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weakly periodic cohomology theory

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Cohomology

cohomology

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Contents

Definition

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

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

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

Compare with the notion of a periodic cohomology theory.

Properties

Relation to formal groups

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

To get a formal group from a weakly periodic, even multiplicative cohomology theory A A^\bullet, we look at the induced map on A A^\bullet from a morphism

i 0:*P i_0 : {*} \to \mathbb{C}P^\infty

and take the kernel

J:=ker(i 0 *:A 0(P )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 SpfA 0(P )Spf A^0(\mathbb{C}P^\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 AA 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 Lie(SpfA 0(P ))Lie(Spf A^0(\mathbb{C}P^\infty)) of the formal group

Lie(SpfA 0(P ))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(*)A^0({*})-module, we can pick a generator tt and this gives an isomorphism

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

if A 0(P )A 0(*)[[t]]A^0(\mathbb{C}P^\infty) A^0({*})[ [t] ] then i 0 *i_0^* “forgets the tt-coordinate”.

Revised on June 19, 2013 23:38:03 by Urs Schreiber (82.169.65.155)