A Lower-Bound on the Hochschild Cohomological Dimension

共形伽利略代数及其中心扩张的上同调刚性

Cohomological rigidity of conformal Galilei algebras and their central extensions

本文研究实共形伽利略代数复化及其中心扩张的第二伴随上同调，利用霍赫希尔德-塞尔分解定理计算相关空间，得出自旋为半整数时d≠2的共形伽利略代数具上同调刚性的结论，完成特定参数范围的形式刚性刻画。

Hochschild cohomology of algebras arising from categories and from bounded quivers

The main objective of this paper is to provide a theory for computing the Hochschild cohomology of algebras arising from a linear category with finitely many objects and zero compositions. For this purpose, we consider such a category using an ad hoc quiver $Q$, with an algebra associated to each vertex and a bimodule to each arrow. The computation relies on cohomological functors that we introduce, and on the combinatorics of the quiver. One point extensions are occurrences of this situation, and Happel's long exact sequence is a particular case of the long exact sequence of cohomology that w…

代数的Hochschild积与整体非交换上同调。应用

Hochschild products and global non-abelian cohomology for algebras. Applications

设$A$是域$k$上的酉结合代数，$E$是向量空间，$π: E \to A$是具有$V = {\rm Ker} (π)$的满射线性映射. $E$上所有使$π: E \to A$成为代数映射的代数结构都由一个显式构造的全局上同调类型对象${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$来描述和分类。任何这样的代数都同构于Hochschild积$A \star V$，一个作为经典构造的推广引入的代数。证明了${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$是所有非交换上同调${\mathbb H}^{2} \, \, (A, \, (V, \cdot))$的余积.计算负责所有共标志代数分类的关键对象${\mathbb G} {\mathbb H}^{2} \, (A, \, k)$。所有的Hochschild产品$A \star k$也分类和自同构群${\rm Aut}_{\rm Alg} (A \star k)$完全确定为一个半直积$A^* \, \ltimes \bigl(k^* \times {\rm Aut}_{\rm Alg} (A) \bigl)$的群的子群。给出了几个例子以及应用到超可解余代数或泊松代数的理论。特别地，对于一个给定的泊松代数$P$，所有的泊松代数有一个泊松代数满射$P$与$1$维核的描述和…

Cup product on Hochschild cohomology of a family of quiver algebras

Let k be a field, q in k. We derive a cup product formula on the Hochschild cohomology ring of a family Lambda_q of quiver algebras. Using this formula, we determine a subalgebra of k[x,y] isomorphic to Hochschild cohomology modulo N, where N is the ideal generated by homogeneous nilpotent elements. We explicitly construct non-nilpotent Hochschild cocycles which cannot be generated by lower homological degree elements, thus disproving the Snashall-Solberg finite generation conjecture.

高遗传代数的Serre函子的熵

Entropies of Serre functors for higher hereditary algebras

该研究计算了高遗传代数的Serre函子相关熵与维数不变量，证明了Hochschild同调熵的Yomdin型不等式，证实了相关问题的肯定答案与Gromov-Yomdin型等式成立。

Galois cohomology of algebraic groups acting on the tensor product of two composition algebras

This is a study of algebras with involution that become isomorphic over a separable closure of the base field to a tensor product of two composition algebras. We classify these algebras, provide criteria for isomorphism and isotopy, and determine their automorphism groups, structure groups, and norm-similitude groups. The most interesting cases of these algebras are the $64$-dimensional bi-octonion algebras, on which groups of absolute type $(G_2 \times G_2)\rtimes \mathbb{Z}/2\mathbb{Z}$ and $\mathbf{Spin}_{14}$ act faithfully by automorphisms and norm-preserving isotopies, respectively. We c…

单项式理想的嵌入形变问题

The embedded deformation problem for monomial ideals

本文研究单项式关系定义的局部/分次环的同调性质，正面回答Avramov问题，建立嵌入形与同伦李代数二次中心元、余法模自由直和项的对应，还给出上同调支撑簇维数下界并分类低维情形的相关子簇。

有限生成域扩张的Brauer $p$ -维数的下界和无穷大判据

Lower bounds and infinity criterion for Brauer $p$-dimensions of finitely-generated field extensions

设$E$是域，$p$是素数，$F/E$是超越度$t$的有限生成扩张。本文证明了：若绝对Galois群$\mathcal{G}_{E}$具有非零上同调$p$ -维数$_{p}(E)$，则域$F$具有BrauerPDMATHABMN-维数$_{p}(F) \ge t$，可能的例外是，在$p = 2$的情况下，$\mathcal{G}_{E}$的Sylowpro-2-子群是2阶的，并且$F$是非实域.证明了当$t \ge 1$和绝对Brauer $p$ -维数abrd $_{p}(E)$都是无穷大时，Brd $_{p}(F)$是无穷大的，而且对于每对具有$1 \le m \le n$的整数对$(m, n)$，存在一个中心除$F$ -代数，其指数为$p ^{m}$，Schur指数为$p ^{n}$ .

QF扩展的理想深度

Ideal depth of QF extensions

一个最小深度d^I（S --> R）被赋给一个环同态S --> R和一个R-R-双模I.当I = R且S --> R是包含映射时，Boltje-Danz-Kuelshammer的文章中关于子环d（S，R）的深度的概念被恢复.理想深度给出了群C-代数对或半单复代数扩张下d（S，R）的下界.如果R| S是有限深度的QF扩张，最小左偶数深度和最小右偶数深度是一致的。若R < S是Frobenius扩张且R是右S-生成元，则其子环深度与塔深度一致。在这个过程中，为Frobenius扩张之上的张量积提供了环、模、Frobenius和Temperley-Lieb结构的公式。深度为3的QF扩张嵌入深度为2的QF扩张中;反过来，某些深度为n的扩张嵌入深度为3的扩张中，如果它们是弗罗贝纽斯扩张或其他特殊的环扩张，在它们的相对Hochschild棒分解群上具有环结构。