A Survey of Elliptic Cohomology - A-equivariant cohomology

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
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This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

see there for background and context.

This entry contains a basic introduction to getting equivariant cohomology from [[derived group scheme]]s.

=--

Previous:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

* [[A Survey of Elliptic Cohomology - elliptic curves]]

* [[A Survey of Elliptic Cohomology - equivariant cohomology]]

* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]

Next:

* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]

> 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** See also at _[[equivariant elliptic cohomology]]_.

***

#Contents#

* toc
{:toc}

#Derived Elliptic Curves#

**Definition** A [[derived elliptic curve]] over an (affine) [[derived scheme]] $\mathrm{Spec} A$ is a commutative [[derived group scheme]] (CDGS) $E /A$ such that $\overline{E} \to \mathrm{Spec} \pi_0 A$ is an [[elliptic curve]].

Let $A$ be an $E_\infty$-ring.  Let $E(A)$ denote the $\infty$-groupoid of oriented elliptic curves over $\mathrm{Spec} A$.  Note that $E(A)$ is in particular a space (we will return to this point later).

The point is to prove the following due to Lurie.

**Theorem** The functor $A \mapsto E(A)$ is representable by a derived Deligne-Mumford stack $\mathcal{M}^{Der} = (\mathcal{M} , O_\mathcal{M} )$.  Further, $\mathcal{M}$ is equivalent to the topos underlying $\mathcal{M}_{1,1}$ and $\pi_0 O_\mathcal{M} = O_{\mathcal{M}_{1,1}}$.  Also, restricting to discrete rings, $O_\mathcal{M}$ provides a lift in sense of Hopkins and Miller.

#$\mathbf{G}$-Equivariant $A$-cohomology
{#equivariant}

##The Strategy##
1. Define $A_{S^1 } (*)$;

1. Extend to $A_{S^1} (X)$ where $X$ is a trivial $S^1$-space;

1. Define $A_T (X)$ where $T$ is a compact abelian Lie group where $X$ is again a trivial $T$-space;

1. Extend to $A_T (X)$ for any (finite enough) $T$-space;

1. Define $A_G (X)$ for $G$ any compact Lie group.

##$S^1$-equivariance##

To accomplish (1) we need a map
$$\sigma : \mathrm{Spf} A^{\mathbb{C} P^\infty} \to \mathbf{G}$$
over $\mathrm{Spec} A$.  Then we can define $A_{S^1} (*) = O (\mathbf{G})$.  Such a map arises from a completion map
$$A_{S^1} \to A^{\mathbb{C}P^\infty}$$
which we may interpret as a preorientation $\sigma \in \mathrm{Map} (BS^1 , \mathbf{G} (A))$.  Recall that such a map $\sigma$ is an orientation if the induced map to the formal completion of $\mathbf{G}$ is an isomorphism.

Recall two facts:

1. There is a bijection $\{ BS^1 \to \mathbf{G} (A) \} \leftrightarrow \{ \mathrm{Spf} A^{BS^1} \to \mathbf{G} \}$;

1. Orientations of the multiplicative group $\mathbf{G}_m$ associated to $A$ are in bijection with maps of $E_\infty$-rings $\{ K \to A\}$, where $K$ is the K-theory spectrum.

**Theorem** We can define equivariant $A$-cohomology using $\mathbf{G}_m$ if and only if $A$ is a $K$-algebra.

##The Abelian Lie Group Case for a Point##

Fix $\mathbf{G}/A$ oriented. Now let $T$ be a compact abelian Lie group.  We construct a commutative [[derived group scheme]] $M_T$ over $A$ whose global sections give $A_T$ which is equipped with an appropriate completion map.

**Definition** Define the Pontryagin dual, $\hat T$ of $T$ by $\hat T := \mathrm{Hom}_\mathrm{Lie} (T, S^1)$.

**Examples**

1. $T=T^n$, the $n$-fold torus.  Then $\hat T = \mathbb{Z}^n$ as
$$\hat T \ni ( \theta_1 , \dots , \theta_n ) \mapsto (k_1 \theta_1 , \dots , k_n \theta_n ).$$

1. If $T = \{ e \}$, then $\hat T = T$.

**Pontryagin Duality** If $T$ is an abelian, locally compact topological group then $\hat \hat T \simeq T$.

**Definition** Let $B$ be an $A$-algebra.  Define $M_T$ by
$$M_T (B) := \mathrm{Hom}_\mathrm{AbTop} (\hat T , \mathbf{G} (B)).$$

Further, $M_T$ is representable.

**Examples**

1. $M_{S^1} (B) = \mathrm{Hom} ( \mathbb{Z} , \mathbf{G} (B)) = \mathbf{G} (B).$

1. $M_{T^n} = \mathbf{G} \times_{\mathrm{Spec} A} \dots \times_{\mathrm{Spec} A} \mathbf{G}.$

1. $M_{\mathbb{Z}/n} = \mathrm{hker} (\times n: \mathbf{G}\to \mathbf{G}).$

1. $M_{\{e\}} (B) = \mathrm{Hom} ( \{e \} , \mathbf{G} (B) = \{e\}$, so $M_{\{e\}}$ is final over $\mathrm{Spec} A$, hence it is isomorphic to $\mathrm{Spec} A$.

How do we get a completion map $\sigma_T : BT \to M_T (A)$ for all $T$ given an orientation $\sigma_{S^1} : BS^1 \to \mathbf{G} (A)$?  By a composition: define
$$ Bev: BT \to \mathrm{Hom}(\hat T , BS^1), \; p \mapsto (f \mapsto Bf(p))$$
then define
$$\sigma_T := \sigma_{S^1} \circ Bev.$$

**Proposition**
There exists a map $\hat M$ such that the assignment $T \mapsto M_T$ factors as $T \mapsto \hat M (BT)$. That is the functor $M$ factors through the category of classifying spaces of compact Abelian Lie groups $B(CALG)$ (considered as orbifolds). Further, such factorizations are in bijection with the preorientations of $\mathbf{G}$.

_Proof._ That such a factorization exists defines $\hat M$ on objects.  Now by choosing a base point in $BT'$ we have

$$ \mathrm{Hom} (BT , BT' ) \simeq BT' \times \mathrm{Hom} (T, T') $$

as spaces.  Now we need a map
$$BT' \to \mathrm{Hom} (M_T , M_{T'}) .$$
Because this map must be functorial in $T$ and $T'$ we can restrict to the universal case where $T$ is trivial and then
$$BT' \to \mathrm{Hom} ( M_{\{e\}} , M_{T'} ) = M_{T'} (A)$$
is just a preorientation $\sigma_{T'}$.

##The Abelian Case for General Spaces##
We will see that $A_T (X)$ is the global sections of a quasi-coherent sheaf on $M_T$.

**Theorem** Let $\mathbf{G}$ be preoriented and $X$ a finite $T$-CW complex.  There exist a unique family of functors $\{ F_T \} $ from finite $T$-spaces to the category of quasi-coherent sheaves on $M_T$ such that

1. $F_T$ maps $T$-equivariant (weak) homotopy equivalences to equivalence of quasi-coherent sheaves;

1. $F_T$ maps finite homotopy colimits to finite homotopy limits of quasi-coherent sheaves;

1. $F_T (*) = O (M_\mathbf{G})$;

1. If $T \subset T'$ and $X' = (X \times T' )/T$ then $F_{T'} (X') \simeq f_* (F_T (X))$, where $f: M_T \to M_{T'}$ is the induced map;

1. The $F_T$ are compatible under finite chains of inclusions of subgroups $T \subset T' \subset T'' \dots$.

_Proof._ Use (2) to reduce to the case where $X$ is a $T$-equivariant cell, i.e. $X = T/ T_0 \times D^k$ for some subgroup $T_0 \subset T$.  Use (1) to reduce to the case where $X = T/T_0$. Use (3) to conclude that $F_T (T/T_0) = f_* F_{T_0} (*)$.  Finally, (4) implies that $F_{T_0} (*) = \hat M (*/T_0 )$, where $\hat M$ is specified by the preorientation.

>For trivial actions there is no dependence on the preorientation.

**Remark**

1. $F_T (X)$ is actually a sheaf of algebras.

1. If $X,Y$ are $T$-spaces then we have maps
$$F_T (X) \to F_T (X \times Y) \leftarrow F_T (Y)$$
and
$$F_T (X) \otimes F_T (Y) \to F_T (X \times Y).$$

1. Define relative version for $X_0 \subset X$ by
$$F_T (X, X_0 ) = hker (F_T (X) \to F_T (X_0 ))$$
and for all $T$-spaces $Y$ we have a map
$$F_T (X, X_0 ) \otimes_{A} F_T (Y) \to F_T (X \times Y , X_0 \times Y).$$

**Definition** $A_T (X) = \Gamma (F_T (X))$ as an $E_\infty$-ring (algebra).

We now verify loop maps on $A_T$.

Recall that in the classical setting $A^n (X)$ is represented by a space $Z_n$ and we have suspension maps $Z_0 \to (S^n \to Z_n)$.  Now we need to consider all possible $T$-equivariant deloopings, that is $T$-maps from $S^n \to Z_n$.

**Theorem** Let $\mathbf{G}$ be oriented, $V$ a finite dimensional unitary representation of $T$.  Denote by $SV \subset BV$ the unit sphere inside of the unit ball.  Define $L_V = F_T (BV, SV)$.  Then

1. $L_V$ is a line bundle on $M_T$, i.e. invertible;

1. For all (finite) $T$-spaces $X$ the map

$$L_V \otimes F_T (X) \to F_T (X \times BV, X \times SV)$$
is an isomorphism.

_Proof for $T = S^1 = U(1)$ and $V = \mathbb{C}$._  Then

$$L_V = hker ( F_T (BV) \to F_T (SV)) .$$

As $BV$ is contractible $F_T (BV) = O (\mathbf{G})$ and by property (3) above $F_T (SV) = f_* (O (\mathrm{Spec} A))$ for $f: \mathrm{Spec} A \to \mathbf{G}$ is the identity section.  As $\mathbf{G}$ is oriented, $\pi_0 \mathbf{G} / \pi_0 \mathrm{Spec} A$ is smooth of relative dimension 1, so $L_V$ can be though of as the invertible sheaf of ideals defining the identity section of $\mathbf{G}$.

