Grobner bases on projective bimodules and the Hochschild cohomology : Part I. Rewriting on vector spaces(Algorithmic problems in algebra, languages and computation systems)
Yuji Kobayashi · Kyoto University Research Information Repository (Kyoto University) · 2006
In this paper we consider an algebra $F$ based on a suitable well-ordered semigroup over a commutative ring $K$ .We develop the theory of Gr\"obner bases on the algebra $F$ as well as Gr\"obner bases on projective $F$ -(bi)modules.We generalize the meth- ods developed in [3] and [4] to construct projective bimodule resolutions of algebras and $(\mathrm{b}\mathrm{i})\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}s$ .It gives an effective way to compute the Hochschild cohomology of algebras and modules.To discuss the three types of Gr\"obner bases above in a uniform way, we consider rewriting systems on free $K$ -modules generated by a well-ordered set in Section 1.The results are applied to algebras based on well-ordered reflexive semigroups in Section 2, projective left modules in Section 3 and projective bimodules in Section 4.1 Rewriting on K-spaces Let $K$ be a commutative ring with 1 and let $(X,$ $\succ)$ be a well ordered set.Let $K\cdot X$ be the free $K$ -module generated by $X$ .An element $f$ of $K\cdot X$ is uniquely written as a finite sum $f= \sum k_{i}x_{i}$ (2.1) with $k_{i}\in K\backslash \{0\}$ and $x_{i}\in X$ , where $x_{i}$ are different.For this $f$ , if $x_{j}\succ x_{i}$ for all $j eq i,$ $k_{i}x_{i}$ is the leading term of $f$ and is denoted by $1\mathrm{t}(f)$ .Set $\mathrm{r}\mathrm{t}(f)=f-1\mathrm{t}(f)$ .We extend the order $\succ$ on $X$ to a partial order $\succ$ on $K\cdot X$ denoted by the same $\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{b}\mathrm{o}\mathrm{l}\succ$ as follows: First, $f\succ \mathrm{O}$ for any $f eq 0$ .Let $f$ and $g$ be nonzero elements in $K\cdot X$ with the leading terms $k\cdot x$ and $\ell\cdot y$ with $k,$ $\ell\in K$ and'This is a preliminary report and the details will appear elsewhere.