On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals
Let \(I\) be a monomial ideal in a polynomial ring \(R=k[X_1,\dots,X_n]\), \(k\) a field. No formula is known for the Castelnuovo-Mumford regularity \(\text{reg}(I)\) of \(I\), except in some (important) cases when one makes some further assumptions on \(I\). -- In this paper, the authors show that \(\text{reg}(I)\leq \text{adeg}(I)\) where the arithmetic degree \(\text{adeg}(I)\) is an invariant defined by Bayer and Mumford as the sum (over the set of associated primes \(P\) of \(I\)) of \(\text{mult}_I(P)\text{deg}(P)\). Here \(\text{mult}_I(P)\) is the length of \(H^0_P(R_p/IR_P)\). The proof is based on a Bezout-type result of \textit{B. Sturmfels, Ngô Viêt Trung} and \textit{W. Vogel} [Math. Ann. 302, No.~3, 417-432 (1995; Zbl 0828.14040)] asserting that for any monomial \(m\) of \(R\) one has \(\text{adeg}(I,m)\leq \text{adeg}(I)\text{deg}(m)\). Assume \(I\) is minimally generated by the monomials \(m_1,\dots,m_s\) with \(\text{deg}(m_1)\geq\dots \geq\text{deg} (m_s)\). Set \(u=\min(s,n)\). Let \(F\) be the least common multiple of \(m_1,\dots,m_s\). Using techniques from \textit{W. Bruns} and \textit{J. Herzog} [Math. Proc. Camb. Philos. Soc. 118, No.~2, 245-257 (1995; Zbl 0888.13004)] the authors show that \(\text{reg}(R/I)\leq\text{deg}(F)-\text{ht}(I)\) and deduce that \(\text{reg}(R/I)\leq \text{deg}(m_1)+\dots+\text{deg}(m_u)-u\), with equality iff \(I\) is a complete intersection. They also improve a result from the above mentioned paper of Sturmfels et al. showing that \(\text{adeg}(I)\leq\text{deg}(m_1)\dots \text{deg}(m_u)-e\), where \(e:=\dim R/I+u-n\).
scientific article; zbMATH DE number 1336837