Leonid Positselski's page

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 was 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 Bezrukavnikov 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" was published by Memoirs of the AMS in 2011. A new November 2014 postpublication version is 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, my paper with Alexander Polishchuk titled "Hochschild (co)homology of the second kind I" was published by Transaction of the AMS in 2012. A slightly older version of this paper is available from the arXiv (see link).

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, my paper with Alexander Efimov titled "Coherent analogues of matrix factorizations and relative singularity categories" was published by Algebra and Number Theory in 2015. A version of this paper is 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. In Appendix B we compute the Hochschild (co)homology of matrix factorization categories.

6. Furthermore, my paper "Mixed Artin-Tate motives with finite coefficients" was 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 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 January 21, 2015. A slightly older version of this paper is 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, my paper titled "Galois cohomology of a number field is Koszul" has been published by Journal of Number Theory in 2014. 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, my short paper titled "The algebra of closed forms in a disk is Koszul" was published by Functional Analysis and its Applications in 2012. A perhaps slightly older version is also available from the arXiv (see link).

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. To continue, a preliminary draft version of my paper "Weakly curved A_infty-algebras over a topological local ring" is now available from this homepage. A slightly older version of this paper is now also available from the arXiv (see link).

As of March 13, 2015, a short Abstract, Table of Contents, the Introduction, Sections 1-7, and Appendices A-B 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.

11. Finally, an incomplete preliminary draft version of my paper "Contraherent cosheaves" is now available from this homepage. A slightly older version of this paper is now also available from the arXiv (see link).

As of October 28, 2015, the Abstact, Table of Contents, Introduction, Sections 1-5, and Appendices A-D have been written up.

Contraherent cosheaves are globalizations of cotorsion (or similar) modules over commutative rings obtained by gluing together over a scheme. The category of contraherent cosheaves over a scheme is an exact category with exact functors of infinite product; over a quasi-compact semi-separated scheme, it also has enough projectives. In this paper we intend to construct the derived co-contra correspondence, meaning an equivalence between appropriate derived categories of quasi-coherent sheaves and contraherent cosheaves, over a quasi-compact semi-separated scheme and, in a different form, over a semi-separated Noetherian scheme with a dualizing complex. The former point of view should allows us to obtain a new construction of the extraordinary inverse image functor f^! for a morphism of quasi-compact semi-separated schemes f. The latter approach would provide an extended version of the covariant Serre-Grothendieck duality theory.

12. And very-very finally, an incomplete preliminary draft version of my new paper "Categorical Bockstein sequences" is now available from this homepage. The current version is dated January 15, 2015. A perhaps slightly older version of this paper is now also available from the arXiv (see link).

We construct the reduction of an exact category with a twist functor with respect to an element of its graded center in presence of an exact-conservative forgetful functor annihilating this central element. The construction uses matrix factorizations in a nontraditional way. We obtain the Bockstein long exact sequences for the Ext groups in the exact categories produced by reduction. Our motivation comes from the theory of Artin--Tate motives and motivic sheaves with finite coefficients, and our key techniques generalize those of Section 4 of the "Mixed Artin-Tate motives ..." paper.

13. And at last, just to finish with it, a preliminary draft version of my overview paper titled "Contramodules" is now available from this homepage. As of April 16, 2015, a short Abstract, a moderate-length Introduction, and Sections 1-3 have been written up. A perhaps slightly older version of this paper is now also available from the arXiv (see link).

Quite independently of the above thirteen works, there are two most unfinished and incomplete manuscripts 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); and

(ii) "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 two 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

Semimodules, Book Announcement on Springer's site

Semimodules on Amazon.com

Semimodules, Link to arXiv

Semi-Infinite Finite-Dimensional, paper in Compositio

Semi-Infinite Finite-Dimensional, Link to arXiv

Koszul Duality, Link to the Memoir

Koszul Duality, Link to AMS Bookstore

Koszul Duality, Link to arXiv

Hochschild, paper in Transactions

Hochschild, Link to arXiv

Matrix Factorizations, paper in ANT

Matrix Factorizations, Link to arXiv

Artin-Tate Motives, Link to MMJ issue

Artin-Tate Motives, Link to arXiv

Motivic Sheaves, Postscript file (880Kb, 33 pages)

Motivic Sheaves, Link to arXiv

Number Field, paper in JNT

Number Field, Link to arXiv

Closed Forms, Link to FAA issue (in Russian)

Closed Forms, Link to arXiv

Curved A_infty, Gzipped Postscript file (740Kb, 169 pages)

Curved A_infty, Link to arXiv

Contraherent, Pdf file (1.4Mb, 255 pages)

Contraherent, Link to arXiv

Bockstein, Postscript file (1Mb, 53 pages)

Bockstein, Link to arXiv

Contramodules, Pdf file (690Kb, 82 pages)

Contramodules, Link to arXiv

D-Ω, Postscript file (350Kb, 11 pages)

Two Exercises, Postscript file (400Kb, 19 pages)

My homepage on the site of my research institute

My homepage on the site of my university

Latest CV with Research Summary, Pdf file (350Kb, 19 pages)

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