This is a sub-entry of
see there for background and context.
This entry discusses the algebraic/homotopy theoretic prerequisites for derived algebraic geometry.
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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
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 $\Omega$-spectrum is a sequence of pointed topological spaces $\{E_n\}$ and base-point preserving maps $\{\sigma_n : E_n \to \Omega E_{n+1}\}$ that are weak homotopy equivalences.
($\Omega E_n$ is the loop space of $E_n$).
if $\{E_n\}$ is an $\Omega$-spectrum, define $h^{-n}(X) := [X, E_n]$ (homotopy classes of continuous maps). Then this $h$ is a generalized (Eilenberg-Steenrod) cohomology theory.
It should be noted that all our spaces are based and $h$ is a reduced cohomology theory. Define $\pi_n(E) := [S^0, E_n]$. $\pi_*(E)$ are the coefficients (i.e. the cohomology over the point of the corresponding unreduced theory) of $E$.
Brown’s representability theory: Any reduced cohomology theory on CW-complexes is represented by an $\Omega$-spectrum.
examples
singular cohomology with coefficients in $A$: the Eilenberg-MacLane spectrum $H A$.
complex K-theory: $K_n = \mathbb{Z} \times BU$ for $n$ even and $\cdots = U$ otherwise
Le $M_{1,1}$ be the moduli stack of all elliptic curves, then $Hom(Spec R, M_{1,1}) = \{elliptic curves over Spec R\}$.
(we will construct this more rigorously later)
If $\phi : Spec R \to M_{1,1}$ is a map that is a flat morphism, then we obtain an elliptic cohomology theory called $A_{\phi}$.
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
Hopkins-Miller: use the multiplicative nature of cohomology theories to solve this, i.e. instead look for a more refined lift
theorem There exists a symmetric monoidal model category $StTop$ 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 $StTop$ and an E-infinity ring is an ordinray commutative monoid there.
So an $E_\infty$-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 $A$ be an A-infinity ring spectrum.
the $\infty$-monoidal structure on the spectrum induces a multiplicative cohomology theory.
$\pi_0(A)$ is a commutative ring
$\pi_n(A)$ is a module over $\pi_0(A)$.
Definition For $A$ an E-infinity ring, $M$ with a map $A \wedge M \to M$ such that the obvious diagrams commute is a module for that E-infinity ring.
Proposition $\pi_*(M)$ is a graded module over $\pi_*(A)$.
Definition for $A$ an E-infinity ring and $M$ an $A$-module, we have that $M$ is flat module if
$\pi_0(M)$ is flat over $\pi_0(A)$ in the ordinary sense
$\forall n : \pi_n(A) \otimes_{\pi_0(A)} \pi_0(M) \to \pi_n(M)$ is an isomorphism of $\pi_0(M)$-modules
definition a morphism $f : A \to B$ of E-infinity rings is flat if $B$ regarded as an $A$-module using this morphism is flat.
Theorem (Goerss-Hopkins-Miller):
A lift $O_{M_{1,1}, der}$ 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:
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 $M_{1,1}$ by its Deligne-Mumford compactification.
recall that we want global sections of the presheaf
(on the left we have something like the etale site of the moduli stack $M_{1,1}$ )
but there is no good notion of gluing in CohomologyTheories (lack of colimits) hence no good notion of sheaves with values in cohomology theories. $CohomologyTheories$ 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
but in the homotopy category the pushout is instead
The result is not even homotopy equivalent. In the homotopy category the pushout does not exist.
So we want to refine $CohomologyTheories$ to the cateory of spectra that they come from by the Brown representability theorem.
In fact, we want to lift $MultiplicativeCohomologyTheories$ to that of E-infinity ring-spectra.
The map
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) .
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 $Sp$
show that $Sp$ is a symmetric monoidal (infinity,1)-category
show that the homotopy category of an (infinity,1)-category of $Sp$ is the stable homotopy category, where the tensor product goes to the smash product of spectra
define an E-infinity ring to be a commutative monoid in an (infinity,1)-category in $Sp$.
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)
a Top-enriched category (essentially simplicially enriched category)
(see there for details). In fact, see chapter 1 of Higher Topos Theory for lots of details.
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 $C$ and its homotopy category $C \to h C$.
A commutative-up-to-homotopy diagram in $C$ is a functor $I \to h C$
for $I$ some diagram category.
to get a homotopy coherent diagram instead take the nerve $N(I)$ of $I$ and then map $N(I) \to C$.
