Topological Hochschild homology is an important invariant, closely related to algebraic K-theory, and can be seen as a non-commutative analogue of de Rham chains.
In this talk, I will describe various computations of the ring/monoid of endomorphisms of THH in different variants, with an approach based on a generalized version of the Dundas-McCarthy theorem.
The global dimension of an associative algebra A over a a field is a measure of the complexity of its representations. It is 0 if A is a matrix algebra. It is 1 if A is a path algebras of quivers without directed cycles. It is infinite if A is the algebra of dual numbers.
I will give a brief introduction to Hochschild homology (1945), in order to explain Han's conjecture (2006): for finite-dimensional algebras, the Hochschild homology should control the finiteness of the global dimension.
Next, I will present some progress made in showing Han's conjecture, using the relative version of Hochschild homology (1956) with respect to a subalgebra B. This theory was little used until recently. Now we have a Jacobi-Zariski long nearly exact sequence which relates the usual and relative versions of Hochschild homology. Its gap to be exact is approximated by a spectral sequence which has Tor functors in its first page, of B-tensor powers of A/B. This tool enables to show, for instance, that the class of algebras verifying Han's conjecture is closed by bounded extensions of algebras.
Modular representation theory of finite groups seeks to understand, and possibly classify, the algebras - called block algebras of finite groups - which arise as indecomposable direct factors of finite group algebras over a complete local principal ideal domain with residue field of prime characteristic p. The expectation is that 'few' algebras should arise in this way, and that this should in turn lead to significant structural connections between finite groups and their block algebras.
The key feature of block algebras of finite groups is the dichotomy of invariants attached to these algebras.
On the one hand, they have all the typical algebra-theoretic invariants - module categories, their derived categories and stable categories, as well as numerical invariants such as the numbers of isomorphism classes of simple modules, and cohomologivcal invariants such as their Hochschild cohomology.
On the other hand, they have p-local invariants, due to their provenance from group algebras - reminiscent of the local structure of a finite group which includes its Sylow p-subgroups and its associated fusion systems.
Essentially all prominent conjectures which drive modular representation theory revolve around the interplay between these two types of invariants. We describe this interplay with a focus on Hochschild cohomology and analogous cohomology rings which are defined p-locally. This involves a variety of angles - Hochschild cohomology is graded commutative, hence methods and notions from commutative algebra will play a role. Hochschild cohomology in positive degree is also a Lie algebra. We will investigate connections between the algebra structure of block algebras and the Lie algebra structure of its first Hochschild cohomology space.
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