David A. Buchsbaum

Schur Functors and Schur Complexes

The study of Schur functors has a relatively long history. Its main impetus derived from representation theory, originally in characteristic zero. Over the years, however, with the development of modular representations, and algebraic geometry over fields of positive characteristic, the need for a theory of universal polynomial functors increased and, since the mid-1960s, approaches to a characteristic-free treatment of Schur functors have been developing (see, for instance, the recent book of Green [ 1 l] in which the treatments by Carter and Lusztig [6], Higman [12], and Towber [20], among others, are discussed). Our own interest in such a treatment was awakened by the work of Lascoux [ 141 on resolutions of determinantal ideals. Although his thesis treated only the characteristic zero case, it suggested that a general and elementary theory of Schur functors could be developed using only the rudiments of multilinear algebra (involving the Hopf algebra structures of the symmetric, exterior, and divided power algebras). Moreover, this elementary development admitted of a natural generalization to the idea of Schur complexes, whose usefulness was demonstrated, for instance, in our construction of a universal minimal

Homological dimension in local rings

Introduction. This paper is devoted primarily to the study of commutative noetherian local rings. The main task is to compare purely algebraic properties with properties of a homological nature. A large part of this paper is an elaboration of [2](2) which contained no proofs. We use [3] as a reference source for homological algebra. We begin with a list of the most important notions and an outline of results. Unless stated otherwise, we assume throughout this paper that all rings are commutative noetherian rings with identity elements and that all modules are finitely generated and unitary. If R is a local ring, we shall denote its maximal ideal by m and the quotient field R/m by F. Given a ring R and an l?-module E we denote by hdRE the integer (finite or + co) which in [3] is denoted by dim# E. We have hdR E^nil there exists an exact sequence

Generic Free Resolutions and a Family of Generically Perfect Ideals

Beginning with Hilbert’s construction of what is now called the Koszul complex [18], the study of finite free resolutions of modules over commutative rings has always proceeded by a study of certain particular generic resolutions. This has led to information about the structure of all finite free resolutions, as in [5] and [II], and to theorems on the structure and deformation of certain classes of “generically perfect” ideals and other ideals whose resolutions are of a known type [5, 6,9, 15, 22, 24, 261. In this paper we will describe some new classes of finite free resolutions and generically perfect ideals. Under “generic” circumstances we will construct the minimal free resolution of the cokernel of a map of the form Ak C# or Sk+, where 4: F --t G is a map between free modules F and G over a noetherian commutative ring, with rank F >, rank G, and where A” and Sk denote the kth exterior and symmetric powers, respectively. We will also describe a family of finite complexes associated with the (n - 1)st order minors of an n x n matrix (a minor is the determinant of a submatrix). Finally, we will consider a class of ideals that is related to inclusions of one ideal generated by an R-sequence in another. This class includes, for example, the ideal defining the singular locus of a projective algebraic variety that is a complete intersection in P. Our main innovation is the construction and use of a (doubly indexed) family of “multilinear” functors LpQ defined on finitely generated free modules, which includes both the symmetric and exterior powers. For q > 1, L,‘J g Aq, while for p > 0, LP1 s S, . These arise naturally in the resolutions of cokernels of the maps A”

Characteristic-free representation theory of the general linear group

and S,

Characteristic-free representation theory of the general linear group II. Homological considerations

; and it turns out more generally that the free modules occuring in the generic minimal free resolutions of the cokernel of a map of the form L,‘+ can all be expressed in terms of tensor products of the form L,qF @I (LrSG)*.

Resolutions of determinantal ideals: The submaximal minors

On etudie systematiquement les Z-formes des representations rationnelles du groupe lineaire generale

LIFTING MODULES AND A THEOREM ON FINITE FREE RESOLUTIONS

In our first paper of this series Cl], we indicated how we were led to consider resolutions of Schur and Weyl modules of a particular form. In order to prove the existence of these resolutions, we were forced to enlarge the family of skew shapes to a class J containing new shapes whose corresponding Schur and Weyl modules had heretofore not been studied. With these shapes in hand, we proved in [ 1 ] the existence of some fundamental exact sequences and described how, from these exact sequences, we could use a mapping cone construction to build up the resolutions we were seeking. For this mapping cone construction, we needed maps, and to provide maps we needed projectivity of tensor products of divided powers. This led to the study of the Schur algebra and its decomposition into orthogonal idempotents. In Sections 1 and 2 we review the information about the Schur algebra that we need to carry out our program. Fortunately there is a very clear and detailed exposition of this subject in the notes of J. A. Green [S] from which we borrowed very heavily.’ In fact, the main function of the first two sections is to condense and translate into our notation and terminology the relevant sections of Green’s notes. In Sections 3 and 4 we define the family of shapes, J, that we will study,

Representations, Resolutions and Intertwining Numbers

For many years there has been considerable interest in finding a resolution of the ideal generated by the minors of order p of a generic m x n matrix. To put the problem more precisely, suppose R, is a commutative ring and X, are variables with 1 < i < m and 1 < j Q n. If we let R = R, [X,] be the polynomial ring over R,, then we have the “generic” matrix (X,) and we may form the ideal ZP in R generated by the p X p minors of this matrix. The problem, then, is to find an explicit free resolution of the ideal ZP over the ring R. It was proved by Eagon and Hochster [lo] that R/I, has a resolution of length (m-p + l)(n -p + l), but their proof consisted in showing that the ideal ZP is perfect; it did not provide a construction of the resolution. In fact, it is not known whether the Betti numbers of the ideal ZP depend on the characteristic of the ground ring R,. In [ 121, Lascoux succeeded in giving an explicit resolution provided that the ground ring R, contained the field, Q, of rational numbers. His construction rests heavily on the theory of Schur functors and the fact that in characteristic zero the Schur functors are the irreducible representations of the general linear group. Over the integers, however, the construction breaks down despite the fact that one can define the Schur functors over an arbitrary commutative ring (see [ 1,2, 14, 151 for various constructions of Schur functors). In analyzing the work of Lascoux and its subsequent reworking by Nielsen [ 131, some basic facts seemed to clamor for attention. One was that within a resolution of R/Z,, there appeared to be two types of boundary maps: one of degree 1 and one of degree p. The maps of degree 1 were maps between sums of Schur functors of fixed Durfee square k (see Section 2 for definitions), while the maps of degree p were from sums of Schur functors of

Letter-Place Methods and Homotopy

Publisher Summary This chapter discusses the lifting modules and a theorem on finite free resolutions. It discusses a new theorem about finite free resolutions over commutative noetherian rings that has proved useful in connection with the lifting problem. The result on free resolutions has some other interesting applications. It yields Burchs theorem on the structure of cyclic modules of homological dimension. In particular, it yields a new proof that any regular local ring is a unique factorization domain. A positive solution to the lifting problem would allow one to reduce the general (complete regular local) case of Serres conjecture to the unramified case. It is found that because of Serres reduction of the conjecture to the cyclic case, it would be sufficient to be able to lift cyclic modules. The chapter presents the results connected with the lifting problem of Grothendieck. The result on lifting cyclic modules S/I, where S/I has finite homological dimension and I is generated by three elements, follows from a broad generalization of Burchs theorem that gives information about the form of a finite free resolution of any length.

Homotopies for resolutions of skew-hook shapes

It may, at first glance, seem inappropriate to be talking about representations of the general linear group at a conference on commutative algebra. We would like, therefore, to offer some observations: one, historical; one, personal; and one, perhaps presumptuous, that might reconcile this apparent anomaly.