Formal Groups (Lecture 11)
April 27, 2010
We begin by recalling our discussion of the Adams-Novikov spectral sequence:
Claim 1. Let X be any spectrum. Then MU (X ) is a module over the commutative ring L = MU, and
can therefore be understood as a quasi-co
MU and Complex Orientations (Lecture 6)
February 4, 2010
In the last lecture, we dened spectra MU(n) = 2n BU (n)/BU (n 1) which form a direct system
MU(0) MU(1) MU(2)
The (homotopy) colimit of this sequence is called the complex bordism spectrum and is d
The Homology of MU (Lecture 7)
February 9, 2010
Last week, we dened the complex bordism spectrum MU and showed that it was a universal complex
oriented cohomology theory. In particular, there is a formal group law f (x, y ) over the ring MU. This
formal g
Complex Bordism (Lecture 5)
February 4, 2010
In this lecture, we will introduce another important example of a complex-oriented cohomology theory:
the cohomology theory MU of complex bordism. In fact, we will show that MU is universal among complexoriente
Complex-Oriented Cohomology Theories (Lecture 4)
February 1, 2010
In this lecture, we will introduce the notion of a complex-oriented cohomology theory E . We will generally
not distinguish between a cohomology theory E and the spectrum that represents it
Lazards Theorem (Lecture 2)
April 27, 2010
Let R be a commutative ring. We recall that a formal group law over R is a power series f (x, y ) R[x, y ]
satisfying the identities
f (x, 0) = f (0, x) = x
f (x, y ) = f (y, x)
f (x, f (y, z ) = f (f (x, y ), z
Lazards Theorem (Continued) (Lecture 3)
January 28, 2010
Our goal in this lecture is to complete the proof of Lazards theorem. In the last lecture, we were reduced
to proving the following result:
Lemma 1. Let : L Z[b1 , b2 , . . .] be the ring homomorphi
The Adams Spectral Sequence (Lecture 8)
April 27, 2010
Recall that our goal this week is to prove the following result:
Theorem 1 (Quillen). The universal complex orientation of the complex bordism spectrum MU determines
a formal group law over MU. This f
The Adams Spectral Sequence for MU (Lecture 9)
February 18, 2010
In this lecture, we will apply the Adams spectral sequence to obtain information about the homotopy
ring MU. Let us begin by recalling the major conclusions of the last lecture:
(1) Let X be
Classication of Formal Groups (Lecture 14)
April 27, 2010
Our goal in this lecture is to prove Lazards theorem, which asserts that a formal group law over an
algebraically closed eld is determined up to isomorphism by its height. We will prove this result
Flat Modules over MFG (Lecture 15)
April 27, 2010
We have seen that if E is a complex oriented cohomology theory, then the coecient ring E has the
structure of an algebra over the Lazard ring L MU. Our next goal is to address the converse: suppose
we are
The Stratication of MFG (Lecture 13)
April 27, 2010
Let p be a prime number, xed throughout this lecture. Our goal is to describe the structure of the
moduli stack MFG Spec Z(p) of formal groups over p-local rings.
We begin by recalling a few denitions fr
Heights of Formal Groups (Lecture 12)
April 27, 2010
Our next goal in this course is to understand the structure of the moduli stack MFG of formal groups.
Our starting point is the following result from Lecture 2:
Proposition 1. Let R be a ring of charact
The Proof of Quillens Theorem (Lecture 10)
February 19, 2010
At the end of the last lecture, we arrived at the following conclusion for the prime p = 2:
Proposition 1. The second page of the mod p Adams spectral sequence for MU is given by
E2 ,
Fp [bi , j
Introduction (Lecture 1)
January 22, 2010
A major goal of algebraic topology is to study topological spaces by means of algebraic invariants (such
as homology or cohomology). There is a balance to be struck here: we would like our invariants to be
simple