On topological methods in homological algebra

This volume presents an elaborated version of lecture notes for two advanced courses: (Re)Emerging methods in Commutative Algebra and Representation Theory and Building Bridges Between Algebra and Topology, held at the CRM in the spring of 2015. Homological algebra is a rich and ubiquitous area; it is both an active field of research and a widespread toolbox for many mathematicians. Together, these notes introduce recent applications and interactions of homological methods in commutative algebra, representation theory and topology, narrowing the gap between specialists from different areas wishing to acquaint themselves with a rapidly growing field. The covered topics range from a fresh introduction to the growing area of support theory for triangulated categories to the striking consequences of the formulation in the homotopy theory of classical concepts in commutative algebra. Moreover, they also include a higher categories view of Hall algebras and an introduction to the use of idempotent functors in algebra and topology.

Covering an exceptional range of topics, this text provides a unique overview of the Maurer—Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a new conceptual treatment of the twisting procedure, guiding the reader through various versions with the help of plentiful motivating examples for graduate students as well as researchers. Topics covered include a novel approach to the twisting procedure for operads leading to Kontsevich graph homology and a description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras using the biggest deformation gauge group ever considered. The book concludes with concise surveys of recent applications in areas including higher category theory and deformation theory.

SAMUEL EILENBERG1. Introduction.The title of this article requires some explanation.The term "abstract algebra" was used to indicate that we shall deal with purely algebraic objects like groups, algebras and Lie algebras rather than topological groups, topological algebras, and so on.The method of study is also purely algebraic but is the replica of an algebraic process which has been widely used in topology, thus the words "toplogical methods" could be replaced by "algebraic methods suggested by algebraic topology."These purely algebraic theories do, however, have several applications in topology.The algebraic process borrowed from topology is the following.Consider a sequence of abelian groups {C q ) and homomorphisms 8,such that ôô = 0.In each group C q two subgroups are distinguished Z q = kernel of 8:C q -* 0 +1 , B q = image of ô:C*~l ~» C q

I. Introduction.This paper is concerned with homological algebra in the category S? of locally compact abelian topological groups.The morphisms of £f are the continuous homomorphisms.However, only certain exact sequences, resolutions etc., are admissible.These are sequences whose continuous homomorphisms are open onto their respective ranges.Such maps are called proper and the corresponding sequences, proper exact.Concomitant with this is the fact that although Sf is an additive category it is not abelian.The material in §11 may be regarded as preparatory, although there may be some independent interest here.Various structural facts are proven (some of which are well known) and basic properties of important dual subcategories of £f are investigated.In §111 the projectives and injeptives of ¿C are computed.It turns out that subgroups of projectives are projective and quotient groups of injectives are injective.Vector groups are characterized by the fact that they are both projective and injective.Finally, necessary and sufficient conditions are given for the existence of proper resolutions.In § §IV and V continuous versions of the functors Horn and (x) (via dualization) are defined on certain subcategories of =5f and their functorial properties, including exactness, are investigated.These extend the usual notions for discrete groups.In order to do this it is necessary to study topological groups of continuous multilinear functions.Sufficient, and in a sense, necessary conditions are given for the various groups to be locally compact.Under these conditions the appropriate functors are shown to be isomorphic.Finally, more or less explicit computations are made for Horn, (x), etc., sharpening some of the earlier results, and their geometric and structural significance is investigated.In §VI Tor and Ext are defined by resolutions as derived functors of (g) and Horn, and their functorial properties are studied.It turns out that Torn and Extn vanish for n g 2, and that Torx and Ex^ are computable.Knowledge of Extj in turn gives information about certain group extensions in ££.Th ,hout this paper, complete duality of all concepts and theorems is obtained, in order to accomplish this some concessions have to be made to the topology at various stages.For this reason the functors Horn, (x), Tor, and Ext are not defined on .£? but only on certain subcategories.

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -- in addition to elegant abstraction -- they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. We thus obtain a comprehensive semantic account of the multiplicative variants of all standard propositional connectives in the bunched logic setting. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the "resource semantics" interpretation underpinning Separation Logic amongst them.

