Exact Category Research Articles

One-sided exact categories

One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller’s axioms defining Quillen’s exact categories. We study one-sided exact additive categories and a stronger version defined by adding the one-sided part of Quillen’s “obscure axiom”. We show that some homological results, such as the Short Five Lemma and the 3×3 Lemma, can be proved in our context. We also note that the derived category of a one-sided exact additive category can be constructed.

K1 of Exact Categories by Mirror Image Sequences

AbstractWe establish a presentation for K1 of any small exact category P, based on the notion of “mirror image sequence,” originally introduced by Grayson in 1979; as part of the proof, we show that every element of K1(P) arises from a mirror image sequence. This provides an alternative to Nenashev's presentation in terms of “double short exact sequences.”

Sato Grassmannians for generalized Tate spaces

We generalize the concept of Sato Grassmannians of locally linearly compact topological vector spaces (Tate spaces) to the Beilinson category of the “locally compact objects”, or Generalized Tate Spaces, of an exact category. This allows us to extend the Kapranov dimensional torsor Dim and determinantal gerbe Det to generalized Tate spaces and unify their treatment in the determinantal torsor. We then introduce a class of exact categories, that we call partially abelian exact, and prove that if the base category is so, then Dim and Det are multiplicative in admissible short exact sequences of generalized Tate spaces. We then give a cohomological interpretation of these results in terms of the Waldhausen K-theoretical space of the Beilinson category. Our approach can be iterated and we define analogous concepts for the successive categories of $n$-dimensional (generalized) Tate spaces. In particular we show that the category of Tate spaces is partially abelian exact, so we can extend the results for Dim and Det obtained for 1-Tate spaces to 2-Tate spaces, and provide a new interpretation in the context of algebraic $K$-theory of results of Kapranov, Arkhipov-Kremnizer and Frenkel-Zhu.

Algebraic 𝐾-theory via binary complexes

Motivated by work of Nenashev on K 1 K_1 , we introduce acyclic binary multicomplexes and use them to provide generators and relations for the Quillen K K -groups of an arbitrary exact category.

LOCALLY COMPACT OBJECTS IN EXACT CATEGORIES

We identify two categories of locally compact objects on an exact category [Formula: see text]. They correspond to the well-known constructions of the Beilinson category [Formula: see text] and the Kato category [Formula: see text]. We study their mutual relations and compare the two constructions. We prove that [Formula: see text] is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants, and study the exact structure of the category [Formula: see text] of Tate spaces. It is natural therefore to consider the Beilinson category [Formula: see text] as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories [Formula: see text], [Formula: see text] of countably indexed ind/pro-objects over any category [Formula: see text] can be described as localizations of categories of diagrams over [Formula: see text].

On exact categories and applications to triangulated adjoints and model structures

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, Jørgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.

Model structures on exact categories

We define model structures on exact categories, which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly idempotent complete, we get Hovey’s one-to-one correspondence between model structures and complete cotorsion pairs. We classify the right and the left homotopy relation in terms of the cotorsion pairs and look at examples of exact model structures. In particular, we see that given any hereditary abelian model category, the full subcategories of cofibrant, fibrant and cofibrant–fibrant subobjects each has natural exact model structures equivalent to the original model structure. These model structures each has interesting characteristics. For example, the cofibrant–fibrant subobjects form a Frobenius category, whose stable category is the same as the homotopy category of its model structure.

Mixed Artin–Tate Motives with Finite Coefficients

The goal of this paper is to give an explicit description of the triangulated categories of Tate and Artin{Tate motives with nite coecients Z=m over a eld K containing a primitive m-root of unity as the derived categories of exact categories of ltered modules over the absolute Galois group of K with certain restrictions on the successive quotients. This description is conditional upon (and its validity is equiv- alent to) certain Koszulity hypotheses about the Milnor K-theory/Galois cohomology of K. This paper also purports to explain what it means for an arbitrary nonnegatively graded ring to be Koszul. Tate motives with integral coecients are discussed in the \Conclusions section. 2010 Mathematics Subject Classification. 12G05, 19D45, 16S37. coecients, Milnor{Bloch{Kato conjecture, Beilinson{Lichtenbaum con- jecture, K(; 1) conjecture, Koszulity conjecture, Koszul algebras, nonat Koszul rings, silly ltrations.

