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…)

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…)

We are in the middle of proving an important result of Lazard:

Theorem 1 The Lazard ring ${L}$ over which the universal formal group law is defined is a polynomial ring in variables ${x_1, x_2, \dots, }$ of degree ${2i}$.

The fact that the Lazard ring is polynomial implies a number of results which are not a priori obvious: for instance, it shows that given a surjection of rings ${ A \twoheadrightarrow B}$, then any formal group law on ${B}$ can be lifted to one over ${A}$.

We began the proof of Lazard’s theorem last time: we produced a map

$\displaystyle L \rightarrow \mathbb{Z}[b_1, b_2, \dots ], \quad \deg b_i = 2i,$

classifying the formal group law obtained from the additive one ${x+y}$ by the “change of coordinates” ${ \exp(x) = \sum b_i x^{i+1}}$. We claimed that the map on indecomposables was injective, and that, in fact the image in the indecomposables of ${\mathbb{Z}[b_1, b_2, \dots ]}$ could be determined completely. I won’t get into the details of this (it was all in the previous post), because the purpose of this post is to prove a result to which we reduced last time.

Let ${A}$ be an abelian group. A symmetric 2-cocycle is a “polynomial” ${P(x,y) \in A[x, y] = A \otimes_{\mathbb{Z}} \mathbb{Z}[x, y]}$ with the properties:

$\displaystyle P(x, y) = P(y,x)$

and

$\displaystyle P(x, y+z) + P(y, z) = P(x,y) + P(x+y, z).$

These symmetric 2-cocycles come up when one tries to classify formal group laws over the ring ${\mathbb{Z} \oplus A}$, as we saw last time: in fact, we can think of them as “deformations” of the additive formal group law.

The main lemma which we stated last time was the following:

Theorem 2 (Symmetric 2-cocycle lemma) A homogeneous symmetric 2-cocycle of degree ${n}$ is a multiple of ${\frac{1}{d} ( ( x+y)^n - x^n - y^n )}$ where ${d =1}$ if ${n}$ is not a power of a prime, and ${d = p}$ if ${n = p^k}$.

For a direct combinatorial proof of this theorem, see Lurie’s notes. I want to describe a longer homological proof, which is apparently due to Mike Hopkins and which appears in the COCTALOS notes. The strategy is to interpret these symmetric 2-cocycles as actual cocycles in a cobar complex computing an ${\mathrm{Ext}}$ group. Then, the strategy is to compute this ${\mathrm{Ext}}$ group independently.

This argument is somewhat longer than the combinatorial one, but it has the benefit (for me) of engaging with some homological algebra (which I need to learn more about), as well as potentially generalizing in other directions.  (more…)

The goal of the next few posts is to compute ${\pi_* MU}$:

Theorem 1 (Milnor) The complex cobordism ring ${\pi_* MU}$ is isomorphic to a polynomial ring ${\mathbb{Z}[c_1, c_2, \dots]}$ where each ${c_i}$ is in degree ${2i}$.

We are also going to work out what the image of the Hurewicz map is on indecomposables. The strategy will be to apply the Adams spectral sequence to ${MU}$, at each prime individually.

1. Change-of-rings theorem

In order to apply the ASS, we’re going to need the groups ${\mathrm{Ext}^{s,t}_{\mathcal{A}_p^{\vee}}(\mathbb{Z}/p, H_*(MU; \mathbb{Z}/p))}$ because the spectral sequence runs

$\displaystyle E_2^{s,t} = \mathrm{Ext}^{s,t}_{\mathcal{A}_p^{\vee}}(\mathbb{Z}/p, H_*(MU; \mathbb{Z}/p)) \implies \widehat{\pi_{t-s}(MU)}.$

The ${\mathrm{Ext}}$ groups are computed in the category of (graded) comodules over ${\mathcal{A}_p^{\vee}}$.

In the previous post, we computed

$\displaystyle H_*(MU; \mathbb{Z}/p) = P \otimes \mathbb{Z}/p[y_i]_{i + 1 \neq p^k},$

as a comodule over ${\mathcal{A}_p^{\vee}}$. In order to compute the ${\mathrm{Ext}}$ groups, we need a general machine. The idea is that ${P \otimes \mathbb{Z}[y_i]_{i + 1 \neq p^k}}$ is almost a coinduced comodule—if it were, the ${\mathrm{Ext}}$ groups would be trivial. It’s not, but the general “change-of-rings” machine will enable us to reduce the calculation of these ${\mathrm{Ext}}$ groups to the calculation of (much simpler) ${\mathrm{Ext}}$ groups over an exterior algebra. (more…)

I’ve been reading Milnor’s paper “The Steenrod algebra and its dual,” and want to talk a little about it today. The starting point of this story is the theory of cohomology operations. Given a cohomology theory ${h^*}$ on spaces (or just CW complexes; one can always Kan extend to all spaces), one can consider cohomology operations on ${h^*}$. Most interesting for our purposes are the stable cohomology operations.

A stable cohomology operation of degree ${k}$ will be a collection of homomorphisms ${h^m(X) \rightarrow h^{m+k}(X)}$ for each ${m}$, which are natural in the space ${X}$, and which commute with the suspension isomorphisms. If we think of ${h^*}$ as represented by a spectrum ${E}$, so that ${h^*(X) = [X, E]}$ is a representable functor (in the stable homotopy category), then a stable cohomology operation comes from a homotopy class of maps ${E \rightarrow E}$ of degree ${k}$.

A stable cohomology operation is additive, because it comes from a spectrum map, and the stable homotopy category is additive. Moreover, the set of all stable cohomology operations becomes a graded ring under composition. It is equivalently the graded ring ${[E, E]}$.

The case where ${E}$ is an Eilenberg-MacLane spectrum, and ${h^*}$ ordinary cohomology, is itself pretty interesting. First off, one has to work in finite characteristic—in characteristic zero, there are no nontrivial stable cohomology operations. In fact, the only (possibly unstable) natural transformations ${H^*(\cdot, \mathbb{Q}) \rightarrow H^*(\cdot, \mathbb{Q})}$ come from taking iterated cup products because ${H^*(K(\mathbb{Q}, n))}$ can be computed, via the spectral sequence, to be a free graded-commutative algebra over ${\mathbb{Q}}$ generated by the universal element. These aren’t stable, so the only stable one has to be zero. So we will work with coefficients ${\mathbb{Z}/p}$ for ${p}$ a prime.

Here the algebra of stable cohomology operations is known and has been known since the 1950’s; it’s called the Steenrod algebra ${\mathcal{A}^*}$. In fact, all unstablecohomology operations are themselves known. Let me state the result for ${p=2}$.

$\displaystyle \mathrm{Sq}^i: H^*(\cdot, \mathbb{Z}/2) \rightarrow H^{* +i}(\mathbb{Z}/2) .$
1. ${\mathrm{Sq}^0}$ is the identity operation.
2. ${\mathrm{Sq}^i}$ on a cohomology class ${x}$ of dimension ${n}$ vanishes for ${i > n}$. For ${i = n}$, ${\mathrm{Sq}^i}$ acts by the cup square on ${x}$.
3. The Steenrod squares behave well with respect to the cohomology cross (and thus cup) product: ${\mathrm{Sq}^i(a \times b) = \sum_{j + k = i} \mathrm{Sq}^j a \times \mathrm{Sq}^k b}$.
4. ${\mathrm{Sq}^1}$ is the Bockstein connecting homomorphism associated to the short exact sequence ${0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0}$.
5. ${\mathrm{Sq}^i}$ commutes with suspension (and thus is a homomorphism). (more…)