Suppose $V$ and $V'$ are representations of $T$ then $L_V \otimes L_{V'} \to L_{V \oplus V'}$ is an equivalence.  So if $W$ is a virtual representation (i.e. $W = U - U'$) then
$L_W = L_U \otimes (L_{U'} )^{-1} .$

**Definition** Let $V$ be a virtual representation of $T$ and define
$$A_T^V (X) = \pi_0 \Gamma (F_T (X) \otimes L_V^{-1}) .$$

>The point is that in order to define equivariant cohomology requires functors $A_G^W$ for all representations of $G$, not just the trivial ones.  In the derived setting we obtain this once we have an orientation of $\mathbf{G}$.

##The Non-Abelian Lie Group Case##
Let $A$ be an $E_\infty$-ring, $\mathbf{G}$ an orientated commutative [[derived group scheme]] over $\mathrm{Spec} A$, and $T$ a (not necessarily Abelian) compact Lie group.

**Theorem** There exists a functor $A_T$ from (finite?) $T$-spaces to Spectra which is uniquely characterized by the following.

1. $A_T$ preserves equivalence;

1. For $T_0 \subset T$, $A_{T_0} (X) = A_T ((X \times G) / T_0 )$;

1. $A_T$ maps homotopy colimits to homotopy limits;

1. If $T$ is Abelian, then $A_T$ is defined as above;

1. For all spaces $X$ the map

$$ A_T (X) \to A_T (X \times E^{ab} T) $$
where $E^{ab} T$ is a $T$-space characterized by the requirement that for all Abelian subgroups $T_0 \subset T$, $(E^{ab} T )^{T_0}$ is contractible and empty for $T_0$ not Abelian.
Further, for Borel equivariant cohomology we require

1. If $T = \{e \}$, then $A_T(X) = A(X) = A^X$;

1. $A_T (X) \to A_T (X \times ET)$ is an isomorphism.

_Proof._ In the case of ordinary equivariant cohomology we can use property (5) to reduce to the case where $X$ has only Abelian stabilizer groups.  Then via (3) we reduce to $X$ being a colimit of $T$-equivariant cells $D^k \times T/T_0$ for $T_0$ Abelian. Via homotopy equivalence (1) we reduce to $X = T / T_0$.  Using property (2) we see $A_T (X) = A_{T_0} (*)$, so (4) yields $A_{T_0} (*) = \hat M (*/T_0 )$


A Survey of Elliptic Cohomology - cohomology theories

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Higher geometry
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[[!include higher geometry - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
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+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

see there for background and context.

This entry reviews basics of [[periodic cohomology theory|periodic]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory|cohomology theories]] and their relation to formal group laws.

=--

Next:

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

## rough notes from a talk##

>the following are rough unpolished notes taken more or less verbatim from some seminar talk -- needs attention

A complex oriented [[cohomology]] theory (meant is here and in all of the following a [[generalized (Eilenberg-Steenrod) cohomology]]) is one with a _good notion of [[Thom class]]es, equivalently first [[Chern class]] for complex [[vector bundle]]_

>(this "good notion" will boil down to certain extra assumptions such as multiplicativity and periodicity etc. What one needs is that the [[cohomology ring]] assigned by the cohomology theory to $\mathbb{C}P^\infty \simeq \mathcal{B}U(1)$ is a power series ring. The formal variable of that is then identified with the universal first Chern class as seen by that theory).

ordinary [[Chern class]] lives in [[integral cohomology]]
$
  H^*(-,\mathbb{Z})
$

or in [[K-theory]] $K^*(-)$ where for a [[vector bundle]] $V$ we would set $c_1(V) := ([V]-1)\beta$ where $\beta$ is the [[Bott generator]].

In the first case we have that under [[tensor product]] of [[vector bundle]]s the class behaves as

$$
  c_1(V\otimes W) = c_1(V) + c_1(W)
$$

whereas in the second case we get

$$
  c_1(V \otimes W) = c_1(V)c_1(W)\beta^{-1} + c_1(V) + c_1(W)
  \,.
$$

In general we will get that the [[Chern class]] of a tensor product is given by a certain [[power series]] $E^*(pt)$

not all [[formal group law]]s arises this way. the [[Landweber criterion]] gives a condition under which there is a cohomology theory

**definition** of complex-orientation

there is an

$$
  x \in \tilde E^2(\mathbb{C}P^\infty)
$$

such that under the map

$$
  \tilde E^2(\mathbb{C}P^\infty)
  \to
  \tilde E^2(\mathbb{C}P^1)
  \simeq
  \tilde E^2(S^1)
  \simeq E^0({*})
$$

induced by

$\mathbb{C}P^1 \to \mathbb{C}P^\infty$

we have $x \mapsto 1$

**remark** this also gives [[Thom class]]es since
$\mathbb{C}P^\infty \to (\mathbb{C}P^\infty)^\gamma$ is a [[homotopy equivalence]]

$$
  \tilde E^2((\mathbb{C}P^\infty)^\gamma)
  \simeq
  \tilde E^2((\mathbb{C}P^\infty))
  \ni X
$$

Thom iso $\tilde H^{*+2}(X^\gamma) \simeq H^*(X)$

...

(here and everywhere the tilde sign is for [[reduced cohomology]])

**definition (Bott element and even periodic cohomology theory)**

* An [[even cohomology theory]] is one whose odd cohomology rings vanish: $E^{2k+1}({*}) = 0$.

* A [[periodic cohomology theory]] is one with a [[Bott element]] $\beta \in E^2({*})$ which is invertible (under multiplication in the [[cohomology ring]] of the point)

  so that gives an isomorphism $(-)\cdot \beta : E^*({*}) \simeq E^{*+2}({*})$

Periodic cohomology theories are complex-orientable. $E^*(\mathbb{C}P^\infty)$ can be calculated using the [[Atiyah-Hirzebruch spectral sequence]]

$$
  H^p(X, E^q({*})) \Rightarrow E^{p+q}(X)
$$

notice that since $\mathbb{C}P^\infty $ is [[homotopy equivalence|homotopy equivalent]] to the [[classifying space]] $\mathcal{B}U(1)$ (which is a topological group) it has a product on it

$$
  \mathbb{C}P^\infty \times  \mathbb{C}P^\infty
  \to
  \mathbb{C}P^\infty
$$

which is the one that induces the [[tensor product]] of [[line bundle]]s classified by maps into $\mathbb{C}P^\infty$.

on (at least on even periodic cohomology theories) this induces a map of the form

$$
  \array{
    \mathbb{C}P^\infty \times  \mathbb{C}P^\infty
    &\to&
    \mathbb{C}P^\infty
    \\
    E({*})[[x,y]]  &\leftarrow& E(*)[[t]]
   \\
   f(x,y) &\leftarrow |& t
  }
$$

this $f$ is called a [[formal group law]] if the following conditions are satisfied

1. **commutativity** $f(x,y) = f(y,x)$

1. **identity** $f(x,0) = x$

1. **associtivity** f(x,f(y,z)) = f(f(x,y),z)

**remark** the second condition implies that the constant term in the power series $f$ is 0, so therefore all these power series are automatically invertible and hence there is no further need to state the existence of inverses in the formal group. So these $f$ always start as

$$
  f(x,y) = x + y + \cdots
$$

The [[Lazard ring]] is the "universal formal group law". it can be presented as by generators $a_{i j}$ with $i,j \in \mathbb{N}$

$$
  L = \mathbb{Z}[a_{i j}] / (relations 1-3 below)
$$

and relatins as follows

1. $a_{i j} = a_{j i}$

1. $a_{10} = a_{01} = 1$; $\forall i \neq 0: a_{i 0} = 0$

1. the obvious associativity relation

the universal formal group law we get from this is the power series in $x,y$ with coefficients in the [[Lazard ring]]

$$
  \ell(x,y) = \sum_{i,j} a_{i j} x^j y^j \in L[[x,y]]
  \,.
$$

**remark** the formal group law is not canonically associated to the cohomology theory, only up to a choice of rescaling of the elements $x$. But the underlying [[formal group]] is independent of this choice and well defined.

For any ring $S$ with formal group law $g(x,y) \in power series in x,y with coefficients in S$ there is a unique morphism $L \to S$ that sends $\ell$ to $g$.

**remark** Quillen's theorem says that the Lazard ring is the ring of complex cobordisms

**some universal cohomology theories** $M U$ is the [[spectrum]] for [[complex cobordism cohomology theory]]. The corresponding [[spectrum]] is in degree $2 n$ given by

$$
  M U(2n) = Thom
  \left(
    standard associated bundle to universal bundle
    \array{
      E U(n) \\ \downarrow \\ B U(n)
    }
  \right)
$$

periodic [[complex cobordism cohomology theory]] is given by

$$
  M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U
$$

we get a canonical [[oriented cohomology theory|orientation]] from

$$
  \omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to}
  M U(1)
  \;\;\;\;
  M U(\mathbb{C}P^\infty)
$$

this is the universal even periodic cohomology theory with orientation

**Theorem (Quillen)** the [[cohomology ring]] $M P(*)$ of periodic [[complex cobordism cohomology theory]] over the point together with its [[formal group law]] is naturally isomorphic to the universal  [[Lazard ring]] with its [[formal group law]] $(L,\ell)$

how one might make a [[formal group law]] $(R,f(x,y))$ into a cohomology theory

use the classifying map $M P({*}) \to R$
to build the [[tensor product]]

$$
  E^n(X) := M P^n(X) \otimes_{M P({*})} R
$$

this construction could however break the left exactness condition. However, $E$ built this way will be left exact of the ring morphism $M P({*}) \to R$ is a flat morphism. This is the [[Landweber exactness]] condition (or maybe slightly stronger).

[[!redirects A Survey of Eilliptic Cohomology - cohomology theories]]


A Survey of Elliptic Cohomology - compactifying the derived moduli stack

[[!redirects A Survey of Elliptic Cohomology - compactifying the derived moduli space]]

<div class="rightHandSide toc">

[[!include higher algebra - contents]]

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>**Abstract** We sketch how to compactify $M^{Der}$ such that the underlying scheme is the Deligne-Mumford compactification of $M_{1,1}$.

