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I have problems to understand a definition from lecture:

Let $R$ be a commutative ring with unit and such that $2$ is invertible in $R$. The free graded commutative algebra in generators $a_1, .., a_k$ with degree $|a_1|,..,|a_k|$ and with $a^2=0$ over $R$ is the graded commutative algebra $\Lambda_R[a_1,..,a_k]$ with elements of degree $N$ are formal sums $$\sum\limits_{i_1|a_1|+..+i_n|a_n|=N, i_1,..,i_n\ge 0, i_j\in \{0,1\}\; \text{if}\; |a_j|\;\text{is odd} }r_{i_1,..,i_n}a_1^{i_1}\cdot ..\cdot a_n^{i_n}.$$

The product is the unique $R$-linear associative graded commutative product such that $a_\lambda^j\cdot a_{\lambda'}^{j'}=a_\lambda^j a_{\lambda'}^{j'}$ if $\lambda <\lambda'$ and $a_\lambda^j\cdot a_{\lambda'}^{j'}=a_\lambda^{j+j'}$ if $|a_\lambda|$ is odd.

My problem is why the description of the product "$a_\lambda^j\cdot a_{\lambda'}^{j'}=a_\lambda^j a_{\lambda'}^{j'}$ if $\lambda <\lambda'$ and $a_\lambda^j\cdot a_{\lambda'}^{j'}=a_\lambda^{j+j'}$ if $|a_\lambda|$ is odd" could be graded commutative. Then I understand, that $a^2=0$ is satisfied for all $a$ with odd degree. It follows that if all $a_i$ have even degree, I have something similar of a polynomial ring in $n$ variables. But The multiplication of polynomials is not graded commutative in general, or am I wrong? If i'm wrong, can you give me a simple example (maybe something like $\mathbb{Z}[x]$)? Can we elaborate 1-2 simple examples of free graded commutative algeras as in the definition? This would be great, I'm frustrated:(. Best.

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  • $\begingroup$ What does "with $a^2=0$" mean? $\endgroup$ – Eric Wofsey Feb 3 '16 at 8:34
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Instead of "$a_\lambda^j\cdot a_{\lambda'}^{j'}=a_\lambda^{j+j'}$ if $|a_\lambda|$ is odd", it should just say "$a_\lambda^j\cdot a_{\lambda}^{j'}=a_\lambda^{j+j'}$ (hopefully this makes more sense). Strictly speaking, you should also clarify that "$a_\lambda^j$" is a shorthand for the formal expression $a_1^{i_1}\dots a_n^{i_n}$ where $i_\lambda=j$ and $i_{\lambda'}=0$ for all $\lambda'\neq\lambda$.

You're right that if $|a_i|$ is even for all $n$, you just get a polynomial ring. A polynomial ring is graded-commutative as long as all the variables have even degree; more generally, a graded ring concentrated in even degrees is graded-commutative iff it is commutative. In general, $\Lambda_R[a_1,..,a_k]$ is naturally isomorphic to the tensor product of an exterior algebra on the $a_i$ of odd degree and a polynomial algebra on the $a_i$ of even degree.

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