Noncommutative supports, local cohomology and spectral sequences

If $k$ is a field and $A$ is a commutative $k\text{-}$algebra, the category $\mathrm{Mod}-A$ of $A\text{-}$modules determines the geometry of the affine scheme $\mathrm{Spec}(A)\text{.}$ If $A$ is no longer commutative, the category $\mathrm{Mod}-A$ of right $A\text{-}$modules plays the role of a noncommutative affine scheme. Suppose that $A$ is a commutative graded $k\text{-}$algebra and $\mathrm{Gr}(A)$ is its category of graded modules. Then, the geometry of the projective scheme $\mathrm{Proj}(A)$ can be understood by means of the category $\mathrm{QGr}(A)$ which is the quotient of $\mathrm{Gr}(A)$ over torsion modules, that is, modules which are obtained as filtered colimits of bounded graded modules (see [5]). Accordingly, the category $\mathrm{QGr}(A)$ plays the role of a noncommutative projective scheme in the theory of Artin and Zhang [5], where $A$ is a noncommutative graded $k\text{-}$algebra. In general, one can approach noncommutative algebraic geometry by adapting notions from commutative rings and the usual theory of schemes to arbitrary Grothendieck categories.

One of the notions that is fundamental in algebraic geometry is that of base change. Let $k$ be a field, and let $A\text{,}$ $R$ be $k\text{-}$algebras. Then, a module over the tensor product algebra $(A{\otimes }_{k}R)$ may be described as an ordinary $A\text{-}$module equipped with an additional $R\text{-}$module action. If we replace the category of $A\text{-}$modules by a general $k\text{-}$linear abelian category $𝒮\text{,}$ we obtain the category ${𝒮}_R$ of “$R\text{-}$module objects in $𝒮\text{.}$” These module objects were introduced by Popescu [21, p. 108] and may be treated as modules over a noncommutative base change of $R$ by the abelian category $𝒮\text{.}$ An $R\text{-}$module object in $𝒮$ consists of an object $ℳ\in 𝒮$ along with a morphism of $k\text{-}$algebras

In [6], Artin and Zhang systematically developed the properties of the abstract module category ${𝒮}_R\text{,}$ where $𝒮$ is a Grothendieck category, and often a locally noetherian or strongly locally noetherian Grothendieck category. The theory developed by Artin and Zhang in [6] includes tensor products, internal hom, localization, versions of the Hilbert basis theorem, the Nakayama lemma, as well as the derived functors $\mathrm{Tor}$ and $\mathrm{Ext}\text{.}$ Several other fundamental constructions on ${𝒮}_R\text{,}$ such as ind-objects, coherence and deformation theory, have been developed by Lowen and Van den Bergh [19]. In [7], we have also studied descent properties in ${𝒮}_R\text{.}$

We note that a number of properties of the category $𝒮$ extend to the module category ${𝒮}_R\text{,}$ following the idea of noncommutative base change. For example, if $𝒮$ is locally finitely generated, so is ${𝒮}_R\text{.}$ The version of the Hilbert basis theorem proved by Artin and Zhang in [6] shows that if $𝒮$ is locally noetherian and $R$ is a finitely generated $k\text{-}$algebra, then ${𝒮}_R$ is also locally noetherian. Since Grothendieck categories abound in nature, the theory of the abstract module category ${𝒮}_R$ is very general and applies to a wide variety of situations. For instance, $𝒮$ could be the category of sheaves of abelian groups on a topological space, comodules over a coalgebra over a field, modules over a ringed space, quasi-coherent sheaves over a scheme or the category of comodules over a flat Hopf algebroid.