+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

see there for background and context.

=--

Here are the entries on the previous sessions:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

* [[A Survey of Elliptic Cohomology - elliptic curves]]

* [[A Survey of Elliptic Cohomology - equivariant cohomology]]

* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]

* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]

* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]

* [[A Survey of Elliptic Cohomology - towards a proof]]

***

# Compactifying $M^{Der}$#
* automatic table of contents goes here
{:toc}

##Introduction##
Let $\mathbf{M}^{Der}$ be the derived Deligne-Mumford moduli stack of oriented elliptic curves.  Recall that the underlying Deligne-Mumford stack $\pi_0 \mathbf{M}^{Der}$ is $\mathbf{M}_{1,1}$ the classical Deligne-Mumford stack which is the fine moduli stack of elliptic curves.  We would like to construct a derived Deligne-Mumford stack $\overline{\mathbf{M}^{Der}}$ such that $\pi_0 \mathbf{M}^{Der}$ is the classical compactification $\overline{\mathbf{M}_{1,1}}$.

Recall that one can define the $E_\infty$-ring $tmf[\Delta^{-1}]$ as the global sections of $\OM_{\mathbf{M}^{Der}}$.  If we take global sections of $\mathbf{O}_{\overline{\mathbf{M}^{Der}}}$ then we get the $E_\infty$-ring $tmf$.

##The Tate Curve##

Let us focus on elliptic curves over $\mathbb{C}$.  The coarse moduli space of such elliptic curves is again $\mathbb{C}$ with the classifying map given by the $j$-invariant.  $\mathbb{C}$ is not compact, but we can compactify the moduli space by allowing curves with nodal singularities (generalized elliptic curves).

For each $q \in \mathbb{C}, \; \lvert q \rvert \lt 1$, there is an elliptic curve over $\mathbb{C}$ defined by a Weierstrass equation
$$y^2 + xy = x^3 +a_4(q) x + a_6 (q) .$$
If $0 \lt \lvert q \rvert \lt 1$ the elliptic curve is isomorphic to $\mathbb{C}^*  /q^\mathbb{Z}$ as a Riemann surface with group structure induced from $\mathbb{C}^*$ (equivalently, this curve corresponds to $\mathbb{C}/ \Lambda_\tau$ where $e^{2\pi i \tau} = q$). As a function of $q$, $a_4$ and $a_6$ are analytic over the open disk and their power series at $q=0$ have integral coefficients hence the Weierstrass equation defines an elliptic curve $T$ over $\mathbb{Z}[ [q] ]$.  We really would like to think of $T$ as an elliptic curve over $\mathbb{Z} ( (q) ):= \mathbb{Z} [ [q] ][q^{-1}]$.

It should be noted that this construction (which goes back to Tate) can be extended to more general fields.

The Tate curve defines a cohomology theory $K_{Tate}$ (an elliptic spectrum).  As a cohomology theory $K_{Tate}$ is just $K$-theory tensored with $\mathbb{Z}( (q) )$.

##Annular Field Theories##
As shown in Pokman Cheung's thesis, the Tate curve has a connection to  supersymmetric field theories as defined by Stolz and Teichner.  The main result of is that a subspace of $2|1$ field theories (annular theories) is the $0th$-space of the spectrum $K_{Tate}$.

Let $SAB$ be the subcategory of $2|1$-EB whose morphisms are annuli. Note that $SAB$ does not contain tori.  $SAB$ in essence is completely determined by the supergroup $\mathbb{R}^{2|1}$.  Let $AFT_n$ be the space of natural transformations analogous to the definition of Stolz and Teichner.

**Theorem (Theorem 2.2.2 of Cheung).** For each $n \in \mathbb{Z}$, we have
$AFT_n \simeq (K_{Tate})_n .$

_Proof._
Let $A_{l,\tau}$ be the cylinder obtained by identifying non-horizontal sides of the parallelogram in the upper-half plane spanned by $l$ and $l\tau$ for $l \in \mathbb{R}_+$ and $\tau \in h$.

Let $E$ be a degree 0 field theory. The family of cylinders $\{A_{l,\tau} \}$ has the properties that $A_{l, \tau+1} = A_{l , \tau}$ and $A_{l, \tau} \circ A_{l, \tau'} = A_{l, \tau + \tau'}$.  For a fixed $l \in \mathbb{R}_+$, $\{E(A_{l,\tau}), \tau \in h \}$ is a commuting family of trace class (and hence compact) operators depending only on $\tau$ modulo $\mathbb{Z}$.  Therefore, by writing $q= e^{2 \pi i \tau}$ we can write $E (A_{l, \tau}) = q^{L} q^{\overline{L}}$, where $L, \overline{L}$ are unbounded operators with discrete spectrum.

**Lemma.** The spectrum of $L-\overline{L} \subseteq \mathbb{Z}$.  Also, $\overline{L} = G^2$, where $G$ is an odd operator.

_Proof of Lemma._  The spectral argument follows from having an $S^1$ action.  To see the second claim note that for fixed $l$, $\mathbb{R}^{2|1}_+ /l \mathbb{Z}$ is a super Lie semi-group and the functor $E$ gives a representation $\mathbb{R}^{2|1}_+ \to \mathrm{End} (E(S^1_l))$ which is compatible with the super-semigroup law on $\mathbb{R}^{2|1}$ given by
$$(z_1 , \overline{z}_1 , \theta_1) , (z_2 , \overline{z}_2 , \theta_2) \mapsto (z_1 + z_2 , \overline{z}_1 +\overline{z}_2 + \theta_1 \theta_2 , \theta_1 + \theta_2) .$$
Now $\mathrm{Lie} (\mathbb{R}^{2|1}_+ ) = \langle \partial_z , \partial_{\overline{z}} , Q \rangle$, where $Q = \partial_\theta + \theta \partial_{\overline{z}}$.  Under $E$ the vector fields map to $L, \overline{L},$ and $G$ respectively.  Further, $[Q,Q] = 2Q^2 = 2 \partial_{\overline{z}},$ hence $G^2 = \overline{L}$.

We see that a degree 0 theory is determined by a pair of operators $(L, G)$.  By analyzing the spectral decomposition of the pair $(G, L-G^2)$ one can construct a weak equivalence of categories $AFT \to V$ (a homotopy equivalence of geometric realizations), where $V$ is a certain category of Clifford-modules.  Following work of Segal it is proved that $|V| \simeq (K_{Tate})_0$. The nonzero degrees follow a similar line of reasoning.

##$\overline{\mathbf{M}^{Der}}$##

We can extend the standard toric variety construction to derived schemes by replacing $\mathbb{C}$ by a fixed $E_\infty$-ring $R$.  That is given a fan $F= \{U_\alpha \}$ we can build a derived scheme $X_F$ where $X_F = \varinjlim \{U_\alpha\}$ and $U_\alpha = \mathrm{Spec} \; R [S_{\sigma_\alpha}]$ is an affine derived scheme.  We will define the (derived) Tate curve as the formal completion of the quotient of a toric variety.

Let $F_0 = \{ \{0\} , \mathbb{Z}_{\ge 0} \}$, so $X_{F_0} = \mathrm{Spec} R [\mathbb{Z}_{\ge 0}] = \mathrm{Spec} R[q]$.  Also, let $F= \langle \sigma_n \rangle_{n \in \mathbb{Z}}$, where
$$\sigma_n = \{ (a,b) \in \mathbb{Z} \times \mathbb{Z} \; | \; na \le b \le (n+1) a\}.$$
Note that by projection onto the first factor we have a map of fans $F \to F_0$ and consequently an induced map on varieties $f: X_F \to \mathrm{Spec} \; R[q]$.

Consider $f: X_F \to \mathrm{Spec} \; R[q]$, one can show that
1. $f^{-1} (q) \cong \mathbf{G}_m$ for $q \neq 0$;
1. $f^{-1} (0)$ is an infinite chain of rational curves, each intersecting the next in a node.

Consider the automorphism $\tau : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ defined by $\tau (a,b) = (a, b+a)$.  Note that $\tau (\sigma_n ) = \tau (\sigma_{n+1})$, so $\tau$ preserves the fan $F$ and consequently is an automorphism of the resulting toric variety $X_F$ which we also denote by $\tau$. Then
1. $\tau$ acts on $f^{-1} (q)$ by multiplication by $q$, for $q \neq 0$;
1. $\tau$ acts freely on $f^{-1}$.

Now define $\widehat{X}_F$ to be the formal completion of $X_F$ along $f^{-1} (0)$.  Similarly, define $R[ [q] ]$ as the formal completion of $R[q]$ along $q=0$. One can show that $\tau^{\mathbb{Z}}$ which is the multiplicative group generated by $\tau$ acts freely on $\widehat{X}_F$.

Define $\widehat{T}$ to be the formal (derived) scheme $\widehat{X}_F / \tau^\mathbb{Z}$.

**Theorem (Lurie/Grothendieck).**
The formal derived scheme $\widehat{T}$ is the completion of a unique derived scheme over $\mathrm{Spec} \; R[ [q] ]$.  That is, there exists a unique derived scheme $T \to \mathrm{Spec} \; R[ [q] ]$ such that $\widehat{T} = \mathrm{Spf} \; T$.

We call the derived scheme $T$ in the theorem above the _Tate curve_.
It is a fact that the restriction of $T$ to the punctured formal disk $\mathrm{Spec} \; R ( (q) )$ is an elliptic curve isomorphic to $\mathbf{G}_m / q^\mathbb{Z}$.

We then see that an orientation of $T$ is equivalent to an orientation of $\mathbf{G}_m$ which is equivalent to working over the complex $K$-theory spectrum $K$. Therefore, the oriented Tate curve is equivalent to a map
$$T: \mathrm{Spec} \; K( (q) ) \to \mathbf{M}^{Der}.$$

Now note that the involution $\alpha : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}$ defined by $\alpha (a,b) = (a, -b)$ preserves the fan $F$ and $(\tau \circ \alpha)^2 = 1$.  We allow $\alpha$ to be complex conjugation on $K((q))$ thought of as a group action of $\{\pm 1\}$, so we have a map $\mathrm{Spec} \; K( (q) ) / \{\pm 1\} \to \mathbf{M}^{Der}$.

