8 CORBETT REDDEN
Example 2.2. As noted above, the formal group law obtained from ordinary
cohomology is F+ = F (x1,x2) = x1 +x2, and is known as the additive formal group
Example 2.3. The multiplicative formal group law is defined by
F×(x1,x2) = x1 + x2 − x1x2.
One can explicitly verify it satisfies the definition of a formal group law. One can
also see that it is obtained from the standard complex orientation of K-theory.
Since K-theory is even periodic, we place the classes c1 in degree 0. The resulting
formal group law is over the ring K0(pt) = Z and involves no grading. (Though,
we could use the Bott element and its inverse to maintain the grading |c1| = 2 if
To the universal line bundle ξ →
we define the universal first Chern class
to be 1−[ξ] ∈
The term 1 is included so that trivial bundles have trivial
first Chern class. Hence, for any line bundle L → X,
c1(L) = 1 − [L] ∈
A simple calculation demonstrates
c1(L1 ⊗ L2) = 1 − L1 ⊗ L2
= (1 − L1) + (1 − L2) − (1 − L1)(1 − L2)
= c1(L1) + c1(L2) − c1(L1)c1(L2),
demonstrating that the multiplicative formal group law is obtained from K-theory.
Any ring homomorphism R → S induces a map of formal group laws FGL(R) →
FGL(S). In fact, there is a universal formal group law Funiv ∈ Runiv x1,x2 such
that any F ∈ FGL(R) is induced by a ring homomorphism Runiv → R. The
existence of Runiv is easy, since one can construct it formally by
Runiv = Z[aij]/ ∼
where aij is the coeﬃcient of x1x2,
and ∼ represents all equivalence relations in-
duced by the three axioms of a formal group law. Though this description is quite
unwieldy, a theorem by Lazard shows that this ring is isomorphic to a polynomial
L := Z[a1,a2,...]
where |ai| = −2i if we include the grading.
A complex orientation of E therefore induces a map L →
defining the formal
group law. Earlier we noted that complex orientations are basically equivalent to
maps of ring spectra MU → E, so MU has a canonical complex orientation given
by the identity map MU → MU . The following important theorem of Quillen
shows that in addition to MU being the universal complex oriented cohomology
theory, it is also the home of the universal formal group law. It also explains the
grading of the Lazard ring.
Theorem 2.4. (Quillen) The map L → MU
induced from the identity map
MU → MU is an isomorphism.