A Buchsbaum theory for Frobenius closure
Kriti Goel, Kyle Maddox, Lance E. Miller, Pham Hung Quy, Austyn Simpson
TL;DR
The paper investigates when the Frobenius-closure-related difference $e(\mathfrak{q})-\ell_R(R/\mathfrak{q}^F)$ remains constant across parameter ideals in excellent equidimensional local rings of prime characteristic. It introduces the $F$-Buchsbaum concept, formulated via $\mathfrak{q}^{F\textrm{-lim}}$ and derived-category truncations $\tau^{<d,F}$, to mirror the classical Buchsbaum/Tight-Buchsbaum theory in the Frobenius setting. The main result establishes equivalences among four conditions—constancy of $e(\mathfrak{q})-\ell_R(R/\mathfrak{q}^{F\textrm{-lim}})$, constancy of $\ell_R(\mathfrak{q}^{F\textrm{-lim}}/\mathfrak{q})$, the inclusion $\mathfrak{m}\mathfrak{q}^{F\textrm{-lim}}\subseteq\mathfrak{q}$, and the finiteness of the $F$-truncation—under assumptions such as $F$-finiteness and weak $F$-nilpotence. The results show $F$-Buchsbaum rings are Buchsbaum, clarify the relation to tight Buchsbaum and CMFI, and provide concrete criteria and examples illustrating the nuanced hierarchy among these notions.
Abstract
We give a partial characterization for when the difference $e(\mathfrak{q})-\ell_R(R/\mathfrak{q}^F)$ is independent of the choice of parameter ideal $\mathfrak{q}\subseteq R$ in an excellent equidimensional local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$. Here, $\mathfrak{q}^F$ is the Frobenius closure of $\mathfrak{q}$ and $e(\mathfrak{q})$ denotes the Hilbert--Samuel multiplicity of $\mathfrak{q}$. In addition to ideal-theoretic equivalences, our characterization involves the derived category and is motivated by Schenzel's criterion of the Buchsbaum property as well as similar results of Ma-Quy in the setting of tight closure.