Gröbner bases of associative algebras and the Hochschild cohomology
Gröbner bases of associative algebras and the Hochschild cohomology
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DOI：
10.1090/s0002-9947-04-03556-1
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发表时间：
2004-07
影响因子：
1.3
通讯作者：
Yuji Kobayashi
中科院分区：
文献类型：
--
作者：
Yuji Kobayashi
本文给出了一种构造允许Grobner基的代数的自由双模归结的算法方法。它使我们能够计算代数的Hochschild（上）同调。设A是自由代数F上的交换环K上的Grobner基G（可能是无限的）的生成代数，即A是F与G生成的F的理想I（G）的商F/I（G）.给定自由A-双模A的A-子双模L的Grobner基H。X . A = A K K。X <$K A由一个集合X生成，我们有一个来自自由A-双模A的A-双模的态射<$。H . A由H生成为A。X . A将生成器[h]发送给元素h ∈ H。我们在F上构造了一个Grobner基C。H . F对于A的A-子双模Ker（Kr）. H .有了这个C，我们就有了自由A-双模A。C .由C生成的A和正合序列A。C . A → A。H . A . A . X . A.把这个构造归纳地应用于A-双模A本身，我们得到A的自由A-双模分解。
We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Grobner base. It enables us to compute the Hochschild (co)homology of the algebra. Let A be a finitely generated algebra over a commutative ring K with a (possibly infinite) Grobner base G on a free algebra F, that is, A is the quotient F/I(G) with the ideal I(G) of F generated by G. Given a Grobner base H for an A-subbimodule L of the free A-bimodule A . X . A = A K ⊗ K . X ⊗ K A generated by a set X, we have a morphism ∂ of A-bimodules from the free A-bimodule A . H . A generated by H to A . X . A sending the generator [h] to the element h ∈ H. We construct a Grobner base C on F . H . F for the A-subbimodule Ker(∂) of A . H . A, and with this C we have the free A-bimodule A . C . A generated by C and an exact sequence A . C . A → A . H . A . A . X . A. Applying this construction inductively to the A-bimodule A itself, we have a free A-bimodule resolution of A.