2.2. The iS-based Adams spectral sequence. We next briefly recall an ap-
proach to the jE?-based Adams spectral sequence that we learnt from Mike Hop-
kins, which is explained in more detail in [Dev97]. This approach is also used by
Franke [Fra96], who attributes it to Brinkmann. Let A be an even Landweber exact
commutative ring spectrum; in our applications, A will be E or E(n). By Proposi-
tion 2.16* A*A is flat as a left module over A*. Similarly, it is flat as a right module.
It follows in the usual way that it is a Hopf algebroid, so we can think about co-
modules over A*A. If/* is an injective module over A* then the extended comodule
A*A &Am £* is injective. It follows that there are enough injective comodules, and
that they can be used to define Ext groups. If J* is an injective comodule then
Brown representability gives a spectrum W such that [X, W] =
for all X. The identity map of W corresponds to a map A*W J*, which we
claim is an isomorphism. Indeed, when X £ £& we know that A*X is free over A*
and using duality we find that
X*W = [DX, W] = H.omAmA(A*DX, J*) = Hom^
(A*, A*X ®Am J*).
By Proposition 2.13, we can write A = mwlim Aa for Aa G S?. By taking X = Aa
and passing to the limit we find that
A*W = HomA,A (A*, A*A gA* J*) = J*.
There is some inconsistency in the literature about what to call spectra such as
W. We will use the following terminology.
Definition 2.23. Let A be a ring spectrum. We say that a spectrum X is A-
injective if it is a retract of A A Y for some Y. Suppose in addition that A* A is flat
as a module over A*. We say that X is strongly A-injective if A*X is an injective
comodule and the natural map [Z,X]
A*X) is an isomorphism
for all Z. Note that if X is A-injective then A*X is injective relative to A*-split
exact sequences, but not necessarily absolutely injective.
If X Xo is any spectrum then we can embed A*Xo in an injective comodule
J* and define W = Wo as above. We then have a map Xo WQ^ and we let X\
be the fibre. Continuing in the obvious way, we get a tower
X = Xo Xi X2 ... .
For any spectrum Y we can apply the functor [Y, —] to get a spectral sequence
= EXtf£A(A.Y,AmX) = » [Y,LAX}t.s.
We call this the modified Adams spectral sequence (MASS) . It need not converge
without additional assumptions. We will prove a convergence result in Proposi-
tion 6.5.
We will also have occasion to use the (unmodified) Adams spectral sequence.
For this, we choose a complex X Jo * h of .E-injective spectra which
becomes a split exact sequence after applying the functor E A (—). Such a complex
is unique up to chain homotopy equivalence under X. It can be converted into
a tower X = Xo X\ ... just as above, and by applying [Y, —] we get a
spectral sequence, called the Adams spectral sequence. If X is J3-nilpotent then this
converges to [Y,X]. If E*Y is projective over E* then the modified and unmodi-
fied Adams spectral sequences coincide from the E2 page onwards [Dev97], and in
particular the E2 page can be identified as an Ext group.
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