Dear SIAGA Reader, It is a great pleasure and privilege to welcome you to SIAM Journal on Applied Algebra and Geometry (SIAGA). This journal, with the help of its editorial board, contributors, and readers and the SIAM staff, follows in the outstanding tradition of high-quality SIAM journals. We anticipate that it will become the leading journal dedicated to algebraic, geometric, and topological methods that have strong connection to applications. The journal began accepting submissions in March 2016. By December, it had received 90 manuscripts submitted for publication. Of these, you will find the first nine papers of the initial all-electronic batch published in February 2017. This first group marks the point at which we have established a regular rate of acceptances, building the foundation for a bright future. SIAGA came to exist thanks to the tremendous effort of numerous people. The impetus came from the SIAM Activity Group on Algebraic Geometry, which had its inaugural meeting in Raleigh, North Carolina, in October 2011. Following the second meeting, at Fort Collins, Colorado, in August 2013, a committee was formed to discuss the idea of a new journal. Under the leadership of Frank Sottile and Thorsten Theobald, a proposal was written and then rewritten numerous times. We had strong support from the Journal Committee and SIAM's Board and Council, and the journal received its approval in December 2015. All our invitations to serve on the editorial board were accepted, and SIAGA is now led by a terrific team composed of three Corresponding Editors and 26 Associate Editors. They are doing an amazing job in handling the growing number of submissions in a timely manner. The SIAM staff is a fantastic asset. In particular, Heather Blythe, Mitch Chernoff, and Frank Kunkle have been critical sources of support. The seven cover pictures of SIAGA have been an inspiration for all of us, and we thank Anna Seigal for her SIAM News articles about the mathematics that is expressed in these diagrams. Please enjoy reading the inaugural SIAGA articles. We look forward to receiving your high-quality submissions in the future.

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry.

Algebraic, Geometric, and Topological Methods in Optimization

This article describes the Hartogs and the Hartogs–Bochner extension phenomena in smooth toric surfaces and the connection with the first cohomology group with compact support. The affirmative and negative results are proved using topological, analytic, and algebraic methods.

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of Ext\ for rings A of global dimension 1.Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC.Applications are given to abelian group theory, including a partial solution (assuming V = L) to a problem of Baer on the splitting of abelian groups.Some independence results in abelian group theory are also proved.Introduction.In a remarkable recent paper [22], S. Shelah proved that Whitehead's problem is unsolvable in ordinary set theory (Zermelo-Frankel set theory with the Axiom of Choice).More precisely, he proved that two different answers to the problem are obtained when set theory is extended by adding two different axioms: the Axiom of Constructibility and Martin's Axiom.(See Theorem 4.1.)Shelah's arguments were specific to Whitehead's problem and nonhomological in character.In this paper we extend Shelah's techniques in order to obtain some general results about the vanishing of ExtA (for modules over a ring A of global dimension 1) in different models of set theory, and we apply them to some specific problems in abelian group theory.Assuming the Axiom of Constructibility we obtain necessary and sufficient conditions for the vanishing of ExtA (Theorems 1.2 and 1.5).To state the result in.a special case: if C and A are abelian groups of cardinality N,, Ext(C, A) -0 if and only if C is the union of an increasing chain of countable subgroups [Cv\v < «,} such that the chain is smooth (i.e., C0 -U"<oC, for aii iimit ordinals a) and satisfies Ext(C0, A) = 0 and E\i(Cv+x/Cv,A) = 0 for all v < K,.Although the sufficiency of this condition is an easy consequence of classical homological algebra (see Theorem 1.2), the proof of necessity makes use of the Axiom of Constructibility (Theorem 1.5).The necessity of this condition is not provable in ordinary set theory; in particular the condition is not necessary when Martin's Axiom is assumed ( §3).