In this paper, we develop a local cohomology theory in the noncommutative algebraic geometry of Artin and Zhang [5, 6], by making use of abstract module categories. We let the $k\text{-}$linear Grothendieck category $𝒮={𝒮}_k$ be strongly locally noetherian, whence it follows that ${𝒮}_R$ is locally noetherian for any noetherian commutative $k\text{-}$algebra $R\text{.}$ We should note that our framework will give new results, even in the case we take $𝒮$ to be the category of modules over a noncommutative algebra $A$ that is strongly locally noetherian, that is, $A{\otimes }_{k}R$ is right noetherian for every commutative noetherian $k\text{-}$algebra $R$ (see [4]). If $C$ is a coalgebra over a field, its category of comodules is locally noetherian (see [14, Section 2.4.8] and [23, Section V.4.3]). As with the theory in [6], we can take $𝒮$ to be the category $\mathrm{Gr}(A)$ of graded modules over a graded algebra $A\text{.}$ The category $\mathrm{Gr}(A)$ is strongly locally noetherian whenever $A$ is strongly noetherian in a graded sense (see [6, Section B5]). We can also take $𝒮=\mathrm{QGr}(A)$ for a noncommutative graded $k\text{-}$algebra $A\text{.}$ These are the noncommutative projective schemes of Artin and Zhang [5]. If $A$ is strongly noetherian in a graded sense (but not necessarily commutative), the category $\mathrm{QGr}(A)$ is strongly locally noetherian and $\mathrm{QGr}{(A)}_R=\mathrm{QGr}(A{\otimes }_{k}R)$ (see [6, Section B8]).

The aims of our paper are threefold. First, we build a theory of associated primes and supports in the abstract module category ${𝒮}_R$ using what we call $R\text{-}$elementary objects in ${𝒮}_R\text{.}$ If $𝒮$ is the category of modules over a commutative $k\text{-}$algebra $A\text{,}$ then an $R\text{-}$elementary object in ${𝒮}_R$ behaves somewhat like a quotient over a prime ideal. Accordingly, the injective hulls of these $R\text{-}$elementary objects in ${𝒮}_R$ play a key role in our theory.

Our second purpose is to develop local cohomology in ${𝒮}_R\text{.}$ In fact, we work with a more general framework, extending the local cohomology with respect to a pair of ideals $I\text{,}$ $J\subseteq R$ introduced by Takahashi, Yoshino and Yoshizawa [24]. One of the key aspects of the theory in [24] is that local cohomology is based on a nonclosed support $𝕎(I,J)\text{.}$ In order to study the two-variable local cohomology objects $H_{I,J}^{\bullet }(\mathrm{\_}\mathrm{\_})$ in ${𝒮}_R\text{,}$ we will need the support theory, associated primes and $R\text{-}$elementary objects in ${𝒮}_R\text{.}$ If $ℳ\in {𝒮}_R$ is a finitely generated object, we show that the set $\mathrm{Ass}(ℳ)$ is finite. For an ideal $I\subseteq R\text{,}$ the question of the finiteness of the set of associated primes of local cohomology objects is very interesting in commutative algebra and is also related to Faltings’ local-global-principle for finiteness dimensions (see [16], and also [13, Section 9.6.2]). In fact, a number of questions related to the supports and associated primes of local cohomology have been studied extensively in the literature (see, for instance, [9-12, 17, 18, 22]). We give a condition for $\mathrm{Ass}(H_{I,J}^i(ℳ)/𝒩)$ to be finite, where $ℳ\in {𝒮}_R$ is finitely generated and $𝒩\subseteq H_{I,J}^i(ℳ)$ is a finitely generated subobject. This generalizes a result of Brodmann and Faghani [12] to the two-variable local cohomology context, as well as to the abstract module category ${𝒮}_R\text{.}$ In particular, by taking $𝒩=0\text{,}$ we have a condition for the collection of associated primes of $H_{I,J}^i(ℳ)$ to be finite.

Finally, using properties of the injective hull of $R\text{-}$elementary objects, we apply our theory to describe a general functorial setup that is similar to the one recently introduced by Àlvarez Montaner, Boix and Zarzuela in [2] (see also [3]). By considering systems of functors on ${𝒮}_R$ satisfying certain conditions with respect to their values on injective hulls of $R\text{-}$elementary objects in ${𝒮}_R\text{,}$ we obtain a general spectral sequence for their derived functors in this context. In particular, this gives us a spectral sequence for two-variable local cohomology objects in ${𝒮}_R\text{.}$ In fact, it is striking that the nonclosed supports $𝕎(I,J)$ from [24] appear naturally in the theory developed in [2].

