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.
The derived category, D(A), of the category Mod(A) of modules over a ring A is an important example of a triangulated category in algebra. It can be obtained as the homotopy category of the category Ch(A) of chain complexes of A-modules equipped with its standard model structure. One can view Ch(A) as the category Fun(Q, Mod(A)) of additive functors from a certain small preadditive category Q to Mod(A). The model structure on Ch(A) = Fun(Q, Mod(A)) is not inherited from a model structure on Mod(A) but arises instead from the "self-injectivity" of the special category Q. We will show that the functor category Fun(Q, Mod(A)) has two interesting model structures for many other self-injective small preadditive categories Q. These two model structures have the same weak equivalences, and the associated homotopy category is what we call the Q-shaped derived category of A. We will also show that it is possible to generalize the homology functors on Ch(A) to homology functors on Fun(Q, Mod(A)) for most self-injective small preadditive categories Q.
The operation of mutation has a long history in representation theory and algebraic geometry, be it in the context of exceptional collections of sheaves or in the combinatorial study of tilting modules. The aim is to create a new object from an old one by changing a designated part of it and keeping the other part. Here I discuss a variant in the context of cosilting objects in compactly generating triangulated categories (which are also known as derived injective cogenerators for t-structures of Grothendieck type). In that case, the operation of mutation corresponds to certain nice tilts of t-structures with respect to torsion pairs. Time permitting, I will explain how this is related to the lattice of torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra, as studied by Demonet, Iyama, Reading, Reiten and Thomas.
Enriques categories are characterized by the property that their Serre functor is the composition of an involutive autoequivalence and the shift by 2. The bounded derived category of an Enriques surface is an example of Enriques category. Other interesting examples are provided by the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids.
In this talk, we study moduli spaces of semistable objects in the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids with respect to Serre-invariant stability conditions. We provide a result of non-emptiness for these moduli spaces, by using the relation with certain moduli spaces on the associated K3 category.
Enriched Grothendieck categories naturally occur in algebraic geometry, where associated abelian categories rarely have projectives but have plenty of information encoded by enriched category theory. In this talk general properties of derived categories for Grothendieck categories of enriched functors and various recollements of such categories will be presented. Applications are given for Voevodsky's triangulated categories of motives.
Continuous actions of real reductive groups are often studied by first linearizing the action to spaces related to functions, then using algebra via Lie algebras and compact groups (cf. Gelfand, Harish-Chandra, Vogan). This paradigm essentially simplifies to the easier problem of studying a complex algebraic group K acting on flag varieties. K-orbit closures are important for representation theory, are generalizations of Schubert varieties, and certain properties are explicitly determined via equivariant resolutions of singularities. In joint work with Anna Romanov, we provide a geometric and algebraic categorification of the Lusztig-Vogan module using the equivariant derived category. Our methods allow us to compute cohomology of all fibres of resolutions constructed quite generally and generalize Soergel bimodule techniques from complex to real reductive algebraic groups.
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.
For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. We then explain several important roles of complexes of flat cotorsion modules and give some applications.
The space of Bridgeland stability conditions on a triangulated category is a complex manifold. We propose a compactification of the stability space via a continuous map to an infinite projective space. Under suitable conditions, we conjecture that the compactification is a real manifold with boundary, on which the action of the autoequivalence group of the category extends continuously. We focus on 2-Calabi-Yau categories associated to quivers, and prove our conjectures in the A2 and affine A1 cases.
In this talk I will introduce a cup-cap duality in the Koszul calculus of N-homogeneous algebras. As an application of this duality, it follows that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. I will also comment on a conceptual approach to this problem that may lead to a proof of the graded commutativity, based on derived categories in the framework of DG-algebras and DG-bimodules.
