Stable Module Category Research Articles

The singularity category of a quadratic monomial algebra*

We exploit singular equivalences between artin algebras, that are induced from certain functors between the stable module categories. Such functors are called pre-triangle equivalences. We construct two pre-triangle equivalences connecting the stable module category over a quadratic monomial algebra and the one over an algebra with radical square zero. Consequently, we obtain an explicit singular equivalence between the two algebras.

Nilpotence and generation in the stable module category

Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG-module M, the tensor M-generation number and the tensor M-ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M.

Realizing orbit categories as stable module categories: a complete classification

We classify all triangulated orbit categories of path-algebras of Dynkin diagrams that are triangle equivalent to a stable module category of a representation-finite self-injective standard algebra. For each triangulated orbit category T we give an explicit description of a representation-finite self-injective standard algebra with stable module category triangle equivalent to T.

Stratification for module categories of finite group schemes

The tensor ideal localizing subcategories of the stable module category of all, including infinite dimensional, representations of a finite group scheme over a field of positive characteristic are classified. Various applications concerning the structure of the stable module category and the behavior of support and cosupport under restriction and induction are presented.

Separable Commutative Rings in the Stable Module Category of Cyclic Groups

We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group.

Colocalising subcategories of modules over finite group schemes

The Hom closed colocalising subcategories of the stable module category of a finite group scheme are classified. This complements the classification of the tensor closed localising subcategories in our previous work. Both classifications involve pi-points in the sense of Friedlander and Pevtsova. We identify for each pi-point an endofinite module which both generates the corresponding minimal localising subcategory and cogenerates the corresponding minimal colocalising subcategory.

Precovers and orthogonality in the stable module category

We show that any module admits a presentation as the quotient of a Gorenstein projective module by a submodule which is itself right orthogonal, with respect to the standard Ext1 pairing, to the class of Gorenstein projective modules of type FP∞. For that purpose, we use the concept of orthogonality in the stable module category and examine the orthogonal pair which is induced therein by the class of completely finitary Gorenstein projective modules.

THE GRADED CENTER OF A TRIANGULATED CATEGORY

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor. We show that such natural transformations that have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

Hereditary abelian model categories

We discuss some recent developments in the theory of abelian model categories. The emphasis is on the hereditary condition and applications to homotopy categories of chain complexes and stable module categories.

The Galois group of a stable homotopy theory

To a “stable homotopy theory” (a presentable, symmetric monoidal stable ∞-category), we naturally associate a category of finite étale algebra objects and, using Grothendieck's categorical machine, a profinite group that we call the Galois group. We then calculate the Galois groups in several examples. For instance, we show that the Galois group of the periodic E∞-algebra of topological modular forms is trivial and that the Galois group of K(n)-local stable homotopy theory is an extended version of the Morava stabilizer group. We also describe the Galois group of the stable module category of a finite group. A fundamental idea throughout is the purely categorical notion of a “descendable” algebra object and an associated analog of faithfully flat descent in this context.

Stable auto-equivalences for local symmetric algebras

We construct nontrivial auto-equivalences of stable module categories for elementary, local symmetric algebras over a field k. These auto-equivalences are modeled after the spherical twists of Seidel and Thomas and the Pn-twists of Huybrechts and Thomas, which yield auto-equivalences of the derived category of coherent sheaves on a variety. For group algebras of p-groups in characteristic p we recover many of the auto-equivalences corresponding to endo-trivial modules. We also obtain analogous auto-equivalences for local algebras of dihedral and semi-dihedral type, which are not group algebras.

Tensor ideals and varieties for modules of quantum elementary abelian groups

In a previous paper we constructed rank and support variety theories for quantum elementary abelian groups, that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor ideals in the stable module category, and to prove a tensor product property for the support varieties.