We can define a new derived Deligne-Mumford stack by forming an appropriate pushout square.

Finally, one can show that the underlying scheme $\pi_0 \overline{\mathbf{M}^{Der}}$ is $\overline{\mathbf{M}_{1,1}}$.

There are many subtleties associated with $\overline{\mathbf{M}^{Der}}$.  For instance, we would like to glue the universal curve $\E$ over $\mathbf{M}^{Der}$ with $T$ to obtain a universal elliptic curve $\overline{\E}$ over $\overline{\mathbf{M}^{Der}}$, however the result is only a _generalized_ elliptic curve; it is not a derived group scheme over $\overline{\mathbf{M}^{Der}}$ as it only has a group structure over the smooth locus of the map $\overline{\E} \to \overline{\mathbf{M}^{Der}}$.
Lurie asserts it is possible to construct the necessary geometric objects over $\overline{\mathbf{M}^{Der}}$ (I guess this will show up in DAG VII or VIII).  The global sections of the structure sheaf thus constructed is the spectrum $tmf$.


<div class="rightHandSide toc">
[[!include cohomology - contents]]

***

[[!include higher algebra - contents]]

</div>

+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

see there for background and context.

This entry contains a basic introduction to derived [[group scheme]]s and their orientations.

=--

Previous:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

* [[A Survey of Elliptic Cohomology - elliptic curves]]

* [[A Survey of Elliptic Cohomology - equivariant cohomology]]

Next:

* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]

* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]

* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]

>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#

* toc
{:toc}

# Introduction #

Recall from last time that given $G$ an algebraic group such that the [[formal spectrum]] $Spf A(\mathbb{C}P^\infty)$ is the completion $\hat G$, we could define $A_{S^1}({*}) := \mathcal{o}_{G}$ then passing to germs gave a completion map

$$
  A_{S^1}({*}) \to A(\mathbb{C}P^\infty) =
  A^{Bor}_{S^1}({*}).
$$

The problem we (begin) to address here is how to extend this equivariant cohomology to other spaces besides the point.  This requires derived algebraic geometry.

# Derived group schemes#

Recall that a commutative [[group scheme]] over a scheme $X$ is a functor
$$
G: \mathrm{Sch} /X^{op} \to \mathrm{Ab}
$$
such that composition with the forgetful functor $F: \mathrm{Ab} \to \mathrm{Set}$ is representable.

We would like extend this definition to the world of [[derived scheme]]s.  There are two problems

1. Because of the higher categorical nature of derived schemes Hom sets are spaces.

1. Everything should in the $\infty$-setting, that is defined only up to homotopy.

We will not worry about the second concern and address the first by replacing the category $\mathrm{Ab}$ with $\mathrm{TopAb}$ and $\mathrm{Set}$ with $\mathrm{Top}$.

> The following definition is somewhat restrictive and really should incorporate more of the $\infty$-structure.

**Definition** A commutative [[derived group scheme]] over a [[derived scheme]] $X$ is a topological functor
$$
G: \mathrm{DSch} / X^{op} \to \mathrm{TopAb}
$$
such that composition with the forgetful functor $F: \mathrm{TopAb} \to \mathrm{Top}$ is representable (up to weak equivalence) by an object which is [[flat object|flat]] over $X$.

**Examples**

1. Let $X$ be a scheme, then we have an associated [[derived scheme]] $\overline{X}$.  The structure sheaf of $\overline{X}$ is obtained by viewing the structure sheaf of $X$ as a presheaf of $E_\infty$-rings and then sheafifying.  We then have an equivalence between commutative [[derived group scheme]]s over $\overline{X}$ and commutative [[group scheme]]s which are flat over $X$.

1. For $X$ a [[derived scheme]] we have a map from commutative [[derived group scheme]]s over $X$ to commutative [[group scheme]]s which are flat over $\pi_0 X$.

#Preorientations#

Throughout $A$ will be an $E_\infty$-ring, $X$ the affine [[derived scheme]] $\mathrm{Spec} A$, $G$ a commutative [[derived group scheme]] over $X$, $A_{S^1}$ the $E_\infty$-ring given by $\Gamma (G)$, and $A^{\mathbb{C}P^\infty}$ the $E_\infty$-ring given by

$$
  A^{\mathbb{C}P^\infty} = \mathrm{Hom}_{E_\infty} (\mathbb{C}P^\infty , A).
$$

**Definition**(Preliminary) A preorientation of $G$ is a morphism of commutative [[derived group scheme]]s over $X$
$$
\sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to G,
$$
where $\mathrm{Spf} A^{\mathbb{C}P^\infty}$ is the completion wrt the ideal $\mathrm{ker} (A^{\mathbb{C}P^\infty} \to A^{pt} = A)$. A preorientation is an orientation if the induced map
$$
\hat \sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to \hat G
$$
is an isomorphism.

Suppose that $G$ is affine, then a map
$$
\sigma : \mathrm{Spf} A^{\mathbb{C}P^\infty} \to G
$$
corresponds to a map $A_{S^1} \to A^{\mathbb{C}P^\infty}$
which is the same as a map
$$
\mathbb{C}P^\infty \to \mathrm{Hom} (X, G) = G(X).
$$
Hence we are led to the following definition.

**Definition** Let $X$ be a [[derived scheme]] and $G$ a commutative [[derived group scheme]] over $X$.  A preorientation of $G$ is a morphism of topological commutative monoids
$$
\mathbb{P} (\mathbb{C} [\alpha]) = \mathbb{C}P^\infty \to G(X).
$$

Notice that $\mathbb{C}P^\infty$ is nearly freely generated.  Indeed it follows from the fundamental theorem of algebra that as a topological monoid $\mathbb{C}P^\infty$ is generated by $\mathbb{C}P^1$ subject to the single relation that $\mathrm{pt} = \mathbb{C}P^0 \subset \mathbb{C}P^1$ is the monoidal unit.

**Proposition** A preorientation up to homotopy is a map
$$S^2 \simeq \mathbb{C}P^1 \to G(X) $$
that is an element of $\pi_2 (G(X))$.

Hence, we always have at least one preorientation: the trivial one which corresponds to $0 \in \pi_2 (G(X))$.

#Orientations#

As motivation recall that a map $s: A \to B$ of 1-dim [[formal group]]s is an isomorphism if and only if $s'$ is invertible.  We would like to encode this in our derived language (without defining $s'$).

**Definition** Let $A$ be an $E_\infty$-ring, $G$ a commutative [[derived group scheme]] over $\mathrm{Spec} A$ and $\sigma : S^2 \to G(A)$ a preorientation.  Then $\sigma$ is an orientation if

1. $\pi_0 G \to \mathrm{Spec} \pi_0 A$ is smooth of relative dimension 1, and

1. The map induced by $\beta : \omega \to \pi_2 A$
$$
\pi_n A \otimes_{\pi_0 A} \omega \to \pi_{n+2} A$$
is an isomorphism for each $n$.

Note that (2) implies that $A$ is weakly periodic.  Conversely, if $A$ is weakly periodic then (2) is equivalent to $\beta$ being an isomorphism.

Before defining $\beta$ and $\omega$ we extend the above definition to [[derived group scheme]]s over an arbitrary [[derived scheme]].

**Definition** Let $X$ be a [[derived scheme]], $G$ a commutative [[derived group scheme]] over $X$ and $\sigma: S^2 \to G(X)$ a preorientation.  Then $\sigma$ is an orientation if $(G, \sigma) |_U$ is an orientation for all $U \subset X$ affine.

We now define the module $\omega$ and the map $\beta$.  Let $A$ be an $E_\infty$-ring, $G$ a commutative [[derived group scheme]] over $\mathrm{Spec} A$ and let $G_0 = \pi_0 G$ a scheme over $\mathrm{Spec} \pi_0 A$. Let $\Omega$ denote the sheaf of differentials on $G_0 / \pi_0 A$.  Then define
$$\omega := i^* \Omega, \; i: \mathrm{Spec} \pi_0 A \to G $$
is the identity section.

Now let $U \hookrightarrow G$ be an open affine subscheme containing the identity section, so $U = \mathrm{Spec} B$ for some $E_\infty$-ring $B$.  Then $\sigma$ induces a map
$B \to A^{S^2}$ which is the same as a map
$$
\pi_0 B \to A(S^2)= \pi_0 A \oplus \pi_2 A .
$$
It is a fact that the map $\pi_0 B \to \pi_2 A$ is a derivation over $\pi_0 A$ and hence has a classifying map which yields a map
$$\beta : \omega \to \pi_2 A .$$

#The Multiplicative Derived Group Scheme#

The naive guess for $G_m$ is $GL_1$, where $GL_1 (A) = A^x = (\pi_0 A)^x$.  It is true that $GL_1$ is a [[derived scheme]] over $\mathrm{Spec} \mathbf{S}$, however it is not flat, nor is $GL_1 (A)$ an Abelian group as $A$ is an $E_\infty$-ring and not an honestly commutative ring.

If $A$ is rational, that is there is a map $H \mathbb{Q} \to A$, then $GL_1 (A)$ can be given an Abelian group structure.  Hence, $GL_1$ is a perfectly good [[group scheme]] defined on the category of rational $E_\infty$-rings, however this category is too small; there are too few rational $E_\infty$-rings.

Recall that for a ring $R$, $R^x = \mathrm{Hom}_R (R [ t, t^{-1} ] , R)$.  Further, recall that for a group $M$ we can form the group algebra $R[M]$ which is really a [[Hopf algebra]].  Then $\mathrm{Spec} R[M]$ is a [[group scheme]] over $\mathrm{Spec} A$.  Further, $R[m]$ is characterized by
$$
\mathrm{Ring} ( R[M] , B) = \mathrm{Ring} (R, B) \times \mathrm{Mon} (M,B)
$$
where $\mathrm{Mon}$ is the category of monoids and the ring $B$ is thought of as a monoid wrt to multiplication. Motivated by these observations we make the following definitions.

**Definition** Let $A$ be an $E_\infty$-ring and $M$ a topological Abelian monoid, then we can define $A[M]$ which is characterized by
$$
\mathrm{Alg}_A (A[M], B) \simeq \mathrm{Alg}_A (A,B) \times \mathrm{TopMon} (M, B^\times ).
$$
Recall that because of the higher categorical nature of things, the hom-sets above are spaces and the symbol $\simeq$ indicates weak equivalence of spaces.

