-module
$I(\mathfrak{p})$ identifies with $T_{\mathfrak{p}}A$, and is an injective
cogenerator of $\operatorname{art}A$, the category of artinian $A$-modules.
Moreover, an $A$-complex $X$ is artinian precisely when the $A$-module
$H^{*}(X)$ is artinian, so
$\operatorname{art}_{A}\mathbf{D}(\operatorname{Mod}A)$ identifies with
$\mathbf{D}^{\mathrm{b}}(\operatorname{art}A)$. Thus, given Example 6.1, it
remains to verify that $A$ is regular if and only if $\operatorname{art}A$ has
finite global dimension. This follows from the result due to Auslander,
Buchsbaum, and Serre that $A$ is regular if and only if
$\operatorname{Ext}_{A}^{n}(k,k)=0$ for $n\gg 0$; see Proposition 1.3.1 and
Corollary 2.2.7 in [19]. ∎

Here is another example of a locally regular category.

######  Example 7.4.

Let $k$ be a field and $A=\Lambda^{*}(V)$ the exterior algebra on a finite
dimensional $k$-vector space $V$ which we view as a dg-algebra generated in
degree $-1$ with trivial differential. Let
$\mathcal{T}=\mathbf{K}(\operatorname{Inj}A)$ denote the homotopy category of
injective $A$-modules. This category is compactly generated and the Hopf
algebra structure gives rise to a tensor product. The category $\mathcal{T}$
is noetherian and generated by the injective resolution of the trivial
$A$-module $k$. The graded endomorphism ring identifies with a symmetric
algebra, so $\operatorname{End}_{\mathcal{T}}^{*}(\mathbf{i}k)\cong
S^{*}(V^{\vee}[2])$. Hence $\mathcal{T}$ is locally regular by Theorem 6.9.

### Descent

The result below is a version of Proposition 6.6 for tensor triangulated
categories. In the statement, the condition that $f$ is strong symmetric
monoidal means that $f(\mathds{1}_{\mathcal{T}})\cong\mathds{1}_{\mathcal{U}}$
and there is a natural isomorphism

$f(X\otimes Y)\cong fX\otimes fY\quad\text{for $X,Y$ in $\mathcal{T}$.}$

The condition on an adjoint pair $(f,g)$ that both functors preserve products
and coproducts arises naturally and has been studied in the context of tensor
triangulated categories and Grothendieck–Neeman duality; see [3, Theorem 1.7].

######  Proposition 7.5.

Let $f\colon\mathcal{T}\to\mathcal{U}$ be an exact functor between rigidly
compactly generated tensor triangulated categories that is strong symmetric
monoidal and preserves coproducts. Suppose also that following conditions
hold:

  1. (1)

$\mathds{1}_{\mathcal{T}}\in\operatorname{thick}(g(\mathds{1}_{\mathcal{U}}))$;

  2. (2)

the canonical map
$\operatorname{End}^{*}_{\mathcal{T}}(\mathds{1})\to\operatorname{End}^{*}_{\mathcal{U}}(\mathds{1})$
is finite.

If $\mathcal{U}$ is locally regular, then so is $\mathcal{T}$.

###### Proof.

Set $R=\operatorname{End}^{*}_{\mathcal{T}}(\mathds{1})$ and
$S=\operatorname{End}^{*}_{\mathcal{U}}(\mathds{1})$, and view $\mathcal{U}$
as an $R$-linear category via the natural map $R\to S$. We claim that, under
the hypothesis of the statement, when the ring $S$ is noetherian so is $R$.