On endotrivial modules for Lie superalgebras

Let g=g0¯⊕g1¯ be a Lie superalgebra over an algebraically closed field, k, of characteristic 0. An endotrivial g-module, M, is a g-supermodule such that Homk(M,M)≅kev⊕P as g-supermodules, where kev is the trivial module concentrated in degree 0¯ and P is a projective g-supermodule. In the stable module category, these modules form a group under the operation of the tensor product. We show that for an endotrivial module M, the syzygies Ωn(M) are also endotrivial, and for certain Lie superalgebras of particular interest, we show that Ω1(kev) and the parity change functor actually generate the group of endotrivials. Additionally, for a broader class of Lie superalgebras, for a fixed n, we show that there are finitely many endotrivial modules of dimension n.

The symmetry, period and Calabi–Yau dimension of finite dimensional mesh algebras

Within the class of finite dimensional mesh algebras, we identify those which are symmetric and those whose stable module category is Calabi–Yau. We also give, in combinatorial terms, explicit formulas for the Ω-period of any such algebra, where Ω is the syzygy functor, and for the Calabi–Yau Frobenius and the stable Calabi–Yau dimensions, when they are defined.

The singularity category of a Nakayama algebra

Let A be a Nakayama algebra. We give a description of the singularity category of A inside its stable module category, which provides a new approach to the singularity category of a Nakayama algebra. We prove that there is a duality between the singularity category of A and the singularity category of its opposite algebra. As a consequence, the resolution quiver of A and the resolution quiver of its opposite algebra have the same number of cycles and the same number of cyclic vertices.

The stable category of a left hereditary ring

The (co)completeness problem for the (projectively) stable module category of an associative ring is studied. (Normal) monomorphisms and (normal) epimorphisms in such a category are characterized. As an application, we give a criterion for the stable category of a left hereditary ring to be abelian. By a structure theorem of Colby–Rutter, this leads to an explicit description of all such rings.

Singularity categories of gentle algebras

We determine the singularity category of an arbitrary finite dimensional gentle algebra $\Lambda$. It is a finite product of $n$-cluster categories of type $\mathbb{A}_{1}$. Equivalently, it may be described as the stable module category of a selfinjective gentle algebra. If $\Lambda$ is a Jacobian algebra arising from a triangulation $\ct$ of an unpunctured marked Riemann surface, then the number of factors equals the number of inner triangles of $\ct$.

Hom-configurations in triangulated categories generated by spherical objects

Hom- and Riedtmann configurations were studied in the context of stable module categories of selfinjective algebras and a certain orbit category C1(Q) of the bounded derived category of a Dynkin quiver Q, which is highly reminiscent of the cluster category. The category C1(Q) is (−1)-Calabi–Yau. In this paper we consider triangulated categories generated by w-spherical objects, for w<0, and higher versions of the orbit category C1(An), denoted by C|w|(An). These categories have also negative Calabi–Yau dimension. We classify the (higher) Hom- and Riedtmann configurations for these categories, and link them with noncrossing partitions in the case w=−1. Along the way, we obtain a geometric model for C|w|(An).

Spectra of tensor triangulated categories over category algebras

Let \({\mathcal{C}}\) be a finite EI category and k be a field. We consider the category algebra \({k\mathcal{C}}\). Suppose \({\sf{K}(\mathcal{C})=\sf{D}^b(k \mathcal{C}-\sf{mod})}\) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category, and we compute its spectrum in the sense of Balmer. When \({\mathcal{C}=G \propto \mathcal{P}}\) is a finite transporter category, the category algebra becomes Gorenstein, so we can define the stable module category \({\underline{\sf{CM}} k(G \propto \mathcal{P})}\), of maximal Cohen–Macaulay modules, as a quotient category of \({{\sf{K}}(G \propto \mathcal{P})}\). Since \({\underline{\sf{CM}} k(G\propto\mathcal{P})}\) is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories.

Simple-minded systems, configurations and mutations for representation-finite self-injective algebras

Simple-minded systems of objects in a stable module category are defined by common properties with the set of simple modules, whose images under stable equivalences do form simple-minded systems. Over a representation-finite self-injective algebra, it is shown that all simple-minded systems are images of simple modules under stable equivalences of Morita type, and that all simple-minded systems can be lifted to Nakayama-stable simple-minded collections in the derived category. In particular, all simple-minded systems can be obtained algorithmically using mutations.