Leonid Positselski's Homepage

I am a mathematician. My field is homological algebra.

I've done some work on quadratic and Koszul algebras, nonhomogeneous quadratic duality and curved DG-algebras, absolute Galois groups and Galois cohomology, mixed motives and motivic sheaves with finite coefficients, semi-infinite homological algebra, derived nonhomogeneous Koszul duality, derived categories and functors of the second kind, quasi-coherent matrix factorizations and relative singularity categories, contramodules and the comodule-contramodule correspondence, contraherent cosheaves, MGM duality and derived adic torsion and completion, locally presentable abelian categories, infinitely generated tilting theory and the tilting-cotilting correspondence, very flat and strongly flat modules, topologically perfect topological rings, semi-infinite algebraic geometry, abelian and exact DG-categories, periodicity theorems in homological algebra, limits of accessible categories and accessibility ranks, etc.

Since September 10, 2018, I am working as a researcher at the Department of Algebra, Geometry and Mathematical Physics of the Institute of Mathematics of the Czech Academy of Sciences in Prague.

My official web page with a photograph, the contact information, CV and publication data is here.

My ORCID page is here, my arXiv preprint feed is here, my Google Scholar profile is here, my MathSciNet profile is here or here, and my ZentralBlatt profile is here. My questions and answers on MathOverflow are accessible from here.

Old manuscript file share

Here are the files of Friedrich Ulmer's preprint "Bialgebras in locally presentable categories" (University of Wuppertal, 1977).

Zipped pdf (4 files): Ulmer chap 0,2,Bib,Index,Front.zip; Ulmer chap3.zip; Ulmer chap4.zip; Ulmer chap 5, 6.zip

Unzipped pdf (9 files): UlmerFront.pdf, Ulmer0.pdf, Ulmer2.pdf, Ulmer3.pdf, Ulmer4.pdf, Ulmer5.pdf, Ulmer6.pdf, UlmerBib.pdf, UlmerIndex.pdf

Notice that Chapter 1 does not seem to exist (see the author's note in the top right corner of the title page).

Books and Memoirs I have written

Quadratic algebras. My first book, written in collaboration with Alexander Polishchuk. A long-term project spanning between 1991-2005, it was published by the American Math. Society in its University Lecture Series in 2005. This book has xii+159 pages.

Homological algebra of semimodules and semicontramodules, with a subtitle Semi-infinite homological algebra of associative algebraic structures. Appendix C written in collaboration with Dmitriy Rumynin; Appendix D written in collaboration with Sergey Arkhipov. Published by Springer/Birkhäuser Basel in their Monografie Matematyczne series in 2010. This book has xxiv+349 pages.

Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Published as a Memoir of the American Math. Society in 2011. This memoir has vi+133 pages.

Weakly curved A∞-algebras over a topological local ring. Published as a Mémoire de la Société math. de France between 2018-19. This memoir has vi+206 pages.

Relative nonhomogeneous Koszul duality. Published by Springer Nature/Birkhäuser Switzerland in their Frontiers in Mathematics series between 2021-22. This book has xxix+278 pages.

Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes, with a subtitle Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product. Published by Springer Nature/Birkhäuser Switzerland in 2023. This book has xix+216 pages.

Long self-surveys

My overview paper titled "Contramodules", based on my work on the topic, was published in the journal Confluentes Mathematici between 2021-22.

My overview paper titled "Differential graded Koszul duality: An introductory survey", mostly based on my AMS Memoir "Two kinds of derived categories …", was published in Bulletin of the London Math. Society in 2023.

Selected journal papers, with an emphasis on more recent ones

Nonhomogeneous quadratic duality and curvature. Functional Analysis and its Applications, 1993.

Koszul property and Bogomolov's conjecture. International Math. Research Notices, 2005.

Hochschild (co)homology of the second kind I. Joint work with Alexander Polishchuk. Transactions of the American Math. Society, 2012.

Galois cohomology of a number field is Koszul. Journal of Number Theory, 2014.

Coherent analogues of matrix factorizations and relative singularity categories. Joint work with Alexander I. Efimov. Algebra and Number Theory, 2015.

Artin-Tate motivic sheaves with finite coefficients over an algebraic variety. Proceedings of the London Math. Society, 2015.

Dedualizing complexes and MGM duality. Journal of Pure and Applied Algebra, 2016.

Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories. Joint work with Jiří Rosický. Journal of Algebra, 2017.

Koszulity of cohomology = K(π,1)-ness + quasi-formality. Journal of Algebra, 2017.

On strongly flat and weakly cotorsion modules. Joint work with Alexander Slávik. Mathematische Zeitschrift, 2019.

Flat morphisms of finite presentation are very flat. Joint work with Alexander Slávik. Annali di Matematica Pura ed Applicata, 2020.

