AI 中文总结
该研究关联有限维代数\(\Lambda\)的霍赫希尔德理论的两种丰富内容，即\(\tau\) - 霍赫希尔德（上）同调与考克斯特自同构\(\sigma_{\Lambda}\)。通过中山函子等方法揭示了相关复形及性质，证明两种细化横向，提出组合不变量等，还探讨了导出不变性问题。
AI 中文摘要
我们关联了有限维代数\(\Lambda\)的霍赫希尔德理论的两种近期的丰富内容：由西比尔斯、兰齐洛塔、马科斯和索洛塔尔基于伊谷山的正则双模的高阶奥斯特兰德 - 赖滕平移构建的\(\tau\) - 霍赫希尔德（上）同调，以及塔马尔金 - 齐甘演算的考克斯特自同构\(\sigma_{\Lambda}\)。我们表明包络代数的中山函子将哈佩尔的极小分解变换为一个表示\(\mathrm{D}\Lambda\otimes_{\Lambda}\mathrm{D}\Lambda\)的复形，塞尔双模的平方的移位生成\(\sigma_{\Lambda}\)，并且\(\tau\) - 平移\(\tau_n\Lambda\)恰好是这个复形的循环双模。这产生了扩张\(0\to\mathcal{B}_n\to\tau_n\Lambda\to\mathrm{Tor}_n^{\Lambda}(\mathrm{D}\Lambda,\mathrm{D}\Lambda)\to0\)，其外部项与\(\mathrm{Ext}^n_{\Lambda^e}(\Lambda,\Lambda^e)\)对偶，内部项是极小模型的严格莫里塔理论的剩余部分。在最高次数\(d = \mathrm{gldim}\Lambda\)时，剩余部分消失，并且\(\tau_d\Lambda\)是伊谷山 - 奥珀曼的\((d + 1)\) - 预投射代数的一次分量的对偶；对于\(\Lambda=\mathbb{k}Q\)为遗传代数，\(\tau_{\Lambda^e}\Lambda\cong\mathrm{D}\Pi(Q)_1\)且\(\mathrm{HH}^1_{\tau}(\mathbb{k}Q)\)是预投射代数的零阶霍赫希尔德同调的一次部分。对于自内射代数，导出部分恒为零，这从结构上解释了布赫维茨 - 格林 - 马德森 - 索洛塔尔代数的\(\tau\) - 上同调的增长。在西比尔斯 - 兰齐洛塔 - 马科斯 - 索洛塔尔维数公式中取欧拉特征恢复了哈佩尔的迹公式\(\sum_i(-1)^i\dim\mathrm{HH}^i(\Lambda)=-\mathrm{tr}\sigma_{\Lambda}\)。我们证明这两种细化是横向的，提出组合的莫里塔不变量，展示具有相同维数但相反合成的有限全局维数的导出等价代数，并提出\(\tau\) - 霍赫希尔德理论在光滑轨迹上的导出不变性问题。
英文摘要
We relate two recent enrichments of the Hochschild theory of a finite-dimensional algebra $\Lm$: the $τ$-Hochschild (co)homology of Cibils, Lanzilotta, Marcos and Solotar, built from Iyama's higher Auslander--Reiten translates of the regular bimodule, and the Coxeter automorphism $σ_\Lm$ of the Tamarkin--Tsygan calculus. We show that the Nakayama functor of the enveloping algebra transforms Happel's minimal resolution into a complex representing $\D\Lm\Ltimes_\Lm \D\Lm$, the square of the Serre bimodule whose shift generates $σ_\Lm$, and that the $τ$-translates $τ_n\Lm$ are precisely the cycle bimodules of this complex. This produces extensions $0\to \B_n\to τ_n\Lm\to \Tor_n^\Lm(\D\Lm,\D\Lm)\to 0$ whose outer term is dual to $\Ext^n_{\Lme}(\Lm,\Lme)$ and whose inner term is a strictly Morita-theoretic residue of the minimal model. In top degree $d=\gldim\Lm$ the residue vanishes and $τ_d\Lm$ is the dual of the degree-one component of the $(d+1)$-preprojective algebra of Iyama--Oppermann; for $\Lm=\kk Q$ hereditary, $τ_{\Lme}\Lm\cong \DΠ(Q)_1$ and $\HH^1_τ(\kk Q)$ is the degree-one part of the zeroth Hochschild homology of the preprojective algebra. For self-injective algebras, the derived part vanishes identically, which explains structurally the growth of $τ$-cohomology for the Buchweitz--Green--Madsen--Solberg algebras. Taking Euler characteristics in the Cibils--Lanzilotta--Marcos--Solotar dimension formulas recovers Happel's trace formula $\sum_i(-1)^i\dim\HH^i(\Lm)=-\trσ_\Lm$. We prove that the two refinements are transversal, propose the combined Morita invariant, exhibit derived-equivalent algebras of finite global dimension whose $τ$-translates have identical dimension but opposite composition, and pose the problem of derived invariance of $τ$-Hochschild theory over the smooth locus.
Comments20 pages
