Sally modules of rank one

Abstract

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 structure of Sally modules of rank one

“…In Section 4 we shall discuss two consequences of Theorem 1.2. The results are more or less known by [2,10,11]. However, thanks to Theorem 1.2, not only the statements of the results but also the proofs are substantially simplified, so that we would like to note the improved statements, and would like to indicate a brief proof of Theorem 1.1 as well.…”

“…The following result is also due to [2], which will enable us to reduce the proof of Theorem 1.2 to the proof of the fact that…”

“…It seems natural to ask what happens, when e 2 = 0, on the ideals I which satisfy the equality e 1 = e 0 − A (A/I) + 1. This long standing question has motivated the recent research [2], where the authors gave several partial answers to the question. The present research is a continuation of [2,10,11] and aims at a simultaneous understanding of the structure of Sally modules of ideals I which satisfy the equality e 1 = e 0 − A (A/I) + 1.…”

“…We shall prove Theorem 1.2 in Section 3. In Section 2 we will pick up from the paper [2] some auxiliary results on Sally modules, all of which are known, but let us note them for the sake of the reader's convenience. In Section 4 we shall discuss two consequences of Theorem 1.2.…”

“…In this section let us review three results of [2,10,11] in order to see how our Theorem 1.2 works to prove or improve them. Let us begin with Theorem 1.1.…”