Serre Subcategories Research Articles

English

We consider a generalization of the notion of torsion theory, which is associated with a Serre subcategory over a commutative Noetherian ring. In 2008 Aghapournahr and Melkersson investigated the question of when local cohomology modules belong to a Serre subcategory of the module category. In their study, the notion of Melkersson condition was defined as a suitable condition in local cohomology theory. One of our purposes in this paper is to show how naturally the concept of Melkersson condition appears in the context of torsion theories.

Representations of Leavitt path algebras

We study representations of a Leavitt path algebra L of a finitely separated digraph Γ over a field. We show that the category of L-modules is equivalent to a full subcategory of quiver representations. When Γ is a (non-separated) row-finite digraph we determine all possible finite dimensional quotients of L after giving a necessary and sufficient graph theoretic criterion for the existence of a nonzero finite dimensional quotient. This criterion is also equivalent to L having UGN (Unbounded Generating Number) as well as being algebraically amenable. We also realize the category of L-modules as a retract, hence a quotient by an explicit Serre subcategory of the category of quiver representations (that is, FΓ-modules) via a new colimit model for M⊗FΓL.

Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof

AbstractWe prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber functor to the category of supervector spaces. This proves Ostrik’s conjecture [25, Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $\mathcal{G}$ over $k$ and a grouplike element $\epsilon \in k\mathcal{G}$ of order $\le 2$, whose action by conjugation on $\mathcal{G}$ coincides with the parity automorphism of $\mathcal{G}$, such that $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(\mathcal{G},\epsilon )$. In particular, when $\mathcal{C}$ is unipotent, the functor lands in $\textrm{Vec}$, so $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of [17] to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p>0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p>0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper [4], and more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $\ne 2$ is always a Serre subcategory.

A classification of torsion classes in abelian categories

We give a classification of torsion classes (or nullity classes) in an abelian category by forming a spectrum of equivalence classes of premonoform objects. This is parallel to Kanda’s classification of Serre subcategories.

The Auslander–Gruson–Jensen recollement

For any ring R, the Auslander–Gruson–Jensen functorDA:fp(R-Mod,Ab)→(mod-R,Ab)op is the exact functor which sends a representable functor (X,−) to the tensor functor −⊗X. We show that this functor admits a fully faithful right adjoint DR and a fully faithful left adjoint DL. That is, we show that DA is part of a recollement of abelian categories. In particular, this shows that DA is a localisation and a colocalisation which gives an equivalence of categoriesfp(R-Mod,Ab){F:DAF=0}≃(mod-R,Ab)op. We show that {F:DAF=0} is the Serre subcategory of fp(R-Mod,Ab) consisting of finitely presented functors which arise from a pure-exact sequence. As an application of our main result, we show that the 0-th right pure-derived functor of a finitely presented functor R-Mod→Ab is also finitely presented.

Types of Serre subcategories of Grothendieck categories

Every Serre subcategory S of an abelian category A is assigned a unique type (m,−n), where m (resp. n) counts how many times one can form left (resp. right) adjoints starting from i and Q, where i:S→A is the inclusion and Q is the quotient functor. The main result gives a complete list of all the types of Serre subcategories of Grothendieck categories:(0,0),(0,−1),(1,−1),(0,−2),(1,−2),(2,−1),(+∞,−∞). Two observations are technically crucial in proving the main result: the exactness of all the functors in a recollement of abelian categories forces the recollement to split; and any left recollement of a Grothendieck category can be extended to a recollement.

On refined filtration by supports for rational Cherednik categories $$\mathcal {O}$$O

For a complex reflection group W with reflection representation $$\mathfrak {h}$$ , we define and study a natural filtration by Serre subcategories of the category $$\mathcal {O}_c(W, \mathfrak {h})$$ of representations of the rational Cherednik algebra $$H_c(W, \mathfrak {h})$$ . This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov–Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of finite-dimensional representations over generalized Hecke algebras. When W is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori–Hecke algebras. This yields a method for counting the number of irreducible objects in $$\mathcal {O}_c(W, \mathfrak {h})$$ of given support. We apply these techniques to count the number of irreducible representations in $$\mathcal {O}_c(W, \mathfrak {h})$$ of given support for all exceptional Coxeter groups W and all parameters c, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of $$\mathcal {O}_c(W, \mathfrak {h})$$ for exceptional Coxeter groups W in many new cases.

