Origin of the sign convention in the Tensor product of graded vector spaces

Question

Suppose $V := \bigoplus_{i \in \mathbb{N}}V_i$ and $W := \bigoplus_{i \in \mathbb{N}}W_i$ are $\mathbb{N}$-graded vector spaces. Then their graded tensor power is defined by $V \bigotimes W := \oplus_{n \in \mathbb{N}} \oplus_{i+j=n} V_i \otimes W_j$ and there is a natural isomorphism $\sigma_{V,W}: V \bigotimes W \to W \bigotimes V$ defined on homogeneous elements by the 'sign convention'

$$\sigma_{V,W}(v \otimes w) = (-1)^{\deg(v)*\deg(w)}w \otimes v$$

What is the origin of this natural isomorphism? Is this the only one or are there others defined by other sign conventions?

Answer 1

Is this the only one or are there others defined by other sign conventions?

As has already been mentioned - there are essentially two but the Koszul one is amongst other things, the only one which makes the tensor product of complexes a complex.

Since I have been interested in this topic of late, I would point you towards the following internal (within MathOverflow) references:

• For a recent discussion see: Tensor product of linear mappings versus chain complexes

• In particular Simon Letner and Qiaochu Yauns' answers provide a nice conceptual backdrop.

• I also found Theo Johnson-Freyd's answer here quite illuminating: References for sign conventions in homological algebra

• Theo Johnson-Freyd's answer in this thread is also instructive: Sign convention for derivations in CDGAs

Answer 2

If you want to find another $\sigma'_{V,W}\colon V\otimes W\to W\otimes V$ so that $\sigma'_{U,V\otimes W}=\sigma'_{U,V}\sigma'_{U,W}$ and assume that on homogeneous elements $\sigma'_{V,W}(v\otimes w)=f(\deg(v),\deg(w))w\otimes v$ for some elements $f(\cdot,\cdot)\colon\mathbb{N}\times\mathbb{N}\to F$, then it is easy to deduce that $f(a,b)=f(1,1)^{ab}$. This, together with the requirement that $\sigma'_{V,W}\sigma'_{W,V}=Id_{V\otimes W}$, instantly shows that $f(1,1)=\pm1$. But of course there were many assumptions made along the way.

As for the origin, I think that in addition to what Mark Grant says, when you have the product on the cohomology $H^*(M)$ (over a field), you then can apply the Künneth formula and say $H^*(M\times N)=H^*(M)\otimes H^*(N)$. In what sense these two are isomorphic as algebras? To define aa product on the tensor product $H^*(M)\otimes H^*(N)$, you need those isomorphisms $\sigma$, so that you can do $H^*(M)\otimes H^*(N)\otimes H^*(M)\otimes H^*(N)\to H^*(M)\otimes H^*(M)\otimes H^*(N)\otimes H^*(N)$ and then compute the product in $H^*(M)$ and in $H^*(N)$. The Koszul convention is precisely the choice for which you obtain an algebra isomorphism.