Indeed, assume the ring $S$ is noetherian. Since $S$ is finite as an
$R$-module, for $I=\operatorname{Ker}(R\to S)$, the ring $R/I$ is noetherian,
by the theorem of Eakin and Nagata [35, Theorem 3.7]. Thus $S$ is noetherian
as an $R$-module. Then, since the adjunction isomorphism
$\operatorname{Hom}^{*}_{\mathcal{T}}(\mathds{1}_{\mathcal{T}},g(\mathds{1}_{\mathcal{U}}))\cong\operatorname{Hom}^{*}_{\mathcal{U}}(\mathds{1}_{\mathcal{U}},\mathds{1}_{\mathcal{U}})=S$
is $R$-linear, the $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(\mathds{1}_{\mathcal{T}},g(\mathds{1}_{\mathcal{U}}))$
is noetherian. Since $\mathds{1}_{\mathcal{T}}$ is in the thick subcategory
generated by $g(\mathds{1}_{\mathcal{U}})$, the $R$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(\mathds{1}_{\mathcal{T}},\mathds{1}_{\mathcal{T}})=R$
is noetherian, as claimed.

Suppose $\mathcal{U}$ is locally regular over $S$; in particular $S$, and
hence also $R$, is noetherian. The map $R\to S$ is finite, so Lemma 6.7 yields
that $\mathcal{U}$ is also locally regular over $R$. We claim Proposition 6.6
applies, so $\mathcal{T}$ is locally regular over $R$.

Indeed, from [3, Theorem 1.3] we get that $f$ and $g$ both preserve products
and coproducts. For $X$ in $\mathcal{T}$ the projection formula from [3,
Theorem 1.3] yields

$X\otimes gf(\mathds{1}_{\mathcal{T}})\cong g(fX\otimes
f\mathds{1}_{\mathcal{T}})\cong g(f(X\otimes\mathds{1}_{\mathcal{T}}))\cong
gfX\,.$

By hypothesis there exists an integer $d$ such that $\mathds{1}_{\mathcal{T}}$
is in $\operatorname{thick}^{d}(gf\mathds{1}_{\mathcal{T}})$. Applying
$X\otimes-$ and using the isomorphism above, we deduce that $X$ is in
$\operatorname{thick}^{d}(gfX)$. ∎

The following example illustrates the result above. In particular it shows
that $\mathcal{T}$ locally regular need not imply $\mathcal{U}$ locally
regular.

######  Example 7.6.

Let $\varphi\colon A\to B$ be a finite map of commutative noetherian rings
such that $B$ is faithful and of finite projective dimension as an $A$-module.
Then the adjoint pair

${\mathbf{D}(A)}$${\mathbf{D}(B)}$$\scriptstyle{\varphi^{*}=B\otimes^{\mathbf{L}}_{A}-}$$\scriptstyle{\varphi_{*}=\mathrm{res}}$

satisfies the hypothesis of Proposition 7.5. Indeed, it is clear that
$\varphi^{*}$ is strong symmetric monoidal, preserving coproducts. Since
$\varphi_{*}B$ is a perfect complex and has full support, the theorem of
Hopkins [27] and Neeman [37] yields that $A$ is in
$\operatorname{thick}(\varphi_{*}B)$, so condition (1) holds. Condition (2)
holds because the endomorphisms of the units in $\mathbf{D}(A)$ and
$\mathbf{D}(B)$ are $A$ and $B$, respectively. Proposition 7.5 subsumes the
well known result that when $B$ is regular so is $A$. The converse need not
hold; consider the case $A$ is a field.

### Dualisable torsion objects

Consider for $\mathfrak{p}$ in $\operatorname{Spec}R$ the category of
$\mathfrak{p}$-local $\mathfrak{p}$-torsion objects and describe its
dualisable objects.

######  Proposition 7.7.

Let $\mathcal{T}$ be a rigidly compactly generated tensor triangulated
category that is locally regular. For $\mathfrak{p}$ in $\operatorname{Spec}R$
and any object $X$ in $\varGamma_{\mathfrak{p}}{\mathcal{T}}$ the following
conditions are equivalent.

