| Version 8 |
Version 7 |
| Let $R$ be a commutative ring, and $M$ an $R$-module. |
Let $R$ be a commutative ring, and $M$ an $R$-module. |
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The \emph{tensor algebra}
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The tensor algebra
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| \[ \mc{T}(M) = \bigoplus_{n=0}^\infty \mc{T}_n(M)\] |
\[ \mc{T}(M) = \bigoplus_{n=0}^\infty \mc{T}_n(M)\] |
| is the graded $R$-algebra with $n$-th |
is the graded $R$-algebra with $n$-th |
| graded component simply the $n$th tensor power: |
graded component simply the $n$th tensor power: |
| \[ \mc{T}_n(M) = M^{\otimes n} =\overbrace{M\otimes \cdots \otimes |
\[ \mc{T}_n(M) = M^{\otimes n} =\overbrace{M\otimes \cdots \otimes |
| M}^{n\text{ times}},\quad n=1,2,\ldots,\] |
M}^{n\text{ times}},\quad n=1,2,\ldots,\] |
| and $\mc{T}_0(M)=R$. |
and $\mc{T}_0(M)=R$. |
| The multiplication $m:\mc{T}(M)\times \mc{T}(M)\to\mc{T}(M)$ is given |
The multiplication $m:\mc{T}(M)\times \mc{T}(M)\to\mc{T}(M)$ is given |
| by the usual tensor product: |
by the usual tensor product: |
| \[ m(a,b)=a\otimes b,\quad a\in M^{\otimes n},\; b\in M^{\otimes m}.\] |
\[ m(a,b)=a\otimes b,\quad a\in M^{\otimes n},\; b\in M^{\otimes m}.\] |
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| \paragraph{Remark 1.} One can generalize the above definition to |
\paragraph{Remark 1.} One can generalize the above definition to |
| cover the case where the ground ring $R$ is non-commutative by |
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 |
requiring that the module $M$ is a bimodule with $R$ acting on both |
| the left and the right. |
the left and the right. |
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| \paragraph{Remark 2.} From the point of view of category theory, one |
\paragraph{Remark 2.} From the point of view of category theory, one |
| can describe the tensor algebra construction as a functor $\mc{T}$ |
can describe the tensor algebra construction as a functor $\mc{T}$ |
| from the category of $R$-module to the category of $R$-algebras that |
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 |
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 |
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 |
module homomorphism $M\to \mc{F}(S)$ extends to a unique algebra |
| homomorphism $\mc{T}(M)\to S$. |
homomorphism $\mc{T}(M)\to S$. |
