Introduction.- Elementary Definitions.- I Basic Constructions.- II Dimension Theory.- III Homological Methods.- Appendices.- Hints and Solutions for Selected Exercises.- References.- Index of Notation.- Index.

Commutative Algebra: with a View Toward Algebraic Geometry

Preface Introduction Conventions and terminology 1. Commutative rings and modules 2. prime ideals 3. Properties of extension rings 4. Valuation rings 5. Dimension theory 6. Regular sequences 7. Regular rings 8. Flatness revisited 9. Derivations 10. I-smoothness 11. Applications of complete local rings Appendices References Index.

Commutative Ring Theory

Part I. General theory: Schemes Group schemes and representations Induction and injective modules Cohomology Quotients and associated sheaves Factor groups Algebras of distributions Representations of finite algebraic groups Representations of Frobenius kernels Reduction mod $p$ Part II. Representations of reductive groups: Reductive group Simple $G$-modules Irreducible representations of the Frobenius kernels Kempf's vanishing theorem The Borel-Bott-Weil theorem and Weyl's character formula The linkage principle The translation functors Filtrations of Weyl modules Representations of $G_rT$ and $G_rB$ Geometric reductivity and other applications of the Steinberg modules Injective $G_r$-modules Cohomology of the Frobenius kernels Schubert schemes Line bundles on Schubert schemes Truncated categories and Schur algebras Results over the integers Lusztig's conjecture and some consequences Radical filtrations and Kazhdan-Lusztig polynomials Tilting modules Frobenius splitting Frobenius splitting and good filtrations Representations of quantum groups References List of notations Index.

Representations of algebraic groups

This book provides a careful and detailed algebraic introduction to Grothendieck’s local cohomology theory, and provides many illustrations of applications of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Castelnuovo–Mumford regularity, the Fulton–Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry.

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Local Cohomology: An Algebraic Introduction with Geometric Applications

Introduction. If P is a ring and M a left P-module, then homological algebra attaches three dimensions to M, projective, weak, and injective(1)By taking the supremum of one of these dimensions as M ranges over all left P-modules, one obtains one of the left \"global\" dimensions of R. Auslander and Buchsbaum [3] and, subsequently, Serre [14], found it relevant and fruitful, in the study of commutative Noetherian rings, to introduce a \"finitistic global\" dimension defined by restricting the supremum of projective dimensions to finitely generated modules of finite projective dimension. The impressive theory developed by these authors prompted Kaplansky to consider, for general commutative rings, a similar finitistic dimension (waiving the restriction, in the above, to finitely generated modules), and, in a seminar at the University of Chicago (1958), he proved the theorem below, characterizing commutative rings R for which this dimension vanishes. This result is the origin of the present paper. Definition. If N is an ideal in a ring P, we say that N is left T-nilpotent i\"T\" ior transfinite) if, given any sequence {at} of elements in N, there exists an re such that ai • • • an = 0. iRight T-nilpotence requires instead that a„ • • • oi = 0.)

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Finitistic dimension and a homological generalization of semi-primary rings

Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3

Schur Functors and Schur Complexes

What Makes a Complex Exact

Codimension and Multiplicity

Introduction. In a previous paper [3], a general complex was described which could be associated with a map of modules over a commutative ring. Also mentioned in that paper were several areas which could be investigated in which this complex would play a relevant role. In this paper, we investigate two of those areas, namely depth and multiplicity. The notion of the I-depth of a module E, where I is an ideal in a commutative, noetherian ring R, was introduced in [1; 7]. Iff :Rm' -* Rn is a map, then we may consider the annihilator of Coker A"f (where Ak denotes kth exterior product), and call this ideal I(f). We then ask how the I(f)-depth of a module may be determined and show that this number may be computed in terms of the homology groups of the Koszul complex associated with the map f (Theorem 2.4). As a result, we obtain another proof of the generalized Cohen-Macaulay unmixedness theorem. As for multiplicity, the idea is to assign a non-negative integer to a module M of finite length, which agrees with the usual multiplicity when M is of the form R /q. We do this by studying a generalized Hilbert characteristic polynomial for the module M which involves representing M as the cokernel of a map f : Rm' - Rn. The study of this polynomial leads to a generalization of the Krull principal ideal theorem (Theorem 3.5). We eventually prove that the Euler-Poincare characteristic of the complex associated with the map f : R' -+ Rn is related to the leading coefficient of the Hilbert polynomial associated with this map (Theorem 4.2). ?1 deals with the complex associated with a map f :A-- B. Although the definition here differs slightly from that given in [3], it boils down to the same thing when A = Rm and B = Rn. All the properties of this complex which are needed in the subsequent sections of this paper are proved in the first section. The second section deals with the notion of depth, yielding the generalized Cohen-Macaulay theorem, while the third section treats in detail the characteristic polynomial of a map, the consequent generalization of the Krull principal ideal theorem, and multiplicity of modules of finite length. In this section, we also introduce the idea of a parameter matrix, generalizing the notion of system of

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David A. Buchsbaum

Papers

Algebra Structures for Finite Free Resolutions, and Some Structure Theorems for Ideals of Codimension 3

Schur Functors and Schur Complexes

What Makes a Complex Exact

Codimension and Multiplicity

A generalized Koszul complex. II. Depth and multiplicity