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Let $M$ be a $\mathbb{Z}$-graded module over a trivial graded ring $R=R_0$. The tensor algebra $T_R(M)$ then becomes a $\mathbb{Z}$-graded module with $i$-th graded component $$T_R(M)_i = \bigoplus_{j_1 + \cdots + j_n=i} M_{j_1}\otimes \cdots\otimes M_{j_n}$$

First of all, are $j_1$ the degrees of the component, i.e, the degrees of the components add up to $i$?

Eisenbud then defines the symmetric algebra to be $\mathcal{S}_R(M)$ to be the $R$-graded algebra $T_R(M)/I$ where $I$ is the ideal generated by skew-commutative relations $$ab -(-1)^{|a||b|}ba.$$

In this case $\mathcal{S}_R(M)$ is such that even-odd and even-even homogenous element commute, while two odd elements will anti-commute.

However, in Eisenbud, I don't see that there is a definition given for the exterior algebra of a graded module.

I imagine in an exterior algebra we would want all element to anti-commute regardless of their degree. Although, I also naively thought that in a symmetric algebra we want all elements to commute regardless of their degree so my first instinct about the exterior algebra is most likely wrong. Furthermore, it seems like the above definition would have worked equally well to define the exterior algebra of a graded module.

What is the definition of the exterior algebra of a graded module??

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  • $\begingroup$ Just change the minus in the generators by a plus. $\endgroup$ – Mariano Suárez-Álvarez Apr 29 '17 at 21:00
  • $\begingroup$ Do you mean $ab+(-1)^{|a||b|}ba$? $\endgroup$ – Jadwiga Apr 29 '17 at 21:10
  • $\begingroup$ You'll also need something of the form $aa$ for all even $a$. $\endgroup$ – darij grinberg Apr 29 '17 at 22:07

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