Castelnuovo–Mumford regularity of deficiency modules

Abstract

Let d ∈ N and let M be a finitely generated graded module of dimension d over a Noetherian homogeneous ring R with local Artinian base ring R 0 . Let beg(M), gendeg(M) and reg(M) respectively denote the beginning, the generating degree and the Castelnuovo-Mumford regularity of M. If i ∈ N 0 and n ∈ Z , let d i M (n) denote the R 0 -length of the n-th graded component of the i-th R + -transform module D i R+ (M) of M and let K i (M) denote the i-th deficiency module of M. Our main result says, that reg(K i (M)) is bounded in terms of beg(M) and the "diagonal values" d j M (− j) with j = 0, . . . ,d − 1. As an application of this we get a number of further bounding results for reg(K i (M)).

Boundedness of cohomology

“…This paper continues our investigation [6], which was driven by the question "What bounds cohomology of a projective scheme? "…”

“…Then there is some non-negative integer h such that d i M (n 0 ) (−i) ≤ h for all pairs (R, M) ∈ C and all i ∈ {0, · · · , d − 1}. By [6,Theorem 5.4] it thus follows that the set of functions…”

“…To prove the implication (iv) ⇒ (i) fix n 0 ∈ Z and assume that the set △ C,n 0 is finite. Then there is some non-negative integer h such that [6,Theorem 5.4] it thus follows that the set of functions…”

“…= {(h 0 N/M (n)) n∈Z | M ∈ P} is finite and that the set of cohomology diagonalsW := {(d i N/M (−i)) d−1 i=0 | M ∈ P} is finite.In view of Theorem[6, Theorem 5.4] the finiteness of W implies that the setU 1 := {(d i N/M (n)) (i,n)∈N 0 ×Z | M ∈ P} is finite. Moreover for all M ∈ P we have end(H 1 R + (N/M)) < reg 1 (N/M) ≤ max{reg 2 (M) − 1, reg 2 (N)} ≤ max{r − 1, reg 1 (N)}; h 1 N/M (n) ≤ d 0 N/M (n) for all n ∈ Z, with equality if n < beg(N).…”