Cohomology ring of a torus
= Cohomology ring of a torus
{title2=$H^*(T^r;R)\cong\Lambda_R(x_1,\ldots,x_r),\quad |x_i|=1$}
Since $T^r$ is a product of circles, the <Künneth theorem> identifies its <cohomology ring> with the <exterior algebra> on the coordinate degree-one classes. Inversion negates each integral degree-one class, so it acts as $(-1)^q$ in degree $q$. Even in characteristic two, each degree-one class has square zero.