  1. (1)

$X$ is dualisable in $\varGamma_{\mathfrak{p}}\mathcal{T}$;

  2. (2)

$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is artinian over
$R_{\mathfrak{p}}$ for each compact object $C$ in $\mathcal{T}$;

  3. (3)

$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ has finite length over
$R_{\mathfrak{p}}$ for each compact object $C$ in
$\varGamma_{\mathfrak{p}}\mathcal{T}$;

  4. (4)

$\operatorname{Hom}^{*}_{\mathcal{T}}(X,C)$ has finite length over
$R_{\mathfrak{p}}$ for each compact object $C$ in
$\varGamma_{\mathfrak{p}}\mathcal{T}$;

  5. (5)

$X$ is in
$\operatorname{thick}(\varGamma_{\mathfrak{p}}{\mathcal{T}}^{\mathrm{c}})$;

  6. (6)

$X\to(D{}^{\wedge}_{\mathfrak{p}})^{2}X$ is an isomorphism.

###### Proof.

The proof combines various results from previous sections; only the
implication (2)$\Rightarrow$(5) uses the local regularity condition.

(1)$\Rightarrow$(2): This follows from Proposition 5.11.

(2)$\Leftrightarrow$(3): This follows from Proposition 4.4.

(3)$\Leftrightarrow$(4): This follows from Lemma 6.5.

(2)$\Rightarrow$(5): This follows from Proposition 6.4.

(5)$\Rightarrow$(1): This follows from Corollary 3.7.

(2)$\Leftrightarrow$(6): For any $C\in{\mathcal{T}}^{\mathrm{c}}$ the
$R{}^{\wedge}_{\mathfrak{p}}$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is $\mathfrak{p}$-torsion, so the
assertion follows from the analogue of Lemma 5.3 for
$D{}^{\wedge}_{\mathfrak{p}}$ and Lemma 7.8 below. ∎

######  Lemma 7.8.

Let $M$ be an $R{}^{\wedge}_{\mathfrak{p}}$-module that is
$\mathfrak{p}$-torsion. Then the canonical map $\delta_{M}\colon
M\to(D{}^{\wedge}_{\mathfrak{p}})^{2}M$ is an isomorphism if and only if $M$
is artinian.

###### Proof.

From Matlis duality it follows that the map $\delta_{M}$ is an isomorphism if
$M$ is artinian. Suppose $\delta_{M}$ is an isomorphism. An application of the
Snake Lemma and the fact that $\delta_{M}$ is always a monomorphism shows that
Matlis reflexive modules are closed under submodules and quotients. In
particular, the socle of $M$ has finite length. This is an essential submodule
since $M$ is $\mathfrak{p}$-torsion, and therefore $M$ is artinian. ∎

### Compactness

Given $\mathfrak{p}\in\operatorname{Spec}R$, we consider in
$\varGamma_{\mathfrak{p}}\mathcal{T}$ the difference between compact and
dualisable objects; in cohomological terms it is the difference between having
finite length and being artinian.

######  Corollary 7.9.

Let $\mathcal{T}$ be as in Proposition 7.7. For $\mathfrak{p}$ in
$\operatorname{Spec}R$ and any object $X$ in
$\varGamma_{\mathfrak{p}}{\mathcal{T}}$ the following conditions are
equivalent.

  1. (1)

$X$ is compact in $\varGamma_{\mathfrak{p}}{\mathcal{T}}$;

  2. (2)

$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ has finite length over
$R_{\mathfrak{p}}$ for each compact object $C$ in
${\mathcal{T}}_{\mathfrak{p}}$;

  3. (3)

$\operatorname{Hom}^{*}_{\mathcal{T}}(X,C)$ has finite length over
$R_{\mathfrak{p}}$ for each compact object $C$ in
${\mathcal{T}}_{\mathfrak{p}}$.

In particular, compact objects and dualisable objects in
$\varGamma_{\mathfrak{p}}\mathcal{T}$ coincide if and only if $\mathfrak{p}$
is minimal in $\operatorname{supp}_{R}\mathcal{T}$

###### Proof.

(1)$\Rightarrow$(2): The $R_{\mathfrak{p}}$-module
$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is noetherian, because $X$ is also
compact in $\mathcal{T}_{\mathfrak{p}}$, and $\mathfrak{p}$-torsion, so it has
finite length.

(2)$\Leftrightarrow$(3): This follows from Lemma 6.5.