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 $C$ be an (infinity,1)-category. Suppose that it has a zero object $0 \in C$, i.e. an object that is both an initial object and a terminal object.
Assume that $C$ admits kernels and cokernels, i.e. all homotopy pullbacks and pushouts with $0$ in one corner.
Then from this we get loop space objects $\Omega X$ and delooping objects $B X$ in $C$ (called suspension objects $\Sigma X$ in this context).
in particular a loop space object $\Omega Y$ is the kernel of the 0-map,while the suspension $\Sigma X$ is the cokernel
One example of this is the (infinity,1)-category of pointed topological spaces.
definition a prespectrum object in an (infinity,1)-category $C$ with the properties as above is a (infinity,1)-functor
such that $X(i,j)$ for $i \neq j$ is zero object 0.
(everything filled with 2-cells aka homotopies)
since we have cokernels we get maps from the universal property
and analogously maps $\beta_n : X(n,n) \to \Omega X(n+1, n+1)$
now $X$ is a spectrum object if the $\beta_n$ are equivalences, for all $n$. (We don’t require $\alpha_n$ to be equivalences.)
so to each (infinity,1)-category $C$ we get another (infinity,1)-category $Sp(C)$, the full subcategory $Fun(N(\mathbb{Z}\times \mathbb{Z}), C)$ on the spectrum objects.
In particular, we set
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: $Sp$ has an essentially unique structure of a symmetric monoidal (infinity,1)-category.
This monoidal structure $\otimes$ is uniquely characterized by the following two properties:
$\otimes$ preserves limits and colimits.
the sphere spectrum is the monoidal unit?/tensor unit wrt $\otimes$.
definition A symmetric monoidal (infinity,1)-category structure on an (infinity,1)-category $C$ is given by the following data:
where $\Gamma$ is Segal's category with objects finite pointed sets and morphisms basepoint preserving functions between sets.
such that $C^\otimes_{\langle 1\rangle} \simeq C$
where $C^\otimes_{\langle 1\rangle}$ is the fiber over $\langle 1\rangle = \{*,1\}$, i.e. the pullback
here should go some pictures that illustarte this. But see the first few pages of Noncommutative Algebra for the intuition and motivation.
so let $C$ now be a symmetric monoidal (infinity,1)-category.
definition A commutative monoid in $C$ is a section $s$ of the structure map mentioned above $C^\otimes \to N(\Gamma) \stackrel{s}{\to} C^\otimes$.
The monoid object itself is the image of $\langle 1 \rangle$ under $s$, $A = s(\langle 1 \rangle)$. (Sort of. I think the whole point is that we don’t ever say something like “this particular $A$ 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 $s$, though.
definition an E-infinity ring spectrum is a commutative monoid in an (infinity,1)-category in the stable (infinity,1)-category of spectra $Sp$.
$E_\infty$-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 $E_\infty$ rings!
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 $A$ be an E-infinity ring.
Define its spectrum of an E-infinity ring? $Spec A$ as the ringed space $(|Spec A|, \mathcal{O}_{Spec A})$ whose underlying topological space is the ordinary spectrum of the degroo-0 ring
and where $\mathcal{O}_{Spec A}$ is given on Zariski-opens $D(f)$ for any $f \in \pi_0 A$ by
Here $A \to A[f^{-1}]$ is characterized by the following equivalent ways:
$\pi_\bullet A \to \pi_*(A[f^{-1}])$ identify $\pi_{\bullet}(A[f^{-1}])$ with $\pi_\bullet$
$\forall$ $E_\infty$-rings the induced map $Hom(A[f^{-1}],B) \to Hom(A,B)$ is a homotopy equivalence of the left hand side with the subspace of the right hand side which takes $f \in \pi_0 A$ to an invertible element of $\pi_0 B$.
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 $X$ s a space and $\mathcal{O}$ a sheaf of E-infinity rings then $(X,\pi_0 \mathcal{O}_X)$ is a classical scheme and $\pi_n \mathcal{O}_X$ is a quasicoherent $\pi_0 \mathcal{O}_X$-module.
theorem there exists a derived Deligne-Mumford stack $(M_{1,1}, \mathcal{O}^{der}_{M_{1,1}})$ such that $(M_{1,1}, \pi_0 \mathcal{O}^{der}_{M_{1,1}})$ is the ordinary DM- moduli stack of elliptic curves.