**Definition** Let $A$ be an $E_\infty$-ring.  We define the multiplicative group corresponding to $A$ as
$$
G_m = \mathrm{Spec} A[ \mathbb{Z}].
$$
$G_m$ is a derived commutative group scheme over $\mathrm{Spec} A$.

Note that $\pi_* ( A[ \mathbb{Z}]) = (\pi_* A) [ \mathbb{Z}]$. Also, the map $\pi_0 G_m \to \pi_0 \mathrm{Spec}  A$ is smooth of relative dimension 1.

##Preorientations of $G_m$##

**Proposition** For any $E_\infty$-ring $A$, we have a bijection (of sets) between preorientations of $G_m$ and maps $\mathbf{S} [ \mathbb{C} P^\infty ] \to A$.

The proof follows from the fact that $\mathbf{S}$ is initial in the category of $E_\infty$-rings and the mapping property of $A [ \mathbb{Z}]$.

**Corollary** $\mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty ]$ is the moduli space of preorientations of $G_m$.  That is, if $G_m$ is defined over $\mathrm{Spec} A$, then a preorientation of $G_m$ is the same as a map $\mathrm{Spec} A \to \mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty ]$.

##Orientations of $G_m$##

We consider the map $\beta : \omega \to \pi_2 A$ where
$$
\omega = i^* \Omega^1_{\pi_0 G / \pi_0 \mathrm{Spec} A}
$$
and $i$ is the identity section.  Note that $\pi_0 G_m = (\pi_0 A ) [t, t^{-1}]$, hence it follows that $\omega$ is canonically trivial, so an orientation is just an element $\beta_\sigma \in \pi_2 A$ such that $\beta_\sigma$ is invertible in $\pi_* A$.

Let $\beta$ denote the (universal) orientation of $\mathbf{S} [ \mathbb{C} P^\infty]$.  Then we have the following.

**Theorem** $\mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty] [ \beta^{-1}]$ is the moduli space of orientations of $G_m$.

It is a theorem of Snaith, that this moduli space has the homotopy type of $KU$ the spectrum of complex K-theory.  Note that by considering the homtopy fixed points of a certain action there is a way to recover $KO$ as well.

##Connection to complex orientation##

Let $A$ be an $E_\infty$-ring, so in particular $A$ defines a cohomology theory.  An orientation of $G_m$ over $\mathrm{Spec} A$ is a map $KU \to A$.  A complex orientation of $A$ is a map $MU \to A$.  Recalling that $KU$ is complex oriented, we see that an orientation of $G_m$ gives a complex orientation by precomposing with the map $MU \to KU$.

#The Additive Derived Group Scheme#

The naive definition of $G_a$ is $\mathbf{A}^1$, where $\mathbf{A}^1 (A)$ is the additive group of $A$.  It is true that $\mathbf{A}^1$ is a [[derived scheme]] over $\mathrm{Spec} \mathbf{S}$, however it is not flat as for an $E_\infty$-ring $A$
$$
\pi_k \mathbf{A}^1_A = \oplus_{n \ge 0} A^{-k} (B \Sigma_n )
$$
where as if it were flat we would have
$$
\pi_k \mathbf{A}^1_A = \pi_k A [x] .
$$
Also, $\mathbf{A}^1$ is not commutative.  $\mathbf{A}^1 (A)$ is an infinite loop space, but not an Abelian monoid.  Again $\mathbf{A}^1$ is a derived group scheme when restricted to rational $E_\infty$-rings.

We no restrict to the category of integral $E_\infty$-rings, i.e. those equipped with a map $H \mathbb{Z} \to A$. Note that in this category $H \mathbb{Z}$ is initial.

**Definition** For $A$ an integral $E_\infty$-ring define
$$
G^A_a = \mathrm{Spec} ( A \otimes_\mathbb{Z} \mathbb{Z} [x] ).
$$

It can be shown that $G^A_a$ is flat and has the correct amount of commutativity.

>Why can't we just use $\mathrm{Spec} A [ \mathbb{N}]$?

**Proposition** For all integral $E_\infty$-rings, preorientations of $G_a^A$ are in bijective correspondence with maps $H \mathbb{Z} [ \mathbb{C} P^\infty] \to A$.  Consequently, $\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty]$ is the moduli space of preorientations of $G_a$.

Now, $\pi_* H \mathbb{Z} [ \mathbb{C} P^\infty] = H_* (\mathbb{C} P^\infty , \mathbb{Z} )$.  The right side is a free divided power series on a generator $\beta$ where $\beta \in \pi_2 H \mathbb{Z} [ \mathbb{C} P^\infty]$.

**Proposition** $\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty ] [ \beta^{-1}]$ is the moduli space of orientations of $G_a$.

**Proposition** $\mathrm{Spec} H \mathbb{Z} [ \mathbb{C} P^\infty ] [ \beta^{-1}] = KU \otimes \mathbb{Q}$.  Hence the Chern character yields an isomorphism with rational periodic cohomology.


A Survey of Elliptic Cohomology - descent ss and coefficients

[[!redirects A Survey of Elliptic Cohomology - compactifying the derived moduli space]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

>**Abstract** This entry discusses the [[descent spectral sequence]] and sheaves in homotopy theory.  Using said spectral sequence we compute $\pi_* tmf_{(3)}$.

+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

see there for background and context.

=--

Here are the entries on the previous sessions:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

* [[A Survey of Elliptic Cohomology - elliptic curves]]

* [[A Survey of Elliptic Cohomology - equivariant cohomology]]

* [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]

* [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]

* [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]

* [[A Survey of Elliptic Cohomology - towards a proof]]

* [[A Survey of Elliptic Cohomology - compactifying the derived moduli stack]]

***

#The Descent Spectral Sequence#
* automatic table of contents goes here
{:toc}

##The spectral sequence##

We would like to understand the following theorem.

**Theorem.** Let $( X, \mathbf{O})$ be a derived Deligne-Mumford stack.  Then there is a spectral sequence
$$H^s (X ; \pi_t \mathbf{O}) \Rightarrow \pi_{t-s} \Gamma (X , \mathbf{O}).$$

###Recalling what is what###
Let $X$ be an $\infty$-topos, heuristically $X$ is ''sheaves of spaces on an $\infty$-category $C$.''  Further $\mathbf{O}$ is a functor $\mathbf{O} : \{ E_\infty \}^{op} \to X$, which for a cover $U$ of $C$ formally assigns
$$A \mapsto (U \mapsto \mathrm{Hom} (A , \mathbf{O} (U))).$$
Via DAG V 2.2.1 we can make sense of global sections and $\Gamma (X , \mathbf{O})$ is an $E_\infty$-ring.

Given an $\infty$-category $C$ we can form the subcategory of $n$ _truncated objects_ $\tau_{\le n} C$ which consists of all objects such that all mapping spaces have trivial homotopy groups above level $n$. Further $\tau_{\le n} : C \to C$ defines a functor which serves the role of the Postnikov decomposition.

Let $X$ be an $\infty$-topos, define $\mathrm{Disc} \; X : = \tau_{\le 0} X$.  Further define functors $\pi_n : X_* \to N (\mathrm{Disc} \; X)$ by
$$Y \mapsto \tau_{\le 0} \mathrm{Map} (S^n , Y) = \pi_n (Y) .$$

**Facts.**

1. For $A \in \mathrm{Disc} \; X$ an abelian group object there exists $K(A,n) \in X$, such that
$$ H^n (X, A) := \pi_0 \mathrm{Map} (1_X , K(A,n)) $$
corresponds to sections of $K(A,n)$ along the identity of $C$.

1. If $C$ is an ordinary site, $H^n (X, A)$ corresponds to ordinary sheaf cohomology (HTT 7.2.2.17).

###The non-derived descent ss###
Let us define a mapping space $\mathrm{Tot} \; X = \mathrm{hom}^\Delta (\Delta , X)$, this is the hom-set as simplicial objects.  Now
$$\mathrm{Tot} \; X = \lim ( \dots \to \mathrm{Tot}^n \; X \to \mathrm{Tot}^{n-1} \; X \to \dots \to \mathrm{Tot}^0 \to * ) ,$$
where $\mathrm{Tot}^n \; X = \mathrm{Tot} (\mathrm{cosk}_n X )$.
We have a homotopy cofiber sequence
$$F_n \to \mathrm{Tot}^n \; X \to \mathrm{Tot}^{n-1} \; X$$
and it is a fact that
$$F_s \simeq \Omega^s ( \prod_{|I|=s+1} \mathbf{O} (U_I )),$$
for the fibered product $U_I$ corresponding to the cover $\{ U_i \to N\}$ of an object $N$ of the etale site of $M_{1,1}$.

Applying $\pi_*$ to the cofiber sequence we obtain an exact couple and hence a spectral sequence with
$$ E^1_{t,s} = \pi_{t-s} F_s \Rightarrow \lim_n \pi_{t-s} \mathrm{Tot}^n X  = \pi_{t-s} \mathrm{Tot} X = \pi_{t-s} \mathbf{O} (N). $$
Note that $\pi_{t-s} F_s$ is the Čech complex of the cover, so the $E^2$-page calculates Čech cohomology.  If we choose an affine cover, hence acyclic and $\lim^1 =0$, then
$$E^2_{t,s} \Rightarrow H^s (N, \pi_t \mathbf{O}) .$$

##Stacks and Hopf Algebroids##

Let $X$ be a (non-derived) Deligne-Mumford stack on $\mathrm{Aff}$ and let $\mathrm{Spec} \; A \to X$ be a faithfully flat cover, then
$$ \mathrm{Spec} \; A \times_X \mathrm{Spec} \; A = \mathrm{Spec} \; \Gamma,$$
for some commutative ring $\Gamma$. Via the projection maps (which are both flat) we have a groupoid in $\mathrm{Aff}$, by definition it is a [[commutative Hopf algebroid]] $(A, \Gamma)$.