Let I be an ideal of a commutative ring R and M an R-module. It is well known that the I-adic completion functor ΛI defined by ΛI(M) = lim←tM/ItM is an additive exact covariant functor on the category of finitely generated R-modules, provided R is Noetherian. Unfortunately, even if R is Noetherian, ΛI is neither left nor right exact on the category of all R-modules. Nevertheless, we can consider the sequence of left derived functors {LIi} of ΛI, in which LI0 is right exact, but in general LI0 ≠ ΛI. Therefore the computation of these functors is in general very difficult. For the case that R is a local Noetherian ring with the maximal ideal [mfr ] and I is generated by a R-regular sequence, Matlis proved in [9, 10] thatwhere D(−) = HomR(−; E(R/[mfr ])) is the Matlis dual functor, and thatIn [18, 5] A.-M. Simon shows that LIo(M) = M and LIi(M) = 0 for i &gt; 0, provided that M is complete with respect to the I-adic topology.Later, Greenlees and May [3] using the homotopy colimit, or telescope, of the cochain of Koszul complexes to define so-called local homology groups of a module M (over a commutative ring R) bywhere x is a finitely generated system of I. Then they showed, under some conditions on x which are satisfied when R is Noetherian, that HI[bull ](M) ≅ LI[bull ](M). Recently, Tarrío, López and Lipman [1] have presented a sheafified derived-category generalization of Greenlees–May results for a quasi-compact separated scheme. The purpose of this paper is to study, with elementary methods of homological and commutative algebra, local homology modules for the category of Artinian modules over Noetherian rings.

In this paper, we show how stabilizing controllers for 2D systems can effectively be computed based on computer algebra methods dedicated to polynomial systems, module theory and homological algebra. The complete chain of algorithms for the computation of stabilizing controllers, implemented in Maple, is illustrated with an explicit example.

Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be identified with the dual of a coalgebra defined using exclusively the root datum of $(G,T)$, a set of simple roots defining $P$ and the formal group law of $h$. In two papers [Push-pull operators on the formal affine Demazure algebra and its dual, arXiv:1312.0019] and [A coproduct structure on the formal affine Demazure algebra, arXiv:1209.1676], we studied the properties of this dual and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper can be viewed as a companion paper, that justifies all the definitions of the algebraic objects and operators by explaining how to match them to equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. Our main tool is the pull-back to the $T$-fixed points of $G/P$ which injects the cohomology ring in question into a direct product of a finite number of copies of the $T$-equivariant oriented cohomology of a point.

The extension problem for abelian groups (as a special case of the general group-theoretical question formulated by O. Schreier) consists in constructing a group from a normal subgroup and the corresponding factor group. The classical way of discussing extensions is via factor sets which we follow in our presentation (simplified for the abelian case). Then we introduce Baer’s group Ext, an extremely important device, and discuss its fundamental properties. The intimate relationship between Hom and Ext has been pointed out by Eilenberg–MacLane [1]; this led to the interpretation of Ext as a derived functor of Hom and has been exploited extensively in Homological Algebra. Another important functor is Pext, the group of pure extensions, which appears unexpectedly as the first Ulm subgroup of Ext.

We show that categories of modules over a ring in homotopy type theory (HoTT) satisfy the internal versions of the AB axioms from homological algebra. The main subtlety lies in proving AB4, which is that coproducts indexed by arbitrary sets are left-exact. To prove this, we replace a set X with the strict category of lists of elements in X. From showing that the latter is filtered, we deduce left-exactness of the coproduct. More generally, we show that exactness of filtered colimits (AB5) implies AB4 for any abelian category in HoTT. Our approach is heavily inspired by Roswitha Harting’s construction of the internal coproduct of abelian groups in an elementary topos with a natural numbers object. To state the AB axioms, we define and study filtered (and sifted) precategories in HoTT. A key result needed is that filtered colimits commute with finite limits of sets. This is a familiar classical result but has not previously been checked in our setting. Finally, we interpret our most central results into an $\infty$ -topos $ {\mathscr{X}} $ . Given a ring R in $ {\tau_{\leq 0}({{\mathscr{X}}})} $ – for example, an ordinary sheaf of rings – we show that the internal category of R-modules in $ {\mathscr{X}} $ represents the presheaf which sends an object $ X \in {\mathscr{X}} $ to the category of $ (X{\times}R) $ -modules in ${\mathscr{X}} / X$ . In general, our results yield a product-preserving left adjoint to base change of modules over X. When X is 0-truncated, this left adjoint is the internal coproduct. By an internalisation procedure, we deduce left-exactness of the internal coproduct as an ordinary functor from its internal left-exactness coming from HoTT.