2024/09/18 by Puthenpurakal, Tony J.
#Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13D02 #Secondary 13C14
paper · doi:10.48550/arxiv.2409.11877
Let (Q, \mathfrakn) be a regular local ring and let f1, …, fc ∈ \mathfrakn2 be a Q-regular sequence. Set (A, \mathfrakm) = (Q/(f), \mathfrakn/(f)). Further assume that the initial forms f1^*, …, fc^* form a G(Q) = \bigoplusn ≥ 0\mathfrakni/\mathfrakni+1-regular sequence. Without loss of any generality assume ordQ(f1) ≥ ordQ(f2) ≥ ⋯ ≥ ordQ(fc). Let M be a finitely generated A-module and let (\mathbbF, ∂) be a minimal free resolution of M. Then we prove that ord(∂i) ≤ ordQ(f1) - 1 for all i ≫ 0. We also construct an MCM A-module M such that ord(∂2i+1) = ordQ(f1) - 1 for all i ≥ 0. We also give a considerably simpler proof regarding the periodicity of ideals of minors of maps in a minimal free resolution of modules over arbitrary complete intersection rings (not necessarily strict).