(3)$\Rightarrow$(1): Passing to $\mathcal{T}_{\mathfrak{p}}$ and
$R_{\mathfrak{p}}$ we can assume $R$ is local. The hypothesis implies $X$ is
dualisable in $\varGamma_{\mathfrak{p}}\mathcal{T}$, by Proposition 7.7. Thus
${X}/\\!\\!/{\mathfrak{p}}\cong X\otimes{\mathds{1}}/\\!\\!/{\mathfrak{p}}$ is
compact in $\varGamma_{\mathfrak{p}}\mathcal{T}$, by Proposition 3.1, and
hence also in $\mathcal{T}$. We claim
$\operatorname{thick}(X)=\operatorname{thick}({X}/\\!\\!/{\mathfrak{p}})$,
which implies $X$ is compact as desired.

For the claim it suffices to show that
$\operatorname{End}^{*}_{\mathcal{T}}(X)$ is $\mathfrak{p}$-torsion, by [10,
Lemma 3.9]. First observe that for any object $C$ there is an isomorphism of
$R$-modules

$\operatorname{Hom}^{*}_{\mathcal{T}}(X,\varGamma_{\mathfrak{p}}C)\cong\operatorname{Hom}^{*}_{\mathcal{T}}(X,C)$

since $X$ is $\mathfrak{p}$-torsion. Thus the hypothesis implies that the
$R$-module $\operatorname{Hom}^{*}_{\mathcal{T}}(X,\varGamma_{\mathfrak{p}}C)$
has finite length for each compact object $C$ in
${\mathcal{T}}_{\mathfrak{p}}$. Since $X$ is dualisable in
$\varGamma_{\mathfrak{p}}\mathcal{T}$, it is finitely built from objects of
the form $\varGamma_{\mathfrak{p}}C$; this is again from Proposition 7.7. Thus
$\operatorname{Hom}^{*}_{\mathcal{T}}(X,X)$ has finite length, and in
particular it is $\mathfrak{p}$-torsion.

The last part of the statement follows from Lemma 4.6, using the cohomological
descriptions of compact and dualisable objects given above. ∎

### Dualisable complete objects

There is an equivalence
$\varGamma_{\mathfrak{p}}\mathcal{T}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\varLambda^{\mathfrak{p}}\mathcal{T}$
of compactly generated tensor triangulated categories, for $\mathfrak{p}$ in
$\operatorname{Spec}R$. Clearly, this equivalence preserves and reflects
dualisability, and restricts to a triangle equivalence

$\operatorname{thick}(\varGamma_{\mathfrak{p}}{\mathcal{T}}^{\mathrm{c}})\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{thick}(\varLambda^{\mathfrak{p}}{\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}})\,.$

This leads to an analogue of Proposition 7.7 for
$\varLambda^{\mathfrak{p}}\mathcal{T}$.

######  Proposition 7.10.

Let $\mathcal{T}$ be a rigidly compactly generated tensor triangulated
category that is locally regular. For $\mathfrak{p}$ in $\operatorname{Spec}R$
and any object $X$ in $\varLambda^{\mathfrak{p}}{\mathcal{T}}$ the following
conditions are equivalent.

  1. (1)

$X$ is dualisable;

  2. (2)

$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ is noetherian over
$R{}^{\wedge}_{\mathfrak{p}}$ for each compact object $C$ in $\mathcal{T}$;

  3. (3)

$\operatorname{Hom}^{*}_{\mathcal{T}}(C,X)$ has finite length over
$R{}^{\wedge}_{\mathfrak{p}}$ for each compact object $C$ in
$\varLambda^{\mathfrak{p}}\mathcal{T}$;

  4. (4)

$\operatorname{Hom}^{*}_{\mathcal{T}}(X,C)$ has finite length over
$R{}^{\wedge}_{\mathfrak{p}}$ for each compact object $C$ in
$\varLambda^{\mathfrak{p}}\mathcal{T}$;

  5. (5)

$X$ is in
$\operatorname{thick}(\varLambda^{\mathfrak{p}}{\mathcal{T}}^{\mathrm{c}}_{\mathfrak{p}})$;

  6. (6)

$X\to(D{}^{\wedge}_{\mathfrak{p}})^{2}X$ is an isomorphism.

