Homological Algebra of Noncommutative 'Spaces' I Introduction A considerable part of this manuscript is based on the notes of a lecture course in noncommutative algebraic geometry given at Kansas State University during the Fall of 2005 and Spring of 2006 and on (the second half of) my lectures delivered at the School on Algebraic K-Theory and Applications which took place at the International Center for Theoretical Physics (ICTP) in Trieste during the last two weeks of May of 2007 (see [R2], or [R3]). The starting point of the actual lectures in Kansas was the homological algebra of exact categories as it is viewed by Keller and Vossieck [KeV]. Besides an optimization of the Quillen's definition of an exact category, they observed that the stable categories of exact categories with enough injectives have a suspension functor and triangles whose properties give a 'one-sided' version of Verdier's triangulated category, which they call a suspended category. A short account on this subject is given in Appendix K. The main body of the text reflects attempts to find natural frameworks for fundamen- tal homological theories which appear in noncommutative algebraic geometry. The first move in this direction is the replacing exact categories with a much wider class of right exact categories. These are categories endowed with a Grothendieck pretopology whose covers are strict epimorphisms. The dual structures, left exact categories, appear naturally and play a crucial role in a version of K-theory sketched in Section 7 of this work. Sections 1 and 2 contain generalities on right exact categories. In Section 1, we intro- duce right exact (not necessarily additive) categories and sketch their basic properties. We define Karoubian right exact categories and prove the existence (under certain conditions) of the Karoubian envelope of a right exact category. We observe that any k-linear right exact category is canonically realized as a subcategory of an exact k-linear category { its exact envelope. In Section 2, we consider right exact categories with initial objects. The existence of initial (resp. final) objects allows to introduce the notion of the kernel (resp. cokernel) of a morphism. Most of the section is devoted to some elementary properties of the kernels of morphisms, which are well known in the abelian case. Section 3 is dedicated to satellites on right exact categories. Its content might be regarded as a non-abelian and non-additive (that is not necessarily abelian or additive) version of the classical theory of derived functors. We introduce @∗-functors and prove the existence of the universal @∗-functors on a given right exact 'space' with values in categories with kernels of morphisms and limits of filtered diagrams. We establish the existence of a universal 'exact' @∗-functor on a given right exact 'space'. The latter subject naturally leads to a general notion of the costable category of a right exact category, which appears in Section 4. We obtain (by turning properties of costable categories into axioms) the notion of a (not necessarily additive) cosuspended category. We introduce the notion of a homological functor on a cosuspended category and prove the existence of a universal homological functor. 1 New Prairie Press, 2014 In Section 5, we introduce projective objects of right exact categories (and injective objects of left exact categories). They play approximately the same role as in the classical case: every universal @∗-functor annihilates pointable projectives (we call this way projec- tives which have morphisms to initial objects); and if the right exact category has enough projectives, then every 'exact' @∗-functor which annihilates all projectives is universal. Starting from Section 6, a (noncommutative) geometric flavor becomes a part of the picture: we interpret svelte right exact categories as dual objects to (noncommutative) right exact 'spaces' and 'exact' functors between them as inverse image functors of morphisms of 'spaces'. We introduce a natural left exact structure on the category of right exact 'spaces'. Inverse image functors of its inflations are certain localizations functors. Section 7 is dedicated to the first applications: the universal K-theory of right exact 'spaces'. We define the functor K0 and then introduce higher K-functors as satellites of ∗ K0. More precisely, the K-functor appears as a universal contravariant @ -functor on a left exact category over the left exact category of right exact 'spaces'. Here `over' means an 'exact' functor to the category left exact 'spaces'. Our K-functor has exactness properties which are expessed by the long 'exact' sequences corresponding to those 'exact' localiza- tions which are inflations. In the abelian case, every exact localization is an inflation. Quillen's localization theorem states that the restriction of his K-functor to abelian cat- egories has a natural structure of a an 'exact' @-functor. It follows from the universality of the K-theory defined here, that there exists a unique morphism from the Quillen's K- Q a functor K• to