If k is a field and A is a commutative k-algebra, the category Mod−A of A-modules determines the geometry of the affine scheme Spec(A). If A is no longer commutative, the category Mod−A of right A-modules plays the role of a noncommutative affine scheme. Suppose that A is a commutative graded k-algebra and Gr(A) is its category of graded modules. Then, the geometry of the projective scheme Proj(A) can be understood by means of the category QGr(A) which is the quotient of Gr(A) over torsion modules, that is, modules which are obtained as filtered colimits of bounded graded modules (see [5]). Accordingly, the category 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-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, R be k-algebras. Then, a module over the tensor product algebra (A⊗kR) may be described as an ordinary A-module equipped with an additional R-module action. If we replace the category of A-modules by a general k-linear abelian category 𝒮, we obtain the category 𝒮R of “R-module objects in 𝒮.” 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. An R-module object in 𝒮 consists of an object ℳ∈𝒮 along with a morphism of k-algebras

ρℳ:R⟶𝒮(ℳ,ℳ).

In [6], Artin and Zhang systematically developed the properties of the abstract module category 𝒮R, 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 Tor and Ext. Several other fundamental constructions on 𝒮R, 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 𝒮R.

We note that a number of properties of the category 𝒮 extend to the module category 𝒮R, following the idea of noncommutative base change. For example, if 𝒮 is locally finitely generated, so is 𝒮R. 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-algebra, then 𝒮R is also locally noetherian. Since Grothendieck categories abound in nature, the theory of the abstract module category 𝒮R 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-linear Grothendieck category 𝒮=𝒮k be strongly locally noetherian, whence it follows that 𝒮R is locally noetherian for any noetherian commutative k-algebra R. 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⊗kR is right noetherian for every commutative noetherian k-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 Gr(A) of graded modules over a graded algebra A. The category Gr(A) is strongly locally noetherian whenever A is strongly noetherian in a graded sense (see [6, Section B5]). We can also take 𝒮=QGr(A) for a noncommutative graded k-algebra A. 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 QGr(A) is strongly locally noetherian and QGr(A)R=QGr(A⊗kR) (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 𝒮R using what we call R-elementary objects in 𝒮R. If 𝒮 is the category of modules over a commutative k-algebra A, then an R-elementary object in 𝒮R behaves somewhat like a quotient over a prime ideal. Accordingly, the injective hulls of these R-elementary objects in 𝒮R play a key role in our theory.

Our second purpose is to develop local cohomology in 𝒮R. In fact, we work with a more general framework, extending the local cohomology with respect to a pair of ideals I, J⊆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). In order to study the two-variable local cohomology objects HI,J∙(__) in 𝒮R, we will need the support theory, associated primes and R-elementary objects in 𝒮R. If ℳ∈𝒮R is a finitely generated object, we show that the set Ass(ℳ) is finite. For an ideal I⊆R, 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 Ass(HI,Ji(ℳ)/𝒩) to be finite, where ℳ∈𝒮R is finitely generated and 𝒩⊆HI,Ji(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 𝒮R. In particular, by taking 𝒩=0, we have a condition for the collection of associated primes of HI,Ji(ℳ) to be finite.

Finally, using properties of the injective hull of R-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 𝒮R satisfying certain conditions with respect to their values on injective hulls of R-elementary objects in 𝒮R, 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 𝒮R. 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

ΓVI:SR⟶SR ℳ↦lim→t≥1Hom_R(ItV,ℳ),

where V is a finitely generated R-module and Hom¯(V,__):𝒮R→𝒮R is the internal hom of an R-module V with respect to an object of 𝒮R. The second is the “generalized Nagata ideal transform” (see [2, Section 4] and [15]) that we extend to 𝒮R as follows:

ΔVI:SR⟶SR ℳ↦lim→t≥1Hom_R(ItV,ℳ),

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

Let k be a field. Throughout, we let 𝒮=𝒮k be a strongly locally noetherian k-linear category. We let R be a commutative and noetherian k-algebra. Accordingly, the category 𝒮R of R-module objects in 𝒮 is locally noetherian. By [6, Section B3], there is a “tensor product” __⊗R__:𝒮R×R−Mod→𝒮R that is right exact in both variables. If V is a flat R-module, then the functor __⊗RV:𝒮R→𝒮R is exact. Since k is a field, the functor ℳ⊗k__:k−Mod→𝒮 is exact for any ℳ∈𝒮. Consequently, (ℳ⊗kR)⊗R__:R−Mod→𝒮R is exact for the k-algebra R (see [6, Section C1]). We will typically write ⊗⁣:=⁣⊗k. We also note that an R-module object in 𝒮 may be described by giving ℳ∈𝒮 and a “structure map” ℳ⊗R→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 rℳ denote the endomorphism induced on ℳ∈𝒮R by r∈R. We set the annihilator Ann(ℳ)⊆R to be the ideal

Ann(ℳ):={r∈R∣rℳ=0:ℳ=ℳ⊗RR →ℳ⊗rℳ⊗RR=ℳ}.

