Building Bridges Between Algebra and Topology

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied for many years by a variety of authors from the fields of combinatorics, representation theory, probability, number theory, and commutative algebra. We call such polynomials cyclotomic generating functions (CGFs). With Konvalinka, we have studied the support and asymptotic distribution of the coefficients of several families of CGFs arising from tableau and forest combinatorics. In this paper, we survey general CGFs from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGFs in combinatorial representation theory; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. As a sample result, we show that CGFs are "generically" asymptotically normal, generalizing a result of Diaconis on $q$-binomial coefficients using work of Hwang-Zacharovas. We include several open problems concerning CGFs.

In the 2012–13 academic year, the Mathematical Sciences Research Institute, Berkeley, hosted programs in Commutative Algebra (Fall 2012 and Spring 2013) and Noncommutative Algebraic Geometry and Representation Theory (Spring 2013). There have been many significant developments in these fields in recent years; what is more, the boundary between them has become increasingly blurred. This was apparent during the MSRI program, where there were a number of joint seminars on subjects of common interest: birational geometry, D-modules, invariant theory, matrix factorizations, noncommutative resolutions, singularity categories, support varieties, and tilting theory, to name a few. These volumes reflect the lively interaction between the subjects witnessed at MSRI. The Introductory Workshops and Connections for Women Workshops for the two programs included lecture series by experts in the field. The volumes include a number of survey articles based on these lectures, along with expository articles and research papers by participants of the programs. Volume 2 focuses on the most recent research.

Written to honor the 80th birthday of William Fulton, the articles collected in this volume (the first of a pair) present substantial contributions to algebraic geometry and related fields, with an emphasis on combinatorial algebraic geometry and intersection theory. Featured topics include commutative algebra, moduli spaces, quantum cohomology, representation theory, Schubert calculus, and toric and tropical geometry. The range of these contributions is a testament to the breadth and depth of Fulton's mathematical influence. The authors are all internationally recognized experts, and include well-established researchers as well as rising stars of a new generation of mathematicians. The text aims to stimulate progress and provide inspiration to graduate students and researchers in the field.

The purpose of these notes is to explain in detail some topics on the intersection of commutative algebra, representation theory and singularity theory. They are based on lectures given in Tokyo, but also contain new research. It is the first cohesive account of the area and will provide a useful synthesis of recent research for algebraists.

Commutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world “resolution” is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations.

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 > 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.

Abstract. In this paper, we show that, under natural homological conditions, Hopf cyclichomology theory has excision. Mathematics Subject Classifications (2000): 16W30, 19D55, 57T05, 58B34. Key words: cyclic homology, Hopf algebras. 1. Introduction Cyclic cohomology theory for Hopf algebras was introduced by Connesand Moscovici in their ground breaking paper on transverse index theory[4] (cf. also [3, 5, 6] for a recent survey and new results). This theory can beregarded as the right noncommutative analogue of Lie algebra homologybecause of existence of a characteristic map for actions of Hopf algebrasand since it reduces to Lie algebra homology for Hopf algebras associ-ated to Lie algebras. This should be compared with the role played byordinary cyclic cohomology of noncommutative algebras and its reductionto de Rham homology for smooth commutative algebras [2]. The alge-braic underpinnings of Hopf cyclic theory however turned out to be morecomplicated. It is now understood as a special case of a general theorydeveloped in the series of papers [11, 12, 15, 16] (with the help of [1])for (co)algebras endowed with a (co)action of a Hopf algebra and where,among other things, the right notion of coefficients was introduced (cf. also[14] for a recent survey). This theory was later extended by the first author[13] to bialgebras and to a much larger class of coefficients. This extensionis rather surprising since existence of a bijective antipode was essential inconstructing the cyclic and cocyclic complexes developed for Hopf cyclichomology of both variants. By bialgebra or Hopf cyclic homology we nowmean the theory expounded in these papers mentioned above.The main results of the present paper are two excision theorems(Theorems 4.13 and 7.7) for Hopf cyclic homology of module coal-gebras and comodule algebras, respectively. A basic property of cyclic(co)homology for algebras, and coalgebras is excision. If by cyclic theory

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.

