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This is a sub-entry of

see there for background and context.

This entry discusses the algebraic/homotopy theoretic prerequisites for derived algebraic geometry.

Previous:

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the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention, meaning: somebody should go through this and polish

contents

part 1 – the sheaf of elliptic cohomology ring spectra

We will talk about a lifting problem that will lead to the formulation of tmf. This requires E-infinity rings and derived algebraic geometry.

Definition

An Ω\Omega-spectrum is a sequence of pointed topological spaces {E n}\{E_n\} and base-point preserving maps {σ n:E nΩE n+1}\{\sigma_n : E_n \to \Omega E_{n+1}\} that are weak homotopy equivalences.

(ΩE n\Omega E_n is the loop space of E nE_n).

if {E n}\{E_n\} is an Ω\Omega-spectrum, define h n(X):=[X,E n]h^{-n}(X) := [X, E_n] (homotopy classes of continuous maps). Then this hh is a generalized (Eilenberg-Steenrod) cohomology theory.

It should be noted that all our spaces are based and hh is a reduced cohomology theory. Define π n(E):=[S 0,E n]\pi_n(E) := [S^0, E_n]. π *(E)\pi_*(E) are the coefficients (i.e. the cohomology over the point of the corresponding unreduced theory) of EE.

Brown’s representability theory: Any reduced cohomology theory on CW-complexes is represented by an Ω\Omega-spectrum.

examples

  1. singular cohomology with coefficients in AA: the Eilenberg-MacLane spectrum HAH A.

  2. complex K-theory: K n=×BUK_n = \mathbb{Z} \times BU for nn even and =U\cdots = U otherwise

Le M 1,1M_{1,1} be the moduli stack of all elliptic curves, then Hom(SpecR,M 1,1)={ellipticcurvesoverSpecR}Hom(Spec R, M_{1,1}) = \{elliptic curves over Spec R\}.

(we will construct this more rigorously later)

If ϕ:SpecRM 1,1\phi : Spec R \to M_{1,1} is a map that is a flat morphism, then we obtain an elliptic cohomology theory called A ϕA_{\phi}.

This assignment is a presheaf of cohomology theories.

To get a single cohomology theory from that we want to take global sections, but there is no good way to say what a global section of a cohomology-theoy valued functor would be. One reason is that there is not a good notion to say what a sheaf of cohomology theorys is.

But if we had an (infinity,1)-category valued functor, then Higher Topos Theory would provide all that technology. So that’s what we try to get now.

goal find lift

Spectra ? represent {ϕ:SpecRM 1,1} CohomologyTheories. \array{ && Spectra \\ & {}^{?}\nearrow & \;\;\;\downarrow^{represent} \\ \{\phi : Spec R \to M_{1,1}\} &\to& CohomologyTheories } \,.

Hopkins-Miller: use the multiplicative nature of cohomology theories to solve this, i.e. instead look for a more refined lift

CommRingSpectra O M der {ϕ:SpecRM 1,1} MultiplicativeCohomologyTheories. \array{ && CommRingSpectra \\ & {}^{O_{M^{der}}}\nearrow & \downarrow \\ \{\phi : Spec R \to M_{1,1}\} &\to& MultiplicativeCohomologyTheories } \,.

theorem There exists a symmetric monoidal model category StTopStTop of spectra such that the homotopy category is the stable homotopy category as a symmetric monoidal category.

This and the following is described in more detail at symmetric monoidal smash product of spectra.

Definition An A-infinity ring is an ordinary monoid in StTopStTop and an E-infinity ring is an ordinray commutative monoid there.

So an E E_\infty-ring is an honest monoid with respect to the funny smash product that makes spectra a symmetric monoidal category, but it is just a monoid up to homotopy with respect to the ordinary product of spaces.

For more on this see (for the time being) the literature referenced at stable homotopy theory.

proposition

Let AA be an A-infinity ring spectrum.

  1. the \infty-monoidal structure on the spectrum induces a multiplicative cohomology theory.

  2. π 0(A)\pi_0(A) is a commutative ring

  3. π n(A)\pi_n(A) is a module over π 0(A)\pi_0(A).

Definition For AA an E-infinity ring, MM with a map AMMA \wedge M \to M such that the obvious diagrams commute is a module for that E-infinity ring.

Proposition π *(M)\pi_*(M) is a graded module over π *(A)\pi_*(A).