###### Proof.

Given the equivalence
$\varGamma_{\mathfrak{p}}\mathcal{T}\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\varLambda^{\mathfrak{p}}\mathcal{T}$,
the result follows from Proposition 7.7. The only condition that does not
translate directly is (2), but this is equivalent to (3) by Proposition 5.10.
∎

######  Corollary 7.11.

Let $\mathcal{T}$ be a rigidly compactly generated tensor triangulated
category that is locally regular. For $\mathfrak{p}$ in $\operatorname{Spec}R$
the functor $D{}^{\wedge}_{\mathfrak{p}}$ restricts to a pair of mutually
quasi-inverse contravariant equivalences
$\mathcal{A}_{\mathfrak{p}}\rightleftarrows\mathcal{N}_{\mathfrak{p}}$,
identifying the dualisable objects in $\varGamma_{\mathfrak{p}}\mathcal{T}$
with the dualisable objects in $\varLambda^{\mathfrak{p}}\mathcal{T}$. Hence
the subcategories of compact objects in $\varGamma_{\mathfrak{p}}\mathcal{T}$
and $\varLambda^{\mathfrak{p}}\mathcal{T}$ agree and equal
$\mathcal{A}_{\mathfrak{p}}\cap\mathcal{N}_{\mathfrak{p}}$.

###### Proof.

The pair of equivalences
$\mathcal{A}_{\mathfrak{p}}\rightleftarrows\mathcal{N}_{\mathfrak{p}}$ has
been established in Theorem 5.8. The statement about dualisable objects then
follows from Propositions 7.7 and 7.10. The statement about compact objects
follows from Corollary 7.9. ∎

### Strong generation

Here is a consequence of Proposition 7.7 and Lemma 7.2.

######  Corollary 7.12.

For each $\mathfrak{p}$ in $\operatorname{Spec}R$, the thick subcategory of
dualisable objects in $\varGamma_{\mathfrak{p}}\mathcal{T}$ and in
$\varLambda^{\mathfrak{p}}\mathcal{T}$ have strong generators. ∎

##  8\. Finite groups

The main result in this section is that for any finite group $G$, the tensor
triangulated category $\mathbf{K}(\operatorname{Inj}kG)$ is locally regular.
Applications include the various characterisations of local dualisable objects
stated in the introduction. With an eye towards future applications, we
establish an intermediate step in the more general setting of finite group
schemes. We begin with recalling some fundamental results about cohomology and
representations of finite group schemes.

### Finite dimensional Hopf algebras

Let $k$ be a field of positive characteristic and $A$ a finite dimension co-
commutative Hopf algebra over $k$. Equivalently, $A\cong kG$ is a group
algebra of some finite group scheme $G$ over $k$, that is to say, the
$k$-linear dual to the coordinate algebra $k[G]$ of $G$. The homotopy category
$\mathbf{K}(\operatorname{Inj}A)$ of injective $A$-modules is a compactly
generated triangulated category, and the quotient functor
$q\colon\mathbf{K}(\operatorname{Inj}A)\to\mathbf{D}(\operatorname{Mod}A)$
induces an equivalence

(8.1) ${\mathbf{K}(\operatorname{Inj}A)}^{\mathrm{c}}\xrightarrow{\
\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}{\mathbf{D}}^{\mathrm{b}}(\operatorname{mod}A)\,;$

its quasi-inverse sends an $A$-complex $M$ to its injective resolution,
$\mathbf{i}M$; see [12].

Endowed with the product given by $X\otimes_{k}Y$ with diagonal $A$-action,
$\mathbf{K}(\operatorname{Inj}A)$ is a tensor triangulated category, with unit
$\mathbf{i}_{A}k$, where $k$ is viewed as an $A$-module via the augmentation
$A\to k$. Moreover $\mathbf{K}(\operatorname{Inj}A)$ is rigidly compactly
generated. For each compact object $C$ in $\mathbf{K}(\operatorname{Inj}A)$,
the $R=\operatorname{Ext}^{*}_{A}(k,k)$-module
$\operatorname{Ext}_{A}^{*}(C,C)$ is noetherian, by [25, Theorem 1.1]. In
particular, the tensor triangulated category $\mathbf{K}(\operatorname{Inj}A)$
is noetherian.

