arXiv subjects
Publications and source records attributed to Henning Krause.
These are the notes from lectures I gave at the Oberwolfach Seminar "Tensor Triangular Geometry and Interactions" which was held in October 2025. The aim of these notes is twofold: We develop notions of support for triangulated categories, and we apply them to classify thick and localising tensor ideals of categories that arise in modular representation theory of finite groups.
A self-contained introduction to infinite dimensional representations over a tame hereditary algebra is provided, assuming a basic knowledge of the category of finite dimensional representations. This includes a complete description of all pure-injective modules. Of particular interest are the torsionfree divisible modules, which are precisely the direct sums of copies of the unique generic module.
For any length category, we establish a set of rules (necessary and sufficient) that ensure a partial order on the isomorphism classes of simple objects such that the category is equivalent to the category of finite dimensional representations of this partially ordered set. Equivalently, we characterise the length categories that arise as categories of modules over a sheaf of division rings on a finite $T_0$-space.
For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences, including Catalan and large Schr\"oder numbers. Surprisingly, in some cases new integer sequences arise. The formulation of this counting problem leads to a universal construction which assigns to any poset a finitely cocomplete additive category; it is abelian when the poset is finite and does not depend on the choice of any ring of coefficients. For a general poset the universal category of representations is abelian if and only if for the lattice of ideals the meet of two compact elements is again compact.
We survey the main results in Jeremy Rickard's seminal papers `Morita theory for derived categories' and `Derived equivalences and derived functors'. These papers catalysed the later development of the Morita theory of (enhanced) compactly generated triangulated categories by Keller in the algebraic setting and by Schwede and Shipley in the topological setting. We also discuss the role of Rickard's notion of splendid equivalence in the context of Brou\'e's abelian defect group conjecture, and indicate an alternative proof of parts of Rickard's Derived Morita Theorem that leverages the notion of completion of a triangulated category.
For a point $\mathfrak{p}$ in the spectrum of the cohomology ring of a finite group $G$ over a field $k$, we calculate the spectrum for the subcategory of dualisable objects inside the tensor triangulated category of $\mathfrak{p}$-local and $\mathfrak{p}$-torsion objects in the (big) stable module category of the group algebra $kG$.
It is shown that any localisation of triangulated categories induces (up to an equivalence) a localisation of abelian categories when one passes to their abelianisations. From this one obtains for any enlargement of Grothendieck universes an example of an abelian category and a Serre subcategory within the smaller universe such that the corresponding quotient does only exist within the bigger universe. The second part of this note provides an analogue for the abelian hull of an arbitrary category.
A notion of support for objects in any Grothendieck category is introduced. This is based on the spectral category of a Grothendieck category and uses its Boolean lattice of localising subcategories. The support provides a classification of all subcategories that are closed under arbitrary coproducts, subobjects, and essential extensions. There is also a notion of exact support which classifies certain thick subcategories. As an application, the coproduct decompositions of objects are described in terms of Boolean lattices. Also, for any ring Crawley-Boevey's correspondence between definable subcategories of modules and closed subsets of the Ziegler spectrum is extended.
The finitistic dimension of a triangulated category is introduced. For the category of perfect complexes over a ring it is shown that this dimension is finite if and only if the small finitistic dimension of the ring is finite.
It is shown that a quiver is left noetherian if and only if the category of quiver representations in any locally noetherian abelian category is again locally noetherian. Here, locally noetherian means that any object is the directed union of its noetherian subobjects. For a quiver to be left noetherian means that the left ideals of paths starting at any fixed vertex satisfy the ascending chain condition. The proof generalises to representations of any small category that admits a Gröbner enrichment.
For an abelian category and a distinguished object with a graded endomorphism ring a necessary and sufficient criterion is given so that the category is equivalent to the abelian quotient of the category of finitely presented graded modules modulo the Serre subcategory of finite length modules. A particular example is the category of coherent sheaves on a projective variety, following a theorem of Serre from 1955. The proof uses Auslander's theory of coherent functors, and there are no noetherianess assumptions. A theorem of Lenzing for representations of hereditary algebras is given as an application.
Matlis duality for modules over commutative rings gives rise to the notion of Matlis reflexivity. It is shown that Matlis reflexive modules form a Krull-Schmidt category. For noetherian rings the absence of infinite direct sums is a characteristic feature of Matlis reflexivity. This leads to a discussion of objects that are extensions of artinian by noetherian objects. Classifications of Matlis reflexive modules are provided for some small examples.
This work concerns the stable module category of a finite group over a field of characteristic dividing the group order. The minimal localising tensor ideals correspond to the non-maximal homogeneous prime ideals in the cohomology ring of the group. Given such a prime ideal, a number of characterisations of the dualisable objects in the corresponding tensor ideal are given. One characterisation of interest is that they are exactly the modules whose restriction along a corresponding $π$-point are finite dimensional plus projective. A key insight is the identification of a special property of the stable module category that controls the cohomological behaviour of local dualisable objects. This property, introduced in this work for general triangulated categories and called local regularity, is related to strong generation. A major part of the paper is devoted to developing this notion and investigating its ramifications for various special classes of objects in tensor triangulated categories.
These notes for a master class at Aarhus University (March 22--24, 2023) provide an introduction to the theory of completion for triangulated categories.
This work concerns representations of a finite flat group scheme $G$, defined over a noetherian commutative ring $R$. The focus is on lattices, namely, finitely generated $G$-modules that are projective as $R$-modules, and on the full subcategory of all $G$-modules projective over $R$ generated by the lattices. The stable category of such $G$-modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of $G$. Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.
The notion of support provides an analogue of Stone duality, relating lattices to topological spaces. This note aims to explain in lattice theoretic terms what has been developed in the context of triangulated categories. In particular, the parallel between support via closed and open sets is addressed in terms of Hochster duality. As an application we indicate some consequences for tensor exact categories.
In the recent paper "The Nakayama functor and its completion for Gorenstein algebras", a class of Gorenstein algebras over commutative noetherian rings was introduced, and duality theorems for various categories of representations were established. The manuscript on hand provides more context to the results presented in the aforementioned work, identifies new classes of Gorenstein algebras, and explores their behaviour under standard operations like taking tensor products and tilting.
We discuss dualisable objects in minimal subcategories of compactly generated tensor triangulated categories, paying special attention to the derived category of a commutative noetherian ring. A cohomological criterion for detecting these local dualisable objects is established. Generalisations to other related contexts are discussed.