2025/05/31 by Souvik Dey, Dey, Souvik, Dipankar Ghosh +7
Mathematics · #13A02 #13A15 (Primary) #13D02 (Secondary) #13D07 #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2506.00529
openalex publication_date 2025/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a Noetherian ring, I1,…,Ir be ideals of R, and N⊆ M be finitely generated R-modules. Let S = \bigoplus_\underlinen ∈ ℕr S_\underlinen be a Noetherian standard ℕr-graded ring with S_\underline0 = R, and M be a finitely generated ℤr-graded S-module. For \underlinen = (n1,…,nr) ∈ ℕr, set G_\underlinen := M_\underlinen or G_\underlinen := M/\bf I^\underlinen N, where \bf I^\underlinen = I1n1 ⋯ Irnr. Suppose F is a coherent functor on the category of finitely generated R-modules. We prove that the set \rmAssR (F(G_\underlinen) ) of associate primes and \rmgrade(J, F(G_\underlinen)) stabilize for all \underlinen ≫ 0, where J is a non-zero ideal of R. Furthermore, if the length λR(F(G_\underlinen)) is finite for all \underlinen ≫ 0, then there exists a polynomial P in r variables over ℚ such that λR(F(G_\underlinen)) = P(\underlinen) for all \underlinen≫ 0. When R is a local ring, and G_\underlinen = M/\bf I^\underlinen N, we give a sharp upper bound of the total degree of P. As applications, when R is a local ring, we show that for each fixed i ≥ 0, the ith Betti number βiR(F(G_\underlinen)) and Bass number μiR(F(G_\underlinen)) are given by polynomials in \underlinen for all \underlinen ≫ 0. Thus, in particular, the projective dimension \rmpdR(F(G_\underlinen)) (resp., injective dimension \rmidR(F(G_\underlinen))) is constant for all \underlinen≫ 0.