The tilting-cotilting correspondence. Joint work with Jan Šťovíček. International Math. Research Notices, 2021.

Topologically semisimple and topologically perfect topological rings. Joint work with Jan Šťovíček. Publicacions Matemàtiques, 2022.

Projective covers of flat contramodules. Joint work with Silvana Bazzoni and Jan Šťovíček. International Math. Research Notices, 2022.

Remarks on derived complete modules and complexes. Mathematische Nachrichten, 2023.

Coderived and contraderived categories of locally presentable abelian DG-categories. Joint work with Jan Šťovíček. Mathematische Zeitschrift, 2024.

Selected unpublished preprints (some of them in preparation for publication)

All strictly exceptional collections in Dbcoh(Pm) consist of vector bundles. arXiv, 1995. 6 pages.

Abelian right perpendicular subcategories in module categories. arXiv, 2017-18. 57 pages.

Philosophy of contraherent cosheaves. arXiv, 2023-24. 65 pages.

Contraherent cosheaves on schemes. arXiv, 2012-24. 457 pages.

The slides of my Beamer presentations (2014-24) are available below

Semi-infinite algebraic geometry. The slides of a 30 min. introductory talk (for a non-algebraist audience) at the external meeting of the Institute of Mathematics in Kubova Huť in October 2024. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Semi-infinite algebraic geometry of quasi-coherent torsion sheaves. The slides of a 60 min. talk at the seminar "New Directions in Group Theory and Triangulated Categories", presented via Zoom on February 8, 2024. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Fp-projective periodicity. The slides of a 20 min. talk at the conference on Categories, Rings and Modules, on the occasion of the retirement of Alberto Facchini, presented in person in Padova, Italy on January 27, 2023. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Abelian and exact DG-categories. Quasi-slides prepared after the talks, based on a 1 hour blackboard talk at the STAR conference in Prague on September 15, 2022, and a 30 min. whiteboard talk at the Seminar Prague-Brno in Brno on September 21, 2022. The quasi-slides were prepared in early October 2022 based on pen-on-paper preparation materials for the two talks. Small pdf file (no \pause commands).

Exotic derived categories. The slides of a 1 hour talk at the Research Seminar on Algebraic Topology of the University of Hamburg, presented via Zoom on May 31, 2021. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Exact category structures and tensor products of topological vector spaces with linear topology. The slides of a 1 hour talk at the Algebra seminar of Charles University in Prague, presented via Zoom on February 1, 2021. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

A description of the second class in the cotilting cotorsion pair in terms of cofiltrations. The slides of a 1 hour talk at the Algebra seminar of Charles University in Prague, presented via Zoom on October 12, 2020. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

A construction of complete cotorsion pairs in the relative context. The slides of a 1 hour talk at the Algebra seminar of Charles University in Prague, presented via Zoom on June 8, 2020. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Koszul algebras and one-dependent random 0-1 sequences. The slides of a 1 hour talk at the seminar on Cohomology in algebra, geometry, physics and statistics of the Institute of Mathematics of the Czech Academy of Sciences in Prague, presented via Zoom on April 1, 2020. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Derived adically complete modules and complexes. The slides of a 1 hour talk at the Algebra seminar of Charles University in Prague, presented via Zoom on March 23, 2020. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Contramodules: their history, and applications in commutative and noncommutative algebra. A mostly historical overview on contramodules, including the prehistory and history of my studies of contramodules. Presented as a 50 min. talk at the external meeting of the Institute of Mathematics in Zvánovice in October 2018. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Covers, direct limits, and pro-perfect topological rings. The slides of a 25 min. talk at the ECI workshop in Třešť, October 2018, about an application of contramodules to the Enochs conjecture on covers and direct limits. Based on my joint work with Silvana Bazzoni. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

∞-tilting theory. The slides of a 25 min. talk at the 18th ICRA conference in Prague, August 2018, about infinitely generated, infinite projective/injective dimension tilting-cotilting theory for abelian categories. Based on my joint work with Jan Šťovíček. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

The very flat conjecture. A discussion of the formulation, a detailed technical overview of the proof, and formulations of some analogous results at the end. Presented as a 50 min. talk at the ECI workshop in Telč in October 2017. Based on my joint work with Alexander Slávik. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Semi-infinite algebraic geometry. A lengthy technical overview of the (at present rather nonexistent or hypothetical) subject; with lots of details, definitions, examples, and particular cases. Presented as a 50 min. talk at the conference "Some Trends in Algebra" in Prague in September 2015. Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).

Comodule-contramodule correspondence. A technical overview of the subject, with a considerable amount of detail. A slightly expanded version of the slides of 50 min. talk at the ECI workshop in Třešť, April 2014 (my first ever projector talk). Big pdf file (compiled with the \pause commands in force); small pdf file (compiled with the \pause commands ignored).