In the next theorem we show for any finite commutative unipotent group scheme,
its homotopy category of injectives is locally regular.

######  Theorem 8.2.

Let $k$ be a field of positive characteristic $p$ and $A$ a finite dimensional
co-commutative Hopf algebra over $k$. Assume that there exist integers
$q_{i}=p^{d_{i}}$ for $1\leq i\leq r$ such that there is an isomorphism of
$k$-algebras

$A\cong\frac{k[z_{1},\dots,z_{r}]}{(z_{1}^{q_{1}},\dots,z_{r}^{q_{r}})}\,.$

Then the tensor triangulated category $\mathbf{K}(\operatorname{Inj}A)$ is
locally regular.

###### Proof.

Set $\mathcal{T}=\mathbf{K}(\operatorname{Inj}A)$ and
$R=\operatorname{End}^{*}_{\mathcal{T}}(\mathbf{i}_{A}k)\cong\operatorname{Ext}^{*}_{A}(k,k)$.
Since $A$ is local, its residue field $k$ generates
${\mathbf{D}}^{\mathrm{b}}(\operatorname{mod}A)$ as a thick subcategory, so
$\mathbf{i}_{A}k$ is a compact generator $\mathcal{T}$, by the equivalence
(8.1). Thus it suffices to prove that
$\operatorname{thick}(\varGamma_{\mathfrak{p}}(\mathbf{i}_{A}k))$ has a strong
generator for each $\mathfrak{p}$ in $\operatorname{Spec}R$.

As is well-understood, the object $\varGamma_{\mathfrak{p}}(\mathbf{i}_{A}k)$
is independent of the coproduct on $A$; see [7, Propostion 4.9]. Thus the same
is true of the local regularity condition. In what follows we assume that the
coproduct on $A$ is the one for which each $z_{i}$ is primitive: $z_{i}\mapsto
z_{i}\otimes 1+1\otimes z_{i}$. Let $B$ be the Koszul complex of $A$ on
elements $z_{1},\dots,z_{r}$, where $B$ is viewed as a dg (=differential
graded) Hopf $A$-algebra, as in [8, §7]; this is where the specific choice of
the coproduct on $A$ is needed. Let $\mathcal{U}$ be the homotopy category of
graded-injective dg $B$-modules; see [8, §4]. Let $\mathbf{i}_{B}k$ be the
semi-injective resolution of $k$ viewed as a dg $B$-module via the
augmentation $B\to k$, and set
$S=\operatorname{End}^{*}_{\mathcal{U}}(\mathbf{i}_{B}k)$; this is the
polynomial ring on $r$ indeterminates over $k$, of degree $2$, see [8, Lemma
7.1]. The triangulated category $\mathcal{U}$ is compactly generated by
$\mathbf{i}_{B}k$ and it is $S$-linear; see [8, §4 and §6] for details. The
global dimension of $S$ is $r$ and so finite. Then, keeping in mind Lemma 7.2,
we apply Theorem 6.9 to conclude that $\mathcal{U}$ is locally regular over
$S$.

Consider the structure map $A\to B$; this is a map of dg Hopf algebras, where
$A$ is viewed as a dg algebra with zero differential. As an $A$-module, $B$ is
finite free so one gets an exact functor $f^{*}\coloneqq
B\otimes_{A}-\colon\mathcal{T}\to\mathcal{U}$ with right adjoint $f_{*}$
induced by restriction along $A\to B$. In particular $f_{*}(\mathbf{i}_{B}k)$
is an injective resolution of $k$ as an $A$-module, and hence a compact
generator of $\mathcal{T}$. Note that
$R=\operatorname{End}^{*}_{\mathcal{T}}(f_{*}(\mathbf{i}_{B}k))$ and let
$\varphi\colon S\to R$ be the natural map induced by $f_{*}$, and view
$\mathcal{T}$ as an $S$-linear category. Since $A\to B$ is a map of Hopf
algebras, the functors $f^{*}$ and $f_{*}$ are $S$-linear, and satisfy the
hypotheses of Proposition 6.6.

