Homological Theory of Exact Categories
This page is describing my habilitation project (version from April 2025).
This thesis is aiming at a reader already familiar with homological algebra for abelian categories who is open-minded to widen his perspective to exact categories. Here is the complete file, 195p (compact layout)
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Why exact categories? pdf
I explain my choice to study exact categories.
Chapter 1: Homologically exact functors pdf
This chapter should be seen as a gentle introduction to homologically exact functors - but we aim at an audience which has seen homological algebra for abelian categories.
What is new? Some Lemmata and the study of exact structures making an additive functor exact are not often considered. It does not seem to be known that a result of Rump shows that there is always a unique maximal one.
Chapter 2: The poset of exact structures pdf
We survey the theory of exact structures on an essentially small idempotent complete additive category. We focus on explicit answers and examples. But we also collect/recall several lattice isomorphisms for the lattice of all exact structures. Several of these isomorphisms are induced by equivalences of 2-categories which we collect in an Appendix.
What is new here? The description of exact structures with enough projectives. The equivalence of 2-categories with tf-Auslander categories (i.e. the subcategory of torsionfree objects in the Auslander exact category) is new. Apart from the Auslander correspondence none of the equivalences of 2-categories are formulated as such in the literature that I know (we treat them for time reasons also somewhat sketchy). Furthermore, we look at all exact substructures in examples (e.g. finitely generated abelian groups) and establish some of their global dimensions.
Chapter 3: The poset of exact subcategories pdf
This is the quest to extend known descriptions of the lattice of exact structures on a given additive category to the much bigger lattice of all exact subcategories. What is new? This question is usually not considered, so everything in this chapter.
Chapter 4: Exact categories represented by morphisms pdf
For every module over a ring there is the associated module category over the endomorphism ring, i.e. a construction of a new module category over a ring.
This has a generalization to exact categories by looking at functors represented by admissible morphisms. Interestingly, we can also choose other classes of morphisms (for example deflations or inflations) and still obtain an exact category. Of course this topic deserves a systematic study that I can not give (due to time constraints). We have already seen in Chapter 2, the Auslander exact category as an instance of this construction, we explain its generalizations which lead to the generality of a contravariantly finite generator M in an arbitrary exact category. We give a short history of ideas in the next section.
Furthermore, we have an exact category with enough projectives always presented by its admissible morphism between the projectives. Tilting subcategories are a generalization of the subcategory of projectives and we start using this construction in tilting theory for exact categories (cf. Chapter 10).
What is new? Functor categories represented by (general) classes of morphisms (in the literature you find either admissible morphisms or deflations). The generator correspondence for exact categories.
Chapter 5: On faithfully balancedness in functor categories pdf
This is a generalization of some results of Ma-Sauter from module categories over artin algebras to more general functor categories (and partly to exact categories). In particular, we generalize the definition of a faithfully balanced module to a faithfully balanced subcategory and find the generalizations of dualities and characterizations from Ma-Sauter. What is new? This generality is new but the results for artin algebras can be found in the joint work.
Chapter 6: Derived categories and functors for exact categories pdf
By now, derived categories and derived functors for abelian categories are standard topics in a course on homological algebra. This is an introduction to derived categories of exact categories assuming that the reader is familiar with the theory for abelian categories.
Our treatment of derived functors is only shortly summarizing the results in Kellers treatment of the same.
What is new? We characterize when derived categories of exact categories are locally small (i.e. Hom-classes are sets)- this is a joint result with Omar Gomez.
Chapter 7: Tilting theory in exact categories pdf
We define tilting subcategories in arbitrary exact categories to achieve the following. Firstly: Unify existing definitions of tilting subcategories to arbitrary exact categories. We discuss standard results for tilting subcategories: Auslander correspondence, Bazzoni description of the perpendicular category.
Secondly: We treat the question when a tilting subcategory induces a derived equivalence to a the functor category with enough projectives given by the tilting subcategory. We prove a generalization of Miyashita's theorem (which is itself a generalization of a well-known theorem of Brenner-Butler) and characterize exact categories with enough projectives such that the before mentioned triangle equivalence can be found.
Chapter 8: An application of tilting theory to infinite type A quiver representations pdf
We show that there are precisely three derived equivalence classes (of finitely presented A-infty quiver representations) for different choices of orientations.
Chapter 9: Realization functors in algebraic triangulated categories (jt. J. Letz) pdf
Let T be an algebraic triangulated category and C an extension-closed, non-negative subcategory. Then c has an exact structure induced from exact triangles in T. Keller and Vossieck state that there is a triangle functor from the bounded derived functor of C to T extending the inclusion.
What is new? We provide the missing details for a complete proof.
Chapter 10: Fragments of derived Morita theory for exact categories with enough projectives pdf
Instead of just tilting subcategories we are looking at tilting subcategories together with a class of morphisms (think of the admissible morphisms from before). This is necessary if we think of the admissibly represented functors as the replacement of the endomorphism ring.
Chapter 11: Noncommutative resolutions of singularities using exact substructures pdf
We introduce (bounded) singularity categories for arbitrary exact categories. An exact category is regular if its singularity category is zero. We recall the known Buchweitz theorem for a Gorenstein exact categories with enough projectives. Then we explore a new concept of a \emph{noncommutative resolution of singularities} (NCR) of a given exact category as an exact substructure which is regular.
There exist various alternative versions of non-commutative resolutions in the literature. Our aims here are:
(1) Partially unify and simplify the theory (singularity categories, non-commutative resolutions of singularities and relative singularity categories) for module categories of rings and for coherent sheaves on a quasi-projective variety. (2)Characterize NCRs corresponding to cluster tilting subcategories (as a candidate for a 'minimal' NCR). What is new? The concept to see NCRs as exact substructures and the generality of our approach.
Chapter 12: The Yoneda category and effaceable functors pdf
For an exact category we introduce its Yoneda category and the category of Yoneda effaceables.
The category of Yoneda effaceables is a Frobenius category. We show that there is a triangle equivalence between the bounded derived category of the effaceable functors and the stable Yoneda effaceables.
As an application, we show that the 2-functor assigning to an exact category its effaceable functors is preserving homological exactness.
What is new? The main result is new in this generality but known for finite-dimensional modules over finite-dimensional algebras.
