Let {k} be an algebraically closed field, and {X} a projective variety over {k}. In the previous two posts, we’ve defined the Picard scheme {\mathrm{Pic}_X}, stated (without proof) the theorem of Grothendieck giving conditions under which it exists, and discussed the infinitesimal structure of {\mathrm{Pic}_X} (or equivalently of the connected component {\mathrm{Pic}^0_X} at the origin).

We saw in particular that the tangent space to the Picard scheme could be computed via

\displaystyle T \mathrm{Pic}^0_X = H^1(X, \mathcal{O}_X),

by studying deformations of a line bundle over the dual numbers. In particular, in characteristic zero, a simply connected smooth variety has trivial {\mathrm{Pic}^0_X}. To get interesting {\mathrm{Pic}_X^0}‘s, we should be looking for non-simply connected varieties: abelian varieties are a natural example.

Let {X} be an abelian variety over {k}. The goal in this post is to describe {\mathrm{Pic}^0_X}, which we’ll call the dual abelian variety (we’ll see that it is in fact smooth). We’ll in particular identify the line bundles that it parametrizes. Most of this material is from David Mumford’s Abelian varieties and Alexander Polischuk’s Abelian varieties, theta functions, and the Fourier transform. I also learned some of it from a class that Xinwen Zhu taught last spring; my (fairly incomplete) notes from that class are here(more…)

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In this post, I’d like to describe a toy analog of the Sullivan conjecture. Recall that the Sullivan conjecture considers (pointed) maps from {BG} into a finite complex, and states that the space of such is contractible if G is finite. The stable version replaces {BG} with the Eilenberg-MacLane spectrum:

 

Theorem 13 Let {H \mathbb{F}_p} be the Eilenberg-MacLane spectrum. Then the mapping spectrum

\displaystyle (S^0)^{H \mathbb{F}_p}

is contractible. In particular, for any finite spectrum {F}, the graded group of maps {[H \mathbb{F}_p, F] = 0}.

 

In the previous post, I sketched a proof (from Ravenel’s “Localization” paper) of this result based on a little chromatic technology. The spectrum {H \mathbb{F}_p} is dissonant: that is, the Morava {K}-theories don’t see it. However, any finite spectrum is harmonic: that is, local with respect to the wedge of Morava {K}-theories. It follows formally that the spectrum of maps {H \mathbb{F}_p \rightarrow S^0} is contractible (and thus the same with {S^0} replaced by any finite spectrum). The non-formal input was the fact that {S^0} is in fact harmonic, which requires a little work.

In this post, I’d like to sketch an earlier proof of the above theorem. This proof is based on the Adams spectral sequence. In fact, the proof runs parallel to Miller’s proof of the Sullivan conjecture. There is a classical Adams spectral sequence for computing {[H \mathbb{F}_p, S^0]}, with {E_2} page given by

\displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}}(\mathbb{F}_p, \mathcal{A}) \implies [ H \mathbb{F}_p, S^0]_{t-s} ,

with {\mathcal{A}} the (mod {p}) Steenrod algebra.

It turns out, however, for purely algebraic reasons, that the {E_2} term is trivial. Miller’s proof of the Sullivan conjecture relies on more complicated algebra to show that the unstable version of all this has the same vanishing property at {E_2}. Most of this material is from Margolis’s Spectra and the Steenrod algebra. (more…)

The next goal of this series of posts (started here) is to analyze the oriented cobordism spectrum {MSO} at the prime 2; the main result is that there is a splitting of {MSO_{(2)}} into a direct sum of copies of {H\mathbb{Z}_{(2)}} (the torsion-free part) and {H \mathbb{Z}/2} (the torsion-part). In particular, it will follow that there is only torsion of order two in the cobordism ring — since we showed last time that there was no odd torsion. We will see this using the Adams spectral sequence at the prime {2}, once we’ve figured out what {H_*(MSO; \mathbb{Z}/2)} looks like as a comodule over the dual Steenrod algebra. This, however, is apparently somewhat tricky to do directly.

In order to get there, we’ll need a bit of algebraic machinery (which we state in a dual context). Recall that a graded vector space {V} is called connected if {V_0} is one-dimensional and {V_i = 0} for {i < 0}. The next result provides a sufficient criterion for a module over a graded, connected Hopf algebra to be free.

Theorem 5 (Milnor-Moore) Let {A} be a connected, graded Hopf algebra over a field {k}, and let {M} be a graded, connected {{A}}-module which is simultaneously a coalgebra (in such a way that {M \rightarrow M \otimes_k M} is an {A}-homomorphism). Let {1 \in M_0} be a generator, and suppose the map of {A}-modules

\displaystyle A \rightarrow M, \quad a \mapsto a . 1

is a monomorphism. Then {M} is a free graded {A}-module.

This is a pretty surprising result, as a relatively minor hypothesis (coalgebra, and the action on {1} is free) leads to freeness of the whole thing. The idea of the proof is going to be to produce generators of {M} by lifting a vector space basis of {\overline{M} = M \otimes_A k}. The fact that these generators are forced to be linearly independent is an unexpected consequence of the coalgebra structure; the graded connectedness will be used to make certain inductive arguments. (more…)

Let {k} be a field of characteristic zero. The intuition is that in this case, a Lie algebra is the same data as a “germ” of a Lie group, or of an algebraic group. This is made precise in the following:

Theorem 1 There is an equivalence of categories between:

  1. Cocommutative Hopf algebras over {k} which are generated by a finite number of primitive elements.
  2. Finite-dimensional Lie algebras.
  3. Infinitesimal formal group schemes over {k} (with finite-dimensional tangent space), i.e. those which are thickenings of one point.
  4. Formal group laws (in many variables).
The result about Hopf algebras is a classical result of Milnor and Moore (of which there is a general version applying in characteristic p); the purpose of this post is (mostly) to describe how it follows from general nonsense about group schemes.  (more…)