Indeed, one has an isomorphism of dg $B$-modules
$\operatorname{Hom}_{A}(B,A)\cong\Sigma^{-r}B$, see [19, Proposition 1.6.10].
Thus $\Sigma^{-r}f^{*}$ is right adjoint to $f_{*}$, and both $f^{*}$ and
$f_{*}$ preserve products and coproducts. Since $A$ is a commutative artinian
local ring and $B$ is the Koszul complex on a finite set of elements in $A$,
one has $A\in\operatorname{thick}^{d}(B)$ for some integer $d$; see [28, Lemma
6.0.9]. Thus for $X\in\mathcal{T}$ we deduce that

$f_{*}f^{*}X=B\otimes_{A}X\in\operatorname{thick}^{d}(A\otimes_{A}X)=\operatorname{thick}^{d}(X)\,.$

Since we already know that $\mathcal{U}$ is locally regular over $R$,
Proposition 6.6 applies and yields that $\mathcal{T}$ is locally regular as an
$S$-linear category. The map $\varphi$ is finite so Lemma 6.7 implies that
$\mathcal{T}$ is also locally regular as an $R$-linear category; in other
words, it is locally regular as a tensor triangulated category. ∎

### Finite groups

Let $G$ be a finite group and $k$ a field whose characteristic is positive and
divides the order of $G$. Let $kG$ be the group algebra; this is a finite
dimensional co-commutative Hopf algebra over $k$, so
$\mathbf{K}(\operatorname{Inj}kG)$ has a natural structure as a tensor
triangulated category, and this is rigidly compactly generated.

######  Theorem 8.3.

The tensor triangulated category $\mathbf{K}(\operatorname{Inj}kG)$ is locally
regular.

###### Proof.

We can assume the characteristic of $k$, say $p$, divides the order of $G$.
Consider first the case where $G=E$ is an elementary abelian $p$-group of rank
$r$. Its group algebra is of the form

$kE\cong\frac{k[z_{1},\dots,z_{r}]}{(z_{1}^{p},\dots,z_{r}^{p})}$

Thus Theorem 8.2 applies, yielding that $\mathbf{K}(\operatorname{Inj}kE)$ is
locally regular as a tensor triangulated category. We deduce the result for an
arbitrary $G$ using Proposition 7.5.

Let $U$ be the (finite) collection of maximal elementary abelian $p$-subgroups
of $G$. For each $E\in U$ we know that the tensor triangulated category
$\mathbf{K}(\operatorname{Inj}kE)$ is locally regular, and this implies that
the tensor triangulated category

$\mathcal{U}\coloneqq\prod_{E\in U}\mathbf{K}(\operatorname{Inj}kE)$

is locally regular as well. For a maximal elementary abelian $p$-subgroup
$E\subset G$ let

$f^{*}_{E}\colon\mathbf{K}(\operatorname{Inj}kG)\to\mathbf{K}(\operatorname{Inj}kE)\quad\text{and}\quad
f_{*}^{E}\colon\mathbf{K}(\operatorname{Inj}kE)\to\mathbf{K}(\operatorname{Inj}kG)$

denote the restriction functor and induction functor, $kG\otimes_{kE}-$,
respectively. Observe that $(f^{*}_{E},f_{*}^{E},f^{*}_{E})$ is an adjoint
triple. Consider functors

$\displaystyle f^{*}\colon$
$\displaystyle\mathbf{K}(\operatorname{Inj}kG)\longrightarrow\prod_{E\in
U}\mathbf{K}(\operatorname{Inj}kE)$ $\displaystyle{\text{where
}X\mapsto(f^{*}_{E}X)}\,,$ $\displaystyle f_{*}\colon$
$\displaystyle\prod_{E\in
U}\mathbf{K}(\operatorname{Inj}kE)\longrightarrow\mathbf{K}(\operatorname{Inj}kG)$
$\displaystyle{\text{where }(Y_{E})\mapsto\bigoplus_{E\in
U}f^{E}_{*}(Y_{E})}\,.$

We claim that the adjoint pair $(f^{*},f_{*})$ satisfies the hypotheses of
Proposition 7.5.

