tensor algebra TensorAlgebra 2002-12-18 18:12:33 2007-03-22 06:12:53 Definition tensor power % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\mc}[1]{\mathcal{#1}} Let $R$ be a commutative ring, and $M$ an $R$-module. The \emph{tensor algebra} \[ \mc{T}(M) = \bigoplus_{n=0}^\infty \mc{T}_n(M)\] is the graded $R$-algebra with $n^{th}$ graded component simply the $n^{th}$ tensor power: \[ \mc{T}_n(M) = M^{\otimes n} =\overbrace{M\otimes \cdots \otimes M}^{n\text{ times}},\quad n=1,2,\ldots,\] and $\mc{T}_0(M)=R$. The multiplication $m:\mc{T}(M)\times \mc{T}(M)\to\mc{T}(M)$ is given by the usual tensor product: \[ m(a,b)=a\otimes b,\quad a\in M^{\otimes n},\; b\in M^{\otimes m}.\] \paragraph{Remark 1.} One can generalize the above definition to cover the case where the ground ring $R$ is non-commutative by requiring that the module $M$ is a bimodule with $R$ acting on both the left and the right. \paragraph{Remark 2.} From the point of view of category theory, one can describe the tensor algebra construction as a functor $\mc{T}$ from the category of $R$-module to the category of $R$-algebras that is left-adjoint to the forgetful functor $\mc{F}$ from algebras to modules. Thus, for $M$ an $R$-module and $S$ an $R$-algebra, every module homomorphism $M\to \mc{F}(S)$ extends to a unique algebra homomorphism $\mc{T}(M)\to S$.