In this paper we study Ulrich ideals of and Ulrich modules over Cohen-Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and classify Ulrich ideals of numerical semigroup rings and rings of finite CM-representation type.

Abstract. The conjecture of Wolmer Vasconcelos [V] on the vanishing of the firstHilbert coefficient e 1 (Q) is solved affirmatively, where Q is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which e 1 (Q) vanishes are derived.The invariance of e 1 (Q) for parameter ideals Q and its relationship to Buchsbaum rings are studied.

The structure of Sally modules of m-primary ideals I in a Cohen-Macaulay local ring (A, m) satisfying the equality e 1 (I) = e 0 (I) − ℓ A (A/I) + 1 is explored, where e 0 (I) and e 1 (I) denote the first two Hilbert coefficients of I.Let B = T /mT which is the polynomial ring with d indeterminates over the field k.Following W. V. Vasconcelos [13], we then define S Q (I) = IR/IT and call it the Sally module of I with respect to Q. We notice that the Sally module S = S Q (I) is a finitely generated graded T -module, since R is a module-finite extension of the graded ring T .

The set of the first Hilbert coefficients of parameter ideals relative to a moduleits Chern coefficients-over a local Noetherian ring codes for considerable information about its structure-noteworthy properties such as that of Cohen-Macaulayness, Buchsbaumness, and of having finitely generated local cohomology. The authors have previously studied the ring case. By developing a robust setting to treat these coefficients for unmixed rings and modules, the case of modules is analyzed in a more transparent manner. Another L. Ghezzi et al. series of integers arise from partial Euler characteristics and are shown to carry similar properties of the module. The technology of homological degree theory is also introduced in order to derive bounds for these two sets of numbers.

The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen-Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.