Three results on Frobenius categories

This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an extension-closed exact subcategory of the Frobenius category formed by Cohen-Macaulay modules over some additive category; this is an analogue of Gabriel-Quillen's embedding theorem for Frobenius categories; (3) we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure, with which the given category becomes Frobenius. Several applications of the results are discussed.

Hermitian K-theory of exact categories

AbstractWe study the theory of higher Grothendieck-Witt groups, alias algebraic hermitian K-theory, of symmetric bilinear forms in exact categories, and prove additivity, cofinality, dévissage and localization theorems – preparing the ground for the theory of higher Grothendieck-Witt groups of schemes as developed in [Sch08a] and [Sch08b]. No assumption on the characteristic is being made.

The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

We prove localization and Zariski-Mayer-Vietoris for higher Gro-thendieck-Witt groups, alias hermitian K-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be singular. Along the way, we prove Additivity, Fibration and Approximation theorems for the hermitian K-theory of exact categories with weak equivalences and duality.

A category of association schemes

We define a category of association schemes and investigate its basic properties. We characterize monomorphisms and epimorphisms in our category. The category is not balanced. The category has kernels, cokernels, and epimorphic images. The category is not an exact category, but we consider exact sequences. Finally, we consider a full subcategory of our category and show that it is equivalent to the category of finite groups.

An Algebraic Proof of Quillen's Resolution Theorem for K1

AbstractIn his 1973 paper [4] Quillen proved a resolution theorem for the K-Theory of an exact category; his proof was homotopic in nature. By using the main result of Nenashev's paper [3], we are able to give an algebraic proof of Quillen's Resolution Theorem for K1 of an exact category. We view this as an advance towards the goal of giving an essentially algebraic subject an algebraic foundation.

Harder–Narasimhan categories

Semistability and the Harder–Narasimhan filtration are important notions in algebraic and arithmetic geometry. Although these notions are associated to mathematical objects of quite different natures, their definition and the proofs of their existence are quite similar. We propose in this article a generalization of Quillen’s exact category and we discuss conditions on such categories under which one can define the notion of the Harder–Narasimhan filtrations and establish its functoriality.

Construction fonctorielle de catégories de Frobenius

Let A , B be exact categories with A karoubian and M be an exact functor. Under suitable adjunction hypotheses for M, we are able to show that the direct factors of the objects of A of the form MY with Y ∈ B make up a Frobenius category which allows us to define an M-stable category for A only by quotienting. In addition, we propose a construction of an M-stable category for A , B triangulated categories and M a triangulated functor. We illustrate this notion with a theorem of Keller and Vossieck (1987) which links the two notions of M-stable category. To cite this article: V. Beck, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Exact categories

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 × 3 -lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel–Quillen embedding theorem in an appendix.

Three extensional models of type theory

We compare three categorical models of type theory with extensional constructs: setoids over extensional type theory; setoids over intensional type theory and a certain free exact category (the free ‘ΠW-pretopos’). By studying the amount of choice available in these categories, we are able show that they are distinct.

Adjoint Functors and Triangulated Categories

We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These categories naturally fit into a framework of relative derived categories, and once we prove that there are decent resolutions of complexes, we are able to prove many familiar results in homological algebra.

A derived functor approach to bounded cohomology

We apply the theory of the derived category of exact categories to the category G – Ban of Banach modules over the discrete group G. Since there are enough injectives in G – Ban , right derived functors exist. The heart of the canonical t-structure on the derived category D ( Ban ) is equivalent to Waelbroeck's Abelian category qBan of quotient Banach spaces. The right derived functor of the functor “submodule of G-invariant vectors” yields a universal δ-functor with values in qBan which allows us to reconstruct the bounded cohomology functors in the sense of Gromov–Brooks–Ivanov–Noskov. To cite this article: T. Bühler, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Sheaves on moment graphs and a localization of Verma flags

To any moment graph G we assign a subcategory V of the category of sheaves on G together with an exact structure. We show that in the case that the graph is associated to a non-critical block of the equivariant category O over a symmetrizable Kac–Moody algebra, V is equivalent (as an exact category) to the subcategory of modules that admit a Verma flag. The projective modules correspond under this equivalence to the intersection cohomology sheaves on the graph, and hence, by a theorem of Braden and MacPherson, to the equivariant intersection cohomologies of Schubert varieties associated to Kac–Moody groups.