We conclude by showing two further applications of the spectral sequence in this formalism. The first is with generalized local cohomology functors given by derived functors of

where $V$ is a finitely generated $R\text{-}$module and $\underset{¯}{\mathrm{Hom}}(V,\mathrm{\_}\mathrm{\_}):{𝒮}_R\to {𝒮}_R$ is the internal hom of an $R\text{-}$module $V$ with respect to an object of ${𝒮}_R\text{.}$ The second is the “generalized Nagata ideal transform” (see [2, Section 4] and [15]) that we extend to ${𝒮}_R$ as follows:

where $V$ is again a finitely generated $R\text{-}$module. In each case, we obtain spectral sequences for their derived functors.

Let $k$ be a field. Throughout, we let $𝒮={𝒮}_k$ be a strongly locally noetherian $k\text{-}$linear category. We let $R$ be a commutative and noetherian $k\text{-}$algebra. Accordingly, the category ${𝒮}_R$ of $R\text{-}$module objects in $𝒮$ is locally noetherian. By [6, Section B3], there is a “tensor product” $\mathrm{\_}\mathrm{\_}{\otimes }_R\mathrm{\_}\mathrm{\_}:{𝒮}_R\times R-\mathrm{Mod}\to {𝒮}_R$ that is right exact in both variables. If $V$ is a flat $R\text{-}$module, then the functor $\mathrm{\_}\mathrm{\_}{\otimes }_{R}V:{𝒮}_R\to {𝒮}_R$ is exact. Since $k$ is a field, the functor $ℳ{\otimes }_k\mathrm{\_}\mathrm{\_}:k-\mathrm{Mod}\to 𝒮$ is exact for any $ℳ\in 𝒮\text{.}$ Consequently, $(ℳ{\otimes }_{k}R){\otimes }_R\mathrm{\_}\mathrm{\_}:R-\mathrm{Mod}\to {𝒮}_R$ is exact for the $k\text{-}$algebra $R$ (see [6, Section C1]). We will typically write $\otimes :={\otimes }_k\text{.}$ We also note that an $R\text{-}$module object in $𝒮$ may be described by giving $ℳ\in 𝒮$ and a “structure map” $ℳ\otimes R\to ℳ$ satisfying analogues of the usual associativity and unit conditions (see [6, Section B3.14]). For notions such as finitely generated objects, noetherian objects and locally noetherian categories that we will need throughout this paper, we refer the reader to [1].

We will now introduce some notation. Let $r_{ℳ}$ denote the endomorphism induced on $ℳ\in {𝒮}_R$ by $r\in R\text{.}$ We set the annihilator $\mathrm{Ann}(ℳ)\subseteq R$ to be the ideal

It is clear that if ${ℳ}^{\prime }\subseteq ℳ\text{,}$ then $\mathrm{Ann}(ℳ)\subseteq \mathrm{Ann}({ℳ}^{\prime })\text{.}$ For any $ℳ\in {𝒮}_R$ and any element $a\in R\text{,}$ we form the short exact sequence

Since ${𝒮}_R$ is a Grothendieck category, an object $ℳ\in {𝒮}_R$ is said to be finitely generated (see, for instance, [23, Section V.3]) if for any directed family ${\{{ℳ}_i\}}_{i\in I}$ of subobjects of $ℳ$ in ${𝒮}_R$ such that ${\sum }_{i\in I}{ℳ}_i=ℳ\text{,}$ there exists some $i_0\in I$ such that ${ℳ}_{i_0}=ℳ\text{.}$ For any $ℳ\in {𝒮}_R\text{,}$ we denote by $\mathrm{fg}(ℳ)$ the collection of finitely generated subobjects of $ℳ$ in ${𝒮}_R\text{.}$ We also let $\mathrm{Pr}(ℳ)$ denote the collection of finitely generated subobjects $0\ne 𝒩\subseteq ℳ$ in ${𝒮}_R$ such that $\mathrm{Ann}(𝒩)$ is a prime ideal in $R\text{.}$

Our first purpose is to show that associated primes exist for every non-zero $ℳ\in {𝒮}_R\text{.}$ For this, we first prove the following result.

Since ${𝒮}_R$ is a Grothendieck category, a subobject $𝒩$ of $ℳ\in {𝒮}_R$ is said to be essential if $𝒩\cap {𝒩}^{\prime }\ne 0$ for every non-zero subobject ${𝒩}^{\prime }\subseteq ℳ$ (see, for instance, [23, Section V.2]).

