Contramodules

Contramodules are module-like algebraic structures endowed with infinite summation (or, occasionally, integration) operations satisfying natural axioms. Introduced originally by Eilenberg and Moore in 1965 in the case of coalgebras over commutative rings, contramodules experience a small renaissance now after being all but forgotten for three decades between 1970-2000. Here we present a review of various definitions and results related to contramodules (drawing mostly from our monographs and preprints arXiv:0708.3398, arXiv:0905.2621, arXiv:1202.2697, arXiv:1209.2995, arXiv:1512.08119, arXiv:1710.02230, arXiv:1705.04960, arXiv:1808.00937) - including contramodules over corings, topological associative rings, topological Lie algebras and topological groups, semicontramodules over semialgebras, and a "contra version" of the Bernstein-Gelfand-Gelfand category O. Several underived manifestations of the comodule-contramodule correspondence phenomenon are discussed.Comment: LaTeX 2e with pb-diagram and xy-pic; 93 pages, 3 commutative diagrams; v.4: updated to account for the development of the theory over the four years since Spring 2015: introduction updated, references added, Remark 2.2 inserted, Section 3.3 rewritten, Sections 3.7-3.8 adde

Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures

We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (and the cotensor product associative). Besides, for a number of technical reasons we mostly have to assume that the basic ring has a finite homological dimension (no such assumptions about the coring and semialgebra are made). In the final sections we construct model category structures on the categories of complexes of semi(contra)modules, and develop relative nonhomogeneous Koszul duality theory for filtered semialgebras and quasi-differential corings. Our motivating examples come from the semi-infinite cohomology theory. Comparison with the semi-infinite (co)homology of Tate Lie algebras and graded associative algebras is established in appendices, and the semi-infinite homology of a locally compact topological group relative to an open profinite subgroup is defined. An application to the correspondence between complexes of representations of an infinite-dimensional Lie algebra on the complementary central charge levels ($c$ and $26-c$ for the Virasoro) is worked out.Comment: Dedicated to the memory of my father. LaTeX 2e, 310 pages. With appendices coauthored by S.Arkhipov and D.Rumynin. v.12: changes in the Introduction, additions to Section 0 and Appendix D, small improvements in Appendix C and elsewhere, subtitle added -- this is intended as the final arXiv version; v.13: abstract updated, LaTeX file unchanged (the publisher's version is more complete

Extensive categories, commutative semirings and Galois theory

We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B

A Vergleichsstellensatz of Strassen's Type for a Noncommutative Preordered Semialgebra through the Semialgebra of its Fractions

Preordered semialgebras and semirings are two kinds of algebraic structures occurring in real algebraic geometry frequently and usually play important roles therein. They have many interesting and promising applications in the fields of real algebraic geometry, probability theory, theoretical computer science, quantum information theory, \emph{etc.}. In these applications, Strassen's Vergleichsstellensatz and its generalized versions, which are analogs of those Positivstellens\"atze in real algebraic geometry, play important roles. While these Vergleichsstellens\"atze accept only a commutative setting (for the semirings in question), we prove in this paper a noncommutative version of one of the generalized Vergleichsstellens\"atze proposed by Fritz [\emph{Comm. Algebra}, 49 (2) (2021), pp. 482-499]. The most crucial step in our proof is to define the semialgebra of the fractions of a noncommutative semialgebra, which generalizes the definitions in the literature. Our new Vergleichsstellensatz characterizes the relaxed preorder on a noncommutative semialgebra induced by all monotone homomorphisms to $\mathbb{R}_+$ by three other equivalent conditions on the semialgebra of its fractions equipped with the derived preorder, which may result in more applications in the future.Comment: 28 page

An explicit self-dual construction of complete cotorsion pairs in the relative context

Let $R\to A$ be a homomorphism of associative rings, and let $(\mathcal F,\mathcal C)$ be a hereditary complete cotorsion pair in $R\mathsf{-Mod}$. Let $(\mathcal F_A,\mathcal C_A)$ be the cotorsion pair in $A\mathsf{-Mod}$ in which $\mathcal F_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $\mathcal F$. Assuming that the $\mathcal F$-resolution dimension of every left $R$-module is finite and the class $\mathcal F$ is preserved by the coinduction functor $\operatorname{Hom}_R(A,-)$, we show that $\mathcal C_A$ is the class of all direct summands of left $A$-modules finitely filtered by $A$-modules coinduced from $R$-modules from $\mathcal C$. Assuming that the class $\mathcal F$ is closed under countable products and preserved by the functor $\operatorname{Hom}_R(A,-)$, we prove that $\mathcal C_A$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $\mathcal C$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $\mathcal F$ have finite $\mathcal F$-resolution dimension bounded by $k$, involves cofiltrations indexed by the ordinal $\omega+k$. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra arXiv:0708.3398. In addition, we discuss the $n$-cotilting and $n$-tilting cotorsion pairs, for which we obtain better results using a suitable version of the classical Bongartz lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.Comment: LaTeX 2e with xy-pic; 53 pages, 2 commutative diagrams; v.3: the discussion of cotilting and tilting cotorsion pairs moved to new Sections 2.3 and 3.3 with much better results, new Theorems 2.11, 2.26, 2.30, 3.10, 3.29, 3.33 inserted; v.4: Section 4 added, Introduction expanded; v.5: title changed, Remarks 2.17 and 3.16 inserted, references adde