In this survey article, we discuss the theory of Koszul algebras and Koszul duality. Koszul property plays an important role in various topics such as commutative and noncommutative algebra, geometry, topology, Lie algebras, representation theory, combinatorics and semigroup rings. Most of the results in this survey article are already known or are easy corollaries of known results. We collect these results and we discuss some proofs. Several survey articles, notably [22, 13, 14, 15, 31], and a book on quadratic algebras [36] contain valuable material to the theory of Koszul algebras or Koszul duality. Keeping these survey articles in mind, the focus has been to collect new results on Koszulness of diagonal subalgebras, its interaction with Polya frequency sequence, and monomial projective curves.

Introduction.Given a field k and a fc-algebra A, Gerstenhaber has introduced the concept of a deformation of A [8] which is a [[/]]-algebra structure on the module A [[/]].One of the main problems of the theory of deformations is to determine those algebras for which the only deformation (modulo an equivalence relation) is the usual A-[[r]]-algebra structure on .4[[*]].Such algebras are said to be rigid.If A is a commutative /c-algebra, we may restrict our attention to those deformations of A which are commutative.This leads us to consider the commutative cohomology groups of A, Hn(A, M)s, defined by Harrison [11].The principal purpose of this paper is to investigate the following conditions on a commutative algebra A over a field k:(1) H2(A, M)s = 0 for all ^-modules M.(2) H\A,f=0.(3) A is a rigid /r-algebra in the commutative deformation theory.(4) The module of differentials of A over k is a projective /4-module.(5) A is a finite product of (not necessarily algebraic) separable extension fields of k.We may show that (1) implies (4) [9, Theorem (20.4.9)] and that (2) implies (3) [8, p. 65].Condition (1) may be expressed by saying that A is a formally smooth fc-algebra [9, 19.4.4].When A is a field, Gerstenhaber has shown that (1), (2), and ( 5) are equivalent and he conjectured that these conditions were equivalent to (3).We show that (1), ( 2), (3), and (5) are equivalent even for A semisimple (Theorem 6.5).Moreover, if A is artinian, then (1) and ( 5) are equivalent (Proposition 5.9).In another direction, if A is a geometric complete intersection, then (1), (2), and (3) are equivalent (Theorem 6.8a).If in addition A is an integral domain with a separable quotient field, then (1), ( 2), (3), and (4) are equivalent (Theorem 6.8c).In the final section, we investigate the above conditions for the group algebra B[G] of an abelian group G which is a direct sum of cyclic groups with B a commutative algebra over a field k of positive characteristic p.In addition, we consider the following condition on G and k:(6) G has no element of order p = char (k).

We give an appropriate extension of the concept of “tower of surjections” to arbitrary inverse systems. We introduce a natural closed model structure (in the sense of D. Quillen) on the category of pro-(Simplicial Abelian Groups) and interpret our condition as the definition of fibrant object.

This paper is dedicated to the memory of Maurice Auslander, whose love of mathematics will always be an inspiration

In 2002, an introductory workshop was held at the Mathematical Sciences Research Institute in Berkeley to survey some of the many directions of the commutative algebra field. Six principal speakers each gave three lectures, accompanied by a help session, describing the interaction of commutative algebra with other areas of mathematics for a broad audience of graduate students and researchers. This book is based on those lectures, together with papers from contributing researchers. David Benson and Srikanth Iyengar present an introduction to the uses and concepts of commutative algebra in the cohomology of groups. Mark Haiman considers the commutative algebra of n points in the plane. Ezra Miller presents an introduction to the Hilbert scheme of points to complement Professor Haiman's paper. Further contributors include David Eisenbud and Jessica Sidman; Melvin Hochster; Graham Leuschke; Rob Lazarsfeld and Manuel Blickle; Bernard Teissier; and Ana Bravo.

We consider a problem of calculating the loop space homology for so-called polyhedral products defined by an arbitrary simplicial complex K. A presentation of this homology algebra is obtained from the homology of the complements of diagonal subspace arrangements, which, in turn, is calculated using an infinite resolution of the exterior Stanley-Reisner algebra. We get an explicit presentation of the loop homology algebra for polyhedral products for classes of simplicial complexes such as flag complexes and the duals of sequentially Cohen-Macaulay complexes in terms of higher commutator products. We give a construction for the iteration of higher products and discuss the relationship between this problem and problems in commutative algebra.