Jan Šťovíček

On exact categories and applications to triangulated adjoints and model structures

Abstract We show that Quillenʼs small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, Jorgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.

All tilting modules are of finite type

We prove that any infinitely generated tilting module is of finite type, namely that its associated tilting class is the Ext-orthogonal of a set of modules possessing a projective resolution consisting of finitely generated projective modules.

Tilting, cotilting, and spectra of commutative noetherian rings

We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated modules of bounded projective dimension are classified. We also relate our results to Hochsters conjecture on the existence of finitely generated maximal Cohen-Macaulay modules.

All n-cotilting modules are pure-injective

We prove that all n-cotilting R-modules are pure-injective for any ring R and any n > 0. To achieve this, we prove that ⊥ 1U is a covering class whenever U is an R-module such that ⊥1 U is closed under products and pure submodules.

Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves

Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations. We describe the relation in terms of pointed modules. We present methods for producing superdecomposable pure-injectives and give some details of recent work of Harland doing this in the context of tubular algebras. 2010 Mathematics Subject Classification. Primary 16G20; 16G60; 03C60.We propose an approach to Geiss-Leclerc-Schroers conjecture on the cluster algebra structure on the coordinate ring of a unipotent subgroup and the dual canonical base. It is based on singular supports of perverse sheaves on the space of representations of a quiver, which give the canonical base.The article covers developments in the representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and theories that ultimately grew out of that result. This includes the theory of one-parameter subgroups and rank varieties for infinitesimal group schemes; the

The telescope conjecture for hereditary rings via Ext-orthogonal pairs

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Derived equivalences induced by big cotilting modules

-points and

On compactly generated torsion pairs and the classification of co--structures for commutative noetherian rings

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Torsion classes, wide subcategories and localisations

-support spaces for finite group schemes, modules of constant rank and constant Jordan type, and construction of bundles on projective varieties associated with cohomology ring of an infinitesimal group scheme

The countable Telescope Conjecture for module categories

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