支持τ-倾斜偏序集与矩阵中心化子代数的Hochschild重构
Support $τ$-tilting posets and Hochschild reconstruction for matrix centralizer algebras
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中文总结 AI 辅助
该研究确定矩阵中心化子代数的支持τ-倾斜偏序集与Hochschild不变量，给出其Morita等价准则，实现对应代数类的Morita重构。
中文摘要 AI 辅助
设R为域，O是Loewy长度为ℓ的有限维交换局部主理想R-代数，满足rad O=(π)。对于非空集合P={p₁<…<p_s}⊆{1,…,ℓ}，记Λ_O(P):=End_O(⊕_{p∈P}O/(π^p))。我们证明Λ_O(P)的支持τ-倾斜偏序集同构于Σ_{s+1}上的左弱序，因此可精确恢复s。还证明Z(Λ_O(P))≃O/(π^{p_s})，作为该中心上的模，HH₀(Λ_O(P))≃⊕_{i=1}^sO/(π^{g_i})，其中g₁=p₁，i≥2时g_i=p_i-p_{i-1}。这些公式分块应用时，可计算任意域上矩阵中心化子代数的对应不变量。最后，多项式主块Morita等价于分裂弦代数当且仅当其定义不可约多项式为线性且指数集为{p}或{p,p+1}；Morita等价于分裂gentle代数当且仅当多项式为线性且指数集为{1}、{2}或{1,2}。这些准则在对应类中产生Morita重构。
英文摘要
Let $R$ be a field and $A$ the endomorphism algebra of a finite direct sum of cyclic modules over a finite-dimensional commutative local principal ideal $R$-algebra. We construct a central quotient showing that the support $τ$-tilting poset of $A$ is isomorphic to the poset of the symmetric group with the weak order. We show that the center of $A$ and degree-zero Hochschild homology, viewed as a module over the center, determine the truncated local algebra and the multiset of successive length gaps. For centralizer matrix algebras, the support $τ$-tilting poset determines the multiset of distinct-exponent counts of the primary blocks. The corresponding algebra--module pair also recovers their local algebras and gap multisets. Combining this reconstruction with the known derived equivalence classification, we characterize derived equivalence by isomorphism of these algebra-module pairs. We apply the results to Morita reconstruction in the string and gentle classes.
发表机构
- School of Mathematics, Hangzhou Normal University（杭州师范大学数学学院）
- School of Mathematics and Statistics, Guizhou University（贵州大学数学与统计学院）
- School of Artificial Intelligence, Jianghan University（江汉大学人工智能学院）
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