Definition for AA an E-infinity ring and MM an AA-module, we have that MM is flat module if

  1. π 0(M)\pi_0(M) is flat over π 0(A)\pi_0(A) in the ordinary sense

  2. n:π n(A) π 0(A)π 0(M)π n(M)\forall n : \pi_n(A) \otimes_{\pi_0(A)} \pi_0(M) \to \pi_n(M) is an isomorphism of π 0(M)\pi_0(M)-modules

definition a morphism f:ABf : A \to B of E-infinity rings is flat if BB regarded as an AA-module using this morphism is flat.

Theorem (Goerss-Hopkins-Miller):

A lift O M 1,1,derO_{M_{1,1}, der} as indicated in the GOAL above (multiplicative version) does exists and is unique up to homotopy equivalence.

The tmf-spectrum is the global sections of this:

tmf[Δ 1]=Γ(O M 1,1,der) tmf[\Delta^{-1}] = \Gamma(O_{M_{1,1}, der})

this is not elliptic (its not even nor has period 2), but is a multiplicative spectrum and hence defines a cohomology theory.

The spectrum tmf is obtained in the same manner by replacing M 1,1M_{1,1} by its Deligne-Mumford compactification.

part 2 - the stable symmetric monoidal (,1)(\infty,1)-category of spectra

recall that we want global sections of the presheaf

{SpecRM 1,1}CohomologyTheories \{Spec R \to M_{1,1}\} \to CohomologyTheories

(on the left we have something like the etale site of the moduli stack M 1,1M_{1,1} )

but there is no good notion of gluing in CohomologyTheories (lack of colimits) hence no good notion of sheaves with values in cohomology theories. CohomologyTheoriesCohomologyTheories is the homotopy category of some other category, to be identified, and passage to homotopy categories may destroy existence of useful colimits. The category of CohomologyTheories “is” the stable homotopy category.

A simple example:

in the (infinity,1)-category Top we have the homotopy pushout

S 1 D 2 D 2 S 2 \array{ S^1 &\to& D^2 \\ \downarrow && \downarrow \\ D^2 &\to& S^2 }

but in the homotopy category the pushout is instead

S 1 D 2 D 2 * \array{ S^1 &\to& D^2 \\ \downarrow && \downarrow \\ D^2 &\to& * }

The result is not even homotopy equivalent. In the homotopy category the pushout does not exist.

So we want to refine CohomologyTheoriesCohomologyTheories to the cateory of spectra that they come from by the Brown representability theorem.

In fact, we want to lift MultiplicativeCohomologyTheoriesMultiplicativeCohomologyTheories to that of E-infinity ring-spectra.

The map

E RingsMultiplicativeCohomologyTheories E_\infty Rings \to MultiplicativeCohomologyTheories

should be that of taking the homotopy category of an (infinity,1)-category.

Approach A (modern but traditional stable homotopy theory) choose a symmetric monoidal simplicial model category whose homotopy category is the stable homotopy category and whose tensor product is the smash product of spectra. For instance use the symmetric monoidal smash product of spectra.

Then define E-infinity ring spectra to be ordinary monoid objects in this symmetric monoidal model category of spectra.

Approach B (Jacob Lurie: be serious about working with (infinity,1)-category instead of just model category theory) .

  1. define (infinity,1)-category (chapter 1 of HTT)

    in this framework we’ll have a stable (infinity,1)-category of spectra, let’s call that SpSp

  2. show that SpSp is a symmetric monoidal (infinity,1)-category

  3. show that the homotopy category of an (infinity,1)-category of SpSp is the stable homotopy category, where the tensor product goes to the smash product of spectra

  4. define an E-infinity ring to be a commutative monoid in an (infinity,1)-category in SpSp.

These two approaches are equivalent is a suitable sense. See Noncommutative Algebra, page 129 and Commutative Algebra, Remark 0.0.2 and paragraph 4.3.

derived algebraic geometry categorifies algebraic geometry

E-infinity ring categoriefies commutative ring

(infinity,1)-category catgeorifies category

Definition An (infinity,1)-category is (for instance modeled by)

use homotopy coherent nerve to go from a simplicially enriched category to its corresponding quasi-category

definition homotopy category of an (infinity,1)-category (see there)

definition morphism of (infinity,1)-categories is, when regarded as a quasi-category, just a morphism of simplicial sets.: this is an (infinity,1)-functor.

