I'll talk about the tmf-based Adams spectral sequence, and how it detects most of the v2-periodic elements in the known range of the 2-primary stable stems. Parts of the material I will discuss are joint with Dominic Culver, Prasit Bhattacharya, JD Quigley, and Mark Mahowald.
This lecture will partly survey branching laws for real and p-adic groups which often is related to period integrals of automorphic representations, discuss some of the more recent developments, focusing attention on homological aspects and the Bernstein decomposition.
When studying the Adams spectral sequence in triangulated categories, one runs into the issue of choosing suitably coherent cofibers in an Adams resolution. Motivated by this, in joint work with Dan Christensen, we develop tools to deal with the limited coherence afforded by the triangulated structure. We use and expand Neeman's work on good morphisms of exact triangles. The talk will include examples from stable module categories of group algebras.
Inspired by the intrinsic formality of graded algebras, we give a characterization of strongly unique DG-enhancements for a large class of algebraic triangulated categories, linear over a commutative ring. We will discuss applications to bounded derived categories and bounded homotopy categories of complexes. For the sake of an example, the bounded derived category of finitely generated abelian groups has a strongly unique enhancement.
Many tensor triangulated categories admit 'residue field functors' that control their large-scale structure. The derived category of a ring is controlled by the residue fields of the ring, the structure of the stable homotopy category is controlled by the Morava K-theories, and in modular representation theory there are the pi-points. Unfortunately, it is not known if every tensor triangulated category has a notion of tensor triangulated residue fields. Homological residue fields were introduced by Balmer, Krause, and Stevenson as an abelian avatar of the putative tensor triangulated residue fields. They exist in complete generality, but they are hard to understand and compute with in general. I will discuss how to connect homological residue fields with the tensor triangulated residue fields that exist in examples. I will show that for the derived category of a ring, homological residue fields are closely related to usual residue fields, and in stable homotopy theory they are closely related to Morava K-theories. In fact, the homological residue fields have even more structure, and can be identified with comodules for a Tor coalgebra which in the case of the stable homotopy category is the coalgebra of coooperations for a Morava K-theory. I will introduce homological residue fields, give some examples, and mention some open problems. This is joint work with Paul Balmer and with Greg Stevenson.
In a recent paper, Jeremy Rickard showed that for any finite-dimensional algebra R over a field, if the R-injectives generate the derived category D(R), then the finitistic dimension of R is finite (recall that the famous finitistic dimension conjecture claims that for any R that is a finite-dimensional algebra over a field, the finitistic dimension of R should be finite). In the same paper, it was noted that there are no known finite-dimensional algebras over fields for which this injective generation property is absent. In this talk, we will consider the case when R is a group algebra (not necessarily a finite-dimensional algebra over a field), and show that for a large class of groups, the finiteness of the finitistic dimension of the group algebra (over any commutative ring of finite global dimension) implies the above injective generation property. We will also show how this question, for group algebras, is very closely connected to some existing conjectures on various cohomological invariants for groups, and that will lead us to a version of the finitistic dimension conjecture for group algebras.
The Bruhat order on a Weyl group has a representation theoretic interpretation in terms of Verma modules. The talk concerns resulting interactions between combinatorics and homological algebra. I will present several questions around the above realization of the Bruhat order and answer them based on a series of recent works, partly joint with Volodymyr Mazorchuk and Rafael Mrden.
Classes of modules closed under transfinite extensions often provide for precovers, and hence fit in the machinery of relative homological algebra. However, there are important exceptions: the Whitehead groups, and flat Mittag-Leffler modules over non-perfect rings. The latter class is just the zero dimensional instance (for T = R and n = 0) of non-precovering of the class of all locally T-free modules, where T is any n-tilting module which is not Σ-pure split. The phenomenon occurs even for finite dimensional algebras, when R is hereditary of infinite representation type, and T is the Lukas tilting module. The key tools here are the tree modules, which have recently been generalized in order to solve Auslander's problem on the existence of almost split sequences.
I'll talk around some joint work with Ivo Dell'Ambrogio and Jan Stovicek on the role of Gorenstein module categories in homological algebra. The idea is to reduce understanding universal coefficient theorems to very concrete questions about when a small category is Gorenstein and how one can detect when a representation has finite projective dimension.
Nakayama algebras are among the best understood representation-finite algebras. They are defined as those algebras such that each indecomposable projective and each indecomposable injective module admits a unique composition series. An equivalent characterisation is that τjS is simple (or zero) for all j ∈ ℤ and every simple module S. Here, τ denotes the Auslander–Reiten translation. Nakayama algebras can be classified by the sequence of lengths of their indecomposable projective modules, called the Kupisch series.
In this talk, we introduce a higher analogue of a Nakayama algebra for each Kupisch series 𝓁 in the sense of Iyama's higher Auslander–Reiten theory. More precisely, (in type A) the higher Nakayama algebra A𝓁(d) is a quotient of the higher Auslander algebra An(d) of type A, constructed by Iyama and studied extensively by Oppermann and Thomas. In type ̃A, one has to use an infinite version of An(d). The higher Nakayama algebra has a d-cluster-tilting module, i.e. a module M with
add(M) = {N | Exti(M,N) = 0 ∀i = 1, . . . , d−1 } = {N | Exti(N,M) = 0 ∀i = 1, . . . , d−1 }.
There are n simple modules in add(M) and they satisfy that τdjS is simple for all j ∈ ℤ and every simple module S in add(M), where τd = τΩd−1 is Iyama's higher Auslander–Reiten translation.
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