Torsion functors of generalized local cohomology modules

ABSTRACTLet R be a commutative Noetherian ring, 𝔞 an ideal of R, M a finite R–module with finite projective dimension, and X an arbitrary R–module. In this paper, for fixed integers s,t and an arbitrary R–module N, we study the membership of in Serre subcategories of the category of R–modules. We present some upper bounds for the flat dimension and the Betti numbers of . We also give some results on cofiniteness, finiteness, vanishing, and Artinianness of generalized local cohomology modules.

On the closedness of taking injective hulls of several Serre subcategories

ABSTRACTBelshoff and Xu showed that every Matlis reflexive module has a Matlis reflexive injective hull if and only if R is complete and has dimension less than or equal to 1. In this paper, we give a characterization of the closedness of taking injective hulls for a Serre subcategory consisting of Minimax modules. In addition, the closedness of taking injective hulls for a Serre subcategory consisting of extension modules of finitely generated modules by modules with finite support is characterized by the number of prime ideals. Our results provide a negative answer to Aghapournahr and Melkersson’s question concerning Melkersson subcategories.

Some Properties of Serre Subcategories in the Graded Local Cohomology Modules

LetR=⊕n≥0Rnbe a standard homogeneous Noetherian ring with local base ring(R0,m0)and letMbe a finitely generated gradedR-module. LetHR+i(M)be theith local cohomology module ofMwith respect toR+=⊕n>0Rn. LetSbe a Serre subcategory of the category ofR-modules and letibe a nonnegative integer. In this paper, ifdim⁡R0≤1,then we investigate some conditions under which theR-modulesR0/m0 ⊗R0 HR+i(M),Γm0R(HR+i(M))andHm0R1(HR+i(M))are inSfor alli≥0. Also, we prove that ifdim⁡R0≤2, then the gradedR-moduleHm01(HR+i(M))is inSfor alli≥0. Finally, we prove that ifnis the biggest integer such thatHai(M)∉S, thenHR+i(M)/m0HR+i(M)∈Sfor alli≥n.

Serre subcategories and the cofiniteness of generalized local cohomology modules with respect to a pair of ideals

We study some properties of the generalized local cohomology modules [Formula: see text] of [Formula: see text] with respect to a pair of ideals [Formula: see text] in Serre subcategories. Some results concerning to [Formula: see text]-cofinite modules are also given in this paper.

Some Results on Local Cohomology Modules with Respect to a Pair of Ideals

We study the finiteness of the sets $\operatorname{Ass}(H^d_{I, J}(M))$ and $\operatorname{Ass}(\operatorname{Hom}_R(R/I, H^d_{I, J}(M)))$ concerning Grothendieck's conjecture. We also show some properties of local cohomology modules $H^i_{I, J}(M)$ from the point of view of Serre subcategories.

Lower Bounds for Local Cohomology Modules with Respect to a Pair of Ideals

Let R be a Noetherian ring, I and J two ideals of R, M an R-module and t an integer. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI, and N be a finitely generated R-module with Supp RN= V (𝔞) for some [Formula: see text]. It is shown that if [Formula: see text] for all i < t and all j < t-i, then [Formula: see text] for all i < t. Let S be the class of all R-modules N with dim R N ≤ k, where k is an integer. It is proved that if [Formula: see text] for all i < t and all [Formula: see text], then [Formula: see text] for all i < t. It follows that [Formula: see text].

Melkersson condition on Serre subcategories

Let $R$ be a commutative noetherian ring, let $\frak a$ and $\frak b$ be two ideals of $R$; and let $\Ss$ be a Serre subcategory of $R$-modules. We give a necessary and sufficient condition by which $\Ss$ satisfies $C_{\frak a}$ and $C_{\frak b}$ conditions. As an conclusion we show that over a artinian local ring, every Serre subcategory satisfies $C_{\frak a}$ condition. We also show that $\Ss_{\frak a}$ is closed under extension of modules. If $\Ss$ is a torsion subcategory, we prove that $S$ satisfies $C_{\frak a}$ condition. We prove that $C_{\frak a}$ condition can be transferred via rings homomorphism. As some applications, we give several results concerning with Serre subcategories in local cohomology theory.

