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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Gröbner bases of associative algebras and the Hochschild cohomology


Author: Yuji Kobayashi
Journal: Trans. Amer. Math. Soc. 357 (2005), 1095-1124
MSC (2000): Primary 16E05, 16E40, 16S15
Published electronically: July 16, 2004
MathSciNet review: 2110434
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Abstract: We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner 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) Gröbner 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 Gröbner base $H$ for an $A$-subbimodule $L$ of the free $A$-bimodule $A \cdot X \cdot A = A_K \otimes K \cdot X \otimes_KA$ generated by a set $X$, we have a morphism $\partial$ of $A$-bimodules from the free $A$-bimodule $A \cdot H \cdot A$ generated by $H$ to $A \cdot X \cdot A$ sending the generator $[h]$to the element $h \in H$. We construct a Gröbner base $C$ on $F \cdot H \cdot F$ for the $A$-subbimodule Ker($\partial$) of $A \cdot H \cdot A$, and with this $C$ we have the free $A$-bimodule $A \cdot C \cdot A$ generated by $C$ and an exact sequence $A \cdot C \cdot A \rightarrow A \cdot H \cdot A \rightarrow A \cdot X \cdot A$. Applying this construction inductively to the $A$-bimodule $A$ itself, we have a free $A$-bimodule resolution of $A$.


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Additional Information

Yuji Kobayashi
Affiliation: Department of Information Science, Toho University, Funabashi 274-8510, Japan
Email: kobayasi@is.sci.toho-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03556-1
PII: S 0002-9947(04)03556-1
Received by editor(s): September 10, 2002
Received by editor(s) in revised form: September 9, 2003
Published electronically: July 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society