A periodic cohomology theory?

Question

I'm studying Lurie's paper, A survey of elliptic cohomology. He defines, in the third page, a periodic cohomology to be a cohomology theory $A$ in which there is an element $\beta$ in $A^{-2}(*)$ so that $\beta$ is invertible in $A^*(*)$. He says that this means $\beta$ has an inverse in $A^2(*)$. I cannot understand why this inverse should be in $A^2(*)$. I'm aslo confused with the definition of periodicity here. Is there a relationship between this definition and other definitions of periodicity in other contexts? (for example, periodic functions ...)

Answer 1

The inverse of $\beta$ lives in $A^2(*)$ purely for degree reasons. We know that is some $\gamma\in A^*(*)$ such that $\beta\gamma=1\in A^0(*)$, so $|\beta|+|\gamma|=0$ where $|\cdot|$ denotes the degree of a (homogeneous) element.

The reason such a theory deserves to be called periodic is that multiplication by $\beta$ is invertible, and so induces an isomorphism $A^k(*)\to A^{k+2}(*)$ for all indices $k$. In fact, for any space $X$, since $A^*(X)$ is a (graded) $A^*(*)$-module, multiplication by $\beta$ induces an isomorphism $A^k(X)\to A^{k+2}(X)$, so $A^*$ is periodic in the usual sense.