The heart fan is a new convex-geometric invariant of an abelian category which captures interesting aspects of the related homological algebra. I will review the construction and some of its key properties, illustrating them through examples. In particular, I will explain how the heart fan can be viewed as a 'universal phase diagram' for Bridgeland stability conditions with the given heart.
The definition of the nucleus was originally formulated in joint work with Carlson and Robinson, to capture the supports of modules with no cohomology. This definition works in various contexts such as finite groups, restricted Lie algebras, and more generally, suitable triangulated categories of modules. In the finite group context it has a characterization in terms of subgroups whose centralizer is not p-nilpotent. In the restricted Lie algebra context, it is described in terms of the Richardson orbit, at least for large primes. Recent work with Greenlees has highlighted a connection with the singularity category of the cochains on the classifying space, in the group theoretic context. My plan is to give an introduction to these ideas.
We discuss support variety theory for quasireductive algebraic supergroups, i.e. supergroups with reductive even part over complex numbers. The corresponding categories of representations are Frobenius and share many properties of representations of finite groups in positive characteristic. It is desirable to describe Balmer spectrum of related triangulated symmetric monoidal categories. Our approach involves so called homological odd elements and certain tensor functors associated to them. On the way we encounter analogues of p-groups and Sylow subgroups for supergroups. We prove projectivity detection for our support theory and present other related results. We also explore connections with homological support theory developed by B. Boe, J. Kujawa and D. Nakano.
In this joint work with Beren Sanders, we analyze the question of (topologically) gluing an open piece and a closed piece of the spectrum of a tt-category, that is, of deciding which points in the open specialize to which points of the closed piece. We show that a critical role is played by the support of the Tate ring associated with this decomposition. I will take this opportunity to remind the audience of the very general notion of support for big objects in tt-categories that follows from the theory of homological residue fields.
Carlson’s Connectedness Theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. For monoidal triangulated categories, the Balmer spectrum is an intrinsic geometric space associated to the category which generalizes the notion of cohomological support for finite groups. In this talk, we will discuss a generalization of the Carlson Connectedness Theorem: that the Balmer support of any indecomposable object in a monoidal triangulated category with a thick generator is a connected subset of the Balmer spectrum. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.
Describing the thick ideals of a monoidal triangulated category is a key component of the analysis of the category. We will show how this can be done by non-commutative tensor triangular geometry (NTTG), thus extending the celebrated Balmer’s theorem from the symmetric case. We will then use NTTG to analyse the stable categories of finite tensor categories, which play an important role in representation theory, mathematical physics and quantum computing. We will present general results linking this approach to the traditional one through cohomological support, based on a notion of categorical centers of cohomology rings of monoidal triangulated categories.
To study topological Hochschild homology as an invariant of stable ∞-categories and endow it with its universal property in this context, Nikolaus introduced the formalism of stable cyclic graphs and trace theories (after Kaledin). On the other hand, Poincaré ∞-categories are a C2-refinement of stable ∞-categories that provide an adequate formalism for studying real and hermitian algebraic K-theory, which should be then well-approximated by the real cyclotomic trace. In this talk, we explain how to systematically provide Poincaré refinements of all the components of Nikolaus's approach to stable trace theories.
This is a talk about the situation in commutative algebra. A homomorphism f: S → R of commutative local rings has a derived fibre F (a differential graded algebra over the residue field k of R) and we say that f is Koszul if F is formal and its homology H(F) = TorS(R,k) is a Koszul algebra in the classical sense. I'll explain why this is a very good definition and how it is satisfied by many many examples.
The main application is the construction of explicit free resolutions over R in the presence of a Koszul homomorphism. These tell you about the asymptotic homological algebra of R, and so the structure of the derived category of R. This construction simultaneously generalizes the resolutions of Priddy over a Koszul algebra, the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring.
Stable equivalences occur frequently in the representation theory of finite-dimensional algebras; however, these equivalences are poorly understood. An interesting class of stable equivalences is obtained by ‘gluing’ two idempotents. More precisely, let A be a finite-dimensional algebra with a simple projective module and a simple injective module. Assume that B is a subalgebra of A having the same Jacobson radical. Then B is constructed by identifying the two idempotents belonging to the simple projective module and to the simple injective module, respectively. In this talk we will compare the first Hochschild cohomology groups of finite-dimensional monomial algebras under gluing two arbitrary idempotents (hence not necessarily inducing a stable equivalence). As a corollary, we will show that stable equivalences obtained by gluing two idempotents provide 'some functoriality' to the first Hochschild cohomology, that is, HH1(A) is isomorphic to a quotient of HH1(B).
In this talk we will present recent results on the category of finite-dimensional modules for map superalgebras. Firstly, we will show a new description of certain irreducible modules. Secondly, we will use this new description to extract homological properties of the category of finite-dimensional modules for map superalgebras, most importantly, its block decomposition.
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