2024/01/31 by Fiorindo, Luca, Ghosh, Dipankar
#13A02 #13A15 #13A30 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.17815
Let R be a commutative Noetherian ℕ-graded ring. Let N⊆ M be finitely generated ℤ-graded R-modules. Let I1,…,Ir be non-zero proper homogeneous ideals of R. Denote \bf I^\underlinen:=I1n1⋯ Irnr for \underlinen=(n1,…,nr)∈ℕr. In this paper, we prove that the (local) Vasconcelos invariant of \bf I^\underlinenM/\bf I^\underlinenN is eventually the minimum of finitely many linear functions in \underlinen. The same holds for M/\bf I^\underlinenN under certain conditions. Some specific examples are provided, where these functions are not eventually linear in \underlinen. However, when R is a polynomial ring over a field, we show that the global Vasconcelos invariants of R/\bf I^\underlinen and \bf I^\underlinen/\bf I^\underlinen+\underline1 are, in fact, asymptotically linear in \underlinen with the leading coefficients given by the initial degrees of I1,…,Ir. The last result is surprising: It differs from the Castelnuovo-Mumford regularity, which is not always linear even over polynomial rings, as shown by Bruns-Conca.