Let ${k}$ be a field of characteristic zero. In the previous post, we introduced the category (i.e., ${\infty}$-category) ${\mathrm{Moduli}_k}$ of formal moduli problems over ${k}$. A formal moduli problem over ${k}$ is a moduli problem, taking values in spaces, that can be evaluated on the class of “derived” artinian ${k}$-algebras with residue field ${k}$: this was the category ${\mathrm{CAlg}_{sm}}$ introduced in the previous post.

In other words, a formal moduli problem was a functor

$\displaystyle F: \mathrm{CAlg}_{sm} \rightarrow \mathcal{S} \ (= \text{spaces}),$

which was required to send ${k}$ itself to a point, and satisfy a certain cohesiveness condition: ${F}$ respects certain pullbacks in ${\mathrm{CAlg}_{sm}}$ (which corresponded geometrically to pushouts of schemes).

The main goal of the series of posts was to sketch a proof of (and define everything in) the following result:

Theorem 7 (Lurie; Pridham) There is an equivalence of categories between ${\mathrm{Moduli}_k}$ and the ${\infty}$-category ${\mathrm{dgLie}}$ of DGLAs over ${k}$.

4. Overview

Here’s a rough sketch of the idea. Given a formal moduli problem ${F}$, we should think of ${F}$ as something like a small space, concentrated at a point but with lots of “infinitesimal” thickening. (Something like a ${\mathrm{Spf}}$.) Moreover, ${F}$ has a canonical basepoint corresponding to the “trivial deformation.” That is, we can think of ${F}$ as taking values in pointed spaces rather than spaces.

It follows that we can form the loop space ${\Omega F = \ast \times_F \ast}$ of ${F}$, which is a new formal moduli problem. However, ${\Omega F}$ has more structure: it’s a group object in the category of formal moduli problems — that is, it’s some sort of derived formal Lie group. Moreover, knowledge of the original ${F}$ is equivalent to knowledge of ${\Omega F}$ together with its group structure: we can recover ${F}$ as ${B \Omega F}$ (modulo connectivity issues that end up not being a problem). This relation between ordinary objects and group objects (via ${B, \Omega}$) is something very specific to the derived or homotopy world, and it’s what leads to phenomena such as Koszul duality. (more…)

There’s a “philosophy” in deformation theory that deformation problems in characteristic zero come from dg-Lie algebras. I’ve been trying to learn a little about this. Precise statements have been given by Lurie and Pridham which consider categories of “derived” deformation problems (i.e., deformation problems that can be evaluated on derived rings) and establish equivalences between them and suitable (higher) categories of dg-Lie algebras. I’ve been reading in particular Lurie’s very enjoyable survey of his approach to the problem, which sketches the equivalence in an abstract categorical context with the essential input arising from Koszul duality between Lie algebras and commutative algebras. In this post, I’d just like to say what a “deformation problem” is in the derived world.

1. Introduction

Let ${\mathcal{M}}$ be a classical moduli problem. Abstractly, we will think of ${\mathcal{M}}$ as a functor

$\displaystyle \mathcal{M}:\mathrm{Ring} \rightarrow \mathrm{Sets},$

such that, for a (commutative) ring ${R}$, the set ${\mathcal{M}(R)}$ will be realized as maps from ${\mathrm{Spec} R}$ into a geometric object—a scheme or maybe an algebraic space.

Example 1${\mathcal{M}}$ could be the functor that sends ${R}$ to the set of closed subschemes of ${\mathbb{P}^n_R}$ which are flat over ${R}$. In this case, ${\mathcal{M}}$ comes from a scheme: the Hilbert scheme.

We want to think of ${\mathcal{M}}$ as some kind of geometric object and, given a point ${x: \mathrm{Spec} k \rightarrow \mathcal{M}}$ for ${k}$ a field (that is, an element of ${\mathcal{M}(k)}$), we’d like to study the local structure of ${\mathcal{M}}$ near ${x}$. (more…)

The purpose of this post and the next is to work through a basic example of intersection theory: intersections of curves on a surface. This is a fundamental and basic example in algebraic geometry, and since I’ve never studied intersection theory, it like seems a reasonable place to start. The references here are chapter 5 of Hartshorne’s Algebraic geometry and Mumford’s Lectures on curves on an algebraic surface.

1. Curves on surfaces

The subject of “curves on a surface” is the subject of Mumford’s book mentioned above; the purpose of this section is simply to set down the definitions.

Let ${k}$ be an algebraically closed field. A surface ${S}$ is a smooth projective surface over ${k}$. There is a classification of surfaces, but let’s just list a couple of basic examples: ${\mathbb{P}^2, \mathbb{P}^1 \times \mathbb{P}^1}$, (smooth) hypersurfaces in ${\mathbb{P}^3}$, and ruled surfaces.

Definition 1 curve on a surface ${S}$ is an (effective) divisor on ${S}$. Equivalently, it is a subscheme ${C \subset S}$ pure of codimension one, so locally cut out by one equation. (But ${C}$ is not necessarily smooth, or even reduced.)

