Chromatic Homotopy Theory

4 Landweber Exact Functor Theorem

Given a ring \(R\) with a formal group law, when is \(R = E_{R*}\) for some complex oriented theory \(E_R\)? One way to try to produce such a cohomology theory would be to define \(E_{R*}(X) = \MU _*(X) \otimes _{\MU _*} E_R\) and hope that it is a cohomology theory. If \(R\) were flat over \(\MU _*\), then the long exact sequence on cohomology would work, so this would produce a ring. However \(\MU _*(X)\) doesn’t really live in the category of \(\MU _*\) modules. Rather it lives in the category of \(\ZZ /2\ZZ \)-graded comodules over the graded Hopf algebroid \((\MU _*, \MU _*\MU )\), which is th same as \(\ZZ /2\ZZ \)-graded quasi-coherent sheaves on the moduli stack of formal groups \(M_{fg}\). Thus it suffices for \(\Spec (R)/\GG _m\) (the \(\GG _m\)-action coming from the grading) to be flat over \(M_{fg}\). Note that we don’t really need \(R\) to be a ring, just a module \(M\) over \(L\).

Let \(M_{fgs}\) be the moduli stack of formal groups with a trivialization of their Lie algebra. The projection \(M_{fgs} \to M_{fg}\) is just the quotient by the \(\GG _m\) action, so it is faithfully flat. Thus it is equivalent that \(M\) is flat over \(M_{fgs}\).

Checking this can be done locally at every prime \(p\), so we will work locally. Then although \(\Spec (L)\) is a faithfully flat cover of \(M_{fgs}\), there is a more efficient cover, coming from the fact that every formal group law has a canonical \(p\)-typicalization. Namely let \(BP_* = \ZZ _{(p)}[v_1,v_2,\dots ]\), be the quotient of \(L = \ZZ _{(p)}[t_1,t_2,\dots ]\) by the ideal \((t_j, j \neq p^k-1)\). The \(v_i\) are canonical representatives of \(t_{p^i-1}\) coming from the \(p\)-series of the formal group law. To show that \(\Spec (BP_*)\) is flat over \(M_{fgs}\) it suffices to pullback to the cover \(\Spec (L)\) and show it there. Since the map \(\Spec (BP_*) \to M_{fgs}\) factors through \(\Spec (L)\), this map can be computed by taking \(L[b_1,\dots ]\) and quotienting by the images of \(t'_i\) for \(j \neq p^k-1\) for the non obvious inclusion \(\ZZ _{(p)}[t'_j] \to L[b_1,\dots ]\). But the proof of Lazard’s theorem shows that the image of these are the \(b_i\) up to decomposables and units, so the quotient is a free algebra over \(L\), in particular faithfully flat. \(\MU _*(X) \otimes _{\MU _*} BP_*\) is Brown-Peterson homology, and the representing spectrum is \(BP\).

Theorem 4.1 (Landweber Exact Functor Theorem). An \(L\)-module \(M\) is flat over \(M_{fgs}\) iff \(p,v_1,v_2,\dots \) forms a regular sequence for all \(p\).

Proof. Since \(BP_*\) is faithfully flat over \(M_{fgs}\) it suffices to show that it is flat after pulling back to \(BP_*\). Let \(M'\) be the pulled back module. It is a module over \(BP_*\otimes _{M_{fg}}L = BP_*[b_1,b_2,\dots ]\). Since the two formal group laws on this ring are isomorphic, it follows that the ideals \((p,v_1,\dots ,v_n)\) and \((p,v'_1,\dots v'_n)\) are the same, where \(v'_n\) is the ones coming from \(L\). Now to show flatness over \(BP_*\), one needs to show that \(Tor_1(M',N) = 0\) for all finitely presented modules \(N\) (every module is a filtered colimit of these). A finitely presented module only involves finitely many of the \(v_i\) so we can assume \(N\) is pulled back from \(\ZZ _{(p)}[v_1,\dots ,v_n]\) for some \(n\).

For any non-zero divisor \(x\) over a commutative ring \(R\), \(M'\) is flat over \(R\) iff \(M'/x\) is flat over \(R/x\), \(x^{-1}M'\) is flat over \(x^{-1}R\), and \(x\) is a nonzero divisor on \(M'\). In particular we see that this being a regular sequence on \(M'\) is necessary, since it is a regular sequence on the ring.

We can apply this using descending induction to the regular sequence \((p,v_1,\dots ,v_n)\) which up to units agrees with \((p,v'_1,\dots v'_n)\) which we have assumed is a regular sequence on \(M'\). So in the base case, we are working over \(\ZZ _{(p)}[v_1,\dots ,v_n]/(p,v_1,\dots ,v_n) = \FF _p\) where everything is flat. In the inductive hypothesis, we have \(M/(p,v_1,\dots v_{k+1})\) is flat over \(\ZZ _{(p)}[v_1,\dots ,v_n]/(p,v_1,\dots ,v_{k+1})\). By our assumption on \(M\) it suffices to show then that \(M'/(p,v_1,\dots v_k)[v_{k+1}^{-1}]\) is flat over \(\ZZ _{(p)}[v_1,\dots ]/(p,v_1,\dots ,v_{k})[v_{k+1}^{-1}]\) But there is a pullback diagram:

Where \(M_{fgs}^{k+1}\) is the moduli stack of formal group laws of height exactly \(k+1\). Thus it suffices to show that \(M/(v_1,\dots ,v_{k})[v_{k+1}^{-1}]\) is flat over \(M_{fgs}^{k+1}\). When \(k+1=0\), this is vecause \(M_{fgs}^0\) is literally \(\Spec \QQ \). For \(k+1>0\), choose a formal group law \(f\) of height \(n\) over \(\FF _p\). For any other formal group law of height \(n\), the universal strict isomorphism with \(F\) is a direct limit of finite etale maps, so it faithfully flat. Thus \(\FF _p\) is faithfully flat over the moduli stack, and since every \(\FF _p\) module is flat, the same is true of the moduli stack. □

Remark 4.2. I think this proof can almost completely be phrased almost completely internally to stacks. \(v_n\) descends to a section of a line bundle on \(M_{fg}\) (the dual of the Lie algebra of the universal formal group), and for any locally presentable stack, you can probably say that if \(f\) is a non zero-divisor on a quasicoherent sheaf \(M\) and \(M/f\) and \(M[f^{-1}]\) are flat over the zero locus and open substacks, then \(M\) is flat. Then since \(M_{fg}\) is the inductive limit of \(M_{fg}^{\leq n}\) it suffices to prove flatness over \(M_{fg}^{\leq n}\), and this can be done inductively using the fact that every quasicoherent sheaf on \(M_{fgs}^{k+1}\) is flat.