Now let $(A,\Gamma)$ be a commutative Hopf algebroid, then the collection of principal bundles form a stack $M_{A,\Gamma}$.  Here a principal bundle is a map of schemes $P \to X$, a $\mathrm{Spec} \; \Gamma$ equivariant map $P \to \mathrm{Spec} \; A$, where the action is given by a map $P \times_{\mathrm{Spec} A} \mathrm{Spec} \; \Gamma \to P$. In this we have an equivalence of 2-categories
$$ \{DM \; Stacks\} \simeq \{Hopf \; Algebroids, \; bibundles\} .$$
and
$$ \{DM \; stacks \; equipped \; with \; cover\} \simeq \{Hopf \; algebroids, \; functors \; of \; groupoids\}. $$

Let $X$ be a scheme then a sheaf of abelian groups is a functor
$$\mathfrak{I} : \mathrm{Aff}/X^{op} \to \mathrm{Ab} .$$
The structure sheaf $\mathbf{O}_X$ is defined by
$$ \mathbf{O}_X ( \mathrm{Spec} \; A \to X) = A .$$
Let $\mathfrak{I}$ be a sheaf of $\mathbf{O}_X$ modules.  $\mathfrak{I}$ is quasi-coherent if for any map $\mathrm{Spec} \; B \to \mathrm{Spec} \; A$ and maps $f: \mathrm{Spec} \; A \to X$, $g: \mathrm{Spec} \; B \to X$ we have
$$ B \otimes_{A} \mathfrak{I} (f) \simeq \mathfrak{I} (g) .$$
We have an equivalence of categories $\mathrm{QCSh}/\mathrm{Spec} \; A \simeq A$-mod via the assignment $\mathfrak{I} \mapsto \mathfrak{I} (1_A)$.

Now consider the stack $M_{A,\Gamma}$ from above.  One can show that quasi-coherent sheaves over $M_{A, \Gamma}$ is nothing but a $(A,\Gamma)$ [[comodule]], that is an $A$-module, $M$, and a coaction map of $A$-modules
$$M \to \Gamma \otimes^{d_1}_{A} M$$
where the right hand side is an $A$-module via the map $d_0$.

##Cohomology of Sheaves##
Recall that sheaf cohomology is obtained by deriving the global sections functor.  If $X$ is a [[noetherian scheme|noetherian]] scheme/stack then we restrict to deriving
$$ \Gamma (-) : \mathrm{QCSh}/X \to \mathrm{Ab} .$$
Suppose further that $X = \mathrm{Spec} \; A$, so $\Gamma$ lands in $A$-modules, however from above we know $\mathrm{QCSh}/\mathrm{Spec} \; A \simeq A$-mod, hence $\Gamma$ is exact and all higher cohomology groups vanish.

Let $\mathfrak{I}_N$ be a quasi-coherent sheaf on a DM stack $M_{A,\Gamma}$.  Then global sections of $\mathfrak{I}_N$ induce global sections $n \in N$ such that the two pullbacks to $\Gamma$ correspond to each other
$$ \Gamma \otimes_A^{d_0} N \to \Gamma \otimes_A^{d_1} N; \; 1 \otimes n \mapsto 1 \otimes n .$$
That is the coaction map $n \mapsto 1 \otimes n$ is well defined and $n: A \to N; \; 1 \mapsto n$ is a map of comodules.  This allows us to interpret global sections as
$$ \mathrm{Hom}_{A,\Gamma} (A, -) : \mathrm{Comod}_{A,\Gamma} \to A-\mathrm{mod} ,$$
so a section is a map from the trivial sheaf to the given sheaf.  It follows that
$$ H^n ( M_{A,\Gamma} , \mathfrak{I}_N ) = \mathrm{Ext}^n_{A,\Gamma} (A,N) .$$
To simplify notation we write the above as $H^n (A, \Gamma ; N)$ and if the $N$ is suppressed it is assumed that $N=A$.  In general we compute these Ext groups via the [[cobar complex]].

###Change of Rings###
Let $(A,\Gamma)$ be a commutative Hopf algebroid and $f: A \to B$  a ring homomorphism.  Define
$$ \Gamma_B = B \otimes_A^{d_0} \Gamma \otimes_A^{d_1} B ,$$
so we have a map of Hopf algebroids $f_* : (A, \Gamma) \to (B, \Gamma_B)$ and of stacks
$$ f^* : M_{B,\Gamma_B} \to M_{A,\Gamma} .$$

**Theorem.** If there exists a ring $R$ and a homomorphism $\Gamma \otimes_A B \to R$ such that
$$A \to \Gamma \otimes_A B \to R $$
is faithfully flat, then $f^*$ is an equivalence of stacks.

###The Weierstrass Stack###
Given $C/S$ an [[elliptic curve]], [[Riemann-Roch theorem|Riemann–Roch]] gives us (locally on $S$) sections $x \in \Gamma (C, \mathbf{O} (2e)) , \; y \in \Gamma (C, \mathbf{O} (3e))$ such that $x^3-y^2 \in \Gamma (C, \mathbf{O}(5e))$ and $C \simeq C_{\underline{a}} \subset \mathbb{P}^2$ is given by
$$ y^2 + a_1 xy + a_3 y = x^3 +a_2 x^2 +a_4 X + a_6 $$
for $a_i \in \mathbf{O}_S$ and $e = [0: 1:0]$.  Such a curve is said to be in Weierstrass form or simply a Weierstrass curve.

Two Weierstrass curves $(C_\underline{a} , e)$ and $(C_{\underline{a}'},e)$ are isomorphic if and only if they are related by a coordinate change of the form
$$ (X,Y) \mapsto (\lambda^{-2} X + r, \lambda^{-3} y + s \lambda^{-2} x +t ) .$$
For instance, this means that $a_1'= \lambda (a_1 +2s)$.  We then build a Hopf algebroid $(A, \Gamma)$ by defining
$$ A = \mathbb{Z} [a_1 , \dots , a_4 , a_6 ] , \; \Gamma = A [r,s,t, \lambda^\pm ] .$$
Further, define the stacks $M_{Weir} = M_{A, \Gamma}$ and $M_{ell} = M_{A[\Delta^{-1}] , \Gamma[\Delta^{-1}]}$.  Note that
$$ M_{ell} \subset \overline{M_{ell}} \subset M_{Weir} .$$

Let $\omega_{C/S} = \pi_* \Omega^1_{C/S}$ (which is locally free) and $\pi_{2n} \mathbf{O} = \omega^n$.  If $C$ is a Weierstrass curve, then $\omega$ is free with generator of degree 2
$$\eta = \frac{dx}{2y + a_1 x +a_3} .$$
Let $\omega^*$ correspond to the graded comodule
$$ A_* = A [ \eta^\pm ] \to \Gamma [\eta^\pm ] = \Gamma_* ; \; \eta \mapsto \lambda^{-1} \eta .$$

It is classical that
$$ H^{0,*} (A, \Gamma ; A_*) = \mathbb{Z} [c_4 , c_6 , \Delta]/ (12^3 \Delta -c_4^3 +c_6^2 ) $$
that is the ring of modular forms.  So we get a map
$$ \pi_* tmf \to \{modular \; forms \} $$
as the edge homomorphism of our spectral sequence
$$ H^{s,t} (A,\Gamma ; A_*) \simeq H^{s,t} (A_* , \Gamma_*) \Rightarrow \pi_{t-s} tmf .$$

It should be noted that we have a comparison map with the Adams-Novikov spectral sequence for $MU$.

##$p$-local Coefficients##
###With 6 inverted###
Note that if 2 is invertible than we can complete the square in the Weierstrass equation to obtain
$$ \overline{y}^2 = x^3 + 1/4 b_2 x^2 + 1/2 b_4 x + 1/4 b_6 $$
and the only automorphisms of the curve are $x \mapsto x+r$.  Now if 3 is invertible we complete the cube and have
$$ \overline{y}^2 = \overline{x}^3 -1/48 c_4 \overline{x} -1/864 c_6 $$
and this curve is rigid.  Define $C= \mathbb{Z} [c_4 , c_6 ]$ and $\Gamma_C = C$, then
$$ H^{s,*} (A_* , \Gamma_* ) [1/6] \simeq H^{s,*} (C,C) [1/6] = \mathbb{Z} [1/6 , c_4 , c_6 ] $$
if $s=0$ and 0 otherwise.

###Localized at 3###
It is true that $H^{s,t} (A_* , \Gamma_* ) = H^{s,t} (B, \Gamma_B)$ where
$$ B = \mathbb{Z}_{(3)} [ b_2 , b_4, b_6 ] \to \Gamma_B = B[r] ; \; b_2 \mapsto b_2 + 12r $$
and the degree of $r$ is 4.
We have the class $\alpha = [r] \in H^{1,4}$ and $\beta = [ -1/2 (r^2 \otimes r + r \otimes r^2)] \in H^{2,12}$.

Let $I = (3 , b_2 , b_4 )$ and consider the Hopf algebroid $(B/I , \Gamma_B /I)$ which by change of rings theorem is equivalent to $(\mathbf{F}_3 , \mathbf{F}_3 [r]/(r^3))$.
A spectral sequence obtained by filtering by powers of $I$ gives:

**Theorem.** $H^{*,*} (B, \Gamma_B) = \mathbb{Z} [c_4 , c_6 , \Delta , \alpha, \beta ]$ subject to the following relations

1. $12^3 \Delta -c_4^3 +c_6^2 = \alpha^2 = 3\alpha = 3 \beta =0;$

1. $c_4 \alpha = c_6 \alpha = c_4 \beta = c_6 \beta = 0.$

By using the comparison map with the Adams-Novikov spectral sequence one can prove the following theorem.

**Theorem.** The edge homomorphism $\pi_* tmf_{(3)} \to \{Modular Forms\}_{(3)}$ has

1. Cokernel given by $\mathbb{Z}/3\mathbb{Z} [\Delta^n]$ for $n \ge 0$ and not divisble by 3;

1. Kernel consisting a copy of $\mathbb{Z}/3\mathbb{Z}$ in degrees 3,10,13,20,27,30,37,40 modulo 72.  This is the 3-torsion in $\pi_* tmf$.

For more see

[[Tilman Bauer]], **Computation of the homotopy of the spectrum tmf**. In _Geom. Topol. Monogr_., 13, 2008.


