Pollitz gave a characterization of complete intersection rings in terms of the triangulated structure of their derived category, akin to the Auslander-Buchsbaum-Serre characterization of regular rings. In this talk, we will explore how to bring this characterization back to the world of modules, and discuss the role of cohomological support varieties in solving this problem.
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.
This talk is based on the book Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes—quasi-coherent torsion sheaves, the semiderived category, and the semitensor product. I will start with some examples serving as special cases of the general theory, such as the tensor structure on the category of unbounded complexes of injective quasi-coherent sheaves on a Noetherian scheme with a dualizing complex. Then I will proceed to explain the setting of a flat affine morphism of ind-schemes into an ind-Noetherian ind-scheme with a dualizing complex, and the main ingredient concepts of quasi-coherent torsion sheaves, pro-quasi-coherent pro-sheaves, and the semiderived category. In the end, I will spell out the construction of the semi-tensor product operation on the semi-derived category of quasi-coherent torsion sheaves, making it a tensor triangulated category.
The derived category of a commutative local noetherian ring and the module category of a modular group algebra are tensor triangulated categories. A dualizable object in such a category is one that has a dual that is compatible with the tensor structure. The question that we address in this paper is whether the subcategory dualizable objects in certain co-local subcategories is the idempotent closure of image of the compact objects under the local cohomology functor associated to the subcategory. In this lecture, I will try to explain what all of these words mean, why one might care about such a question and how we get a negative answer is certain cases.
Recall that a noetherian ring R is regular if every finitely generated R-module has finite projective dimension. In a paper from 2009, Iacob and Iyengar characterize the regularity of R in terms of properties of (unbounded) R-complexes. Their proofs build on results of Jorgensen, Krause, and Neeman on compact generation of the homotopy categories of complexes of projective/injective/flat modules. In the commutative case, these results can be obtained with derived category methods in local algebra. I will illustrate how this is done by proving that the following conditions are equivalent for a commutative noetherian ring R:
1) R is regular.
2) Every complex of finitely generated projective R-modules is semi-projective.
3) Every complex of projective R-modules is semi-projective.
4) Every acyclic complex of projective R-modules is contractible.
The second condition is new, compared to the 2009 results, and relating it to the regularity of R is the novel part of the proof. This argument also plays a central role in the new proof of the corresponding results for complexes of injective modules and complexes of flat modules.
Polarised abelian surfaces vary in 3-dimensional families. In contrast, the derived category of an abelian surface A has a 6-dimensional space of deformations; moreover, based on general principles, one should expect to get 'algebraic families' of their categories over 4-dimensional bases. Generalized Kummer varieties (GKV) are hyperkähler varieties arising from moduli spaces of stable sheaves on abelian surfaces. Polarised GKVs have 4-dimensional moduli spaces, yet arise from moduli spaces of stable sheaves on abelian surfaces only over 3-dimensional subvarieties.
I present a construction that addresses both issues. We construct 4-dimensional families of categories that are deformations of Db(A) over an algebraic space. Moreover, each category admits a Bridgeland stability condition, and from the associated moduli spaces of stable objects one can obtain every general polarised GKV, for every possible polarisation type of GKVs. Our categories are obtained from ℤ/2-actions on derived categories of K3 surfaces.
A guiding principle of non-commutative algebraic geometry is that geometric objects (i.e. rings and schemes) are replaced by categories of modules/sheaves thereof. In order to keep track of the homological information, we actually take derived categories of such modules/sheaves. From this point of view, we are now interested in understanding typical geometric concepts directly in this categorical framework. A key example is given by deformations. In this talk, I will report on joint work with W. Lowen and M. Van den Bergh, where we attempt to define and study deformations categorically, in the framework of (enhanced) triangulated categories with a t-structure. This will also shed light on Hochschild cohomology.
We introduce tilting subcategories for arbitrary exact categories and discuss the question when one can get a bounded derived equivalence to a functor category over it.
This talk is concerned with generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic p when the Frobenius morphism is finite. It is shown that for any compact generator G of D(X), the e-th Frobenius pushforward of G classically generates the bounded derived category whenever pe is larger than the codepth of X, an invariant that is a measure of the singularity of X. From this, we can establish a canonical choice of strong generator when X is separated. The work is joint with Matthew R. Ballard, Srikanth B. Iyengar, Alapan Mukhopadhyay, and Josh Pollitz.
In the 1960's, Grothendieck showed that a commutative noetherian ring which admits a dualizing complex has finite Krull dimension. In 2018, Rickard showed that a finite-dimensional algebra A for which the localizing subcategory generated by the injective modules is equal to D(A) satisfies the finitistic dimension conjecture.
In this talk we explain how to view both of these results as special cases of a single result which is valid for any noncommutative noetherian ring which admits a dualizing complex.
You must be logged in to post a comment.