14

M. HOVEY AND N. P. STRICKLAND

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] =

HOIU^ACAJCX,

«/*)

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*, 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] —

HOITIA^A(A^Z,

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

Es/

= 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.