Jim Williams died in 1983. The surviving three authors are pleased to dedicate this paper to his memory. Abstract. Let C denote the category of Hilbert modules which are similar to con-tractive Hilbert modules. It is proved that if H 0 ; H 2 C and if H 1 is similar to an isometric Hilbert module, then the sequence 0 ! H 0 ! H ! H 1 ! 0 splits. Thus the isometric Hilbert modules are projective i...

Recently Varagnolo and Vasserot established that the q-deformed Fock spaces due to Hayashi, and Kashiwara, Miwa and Stern, admit actions of the quantum toroidal algebra U ′ q(sln,tor) (n ≥ 3) with the level (0, 1). In the present article we propose a more detailed proof of this fact then the one given by Varagnolo and Vasserot. The proof is based on certain non-trivial properties of Cherednik’s...

Given an object in a category, we study its orbit algebra with respect to an endofunctor. We show that if the object is periodic, then its orbit algebra modulo nilpotence is a polynomial ring in one variable. This specializes to a result on Ext-algebras of periodic modules over Gorenstein algebras. We also obtain a criterion for an algebra to be of wild representation

We show that the operadic cohomology for any type of algebras over a non-symmetric operad A can be computed as Ext in the category of operadic A-modules. We use this principle to prove that the Gerstenhaber–Schack diagram cohomology is operadic cohomology.

Let $(A, I)$ be a bounded prism, and $X$ smooth $p$-adic formal scheme over $\Spf(A/I)$. We consider the notion of crystals on Bhatt--Scholze's prismatic site $(X/A)_{\prism}$ relative to $A$. prove that if is proper $\Spf(A/I)$ dimension $n$, then cohomology crystal perfect complex $A$-modules with tor-amplitude in degrees $[0,2n]$. also establish Poincar\'e duality for reduced crystals, i.e. ...

Let be a left and right noetherian ring and mod the category of finitely generated left -modules. In this article, we show the following results. 1 For a positive integer k, the condition that the subcategory of mod consisting of i-torsionfree modules coincides with the subcategory of mod consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. 2 If is an -Gorenstein ring and N...

Let R be a Cohen-Macaulay ring and M a maximal CohenMacaulay R-module. Inspired by recent striking work by Iyama, BurbanIyama-Keller-Reiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension versio...

Let $A$ be a commutative noetherian ring, $\mathfrak{a}$ an ideal of $A$, and $m,n$ non-negative integers. $M$ $A$-module such that $\operatorname{Ext}^i\_A(A/\mathfrak{a},M)$ is finitely generated for all $i\leq m+n$. We define class $\mathcal{S}n(\mathfrak{a})$ modules we assume $H{\mathfrak{a}}^s(M)\in\mathcal{S}{n}(\mathfrak{a})$ $s\leq m$. show $H{\mathfrak{a}}^s(M)$ $\mathfrak{a}$-cofinit...