A Survey of Elliptic Cohomology - E-infinity rings and derived schemes

<div class="rightHandSide toc">
[[!include cohomology - contents]]

***

[[!include higher algebra - contents]]

</div>

+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

see there for background and context.

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

=--

Previous:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

Next:

* [[A Survey of Elliptic Cohomology - elliptic curves]]

>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 #

* table of contents
{: toc}

# 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 ring]]s and [[derived algebraic geometry]].

**Definition**

An $\Omega$-[[spectrum]] is a sequence of pointed [[topological space]]s $\{E_n\}$ and base-point preserving maps $\{\sigma_n : E_n \to \Omega E_{n+1}\}$ that are [[weak homotopy equivalence]]s.

($\Omega E_n$ 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-complex]]es is [[representable functor|represented]] by an $\Omega$-[[spectrum]].

**examples**

1. [[singular cohomology]] with coefficients in $A$: the [[Eilenberg-MacLane spectrum]] $H A$.

1. 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 curve]]s, 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 theory|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 theory]]s 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

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

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

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

**theorem** There exists a [[symmetric monoidal category|symmetric monoidal]] [[model category]] $StTop$ of [[spectrum|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]].

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

1. $\pi_0(A)$ is a commutative ring

1. $\pi_n(A)$ 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

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

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

**definition** a morphism $f : A \to B$ of [[E-infinity ring]]s 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:

   $$
     tmf[\Delta^{-1}] = \Gamma(O_{M_{1,1}, der})
   $$

   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 $M_{1,1}$ by its [[Deligne-Mumford compactification]].

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

recall that we want global sections of the [[presheaf]]

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

(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

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

but in the [[homotopy category]] the pushout is instead

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

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

So we want to refine $CohomologyTheories$ to the cateory of [[spectrum|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

$$
  E_\infty Rings \to MultiplicativeCohomologyTheories
$$

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 category|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]] ([[Higher Topos Theory|chapter 1 of HTT]])

   in this framework we'll have a [[stable (infinity,1)-category of spectra]], let's call that $Sp$

2. show that $Sp$ is a [[symmetric monoidal (infinity,1)-category]]

3. 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]]

4. 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]])

* a [[quasi-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)-category|(infinity,1)-categories]] is, when regarded as a [[quasi-category]], just a morphism of [[simplicial set]]s.: 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 $I \to h C$

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

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 [[kernel]]s and [[cokernel]]s, i.e. all [[homotopy pullback]]s and pushouts with $0$ in one corner.

Then from this we get [[loop space object]]s $\Omega X$ and [[delooping]] objects $B X$ in $C$ (called suspension objects $\Sigma X$ in this context).

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

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

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

One example of this is the [[(infinity,1)-category]] of [[pointed object|pointed]] [[topological space]]s.

**definition** a [[prespectrum object]] in an [[(infinity,1)-category]] $C$ with the properties as above is a [[(infinity,1)-functor]]

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

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

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

(everything filled with 2-cells aka homotopies)

since we have cokernels we get maps from the universal property

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

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

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 object]]s.

In particular, we set

$$
  Sp := Sp(Top)
$$

the [[stable (infinity,1)-category of spectra]] is the stabilization of the [[(infinity,1)-category]] [[Top]] of [[topological space]]s.

> 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:

1. $\otimes$ preserves limits and colimits.

1. 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:

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

  where $\Gamma$ is [[Segal's category]] with objects finite [[pointed object|pointed]] [[set]]s and morphisms basepoint preserving [[function]]s 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

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

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

so let $C$ now be a [[symmetric monoidal (infinity,1)-category]].

**definition** A [[commutative monoid in an (infinity,1)-category|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 [[limit]]s and [[colimit]]s (see DAG III 2.1, 2.7), so we can talk about sheaves of $E_\infty$ rings!

# part 3 - brave new schemes #

Now the theory of [[scheme]]s and [[derived scheme]]s, but not over [[simplicial commutative ring]]s, but over [[E-infinity ring]]s.

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

* [[Jacob Lurie]], [[Spectral Schemes]].

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 degree-0 ring

$$
  |Spec A| := Spec \pi_0 A
$$

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

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

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

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

1. $\forall$ $E_\infty$-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 ring]]s is in [[spectral algebraic geometry]]/[[brave new algebra|brave new algebraic geometry]].

The analog for [[simplicial commutative ring]]s 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 ring]]s 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 [[Deligne-Mumford stack|DM-]] [[moduli stack]] of [[elliptic curve]]s.

#References#

* [[Paul Goerss]], [[Topological Algebraic Geometry - A Workshop]]


A Survey of Elliptic Cohomology - elliptic curves

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

+-- {: .standout}

This is a sub-entry of

* [[A Survey of Elliptic Cohomology]]

and

* [[Gromov-Witten invariants]]

see there for background and context.

This entry contains a basic introduction to [[elliptic curve]]s and their [[moduli space]]s.

=--

Previous:

* [[A Survey of Elliptic Cohomology - cohomology theories]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

* [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

Next:

* [[A Survey of Elliptic Cohomology - equivariant cohomology]]

>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#

* toc
{:toc}

#elliptic curves#

**Definition** An [[elliptic curve]] over $\mathbb{C}$ is equivalently

* a [[Riemann surface]] $X$ of [[genus]] 1 with a fixed point $P \in X$

* a quotient $\mathbb{C}/\Lambda$ where $\Lambda$ is a [[lattice]] in $\mathbb{C}$;

* a compact complex [[Lie group]] of dimension 1.

* a [[smooth scheme|smooth]] [[algebraic curve]] of degree 3 in $\mathcal{P}$.

**Remark**  The third definition is the one that is easiest to generalize. For our simple purposes, though, the second one will be the most convenient.

From the second definition it follows that to study the
[[moduli space of elliptic curves]] it suffices to study the [[moduli space]] of [[lattice]]s in $\mathbb{C}$.

**Definition** A **[[framed elliptic curve]]** is an [[elliptic curve]] $(X,P)$ (in the sense of the first definition above) together with an ordered basis $(a,b)$ of $H_1(X, \mathbb{Z})$ with $(a \cdot b) = 1 $

A **framed lattice** in $\mathbb{C}$ is a lattice $\Lambda$ together with an ordered basis $(\lambda_1, \lambda_2)$ of $\Lambda$ such that $Im(\lambda_2/\lambda_1) \gt 0$.

(So turning $\lambda_1$ to $\lambda_2$ in the plane means going counterclockwise).

#moduli spaces of elliptic curves#

this implies that

the [[upper half plane]] $\mathfrak{h}$ is in bijection with framed lattices in $\mathbb{C}$ which in turn is in bijection with [[isomorphism]] classes of framed elliptic curves over $\mathbb{C}$

$$
  \mathfrak{h}
  \simeq
  \{framed~lattices~in~\mathbb{C}\}
  \simeq
  \{framed~elliptic~curves~over~\mathbb{C}\}/_\sim
$$

and we have

$$
  \{elliptic~curves~over~\mathbb{C}\}_\sim
  \simeq
  \mathfrak{h}/{SL_2(\mathbb{Z})}
$$

where $SL_2(\mathbb{Z}) = \left\{ \left(\array{a & b \\ c & d }\right)| a d - c d = 1\right\}$ acts  by

$$
  \tau \mapsto \frac{a \tau + b}{c \tau + d}
$$

**Claim** the quotient $\mathfrak{h}/_{SL_2(\mathbb{Z})}$ is [[biholomorphic map|biholomorphic]] to the disk and has a unique structure of a [[Riemann surface]] which makes the quotient map
$\mathfrak{h} \to \mathfrak{h}/SL_2(\mathbb{Z})$ a [[holomorphic map]]

>**warning** possibly something wrong here, audience doesn't believe the bit about the disk

**definition** write $M_{1,1} := \mathfrak{h}/SL_2(\mathbb{Z})$

**definition** a **homolorphic family of [[elliptic curve]]s** over a [[complex manifold]] $T$ is

* a holomorphic map $\pi : X \to T$

* together with a [[section]] $s : T \to X$ of $\pi$ such that for any $t \in T$ the pair $(X_t, s(t))$ is an [[elliptic curve]] (using the first definition above).

For every family

$$
  \array{
    X \\ \downarrow^\pi \\ T
  }
$$

we would like to have $F \to M_{1,1}$
$$
  \array{
    X \simeq \phi^* F &\to& F
    \\
    \downarrow & & \downarrow
    \\
    T &\to& M_{1,1}
  }
$$

where

$$
  \phi:   t \mapsto [X_t, s(t)]
$$

such that

* $\phi : T \to M_{1,1}$ is a [[holomorphic map]]

* every [[holomorphic map]] $T \to M_{1,1}$ corresponds to a family over $t$;

* there is a universal family over $M_{1,1}$

This is _impossible_ . One can construct explicit counterexamples. These counterexamples involved [[elliptic curve]]s with nontrivial [[automorphism]]s.

For instance

$$
  \{
    (x,y,z) \in \mathbb{P}^2 \times X :
    y^2 = x(x-1)(x-\lambda)
  \}
  \to
  X := \mathbb{P}^1 - \{0,1,\infty\}
$$

> but see the discussion at [[moduli space]] for a discussion of the statement "it's te automorphisms that prevent the [[moduli space]] from existing"

consider

$$
  \mathbb{Z}^2 \hookrightarrow \mathbb{C} \times \mathfrak{h}
$$

given by

$$
  (m,n) :
  (z,\tau)
  \mapsto
  (z + m \tau + n, \tau)
$$

Then consider the family

$$
  \array{
     E := \mathbb{C}/_{\mathbb{Z}^2} \times \mathfrak{h}
    \\ \downarrow \\ \mathfrak{h}
  }
$$

is a family of [[elliptic curve]]s over $\mathfrak{h}$

and $E_\tau = \mathbb{C}/{\Lambda_\tau}$ with

$$
  \Lambda_{\tau} := \mathbb{Z}\cdot 1 \oplus \mathbb{Z}\cdot \tau
$$

is a family of framed elliptic curves.

**fact** the space $\mathfrak{h}$ with the family $E \to \mathfrak{h}$ is a [[fine moduli space]] for [[framed elliptic curve]]s.

