Let R be a finitely generated N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P ∈ Ass(R/I) one defines the v-invariant vP(I) of I at P as the least c ∈ N such that P = I : f for some f ∈ Rc. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that Ass(R/In) is constant for large n. So it makes sense to consider a prime ideal P ∈ Ass(R/In) for all the large n and investigate how vP(In) depends on n. We prove that vP(In) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [Asymptotic behaviour of the v-number of homogeneous ideals, https://arxiv.org/abs/2306.14243, 2023].
A NOTE ON THE V-INVARIANT
Conca A.
2024-01-01
Abstract
Let R be a finitely generated N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P ∈ Ass(R/I) one defines the v-invariant vP(I) of I at P as the least c ∈ N such that P = I : f for some f ∈ Rc. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that Ass(R/In) is constant for large n. So it makes sense to consider a prime ideal P ∈ Ass(R/In) for all the large n and investigate how vP(In) depends on n. We prove that vP(In) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [Asymptotic behaviour of the v-number of homogeneous ideals, https://arxiv.org/abs/2306.14243, 2023].
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