链复形的几何与导出等价下的外自同构
Geometry of chain complexes and outer automorphisms under derived equivalence
摘要：
证明了两个主要定理：若$A$是代数闭域上的有限维代数，则$A$的外自同构代数群的单位分支在导出等价下不变。这种不变性是作为以下推广的结果福格特。也就是说，给定具有维数固定序列$\bold d$的有限$A$ -模复形族的适当几何化$\text{Comp}^A_{\bold d}$和"几乎投射“复形$X\in \text{Comp}^A_{\bold d}$，存在规范向量空间嵌入$$T_{X}(\text{Comp}^A_{\bold d}) / T_{X}(G.X) \ \longrightarrow \text{Hom}_{D^b (A\text{-Mod})}(X, X[1]),$$，其中$G$是作用在$\text{Comp}^A_{\bold d}$上的一般线性群的相关乘积，$X$处的切空间用$T_X(-)$表示，并且$X$用其在导出的类别$D^b (A\text{-Mod})$中的图像来标识。
英文摘要：
The two main theorems proved here are as follows: If $A$ is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of $A$ is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization $\text{Comp}^A_{\bold d}$ of the family of finite $A$-module complexes with fixed sequence $\bold d$ of dimensions and an ``almost projective'' complex $X\in \text{Comp}^A_{\bold d}$, there exists a canonical vector space embedding $$T_{X}(\text{Comp}^A_{\bold d}) / T_{X}(G.X) \ \longrightarrow \text{Hom}_{D^b (A\text{-Mod})}(X, X[1]),$$ where $G$ is the pertinent product of general linear groups acting on $\text{Comp}^A_{\bold d}$, tangent spaces at $X$ are denoted by $T_X(-)$, and $X$ is identified with its image in the derived category $D^b (A\text{-Mod})$.
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