
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 semiinfinite 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 semiinfinite cohomology of finitedimensional 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 semiinfinite cohomology (but not homology) of finitedimensional (only) algebras alternative to the one developed in the monograph.
3. In addition, my long paper "Two kinds of derived categories, Koszul duality, and comodulecontramodule 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 CDGmodules, 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 CDGcategories using the techniques developed in the previous long paper, together with some more elementary ideas about QDGmodules.
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 pullbacks and pushforwards of matrix factorizations, in the context of a general study of quasicoherent CDGmodules 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 ArtinTate 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 ArtinTate motives with finite coefficients Z/m over a field F containing a primitive mroot 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 Ktheory/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 "ArtinTate 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 ArtinTate 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/mmodules 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 quasifinite 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 ArtinTate motivic sheaves in the triangulated category DM(X,Z/m). For a smooth variety X, an isomorphism of the Z/mmodules Ext between the Tate motives Z/m(j) in F_X^m with the motivic cohomology modules predicted by the BeilinsonLichtenbaum etale descent conjecture (proven by Voevodsky, Rost, et al.) holds if and only if a similar isomorphism holds for ArtinTate motives over fields containing K. When K contains a primitive mroot 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 (onedimensional) 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 PBWbases).
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 HodgeTate structures is discussed in the introduction.
10. To continue, a preliminary draft version of my paper "Weakly curved A_inftyalgebras 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 17, and Appendices AB 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_inftyalgebras complete over a proArtinian 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 15, and Appendices AD 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 quasicompact semiseparated scheme, it also has enough projectives. In this paper we intend to construct the derived cocontra correspondence, meaning an equivalence between appropriate derived categories of quasicoherent sheaves and contraherent cosheaves, over a quasicompact semiseparated scheme and, in a different form, over a semiseparated 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 quasicompact semiseparated schemes f. The latter approach would provide an extended version of the covariant SerreGrothendieck duality theory.
12. And veryvery 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 exactconservative 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 ArtinTate motives and motivic sheaves with finite coefficients, and our key techniques generalize those of Section 4 of
the "Mixed ArtinTate 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 moderatelength Introduction, and Sections 13 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 DOmega" (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 ArtinTate motives); and
(ii) "Two exercises in derived Koszul duality" (the version of January 15, 2011; contains only the Introduction, Table of Contents, and Sections 12. The two originally intended exercises were about ArtinSchelter Gorenstein coalgebras and Golodtype 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 email 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
SemiInfinite FiniteDimensional, paper in Compositio
SemiInfinite FiniteDimensional, 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
ArtinTate Motives, Link to MMJ issue
ArtinTate 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
DΩ, 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)
