标题： 比较布尔复形与其分半细分的面环 显示英文标题

标题： Comparing the face rings of a boolean complex and its barycentric subdivision

摘要： 我们考虑单纯复形或布尔复形$\Delta$的 Stanley-Reisner 环（也称为面环）与其分法细分的 Stanley-Reisner 环之间的关系。这些环共享一个显著的参数子环。S. Murai 询问它们是否在自同构群$\operatorname{Aut}(\Delta)$的作用下，作为该参数子环上的模是同构的。我们证明，在一般情况下，答案是否定的，但对于特征与$|\operatorname{Aut}(\Delta)|$互质的 Cohen-Macaulay 复形，答案是肯定的，并且我们给出了一个显式构造的同构。为了给出这个构造，我们改编并推广了 A. Garsia 在 1980 年引入的一对工具。第一个工具将基从一个 Stanley-Reisner 环转移到其 Gröbner 退化相关的紧密相关环，第二个工具则识别要转移的基。 显示英文摘要

摘要： We consider the relationship between the Stanley-Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $\Delta$ and that of its barycentric subdivision. These rings share a distinguished parameter subring. S. Murai asked if they are isomorphic, equivariantly with respect to the automorphism group $\operatorname{Aut}(\Delta)$, as modules over this parameter subring. We show that, in general, the answer is no, but for Cohen-Macaulay complexes in characteristic coprime to $|\operatorname{Aut}(\Delta)|$, it is yes, and we give an explicit construction of an isomorphism. To give this construction, we adapt and generalize a pair of tools introduced by A. Garsia in 1980. The first one transfers bases from a Stanley-Reisner ring to closely related rings of which it is a Gr\"obner degeneration, and the second identifies bases to transfer.