Local cohomology of Du Bois singularities and applications to families

Abstract

In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,m)$ be a local ring, we prove that if $R_{red}$ is Du Bois, then $H_m^i(R)\to H_m^i(R_{red})$ is surjective for every $i$. We find many applications of this result. For example we answer a question of Kov\'acs and the second author on the Cohen-Macaulay property of Du Bois singularities. We obtain results on the injectivity of $Ext$ that provide substantial partial answers of questions of Eisenbud-Mustata-Stillman in characteristic $0$, and these results can be viewed as generalizations of the Kodaira vanishing theorem for Cohen-Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen-Macaulayness of the defining ideal of Du Bois singularities, which are characteristic $0$ analog of results of Singh-Walther and answer some of their questions. We extend results of Hochster-Roberts on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities, see Hochster-Roberts. We also prove that singularities of dense $F$-injective type deform.Comment: 24 pages, minor changes. To appear in Compositio Mathematic

F-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p> 0

“…It is shown in [27] that if (R, m) is a local ring essentially of finite type over C such that R/xR is of dense F -injective type for a regular element x ∈ m, then R is of dense F -injective type.…”

Frobenius Actions on Local Cohomology Modules and Deformation

“…It is well known that -rationality always deforms while -regularity and -purity do not deform in general [22, 23]. Whether -injectivity deforms is a long- standing open problem [6] (for recent developments, we refer to [11, 18]). Recall that the Frobenius endomorphism induces a natural Frobenius action on every local cohomology module, : .…”

“…We introduce two conditions: -full and -anti-nilpotent singularities, in terms of the Frobenius actions on local cohomology modules of (we refer to Section 2 for detailed definitions). The first condition is motivated by recent results on Du Bois singularities [18]. The second condition has been studied in [5, 16], and is known to be equivalent to stably FH-finite , which means all local cohomology modules of and supported at the maximal ideals have only finitely many Frobenius stable submodules.…”

F-THRESHOLDS OF GRADED RINGS

“… 1 Very recently, Ma, Schwede, and Shimomoto answered the question about projective dimension for Du Bois singularities [MSS16, Corollary 4.3]. …”