24
Jack Palmer Sanders
M is a ring spectrum if there are a pairing w: (M,M) - - M and a map
of spectra U: S -*M such that the following diagrams are homotopy
commutative for all p, q.
1
A
u
p + q
W A 1
M A M A M ^ z S
p q r
M A M
p + q r
1 A W
q , r
p + q , r
p . q + r
M A M ^-^ M
p q + r p + q + r
M A M w
p q _ p , q
p + q
M A M w
q p q / P
Here T is the usual switch map. If M is a ring spectrum, then TT^ (M)
is an anticommutative ring with unit.
N is a module over the ring spectrum M if there is a pairing
g: (M,N) -*N such that
U
A
1
S AN ~ M A N
p q p q
P/q
P/q
p+q
homotopy commutes for all p, q. If N is a module over M, then TT^ (N)
is a left IT (M) -module.
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