It is clear that if ℳ′⊆ℳ, then Ann(ℳ)⊆Ann(ℳ′). For any ℳ∈𝒮R and any element a∈R, we form the short exact sequence

0⟶Ker(aℳ)⟶ℳ⟶aℳ=Im(aℳ)⟶0.

Since 𝒮R is a Grothendieck category, an object ℳ∈𝒮R is said to be finitely generated (see, for instance, [23, Section V.3]) if for any directed family {ℳi}i∈I of subobjects of ℳ in 𝒮R such that ∑i∈Iℳi=ℳ, there exists some i0∈I such that ℳi0=ℳ. For any ℳ∈𝒮R, we denote by fg(ℳ) the collection of finitely generated subobjects of ℳ in 𝒮R. We also let Pr(ℳ) denote the collection of finitely generated subobjects 0≠𝒩⊆ℳ in 𝒮R such that Ann(𝒩) is a prime ideal in R.

Our first purpose is to show that associated primes exist for every non-zero ℳ∈𝒮R. For this, we first prove the following result.

Since 𝒮R is a Grothendieck category, a subobject 𝒩 of ℳ∈𝒮R is said to be essential if 𝒩∩𝒩′≠0 for every non-zero subobject 𝒩′⊆ℳ (see, for instance, [23, Section V.2]).

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

If T⊆R is a multiplicatively closed set, the localization ℳT of an object ℳ∈𝒮R with respect to T is given by ℳ⊗RR[T−1] (see [6, Section B6]). The localization ℳT may also be expressed as the filtered colimit of copies of ℳ connected by morphisms tℳ:ℳ→ℳ over all t∈T (see [6, Section B6.1]). Accordingly, the kernel of the canonical map ℳ→ℳT is given by the filtered colimit (see [6, Proposition B6.2])

(2.2) Ker(ℳ⟶ℳT)=⋃t∈TKer(tℳ:ℳ⟶ℳ).

Moreover, since R[T−1] is a flat R-module, it follows from [6, Proposition C1.7] that localization is an exact functor on 𝒮R. If 𝔭 is a prime ideal, we denote by ℳ𝔭 the localization with respect to R\𝔭. We now have the following definition.

We conclude this section by establishing some properties of injective hulls in 𝒮R 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, J be ideals in R. If ℳ∈𝒮R and 𝔊 is any family of subobjects of ℳ, we write ⨁𝔊 (resp. ∑𝔊) for the direct sum (resp. the sum in ℳ) of all the objects in 𝔊. We now define

(3.1) |ΓI,J(ℳ)| :={𝒩∈fg(ℳ)∣In⊆Ann(𝒩)+Jfor n≫1}, ΓI,J(ℳ) :=∑|ΓI,J(ℳ)|.

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∈𝒮R is (I,J)-torsion if ΓI,J(ℳ)=ℳ. We will say that ℳ∈𝒮R is (I,J)-torsion free if ΓI,J(ℳ)=0. When J=0, we will denote ΓI,J simply by ΓI.

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

𝕎(I,J)={𝔭∈Spec(R)∣In⊆𝔭+Jfor n≫1}.

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

It follows from Lemma 3.2 that ΓI,J(ℳ) is the sum of all finitely generated subobjects 𝒩∈fg(ℳ) satisfying Supp(𝒩)⊆𝕎(I,J). We note that this implies Supp(ΓI,J(ℳ))⊆𝕎(I,J). We will now describe the associated primes of ΓI,J(ℳ).

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∈𝒯, Y∈ℱ and any Z∈𝒜 fits into a short exact sequence 0→Z𝒯→Z→Zℱ→0 with Z𝒯∈𝒯 and Zℱ∈ℱ. Additionally, (𝒯,ℱ) is said to be hereditary if the torsion class 𝒯 is closed under subobjects.

We will now show that if ℳ∈𝒮R is (I,J)-torsion, then HI,Ji(ℳ)=0 for i>0 0\)]]>. This will be achieved by a sequence of steps. We will start by showing that the torsion theory (𝒯(𝒮)I,J,ℱ(𝒮)I,J) satisfies the property that the torsion part of an injective in 𝒮R is still an injective.