Remark 10.75.6. An interesting case occurs when $M = N$ in the above. In this case we get a canonical map $\text{Tor}_ i^ R(M, M) \to \text{Tor}_ i^ R(M, M)$. Note that this map is not the identity, because even when $i = 0$ this map is not the identity! For example, if $V$ is a vector space of dimension $n$ over a field, then the switch map $V \otimes _ k V \to V \otimes _ k V$ has $(n^2 + n)/2$ eigenvalues $+1$ and $(n^2-n)/2$ eigenvalues $-1$. In characteristic $2$ it is not even diagonalizable. Note that even changing the sign of the map will not get rid of this.

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