Indeed, it is clear that each $f^{*}_{E}$, and hence $f^{*}$, is strong
symmetric monoidal and preserves coproducts. The unit of the tensor
triangulated category $\mathcal{U}$ is $\prod_{E\in U}\mathbf{i}_{E}k$, where
$\mathbf{i}_{E}k$ is the injective resolution of $k$ as a $kE$-module. It
follows from [20, Theorem 2.1] that $k$ is in the thick subcategory of
${\mathbf{D}}^{\mathrm{b}}(\operatorname{mod}kG)$ generated by
$\\{f^{E}_{*}(k)\\}_{E\in U}$, that is to say that, $\mathbf{i}_{G}k$ is in
$\operatorname{thick}(f_{*}(\mathds{1}_{\mathcal{U}}))$.

The graded endomorphism rings of $\mathds{1}_{\mathcal{T}}$ and
$\mathds{1}_{\mathcal{U}}$ are $H^{*}(G,k)$ and $\prod_{E\in U}H^{*}(E,k)$,
respectively. For each $E$ the natural map $H^{*}(G,k)\to H^{*}(E,k)$ obtained
by restriction is finite, so the following map is finite as well:

$H^{*}(G,k)\to\prod_{E\in U}H^{*}(E,k)\,.$

Thus Proposition 7.5 applies to yield that $\mathbf{K}(\operatorname{Inj}kG)$
is locally regular. ∎

### A recollement at the irrelevant ideal

Set $\mathfrak{m}=H^{\geqslant 1}(G,k)$, the closed point of
$\operatorname{Spec}H^{*}(G,k)$. Then the recollement (2.1) given by
$V=\\{\mathfrak{m}\\}$ reads:

${\operatorname{StMod}{kG}}$${\mathbf{K}(\operatorname{Inj}kG)}$${\mathbf{D}(\operatorname{Mod}kG)}$$\scriptstyle{q}$$\scriptstyle{q_{\lambda}}$$\scriptstyle{q_{\rho}}$

We have $\varGamma_{\mathfrak{m}}=q_{\lambda}\circ q$, and therefore $q$
induces an equivalence

$\varGamma_{\mathfrak{m}}\mathbf{K}(\operatorname{Inj}kG)\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\mathbf{D}(\operatorname{Mod}kG)\,.$

The left adjoint $q_{\lambda}$ sends $M$ in
${\mathbf{D}}^{\mathrm{b}}(\operatorname{mod}kG)$ to $\mathbf{p}M$, the
projective resolution of $M$, which is
$\varGamma_{\mathfrak{m}}(\mathbf{i}M)$; see [15, §6]. In particular, $q$
restricts to an equivalence

(8.4)
$\varGamma_{\mathfrak{m}}({\mathbf{K}(\operatorname{Inj}kG)}^{\mathrm{c}})\xrightarrow{\raisebox{-2.58334pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}{\mathbf{D}}^{\mathrm{b}}(\operatorname{mod}kG)\,.$

The triangulated category ${\mathbf{D}}^{\mathrm{b}}(\operatorname{mod}kG)$
has a strong generator, namely, the direct sum of representatives of
isomorphism classes of simple $kG$-modules, viewed as a complex concentrated
in a single degree. Thus if $M$ is a generator of
${\mathbf{D}}^{\mathrm{b}}(\operatorname{mod}kG)$, the complex
$\varGamma_{\mathfrak{m}}(\mathbf{i}M)$ is a strong generator for
$\operatorname{thick}(\varGamma_{\mathfrak{m}}(\mathbf{i}M))$.

### Local dualisable objects