This homepage contains the files of and links to some of my recent works (starting approximately from 2007).

1. My monograph on the semi-infinite homological algebra of associative algebraic structures has been published in Birkhaeuser`s series Monografie Matematyczne, vol.70 in 2010. An older version is available from the arXiv, but the book version is better a number of ways; most importantly, it cointains an Index and a List of Notation.

2. The monograph is complemented by our short paper with Roman Bezrukavni-kov titled "On the semi-infinite cohomology of finite-dimensional graded algebras", which was published by Compositio Math. in 2010. A slightly older version of this paper is available from the arXiv (see link).

This short paper suggests a categorical interpretation of the semi-infinite cohomology (but not homology) of finite-dimensional (only) algebras alternative to the one developed in the monograph.

3. In addition, my long paper "Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence" has been published by Memoirs of the AMS in 2011. A slightly older version is also available from the arXiv (see link).

This long paper is intended, in particular, to serve as an extended introduction to the monograph. Indeed, most of the content of this paper was invented much earlier than the "Homological algebra of semimodules..." stuff. Nevertheless, this paper contains many results not covered by "Homological algebra...", because it is written in the generality of DG- and CDG-modules, comodules, and contramodules, while the monograph deals with nondifferential semimodules and semicontramodules most of the time.

4. Moreover, an essentially final version of my paper with Alexander Polish-chuk titled "Hochschild (co)homology of the second kind I" is available from this homepage. The current version is dated May 3, 2011. The same version is also available from the arXiv. This paper has been accepted by Transaction of the AMS.

In this paper we study the Hochschild (co)homology of CDG-categories using the techniques developed in the previous long paper, together with some more elementary ideas about QDG-modules.

5. On top of that, a largely complete preliminary draft version of my paper "Coherent analogues of matrix factorizations and relative singularity categories" is now available from this homepage. The current version is dated December 20, 2011. A perhaps slightly older version of this paper is now also available from the arXiv (see link).

The aim of this paper is to provide an alternative proof of D.Orlov`s theorem connecting matrix factorizations over a nonaffine regular scheme with the triangulated category of singularities of the zero locus of the superpotential. In fact, Orlov also generalizes his result to singular schemes, and so do we; but our generalization is different from, and implies, his one. While his approach leads to a fully faithful functor, we obtain an equivalence of categories in the singular case. We also describe the image of Orlov`s fully faithful functor; and have a detailed discussion of the derived pull-backs and push-forwards of matrix factorizations, in the context of a general study of quasi-coherent CDG-modules and their derived categories of the second kind.

6. Furthermore, my paper "Mixed Artin-Tate motives with finite coefficients" has been published by the Moscow Mathematical Journal in 2011. A version of this paper is also available from the arXiv (see link).

The goal of this paper is to give an explicit description of the triangulated categories of Tate and Artin-Tate motives with finite coefficients Z/m over a field F containing a primitive m-root of unity as the derived categories of exact categories of filtered modules over the absolute Galois group of F with certain restrictions on the successive quotients. This description is conditional upon (and its validity is equivalent to) certain Koszulity hypotheses about the Milnor K-theory/Galois cohomology of F. This paper also purports to explain what it means for an arbitrary nonnegatively graded ring to be Koszul. Exact categories, silly filtrations, and the K(pi,1)-conjecture are discussed in the appendices. Tate motives with integral coefficients are considered in the "Conclusions" section.

7. As a complement to the latter paper, a largely complete preliminary draft version of my paper "Artin-Tate motivic sheaves with finite coefficients over an algebraic variety" is now available from this homepage. The current version is dated October 26, 2011. A slightly older version of this paper is now also available from the arXiv (see link).

In this paper we propose a construction of a tensor exact category F_X^m of Artin-Tate motivic sheaves with finite coefficients Z/m over an algebraic variety X (over a field K of characteristic prime to m) in terms of etale sheaves of Z/m-modules over X. Among the objects of F_X^m, in addition to the Tate motives Z/m(j), there are the cohomological relative motives with compact support of varieties over K quasi-finite over X. Assuming the existence of triangulated categories of motivic sheaves with coefficients Z/m over algebraic varieties over K and a weak version of the "six operations" in these categories, we identify F_X^m with the full exact subcategory of Artin-Tate motivic sheaves in the triangulated category DM(X,Z/m). For a smooth variety X, an isomorphism of the Z/m-modules Ext between the Tate motives Z/m(j) in F_X^m with the motivic cohomology modules predicted by the Beilinson-Lichtenbaum etale descent conjecture (proven by Voevodsky, Rost, et al.) holds if and only if a similar isomorphism holds for Artin-Tate motives over fields containing K. When K contains a primitive m-root of unity, the latter condition is equivalent to a certain Koszulity hypothesis, as shown in the previous paper (see above).

8. Besides, a largely complete version of my paper titled "Galois cohomology of a number field is Koszul" is now available from this homepage. The current version is dated July 31, 2010. A perhaps slightly older version of this paper is also available from the arXiv (see link).

In this paper we prove that the Milnor ring of any (one-dimensional) local or global field K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions that are only needed in the case l=2, we also prove various module Koszulity properties of this algebra. This provides evidence in support of Koszulity conjectures that were proposed in our previous papers. The proofs are based on the Class Field Theory and computations with quadratic commutative Groebner bases (commutative PBW-bases).

9. Also, an essentially complete version of my short note "The algebra of closed forms in a disk is Koszul" is now available from this homepage. The current version is dated June 21, 2011. A perhaps slightly older version of this note is now also available from the arXiv (see link). This note has been accepted by Functional Analysis and its Applications.

In this note we show that the algebra of closed differential forms in an (algebraic, formal, or analytic) disk with logarithmic singularities along several coordinate hyperplanes is (both topologically and nontopologically) Koszul. The connection with variations of mixed Hodge-Tate structures is discussed in the introduction.

10. Finally, a most incomplete draft version of my paper "Weakly curved A_infty-algebras over a topological local ring" is now available from this homepage. As of January 29, 2012, a short Abstract, Table of Contents, the Introduction, Sections 1-6, and the Appendix have been written up.

The aim of this paper is to define and study the derived categories of the first kind for modules over CDG- and curved A_infty-algebras complete over a pro-Artinian local ring with the curvature elements divisible by the maximal ideal of the local ring.

Quite independently of the above ten works, there are three most unfinished and incomplete manuscripts now available from this homepage. These are called

(i) "Nonhomogeneous Koszul duality and D-Omega" (the version of May 30, 2005; this is now largely superseded by section 0.4 and chapter 11 of the above monograph, appendix B to the above long paper on Koszul duality, and sections 6.2 and 7.4 of the above paper on Artin-Tate motives);

(ii) "Koszulity and t-structures" (the version of February 19, 2010; contains only the Abstract and the Introduction; this is now largely superseded by section 8 of the above paper on Artin-Tate motives); and

(iii) "Two exercises in derived Koszul duality" (the version of January 15, 2011; contains only the Introduction, Table of Contents, and Sections 1-2. The two originally intended exercises were about Artin-Schelter Gorenstein coalgebras and Golod-type properties of homomorphisms of cocommutative conilpotent coalgebras).

The latter three manuscripts contain only small fractions of the material that was originally intented to be included into them.

Contact information: my e-mail is posic at mccme dot ru or positselski at yandex dot ru, my blog (in Russian) is posic.livejournal.com