Workshop on Finitistic Dimensions – Abstracts
Rudradip Biswas (Bielefeld)
Bounded t-structures and the finitistic dimension
We will prove that for a finite dimensional algebra over a field, if the finitistic dimension is known to be finite, then the derived perfect category admits a bounded t-structure iff the global dimension of the algebra is finite. To prove this, we follow Amnon Neeman's recent work on proving a similar result for schemes. Neeman's work introduces the notion of equivalence between t-structures, and a key ingredient for our proof is that for the derived unbounded category of an associative (not necessarily commutative) ring, the standard t-structure is equivalent to the t-structure generated by any compact generator.
Ben Briggs (Copenhagen)
Hochschild cohomology and the topology of finite dimensional algebras
Martínez-Villa and de la Peña have associated a fundamental group to each presentation of a finite dimensional algebra A, and Bustamante extended this to construct an entire topological space from each presentation. We prove that the first cohomology of this space embeds into the first Hochschild cohomology group of A, and that every maximal torus occurs in this way. We think of this as a map from the moduli of presentations of A to the moduli of tori in Hochschild cohomology. Thinking along these lines uncovers some useful derived invariants: for example, we find that there are only finitely many monomial algebras within any derived equivalence class. This is all joint with Lleonard Rubio y Degrassi.
Charley Cummings (Aarhus)
Left-right symmetry of finite finitistic dimension
The dominant dimension of an algebra is equal to that of its opposite algebra. One consequence of this is that an algebra satisfies the Nakayama conjecture if and only if its opposite algebra also does so. It is unknown whether the analogous statement holds for the finitistic dimension conjecture. In this talk, we will see that, heuristically, it is unknown for good reason, the analogous statement is equivalent to the finitistic dimension conjecture itself.
Özgür Esentepe (Leeds)
Cohen-Macaulay dominant dimension
Gorenstein rings are ubiquitous. Therefore, there are several generalizations of them to the noncommutative setting. In this talk, I will talk about Gorenstein orders over Cohen-Macaulay local rings. I will define Cohen-Macaulay dominant dimension, formulate the Cohen-Macaulay version of the Nakayama conjecture and give some examples.
Delooping level for Artin algebras
In studying the finitistic dimensions, it is often fruitful to relate them to more tractable invariants, either as bounds or as proxy values. This talk will concentrate on a recent such invariant, the 'delooping level', surveying its properties and recent developments.
Rene Marczinzik (Bonn)
Cohen-Macaulay Artin algebras and the homological conjectures
We show that contracted preprojective algebras of Dynkin type are Cohen-Macaulay Artin algebras in the sense of Auslander and Reiten. As an application we use this class of algebras to answer a question of Auslander and Reiten and show that this class of algebras has finitistic dimension at most two. This is joint work with Aaron Chan and Osamu Iyama. If time permits we survey some recent results on homological problems and conjectures.
Jeremy Rickard (Bristol)
A finite dimensional algebra with infinite delooping level
I will talk about recent joint work with Luke Kershaw. Based on an example of a "semi-Gorenstein-projective" module constructed by Ringel and Zhang, we construct what we believe to be the first known example of a finite dimensional algebra with infinite delooping level in the sense of Gélinas.
Liran Shaul (Prague)
Acyclic complexes and finitistic dimensions
Over any ring, any bounded above acyclic cochain complex of projectives is null-homotopic,
and any bounded below acyclic cochain complex of injectives is null-homotopic.
In this talk we consider questions dual to these,
and show that over a two-sided noetherian ring with a dualizing complex,
they are equivalent to the question whether its big finitistic projective dimension is finite.
Jan Stovicek (Prague)
Finiteness of the big finitistic dimension via singularity categories
For a finite dimensional algebra A, we show that the big left finitistic dimension of A is finite if and only if the right singularity category can be generated by a single object in terms of extensions and summands (but not using suspensions).
Jordan Williamson (Prague)
Finitistic dimensions for commutative DGAs
DGAs (differential graded algebras) provide the affine theory of derived algebraic geometry as well as supplying applications to ordinary ungraded algebra through their use as resolutions of ordinary rings. Bass and Raynaud-Gruson proved that the big finitistic dimension of a commutative Noetherian ring is its Krull dimension, and in this talk I will explain the analogous result for commutative DGAs with finite amplitude. The answer shares several similarities with the classical case, but also has some important differences which I'll highlight. This is joint work with Isaac Bird, Liran Shaul, and Prashanth Sridhar.