ANALYTIC SPREAD OF $\mathcal{F}$-GOOD BIFILTRATIONS OF MODULES

Source abstract

Let $\mathcal{A}$ be a unitary commutative ring, $M$ be an $\mathcal{A}$-module and Hilbert function, quasi-polynomial function, analytic spread. Deval BECHE and Daouda SANGARE, Analytic spread of $\mathcal{F}$-good bifiltrations of modules, Far East Journal of Mathematical Sciences (FJMS) 143(12) (2026), 4295-4309. $\mathcal{F}=\left(I_{m, n}\right)_{(m, n) \in \mathbb{Z}^{2}}$ be a bifiltration of $\mathcal{A}$. In a previous work, we introduced the notion of an $\mathcal{F}$-good bifiltration $\Phi=\left(M_{m, n}\right)_{(m, n) \in \mathbb{Z}^{2}}$ of $M$ and characterized it through the finiteness of the type of the generalized Rees module $\mathcal{R}(M, \Phi)$ over the generalized Rees ring $\mathcal{R}(\mathcal{A}, \mathcal{F})$. In the present paper, we pursue this study in the local setting, where $(\mathcal{A}, \mathfrak{M}, k)$ is a Noetherian local ring. We study the fiber cone $G(\mathcal{A}, \mathfrak{M}, F)$ and the associated bigraded module $G(M, \mathfrak{M}, \Phi)$ and we show that the bigraded Hilbert function $(m, n) \mapsto \ell_{k}\left(\frac{M_{m, n}}{\mathfrak{M} M_{m, n}}\right)$ is of quasi-polynomial type for $m, n \gg 0$, which is a bivariate analogue of the Hilbert-Serre theorem. The degree of this quasi-polynomial allows us to define the analytic spread $\lambda(\Phi)$ of the bifiltration $\Phi$, which generalizes to the bivariate setting the classical analytic spread of an ideal introduced by Rees.