Related papers: Ind-\'etale vs Formally \'etale

We show that the weakly \'etale morphisms, used to define the pro-\'etale site of a scheme, are characterized by a lifting property similar to the one which characterizes formally \'etale morphisms. In order to prove this, we prove a…

Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a…

In 1989 Happel conjectured that for a finite-dimensional algebra $A$ over an algebraically closed field $k$, $\gl A< \infty$ if and only if $\hch A < \infty$. Recently Buchweitz-Green-Madsen-Solberg gave a counterexample to Happel's…

Khovanov-Lauda-Rouquier algebras $R_\theta$ of finite Lie type are affine quasihereditary with standard modules $\Delta(\pi)$ labeled by Kostant partitions of $\theta$. Let $\Delta$ be the direct sum of all standard modules. It is known…

For a flat commutative $k$-algebra $A$ such that the enveloping algebra $A\otimes_k A$ is noetherian, given a finitely generated bimodule $M$, we show that the adic completion of the Hochschild cohomology module $HH^n(A/k,M)$ is naturally…

For a finite group $\Gamma$, acting on a finite group $G,$ we find necessary conditions for which the first $\Gamma_0$-equivariant Hochschild cohomology of the group algebra $kG$ is non-trivial, where $k$ is a field of characteristic $p$…

We prove that $\delta$-derivations of a simple finite-dimensional Lie algebra over a field of characteristic zero, with values in a finite-dimensional module, are either inner derivations, or, in the case of adjoint module, multiplications…

We compute the Hochschild cohomology ring of the algebras $A= k\langle X, Y\rangle/ (X^a, XY-qYX, Y^a)$ over a field $k$ where $a\geq 2$ and where $q\in k$ is a primitive $a$-th root of unity. We find the the dimension of $\mathrm{HH}^n(A)$…

We prove that for any proper smooth formal scheme $\frak X$ over $\mathcal O_K$, where $\mathcal O_K$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of $\mathbb Q_p$ with perfect residue field $k$ and…

Let $(A,\mathfrak{m})$ be a Noetherian local ring, $M$ a finite $A$-module and $x_1,...,x_n\in \m$ such that $\lambda (M/\x M)$ is finite. Serre proved that all partial Euler characteristics of $M$ with respect to $\x$ is non-negative. This…

Let $R$ be a commutative Noetherian ring with identity (not necessarily local) and $\frak a$ a proper ideal of $R$. We study the invariance of some classes of $\frak a$-relative Cohen-Macaulay modules under pure ring homomorphisms and ring…

Let $k$ be a field of characteristic $2$ and let $H$ be a finite group or group scheme. We show that the negative Tate cohomology ring $\widehat{\text{H}}^{\leq 0}(H,k)$ can be realized as the endomorphism ring of the trivial module in a…

Let $S^{\cdot}$ be a noetherian graded algebra over a commutative $k$-algebra $A$, where $k$ is a commutative ring, and assume it is a module over a Lie algebroid ${\mathfrak g}_{A/k}$. If $S^\cdot$ is semi-simple over ${\mathfrak g}_{A/k}$…

The rational homotopy type of a differential graded algebra (DGA) can be represented by a family of tensors on its cohomology, which constitute an $A_\infty$-minimal model of this DGA. When only the cohomology is needed to determine the…

We study weakly symmetric special biserial algebras of infinite representation type. We show that usually some socle deformation of such an algebra has non-periodic bounded modules. The exceptions are precisely the algebras whose Brauer…

We study the properties of level zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by Akasaka and the present author is given. Several properties of modules generated by extremal…

In this paper we calculate the Hochschild cohomology of gentle $A_\infty$-algebras of arc collections on marked surfaces without boundary components. When the underlying arc collection has no loops or two-cycles, we show that the dgla…

Let $\A$ be a $\k$-algebra where $\k$ a field of arbitrary characteristic, and let $\mathscr{A}_\k$ be a full subcategory of $\A$-Mod, the abelian category of left $\A$-modules.Following M. Kleiner and I. Reiten, $\mathscr{A}_\k$ is {\it…

In this paper we will study deformations of A-infinity algebras. We will also answer questions relating to Moore algebras which are one of the simplest nontrivial examples of an A-infinity algebra. We will compute the Hochschild cohomology…

Let $R$ be an algebra essentially of finite type over a field $k$ and let $\Omega_k(R)$ be its module of K\"ahler differentials over $k$. If $R$ is a homogeneous complete intersection and $\mathrm{char}(k)=0$, we prove that $\Omega_k(R)$ is…