Cohomology ring of a torus ($H^{\ast }(T^r;R)\cong {\Lambda }_R(x_1,\dots ,x_r),|x_i|=1$)

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.