The goal of this post and the next is to set up a basic intersection theory for curves on surfaces. Given two curves ${C, D \subset S}$, we’d like to define the intersection product ${C.D}$. There is one case where it is easy: suppose ${C}$ and ${D}$ meet only transversely. In other words, for each ${p \in C \cap D}$, we choose local equations ${f,g \in \mathfrak{m}_{S, p} \subset\mathcal{O}_{S, p}}$ for the subschemes ${C, D}$, and

$\displaystyle (f,g) = \mathfrak{m}_{S, p}.$

In particular, this implies that ${C, D}$ are nonsingular at all points of intersection. In this case, we would like to require

$\displaystyle C.D = \sum_{p \in C \cap D} 1 \quad (\text{if transverse intersection}). \ \ \ \ \ (1)$

Once we require the above condition and two more natural conditions, we will prove that the intersection product is uniquely determined:

• The equation (1) holds under transversality assumptions and if ${C, D}$ are smooth.
• The intersection product is additive. That is, given curves ${C_1, C_2, D}$, we have

$\displaystyle (C_1 + C_2). D = C_1.D + C_2.D,$

where ${C_1+C_2}$ is treated as an effective Cartier divisor.

• The intersection product is invariant under linear equivalence. If ${C, C'}$ are linearly equivalent curves, we want

$\displaystyle C. D = C'.D,$

so that the intersection product is invariant under deformation. In particular, this and the previous item show that the intersection product only depends on the line bundle associated to a divisor (and can make sense for any divisor, not necessarily effective).

Our goal is to prove:

Theorem 2 There is a unique pairing

$\displaystyle \mathrm{Pic}(S) \times \mathrm{Pic}(S) \rightarrow \mathbb{Z}$

satisfying the above three conditions. (more…)

One of the really nice pictures in homotopy theory is the “chromatic” one, relating the structure of the stable homotopy category to the geometry of formal groups (or rather, the geometry of the moduli stack of formal groups). A while back, I did a series of posts trying to understand a little about the relationship between formal groups and complex cobordism; the main result I was able to get to was Quillen’s theorem on the formal group of $MU$. I didn’t understand too much of the picture then, but I spent the summer engaging with it and think I have a slightly better feel for it now. In this post, I’ll try to give a description of how a natural attack on the homotopy groups of a spectrum via descent leads very naturally to the moduli stack of formal groups and to the Adams-Novikov spectral sequence. (There are other approaches to Adams-type spectral sequences, for instance in these notes of Haynes Miller.)

1. Descent

Let’s start with some high-powered generalities that I don’t really understand, and then come back to earth. Consider an ${E_\infty}$-ring ${R}$; the most important examples will be ${R = H \mathbb{Z}/2}$ or ${R = MU}$. There is a map of ${E_\infty}$-rings ${S \rightarrow R}$, where ${S}$ is the sphere spectrum.

Let ${X}$ be a plain spectrum. Then, equivalently, ${X}$ is a module over ${S}$. Tensoring with ${R}$ gives an ${R}$-module spectrum ${ R \otimes X}$, where the smash product of spectra is written ${\otimes}$. In fact, we have an adjunction

$\displaystyle \mathrm{Mod}(S) \rightleftarrows \mathrm{Mod}(R)$

between ${R \otimes }$ and forgetting the ${R}$-module structure. As in ordinary algebra, we might try to apply the methods of flat descent to this adjunction. In other words, given a spectrum ${X}$, we might try to recover ${X}$ from the ${R}$-module ${R \otimes X}$ together with the “descent data” on ${X}$. The benefit is that while the homotopy groups ${\pi_* X}$ may be intractable, those of ${R \otimes X}$ are likely to be much easier to compute: they are the ${R}$-homology groups of ${X}$.

Let’s recall how this works in algebra. Given a faithfully flat morphism of rings ${A \rightarrow B}$ and an ${A}$-module ${M}$, then we can recover ${M}$ as the equalizer of

$\displaystyle M \otimes_A B \rightrightarrows M \otimes_A B \otimes_A B.$

How does one imitate this construction in homotopy? One then has a cosimplicial ${E_\infty}$-ring given by the cobar construction

$\displaystyle R \rightrightarrows R \otimes R \dots .$

The ${\mathrm{Tot}}$ (homotopy limit) of a cosimplicial object is the homotopyish version of the 1-categorical notion of an equalizer. In particular, we might expect that we can recover the spectrum ${X}$ as the homotopy limit of the cosimplicial diagram

$\displaystyle R \otimes X \rightrightarrows R \otimes R \otimes X \dots .$ (more…)

To anyone in the Cambridge, MA area: a bunch of us will be organizing a learning seminar on higher categories and derived algebraic geometry at Harvard. Our goal is to understand some of the topics in the book “Higher Topos Theory” and some of the DAG papers. We will be having an organizational meeting (where we figure out what our goals and format will be) next Tuesday at 4:30. Let me know (at amathew (at) college (dot) harvard (dot) edu) if you are interested and can make it!