The dualizing complex of F-injective and Du Bois singularities

Abstract

Abstract. Let (R, m, k) be an excellent local ring of equal characteristic. Let j be a positive integer such that H i m (R) has finite length for every 0 ≤ i < j. We prove that if R is F -injective in characteristic p > 0 or Du Bois in characteristic 0, then the truncated dualizing complex τ >−j ω q R is quasi-isomorphic to a complex of k-vector spaces. As a consequence, Finjective or Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum. Moreover, when R has F -rational or rational singularities on the punctured spectrum, we obtain stronger results generalizing [Ma15] and [Ish84].

A Buchsbaum theory for tight closure

“…For example, using Theorem 2.7, it is proved in [2] that F-injective generalized Cohen-Macualay rings are Buchsbaum (this was conjectured by Takagi and was first proved in [24] using other methods). The full results in [2] are stronger, and it yields a tight closure analog of the corresponding statement. To explain this, we recall that in [2], the * -truncation (or tight closure truncation) of RΓ m R is defined as the object in D(R) such that we have an exact triangle:…”

“…The full results in [2] are stronger, and it yields a tight closure analog of the corresponding statement. To explain this, we recall that in [2], the * -truncation (or tight closure truncation) of RΓ m R is defined as the object in D(R) such that we have an exact triangle:…”

Tight closure of parameter ideals in local rings $F$-rational on the punctured spectrum

“…Definition 2.7 ( [11,12,13]). Let R • = R \ ∪ p∈MinR p. Then for any ideal I of R we define (1) The Frobenius closure of I, I F = {x | x q ∈ I [q] for some q = p e }, where q] for some c ∈ R • and for all q ≫ 0}.…”

Nilpotence of Frobenius action and the Hodge filtration on local cohomology

“…It is conjectured that a singularity in characteristic zero is a Du Bois singularity, which has its origin in Hodge theory, if and only if its modulo p reduction is F -injective for infinitely many p (the "if" part was proved by Schwede [30]). This conjecture is open even in dimension two, and it has recently turned out in [3] (see also [25]) that it is equivalent to another more arithmetic and wide open conjecture, the so-called weak ordinarity conjecture (see Conjecture 1.9 for the precise statement). Since we do not know how to prove these conjectures at this point, we shift our focus on another class of F -singularities, F -nilpotence, which is also a broader class of singularities than F -rationality.…”