Some Results on Generalized Local Cohomology Modules

Let R be a commutative Noetherian ring with nonzero identity, 𝔞 an ideal of R, M a finite R–module, X an arbitrary R–module, and n a non-negative integer. Here, we show that, in the Serre subcategories of the category of R–modules, how the generalized local cohomology modules, the ordinary local cohomology modules, and the extension modules behave similarly at the initial points i ≤ n. We conclude with some Artinianness and cofiniteness results for , and some finiteness results for and .

Monadicity Theorem and Weighted Projective Lines of Tubular Type

We formulate a version of Beck's monadicity theorem for abelian categories, which is applied to the equivariantization of abelian categories with respect to a finite group action. We prove that the equivariantization is compatible with the construction of quotient abelian categories by Serre subcategories. We prove that the equivariantization of the graded module category over a graded ring is equivalent to the graded module category over the same ring but with a different grading. We deduce from these results two equivalences between the category of (equivariant) coherent sheaves on a weighted projective line of tubular type and that on an elliptic curve, where the acting groups are cyclic and the two equivalences are somehow adjoint to each other.

Deriving Auslander's formula

Auslander's formula shows that any abelian category \mathsf C is equivalent to the category of coherent functors on \mathsf C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of \mathsf C ) is compactly generated and that the full subcategory of compact objects is equivalent to the bounded derived category of \mathsf C . The same approach shows for an arbitrary Grothendieck abelian category that its derived category and the homotopy category of injective objects are well-generated triangulated categories. For sufficiently large cardinals \alpha we identify their \alpha -compact objects and compare them.

Quantized multiplicative quiver varieties

Beginning with the data of a quiver Q, and its dimension vector d, we construct an algebra Dq=Dq(Matd(Q)), which is a flat q-deformation of the algebra of differential operators on the affine space Matd(Q). The algebra Dq is equivariant for an action by a product of quantum general linear groups, acting by conjugation at each vertex. We construct a quantum moment map for this action, and subsequently define the Hamiltonian reduction Adλ(Q) of Dq with moment parameter λ. We show that Adλ(Q) is a flat formal deformation of Lusztigʼs quiver varieties, and their multiplicative counterparts, for all dimension vectors satisfying a flatness condition of Crawley-Boevey: indeed the product on Adλ(Q) yields a Fedosov quantization the of symplectic structure on multiplicative quiver varieties. As an application, we give a description of the category of representations of the spherical double affine Hecke algebra of type An−1, and its generalization constructed by Etingof, Oblomkov, and Rains, in terms of a quotient of the category of equivariant Dq-modules by a Serre subcategory of aspherical modules.

On Multigraded Generalizations of Kirillov–Reshetikhin Modules

We study the category of \(\mathbb Z^\ell\)-graded modules with finite-dimensional graded pieces for certain \(\mathbb Z^\ell_+\)-graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov–Reshetikhin modules and give a recursive formula for computing their graded characters.

FINITENESS PROPERTIES GENERALIZED LOCAL COHOMOLOGY WITH RESPECT TO AN IDEAL CONTAINING THE IRRELEVANT IDEAL

The membership of the generalized local cohomology modules <TEX>$H_a^i$</TEX>(M,N) of two R-modules M and N with respect to an ideal a in certain Serre subcategories of the category of modules is studied from below (<TEX>$i</TEX><TEX><</TEX><TEX>t$</TEX>). Furthermore, the behaviour of the <TEX>$n$</TEX>th graded component <TEX>$H_a^i(M,N)_n$</TEX> of the generalized local cohomology modules with respect to an ideal containing the irrelevant ideal as <TEX>$n{\rightarrow}-{\infty}$</TEX> is investigated by using the above result, in certain graded situations.