预投射代数的广义Gorensteinness和一个同调行列式
Generalized Gorensteinness and a homological determinant for preprojective algebras
摘要：
交换多项式环上群作用不变量的研究促进了交换代数和代数几何的发展。它一直特别感兴趣的是了解什么条件下的组的结果在一个不变的环满足有用的性质。特别地，Watanabe定理指出$k[x_1,\ldots,x_n]$的不变子环在$SL_n(k)$的有限子群的自然作用下总是Gorenstein。本文在非交换非连通代数$A$上群作用的更一般的背景下研究了这个问题。本文提出了$A$的自同构的同调行列式的概念，并利用同调行列式研究了有限群$G$在$A$上的作用.给出了不变环$A^G$具有有限内射维数且满足广义Gorenstein条件的一个充分条件.更精确地说，设$A$是一个整体维数有限的Noether $\mathbb{N}$ -分次广义Gorenstein代数.设$G$的所有元素都固定了$A$的幂等元，并与平凡同调行列式作用。然后将不变环$A^G$推广为Gorenstein环.
英文摘要：
The study of invariants of group actions on commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. It has been of particular interest to understand what conditions on the group result in an invariant ring satisfying useful properties. In particular, Watanabe's Theorem states that the invariant subring of $k[x_1,\ldots,x_n]$ under the natural action of a finite subgroup of $SL_n(k)$ is always Gorenstein. In this paper, we study this question in the more general setting of group actions on noncommutative non-connected algebras $A$. We develop the notion of a homological determinant of an automorphism of $A$, then use the homological determinant to study actions of finite groups $G$ on $A$. We give a sufficient condition so that the invariant ring $A^G$ has finite injective dimension and satisfies the generalized Gorenstein condition. More precisely, let $A$ be a noetherian $\mathbb{N}$-graded generalized Gorenstein algebra with finite global dimension. Suppose all elements of $G$ fix the idempotents of $A$ and act with trivial homological determinant. Then the invariant ring $A^G$ is generalized Gorenstein.
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