the universal K-functor K• defined on the left exact category of 'spaces' represented by abelian categories. In Section 8, we introduce infinitesimal 'spaces'. We establish some general facts about satellites and then, as an application, obtain the devis- sage theorem in K-theory. It is worth to mention that infinitesimal 'spaces' is a serious issue in noncommutative (and commutative) geometry: they serve as a base of a noncom- mutative version of Grothendieck-Berthelot crystalline theory and are of big importance for the D-module theory on noncommutative 'spaces'. We make here only a very little use of them leaving a more ample development to consequent papers. The remaining five sections appear under the general title \complementary facts". In Section C1 (which complements Section 3), we look at some examples, which acquire importance somewhere in the text. In Section C2, we pay tribute to standard techniques of homological algebra by expanding the most popular facts on diagram chasing to right exact categories. They appear here mainly as a curiosity and are used only once in the main body of the manuscript. Section C3 is dedicated to localizations of exact and (co)suspended cat- egories. In particular, t-structures of (co)suspended categories appear on the scene. Again, a work by Keller and Vossieck, [KV1], suggested the notions. Section C4 is dedicated to cohomological functors on suspended categories and can be regarded as a natural next step after the works [KeV] and [Ke1]. It is heavily relied on Appendix K, where the basic facts on exact and suspended categories are gathered, following the approach of B. Keller and D. Vossieck [KeV], [KV1], [Ke2], except for some complements and most of proofs, which are made more relevant to the rest of the work. We consider cohomological func- tors on suspended categories with values in exact categories and prove the existence of a universal cohomological functor. The construction of the universal functor gives, among other consequences, an equivalence between the bicategory of Karoubian suspended svelte 2 categories with triangle functors as 1-morphisms and the bicategory of exact svelte Z+- categories with enough injectives whose 1-morphisms are 'exact' functors. We show that if the suspended category is triangulated, then the universal cohomological functor takes val- ues in an abelian category, and our construction recovers the abelianization of triangulated categories by Verdier [Ve2]. It is also observed that the triangulation of suspended cate- gories induces an abelianization of the corresponding exact Z+-categories. We conclude with a discussion of homological dimension and resolutions of suspended categories and exact categories with enough injectives. These resolutions suggest that the 'right' objects to consider from the very beginning are exact (resp. abelian) and (co)suspended (resp. n triangulated) Z+-categories. All the previously discussed facts (including the content of Appendix K) extend easily to this setting. In Section C5, we define the weak costable cat- egory of a right exact category as the localization of the right exact category at a certain class of arrows related with its projectives. If the right exact category in question is exact, then its costable category is isomorphic to the costable category in the conventional sense (reminded in Appendix K). If a right exact category has enough pointable projectives (in which case all its projectives are pointable), then its weak costable category is naturally equivalent to the costable category of this right exact category defined in Section 4. We study right exact categories of modules over monads and associated stable and costable categories. The general constructions acquire here a concrete shape. We introduce the notion of a Frobenius monad. The category of modules over a Frobenius monad is a Frobe- nius category, hence its stable category is triangulated. We consider the case of modules over an augmented monad which includes as special cases most of standard homological algebra based on complexes and their homotopy and derived categories. A large part of this manuscript was written during my visiting Max Planck Institut f¨urMathematik in Bonn and IHES. I would like to express my gratitude for hospitality and excellent working conditions. 3 1. Right exact categories. 1.1. Right exact categories and (right) 'exact' functors. We define a right exact category as a pair (CX ; EX ), where CX is a category and EX is a pretopology on CX whose covers are strict epimorphisms; that is for any element M −! L of E ({ a cover), −! the diagram M ×L M −! M −! L is exact. This requirement means precisely that the pretopology EX is subcanonical; i.e. every representable presheaf is a sheaf. We call the elements of EX deflations and assume that all isomorphisms are deflations. 1.1.1. The coarsest and the finest right exact structures. The coarsest right exact structure on a category CX is the discrete pretopology: the class of deflations coin- cides with the class Iso(CX ) of all isomorphisms of the category CX .