Let $(R,\mathfrak m)$ be a Noetherian local ring admitting a canonical module $\omega_R$. We prove that $R$ is quasi-Gorenstein if and only if $\operatorname{Fitt}_1^R(\omega_R)\cong\omega_R$. When $R$ is Cohen-Macaulay, the same condition characterizes the Gorenstein property, confirming an expectation of Eisenbud, Ficarra, Herzog, and Moradi and extending their result for canonical ideals in local Cohen--Macaulay domains of type at most two.
We construct a complete local Gorenstein ring $R$ of dimension $1$ with a prime ideal $\mathfrak{p}\in \operatorname{Spec}(R)$ such that $R$ is an AB ring, but the localization $R_{\mathfrak{p}}$ is not an AB ring. This settles the localization problem for AB rings posed by Huneke and Jorgensen in the negative.
Let $(R,\mathfrak m)$ be a Noetherian local ring and $J$ be an arbitrary ideal of $R$. Suppose that $f_1,\ldots,f_r$ is a $J$-filter regular sequence in $R$ and $I=(f_1,\ldots,f_r)$. In this paper, we establish an improved upper bound for the perturbation index of the associated graded ring $\mathrm{gr}_J(R/I)$, refini...
Let $p, N \geq 5$ be primes such that $N \equiv 1 \bmod p$. We prove modularity theorems at levels $N$ and $N^2$, showing that suitable Eisenstein localizations of the weight-$2$ $p$-adic Hecke algebra at these levels are isomorphic to certain natural quotients of a universal pseudodeformation ring. This universal ring...
Let $X$ be a complex $\mathbb{Q}$-factorial Gorenstein Fano fourfold of Picard number one. We prove that, if $X$ has terminal singularities, then $(-K_X)^4\le648$, with equality precisely for $\mathbb{P}(1,1,1,1,2)$. If the singularities are canonical, we obtain $(-K_X)^4\le7332$. Under the additional assumption that a...
Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitl...
We study the following general situation. Let $R\to S$ be a finite flat morphism of regular rings of zero (resp. prime) characteristic. For $P\in Spec R$, put $A=R/P, C=S/PS,$ and $B=S/\sqrt{PS}=C_{\mathrm{red}}.$ The basic question is whether a property of $B$ forces the same property of $A$. The main point is that or...
Mohsen Asgharzadeh· 0 citations
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