Chapter 8: Goerss–Hopkins obstruction theory. Goerss–Hopkins ob-
struction theory is a technical apparatus for approaching questions such as the fol-
lowing: for a ring spectrum E and a commutative E∗-algebra A in E∗E-comodules,
is there an E∞-ring spectrum X such that E∗X is equivalent to A? What is the
homotopy type of the space of all such E∞-ring spectra X?
That space is called the realization space of A and is denoted BR(A). There
is an obstruction theory for specifying points of BR(A), and the obstructions
live in certain Andr´ e–Quillen cohomology groups of A. More precisely, there is
a Postnikov-type tower
. . . BRn(A) BRn−1(A) . . . BR0(A)
with inverse limit BR(A) whose layers are controlled by the Andr´ e–Quillen co-
homology groups of A, as follows. If we let Hn+2(A;ΩnA) be the Andr´e–Quillen
cohomology space (the Eilenberg–MacLane space for the Andr´ e–Quillen cohomol-
ogy group) of the algebra A with coefficients in the nth desuspension of A, then
Hn+2(A;ΩnA) is acted on by the automorphism group of the pair (A, ΩnA) and
we can form, by the Borel construction, a space Hn+2(A;ΩnA) over the classifying
space of Aut(A, ΩnA). This is a bundle of pointed spaces and the base points pro-
vide a section BAut(A,

The spaces BRn(A) then fit into
homotopy pullback squares
BRn(A) BAut(A,
BRn−1(A) Hn+2(A;ΩnA).
Chapter 9: From spectra to stacks. We have focussed on constructing
spectra using stacks, but one can also go the other way, associating stacks to spectra.
Given a commutative ring spectrum X, let MX be the stack associated to the Hopf
(MU∗X, MU∗MU ⊗MU∗ MU∗X).
If X is complex orientable, then MX is the scheme Spec(π∗X)—the stackiness of
MX therefore measures the failure of complex orientability of X. The canonical
Hopf algebroid map (MU∗,MU∗MU) (MU∗X, MU∗MU ⊗MU∗ MU∗X) induces
a map of stacks from MX to MFG,
the moduli stack of formal groups with first
order coordinate. Moreover, under good circumstances, the stack associated to a
smash product of two ring spectra is the fiber product over

= MX ×
MY .
It will be instructive to apply the above isomorphism to the case when Y is
tmf , and X is one of the spectra in a filtration
= X(1) X(2) · · · X(n) · · · MU
of the complex cobordism spectrum. By definition, X(n) is the Thom spectrum
associated to the subspace ΩSU(n) of ΩSU BU; the spectrum X(n) is an E2-
ring spectrum because ΩSU(n) is a double loop space. Recall that for a complex
orientable theory R, multiplicative maps MU R correspond to coordinates on
the formal group of R. There is a similar story with X(n) in place of MU, where the
formal groups are now only defined modulo terms of degree n+1, and multiplicative
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