Special and general types
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
A cohomology theory is called multiplicative if each graded abelian -cohomology group is compatibly equippd with the structure of a graded ring.
Let be three unreduced generalized cohomology theories (def.). A pairing of cohomology theories
is a natural transformation (of functors on ) of the form
such that this is compatible with the connecting homomorphisms of , in that the following are commuting squares
where the isomorphisms in the bottom left are the excision isomorphisms.
An (unreduced) multiplicative cohomology theory is an unreduced generalized cohomology theory theory (def.) equipped with
(external multiplication) a pairing (def. 1) of the form ;
(unit) an element
(unitality) for all .
The mulitplicative cohomology theory is called commutative (often considered by default) if in addition
Given a multiplicative cohomology theory , its cup product is the composite of the above external multiplication with pullback along the diagonal maps ;
e.g. (Tamaki-Kono 06, II.6)
Ring and module structure on cohomology groups
Let be a multiplicative cohomology theory, def. 2. Then
For every space the cup product gives the structure of a -graded ring, which is graded-commutative if is commutative.
For every pair the external multiplication gives the structure of a left and right module over the graded ring .
All pullback morphisms respect the left and right action of and the connecting homomorphisms respect the right action and the left action up to multiplication by
Regarding the third point:
For pullback maps this is the naturality of the external product: let be a morphism in then naturality says that the following square commutes:
For connecting homomorphisms this is the (graded) commutativity of the squares in def. 2:
Multiplicative Atiyah-Hirzebruch spectral sequences
A proof of prop. 2 via Cartan-Eilenberg systems is given at multiplicative spectral sequence. A proof arguing via representing ring spectra is in (Kochman 96, prop. 4.2.9).
Given a multiplicative cohomology theory (def. 2), then for every Serre fibration all the differentials in the corresponding Atiyah-Hirzebruch spectral sequence
are linear over .
By construction (here) the differentials are those induced by the exact couple
consisting of the pullback homomorphisms and the connecting homomorphisms of .
By the nature of spectral sequences induced from exact couples (this prop.) its differentials on page are the composites of one pullback homomorphism, the preimage of pullback homomorphisms, and one connecting homomorphism of . Hence the statement follows with prop. 1.
Brown representability by ring spectra
A multiplicative structure on a generalized (Eilenberg-Steenrod) cohomology theory is the structure of a ring spectrum on the spectrum that represents it.
e.g. (Tamaki-Kono 06, appendix C, Lurie 10, lecture 4)
In particular every E-∞ ring is a ring spectrum, hence represents a multiplicative cohomology theory, but the converse is in general false.