There is an (infinity,1)-category of (infinity,1)-functors between two (infinity,1)-categories

why simplicial sets?

because they provide a convenient calculus for doing homotopy coherent category theory.

suppose some (infinity,1)-category CC and its homotopy category ChCC \to h C.

A commutative-up-to-homotopy diagram in CC is a functor IhCI \to h C

C I hC \array{ && C \\ && \downarrow \\ I &\to& h C }

for II some diagram category.

to get a homotopy coherent diagram instead take the nerve N(I)N(I) of II and then map N(I)CN(I) \to C.

The nerve automatically encodes the homotopy coherence. See Higher Topos Theory pages 37, 38 (but the general idea is well known from simplicial model category theory).

Now let CC be an (infinity,1)-category. Suppose that it has a zero object 0C0 \in C, i.e. an object that is both an initial object and a terminal object.

Assume that CC admits kernels and cokernels, i.e. all homotopy pullbacks and pushouts with 00 in one corner.

Then from this we get loop space objects ΩX\Omega X and delooping objects BXB X in CC (called suspension objects ΣX\Sigma X in this context).

X f Y 0 cokerfker(g) X g 0 Y \array{ X &\stackrel{f}{\to}& Y \\ \downarrow &\Downarrow& \downarrow \\ 0 &\to& coker f } \;\;\;\; \array{ ker(g) &\stackrel{}{\to}& X \\ \downarrow &\Downarrow& \downarrow^g \\ 0 &\to& Y }

in particular a loop space object ΩY\Omega Y is the kernel of the 0-map,while the suspension ΣX\Sigma X is the cokernel

X f 0 0 ΣXΩY 0 g 0 Y \array{ X &\stackrel{f}{\to}& 0 \\ \downarrow &\Downarrow& \downarrow \\ 0 &\to& \Sigma X } \;\;\;\; \array{ \Omega Y &\stackrel{}{\to}& 0 \\ \downarrow &\Downarrow& \downarrow^g \\ 0 &\to& Y }

One example of this is the (infinity,1)-category of pointed topological spaces.

definition a prespectrum object in an (infinity,1)-category CC with the properties as above is a (infinity,1)-functor

X:N(×)C X : N(\mathbb{Z} \times \mathbb{Z}) \to C

such that X(i,j)X(i,j) for iji \neq j is zero object 0.

X(n,n) X(n,n+1)0 g X(n+1,n)0 X(n+1,n+1) \array{ X(n,n) &\stackrel{}{\to}& X(n,n+1) \simeq 0 \\ \downarrow &\searrow& \downarrow^g \\ X(n+1,n) \simeq 0 &\to& X(n+1,n+1) }

(everything filled with 2-cells aka homotopies)

since we have cokernels we get maps from the universal property

X(n,n) X(n,n+1)0 g 0 ΣX(n,n) α n X(n+1,n+1) \array{ X(n,n) &\stackrel{}{\to}& X(n,n+1) \simeq 0 \\ \downarrow &\searrow& \downarrow^g \\ 0 &\to& \Sigma X(n,n) \\ &&& \searrow^{\alpha_n} \\ &&&& X(n+1,n+1) }

and analogously maps β n:X(n,n)ΩX(n+1,n+1)\beta_n : X(n,n) \to \Omega X(n+1, n+1)

now XX is a spectrum object if the β n\beta_n are equivalences, for all nn. (We don’t require α n\alpha_n to be equivalences.)

so to each (infinity,1)-category CC we get another (infinity,1)-category Sp(C)Sp(C), the full subcategory Fun(N(×),C)Fun(N(\mathbb{Z}\times \mathbb{Z}), C) on the spectrum objects.

In particular, we set

Sp:=Sp(Top) Sp := Sp(Top)

the stable (infinity,1)-category of spectra is the stabilization of the (infinity,1)-category Top of topological spaces.

I think we need pointed topological spaces here?

Fact: SpSp has an essentially unique structure of a symmetric monoidal (infinity,1)-category.

This monoidal structure \otimes is uniquely characterized by the following two properties:

  1. \otimes preserves limits and colimits.

