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weakly periodic cohomology theory

Definition

A multiplicative cohomology theory $A$ is weakly periodic if the natural map

$$A^2(*){\otimes }_{A^0(*)}{A}^n(*)\stackrel{\simeq }{\to }{A}^{n+2}(*)$$A^2({*}) \otimes_{A^0({*})} A^n({*}) \stackrel{\simeq}{\to} A^{n+2}({*})

is an isomorphism for all $n\in \mathbb{Z}$.

Compare with the notion of a periodic cohomology theory.

Relation to formal groups

One reason why weakly periodic cohomology theories are of interest is that their cohomology ring over the space $ℂP^{\mathrm{\infty }}$ defines a formal group.

To get a formal group from a weakly periodic, even multiplicative cohomology theory $A^{•}$, we look at the induced map on $A^{•}$ from a morphism

$$i_0:*\to ℂP^{\mathrm{\infty }}$$i_0 : {*} \to \mathbb{C}P^\infty

and take the kernel

$$J:=\ker (i_0^{*}:A^0(ℂP^{\mathrm{\infty }})\to A^0(*))$$J := ker(i_0^* : A^0(\mathbb{C}P^\infty) \to A^0({*}))

to be the ideal that we complete along to define the formal scheme $\mathrm{Spf}{A}^0(ℂP^{\mathrm{\infty }})$ (see there for details).

Notice that the map from the point is unique only up to homotopy, so accordingly there are lots of chocies here, which however all lead to the same result.

The fact that $A$ is weakly periodic allows to reconstruct the cohomology theory essentially from this formal scheme.

To get a formal group law from this we proceed as follows: if the Lie algebra $\mathrm{Lie}(\mathrm{Spf}{A}^0(ℂP^{\mathrm{\infty }}))$ of the formal group

$$\mathrm{Lie}(\mathrm{Spf}{A}^0(ℂP^{\mathrm{\infty }}))\simeq \ker (i_0^{*})/\ker (i_0^{*}{)}^2$$Lie(Spf A^0(\mathbb{C}P^\infty)) \simeq ker(i_0^*)/ker(i_0^*)^2

is a free $A^0(*)$-module, we can pick a generator $t$ and this gives an isomorphism

$$\mathrm{Spf}(A^0(ℂP^{\mathrm{\infty }}))\simeq \mathrm{Spf}(A^0(*)[[t]])$$Spf(A^0(\mathbb{C}P^\infty)) \simeq Spf(A^0({*})[[t]])

if $A^0(ℂP^{\mathrm{\infty }})A^0(*)[[t]]$ then $i_0^{*}$ “forgets the $t$-coordinate”.

Related entries

• A Survey of Elliptic Cohomology - formal groups and cohomology