Since ${𝒮}_R$ is a Grothendieck category, every object $ℳ\in {𝒮}_R$ has an injective hull, which we will denote by $ℰ(ℳ)\text{.}$ We will now prove the main result of this section, which describes the injectives of ${𝒮}_R$ in terms of injective envelopes of $R\text{-}$elementary objects.

If $T\subseteq R$ is a multiplicatively closed set, the localization ${ℳ}_T$ of an object $ℳ\in {𝒮}_R$ with respect to $T$ is given by $ℳ{\otimes }_{R}R[T^{-1}]$ (see [6, Section B6]). The localization ${ℳ}_T$ may also be expressed as the filtered colimit of copies of $ℳ$ connected by morphisms $t_{ℳ}:ℳ\to ℳ$ over all $t\in T$ (see [6, Section B6.1]). Accordingly, the kernel of the canonical map $ℳ\to {ℳ}_T$ is given by the filtered colimit (see [6, Proposition B6.2])

Moreover, since $R[T^{-1}]$ is a flat $R\text{-}$module, it follows from [6, Proposition C1.7] that localization is an exact functor on ${𝒮}_R\text{.}$ If $𝔭$ is a prime ideal, we denote by ${ℳ}_{𝔭}$ the localization with respect to $R\backslash 𝔭\text{.}$ We now have the following definition.

We conclude this section by establishing some properties of injective hulls in ${𝒮}_R$ that we will need throughout this paper. We begin with the following result.

We now need the following fact, which does not seem to have appeared before in the literature.

Let $I\text{,}$ $J$ be ideals in $R\text{.}$ If $ℳ\in {𝒮}_R$ and $𝔊$ is any family of subobjects of $ℳ\text{,}$ we write $⨁𝔊$ (resp. $\sum 𝔊$) for the direct sum (resp. the sum in $ℳ$) of all the objects in $𝔊\text{.}$ We now define

The definition in (3.1) is motivated by the notion of local cohomology for modules with respect to a pair of ideals introduced by Takahashi, Yoshino and Yoshizawa in [24]. We will say that $ℳ\in {𝒮}_R$ is $(I,J)\text{-}$torsion if ${\mathrm{\Gamma }}_{I,J}(ℳ)=ℳ\text{.}$ We will say that $ℳ\in {𝒮}_R$ is $(I,J)\text{-}$torsion free if ${\mathrm{\Gamma }}_{I,J}(ℳ)=0\text{.}$ When $J=0\text{,}$ we will denote ${\mathrm{\Gamma }}_{I,J}$ simply by ${\mathrm{\Gamma }}_I\text{.}$

As defined in [24], we now consider for the pair of ideals $(I,J)$ the set

When $J=0\text{,}$ we note that $𝕎(I,J)=𝕍(I)\text{,}$ the set of prime ideals containing $I\text{.}$

It follows from Lemma 3.2 that ${\mathrm{\Gamma }}_{I,J}(ℳ)$ is the sum of all finitely generated subobjects $𝒩\in \mathrm{fg}(ℳ)$ satisfying $\mathrm{Supp}(𝒩)\subseteq 𝕎(I,J)\text{.}$ We note that this implies $\mathrm{Supp}({\mathrm{\Gamma }}_{I,J}(ℳ))\subseteq 𝕎(I,J)\text{.}$ We will now describe the associated primes of ${\mathrm{\Gamma }}_{I,J}(ℳ)\text{.}$

If $𝒜$ is an abelian category, we recall (see, for instance, [8, Section 1.1]) that a torsion theory on $𝒜$ consists of a pair $(𝒯,ℱ)$ of strict and full subcategories such that $𝒜(X,Y)=0$ for any $X\in 𝒯\text{,}$ $Y\in ℱ$ and any $Z\in 𝒜$ fits into a short exact sequence $0\to Z^{𝒯}\to Z\to Z^{ℱ}\to 0$ with $Z^{𝒯}\in 𝒯$ and $Z^{ℱ}\in ℱ\text{.}$ Additionally, $(𝒯,ℱ)$ is said to be hereditary if the torsion class $𝒯$ is closed under subobjects.