This introductory account of commutative algebra is aimed at advanced undergraduates and first year graduate students. Assuming only basic abstract algebra, it provides a good foundation in commutative ring theory, from which the reader can proceed to more advanced works in commutative algebra and algebraic geometry. The style throughout is rigorous but concrete, with exercises and examples given within chapters, and hints provided for the more challenging problems used in the subsequent development. After reminders about basic material on commutative rings, ideals and modules are extensively discussed, with applications including to canonical forms for square matrices. The core of the book discusses the fundamental theory of commutative Noetherian rings. Affine algebras over fields, dimension theory and regular local rings are also treated, and for this second edition two further chapters, on regular sequences and Cohen–Macaulay rings, have been added. This book is ideal as a route into commutative algebra.

Abstract. Let A be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of A itself, but with respect to an arbitrary ideal of A. The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that A is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true.

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.

t (a ...+e(b)s d .(The reader will also note that this formula is true when r = 0 and s > 0, and also when r > 0 and s = 0.) In particular, on taking r = s = 1, we see that A comparison of this expression with the binomial expansion for (e(a) l/d +e(b) i/d ) dled Teissier to make the following conjecture. TEISSIER'S 1ST CONJECTURE [11; Chapitre 1, §2]. e;(a|b) d < e(a) d " f e(b)'/or all

Abstract. This paper is concerned with the tight closure of an ideal a in a commutative Noetherian local ring R of prime characteristic p. Several authors, including R. Fedder, K-i. Watanabe, K. E. Smith, N. Hara and F. Enescu, have used the natural Frobenius action on the top local cohomology module of such an R to good effect in the study of tight closure, and this paper uses that device. The main part of the paper develops a theory of what are here called 'special annihilator submodules' of a left module over the Frobenius skew polynomial ring associated to R; this theory is then applied in the later sections of the paper to the top local cohomology module of R and used to show that, if R is Cohen-Macaulay, then it must have a weak parameter test element, even if it is not excellent. IntroductionThroughout the paper, R will denote a commutative Noetherian ring of prime characteristic p. We shall always denote by f : R −→ R the Frobenius homomorphism, for which f (r) = r p for all r ∈ R. Let a be an ideal of R. The n-th Frobenius power a [p n ] of a is the ideal of R generated by all p n -th powers of elements of a. We use R• to denote the complement in R of the union of the minimal prime ideals of R. An element r ∈ R belongs to the tight closure a * of a if and only if there exists c ∈ R• such that crWe say that a is tightly closed precisely when a * = a. The theory of tight closure was invented by M. Hochster and C. Huneke [8], and many applications have been found for the theory; see [10] and [11], for example.In the case when R is local, several authors have used, as an aid to the study of tight closure, the natural Frobenius action on the top local cohomology module of R; see, for example, R.

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