有限维代数的Serre--Hochschild平面
The Serre--Hochschild plane of a finite-dimensional algebra
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中文总结 AI 辅助
本文为有限维代数定义了Serre--Hochschild平面，统一了多个Hochschild理论不变量，并证明了梯子定理、谱序列及Han猜想的完整分次Tate判据，在对称和局部周期代数上验证了猜想。
中文摘要 AI 辅助
对于域上的有限维代数$A$，我们将$A$的Hochschild上同调（系数取Serre双模$\omega=\D A$的所有导出张量幂）组织成一个单一的双次数导出不变量，即定义在整个$\mathbb Z^2$上且不反转$\omega$的\emph{Serre--Hochschild平面}$\T^{p,m}(A)$。第$m=0$列是Tamarkin--Tsygan演算，$m=1$列是对偶Hochschild同调（控制Han猜想的列），$m=2$列恢复$\tau$-Hochschild影子，$m=-1$列是Keller的Calabi--Yau完备化。我们证明了梯子定理，使得每一列都是该演算上的对称模；还证明了系数谱序列、固定Han列的Serre反射、将几何代数的平面与光滑射影簇的扭曲多重向量上同调等同起来的字典、在$\Fg$条件下同调列的有限生成性及$\dim\HH_n(A)$的增长三分法，以及关于周期代数上Han猜想的完整分次Tate判据。在任意特征的对称周期代数和特征零的局部周期代数上，Han猜想成立。我们给出了第一个稳定无迹周期代数的例子，表明该分次判据是尖锐的。
英文摘要
For a finite-dimensional algebra $A$ over a field we organize the Hochschild cohomology of $A$ with coefficients in all derived tensor powers of the Serre bimodule $ω=\D A$ into a single bigraded derived invariant, the \emph{Serre--Hochschild plane} $\T^{p,m}(A)$, defined over all of $\mathbb Z^2$ without inverting $ω$. The column $m=0$ is the Tamarkin--Tsygan calculus, $m=1$ is dual Hochschild homology (the column governing Han's conjecture), $m=2$ recovers the $τ$-Hochschild shadow, and $m=-1$ Keller's Calabi--Yau completions. We prove a ladder theorem making every column a symmetric module over the calculus, a coefficient spectral sequence, a Serre reflection fixing the Han column, a dictionary identifying the plane of a geometric algebra with twisted polyvector cohomology of a smooth projective variety, finite generation of the homology column under the condition $\Fg$ with a growth trichotomy for $\dim\HH_n(A)$, and a complete graded Tate criterion for Han's conjecture on periodic algebras. Han's conjecture follows for symmetric periodic algebras in every characteristic and for local periodic algebras in characteristic zero. We exhibit the first example of a stably traceless periodic algebra, showing that the graded criterion is sharp.