  2. the sphere spectrum is the monoidal unit?/tensor unit wrt \otimes.

definition A symmetric monoidal (infinity,1)-category structure on an (infinity,1)-category CC is given by the following data:

  1. another (infinity,1)-category C C^\otimes with an (infinity,1)-functor C N(Γ)C^\otimes \to N(\Gamma) that is a coCartesian fibration

where Γ\Gamma is Segal's category with objects finite pointed sets and morphisms basepoint preserving functions between sets.

such that C 1 CC^\otimes_{\langle 1\rangle} \simeq C

where C 1 C^\otimes_{\langle 1\rangle} is the fiber over 1={*,1}\langle 1\rangle = \{*,1\}, i.e. the pullback

C 1 C pullback {1} N(Γ) \array{ C^\otimes_{\langle 1\rangle} &\to& C^\otimes \\ \downarrow &pullback& \downarrow \\ \{\langle 1\rangle\} &\to& N(\Gamma) }

here should go some pictures that illustarte this. But see the first few pages of Noncommutative Algebra for the intuition and motivation.

so let CC now be a symmetric monoidal (infinity,1)-category.

definition A commutative monoid in CC is a section ss of the structure map mentioned above C N(Γ)sC C^\otimes \to N(\Gamma) \stackrel{s}{\to} C^\otimes.

The monoid object itself is the image of 1\langle 1 \rangle under ss, A=s(1)A = s(\langle 1 \rangle). (Sort of. I think the whole point is that we don’t ever say something like “this particular AA is the monoid object”. Rather, the picture should roughly be that we have all of the standard diagrams describing a commutative monoid object, except that the various objects in the diagrams are not necessarily the same object. However, these a priori different objects will be a fortiori homotopy equivalent, so that in particular the usual picture will reappear in the homotopy category. Moreover, of course, these diagrams will not be strictly commutative, but commutative up to coherent homotopy, so that in particular the usual strict commutativity reappears after passage to the homotopy category.)

There is one more condition on ss, though.

definition an E-infinity ring spectrum is a commutative monoid in an (infinity,1)-category in the stable (infinity,1)-category of spectra SpSp.

E E_\infty-rings themselves form an (infinity,1)-category. And this has all limits and colimits (see DAG III 2.1, 2.7), so we can talk about sheaves of E E_\infty rings!

part 3 - brave new schemes

Now the theory of schemes and derived schemes, but not over simplicial commutative rings, but over E-infinity rings.

So we are trying to guess the content of the not-yet-existsting

Let AA be an E-infinity ring.

Define its spectrum of an E-infinity ring? SpecASpec A as the ringed space (|SpecA|,𝒪 SpecA)(|Spec A|, \mathcal{O}_{Spec A}) whose underlying topological space is the ordinary spectrum of the degroo-0 ring

|SpecA|:=Specπ 0A |Spec A| := Spec \pi_0 A

and where 𝒪 SpecA\mathcal{O}_{Spec A} is given on Zariski-opens D(f)D(f) for any fπ 0Af \in \pi_0 A by

𝒪 SpecA(D(f)):=A[f 1]. \mathcal{O}_{Spec A}(D(f)) := A[f^{-1}] \,.

Here AA[f 1]A \to A[f^{-1}] is characterized by the following equivalent ways:

  1. π Aπ *(A[f 1])\pi_\bullet A \to \pi_*(A[f^{-1}]) identify π (A[f 1])\pi_{\bullet}(A[f^{-1}]) with π \pi_\bullet

  2. \forall E E_\infty-rings the induced map Hom(A[f 1],B)Hom(A,B)Hom(A[f^{-1}],B) \to Hom(A,B) is a homotopy equivalence of the left hand side with the subspace of the right hand side which takes fπ 0Af \in \pi_0 A to an invertible element of π 0B\pi_0 B.

This geometry over E-infinity rings is in spectral algebraic geometry/brave new algebraic geometry?.

The analog for simplicial commutative rings instead of is what is discussed at derived scheme.

theorem (Jacob Lurie)

If XX s a space and 𝒪\mathcal{O} a sheaf of E-infinity rings then (X,π 0𝒪 X)(X,\pi_0 \mathcal{O}_X) is a classical scheme and π n𝒪 X\pi_n \mathcal{O}_X is a quasicoherent π 0𝒪 X\pi_0 \mathcal{O}_X-module.

theorem there exists a derived Deligne-Mumford stack (M 1,1,𝒪 M 1,1 der)(M_{1,1}, \mathcal{O}^{der}_{M_{1,1}}) such that (M 1,1,π 0𝒪 M 1,1 der)(M_{1,1}, \pi_0 \mathcal{O}^{der}_{M_{1,1}}) is the ordinary DM- moduli stack of elliptic curves.

References

Revised on October 27, 2009 22:15:45 by Ryan Grady (195.37.209.182)