Consider any map $\phi : T \to \mathfrak{h}$

with pullback of the universal family

$$
  \array{
     X \stackrel{?}{\to} \phi^* E &\to & E
    \\
    \downarrow && \downarrow
    \\
    T &\stackrel{\phi}{\to}&
    \mathfrak{h}
  }
$$

**claim** for every point $t \in T$ there is an open neighbourhood $t_0 \in U \hookrightarrow T$ such that one can choose [[differential form|1-forms]] $\omega_t$ on $X_\tau$ which vary holomorphically with respect to $t$.

Notice that _locally_ every family of elliptic curves is framed (since we can locally extend a choice of basis for $H_1$).  So

$$
  \array{
    && \mathfrak{h}
    \\
    && \downarrow^{SL_2(\mathbb{Z})}
    \\
    M_{1,1} &\stackrel{Id}{\to}&
    M_{1,1}
  }
$$

at $i$ and $\rho = e^{2\pi i/6}$ , $C = \{\pm I\}$

isn't locally liftable at $i$ and $\rho$ so it is not a univresal family of unframed curves.

#orbifolds#

**definition** A **basic pointed orbifold** (basic meaning global) is a triple $X//\Gamma := (X,\Gamma,\rho)$, where

* $X$ is a [[connected space|connected]] and [[simply connected space|simply connected]] [[topological space]] (or in other variants a [[complex manifold]] or whatever is under consideration)

* $\Gamma$ is a discrete [[group]]

* $\rho : \Gamma \to Aut(X)$ is a group homomorphism

(here "pointed" because we specified the action $\rho$ instead of its iso-class under the following morphisms)

A morphism from  $(X,\gamma, \rho)$ to $(X', \Gamma', \rho')$ is a pair

$$
  (f,\phi)
$$

where

* $f : X \to X'$ is a continuous map

* $\phi : \Gamma \to \Gamma'$ is a group homomorphism

such that for all $\gamma \in \Gamma$

$$
  \array{
    X &\stackrel{f}{\to}& X'
    \\
    \downarrow^{\gamma} && \downarrow^{\phi(\gamma)}
    \\
    X &\stackrel{f}{\to}& X'
  }
$$

This really leads an enlargement of the plain category of spaces:

**remark** We have a faithful embedding of spaxces into orbifolds defined this way: for any connected semi-locally simply connected space $X$ with [[universal cover]] $\tilde X$
we have

$$
  X \mapsto \tilde X //\pi_1(X)
$$

> **warning** notice all the simply-connectedness assumoptions above for making sense of this

**remark** let $X$ be a [[nice topological space]]. Let $G = \pi_1(X)$ be its first [[homotopy group]] and let a discrete group $\Gamma$ action on $X$.

then define

$$
  \tilde \Gamma :=
  \left\{
    (\gamma,g)
    \in
    \Gamma \times Aut(\tilde X)
    |
    \array{
       \tilde X &\stackrel{g}{\to}& \tilde X
       \\
       \downarrow^p && \downarrow^p
       \\
       X &\stackrel{\gamma}{\to}& X
    }
  \right\}
$$

then we have an exact sequence

$$
  1 \to G \to \tilde \Gamma \to \Gamma \to 1
$$

where $G \to \tilde \Gamma$ is given by $(g \mapsto (1,g))$ and $\tilde \Gamma \to \Gamma$ by $(\gamma,g) \mapsto \gamma$.

**definition**

For an [[orbifold]] $(X,\Gamma,\rho)$ write $I \times (X,\Gamma,\rho) := (I \times X, \Gamma, \rho)$.

Then a **homotopy** from $(f,\phi)$ to $(f',\phi') : (X,\Gamma, \rho) \to (X',\Gamma', \rho')$

is a map

$$
  (F,\Psi) : I \times (X, \Gamma, \rho) \to (X', \Gamma', \rho')
$$

such that

* $\Psi = \phi = \phi'$

* $f(-) = F(0,-)$, $f'(-) = F(1,-)$

now write

$$
  S^1 := (\mathbb{R}, \mathbb{Z})
$$

(the circle regarded as a global orbifold)

**definition**

The first [[homotopy group]] for our definition of orbifold is:

$$
  \pi_1(X//\gamma)
  \simeq
  \{
    homotopy~classes~of~maps~S^1 \to X//\Gamma
  \}
$$

**exercise** show that this is $\cdots \simeq \Gamma$

>(recall again the simply-connectness assumoption!!)

*definition** A morphism

$$
  (f,\phi) : (X,\Gamma) \to (X',\Gamma')
$$

is a [[weak homotopy equivalence]] if $\phi$ is an ismorphism and $H_\bullet(f) : H_\bullet(X) \to X_\bullet(X')$.

**note** Let $E \Gamma$ be a contractible space on which $\Gamma$ acts properly, dic. and free, then

$$
  (E \Gamma \times X, \Gamma)
  \stackrel{(f,\phi)}{\to}
  (X,\Gamma)
$$

with $\phi = Id_\Gamma$ and $f$ the projection is a [[weak homotopy equivalence]].

**definition** a [[local system]] $V$ on $(X,\Gamma)$ with fiber $V$ a group homomorphism $\Gamma \to Aut(V)$ with

**definition**

Introduce the following notation for [[homotopy group]]s, [[homology]] and [[integral cohomology]] of our [[orbifold]]s with coefficients in a local system:

* $\pi_n(X//\Gamma)$ := \pi_n(X)$ for $n \geq 2$

* $H_\bullet(X//\Gamma, V) := H_\bullet(E \Gamma \times_\Gama X, V)$

* $H^\bullet(X//\Gamma, V) := H^\bullet(E \Gamma \times_\Gama X, V)$

**example** ${*}//\Gamma$ has a [[weak homotopy equivalence]] to the [[classifying space]] $\mathcal{B}\Gamma$

it follows that for local system $V$ we have

$$
  H^\bullet({*}//\Gamma, V) = H^\bullet_{gp}(\Gamma,V)
$$

where on the right we have [[group cohomology]]

***

We have all kinds of constructions on orbifolds by saying they are structures on $X$ with suitable extension of the action of $\Gamma$ to them

A **[[vector bundle]] on an orbifold** $V \to X//\Gamma$ is a vector bundle $V \to X$ with isomorphism action by $\Gamma$ specified, covering that on $X$.

for instance the [[tangent bundle]] of $X //\Gamma$ is given by $(T X)//\Gamma \to X//\Gamma $ in the obvious way.

**definition** say that $\Gamma$ acts virtually freely if $\exists$ a finite index subgroup $\Gamma'$ of $\Gamma$ which acts freely on $X$.

**note** $SL_2(\mathbb{Z})$ acts virtually freely on $\mathfrak{h}$

$$
  SL_2(\mathbb{Z})[m]
  \to
  SL_2(\mathbb{Z})
  \to SL_2(\mathbb{Z}/ m \mathbb{Z})
$$

Let $\Gamma' \lt \Gamma$ be a finite index subgroup which acts freely on $X$.

set

$$
  X' := X// \Gamma';
$$

the map

$$
  X/\Gamma' \to X//\Gamma
$$

**must be viewed as an unramified covering of degree $[\Gamma:\Gamma']$.**

> supposedly important statement

**definition**

the [[Euler characteristic]] of a global orbifold is

$$
  \chi(X//\Gamma) :=
  \frac{1}{[\Gamma: \Gamma']} \chi(X/\Gamma')
$$

> compare [[groupoid cardinality]]

#moduli stack/orbifold of elliptic curves#

**definition**

Define now the global orbifold

$$
  \mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})
$$

**proposition**

$$
  H_1(\mathcal{M}_{1,1}, \mathbb{Z})
  = \mathbb{Z}/12\mathbb{Z}
$$

$$
  H^1(\mathcal{M}_{1,1}, \mathbb{Z}) = 0
$$

$$
  H^2(\mathcal{M}_{1,1}, \mathbb{Z})
  = \mathbb{Z}/12 \mathbb{Z}
$$

$$
  H_\bullet(\mathcal{M}_{1,1}, \mathbb{Q})
  \simeq
  H_\bullet(M_{1,1}, \mathbb{Q})
$$

and similarly for [[integral cohomology]]

$$
  \chi(\mathcal{M}_{1,1}) = -\frac{1}{12}
$$

$$
  Pic(\mathcal{M}_{1,1}) \simeq \mathbb{Z}/12\mathbb{Z}
$$

# topological invariants of the moduli stack #

Since the upper half plane is contractible, the homotopy type of $\mathfrak{h}//\mathbb{Z}_2$ are the same as that of $* // \mathbb{Z}_2$ and similarly for the (group)cohomology

$$
  H^\bullet(\mathcal{M}_{1,1}, \mathbb{Z})
  = H\bullet(SL_2(\mathbb{Z}), \mathbb{Z})
$$

and similalry for homology.

In particular

$$
  H_1(\mathcal{M}_{1,1}, \mathbb{Z})
  \simeq
  SL_2(\mathbb{Z})^{ab} \simeq \mathbb{Z}/12\mathbb{Z}
$$

for all $m \in \mathbb{N}$ then

$$
  1 \to SL_2(\mathbb{Z})[m]
  \to
  SL_2(\mathbb{Z})
  \to
  SL_2(\mathbb{Z}/m \mathbb{Z})
  \to
  1
$$

so that

$$
   H^1(\mathcal{M}_{1,1}, \mathbb{Z})
  =
  0
$$

**fact** the group $SL_2(\mathbb{Z})[m]$ is free for $m \gt 2$.

so far all $\mathbb{Q}$-representations $V$ we have

$$
  H^k(SL_2(\mathbb{Z}), V)
  \simeq
  H^l(SL_2(\mathbb{Z}), V)^{SL_2(\mathbb{Z}/m\mathbb{Z})}
$$

due to the freeness we have also that

$$
  H^k(SL_2(\mathbb{Z}), V) = 0
$$

for $k \geq 2$

and hence

$$
  H^2(SL_2(\mathbb{Z}), \mathbb{Z})
$$

is [[torsion]]

$$
  \cdots \simeq
  Hom(H_1(SL_2(\mathbb{Z}), \mathbb{Z}), \mathbb{Q}/\mathbb{Z})
  \simeq
  \mathbb{Z}/12 \mathbb{Z}
  \,.
$$

**proposition**

as [[orbifold]]s, we have an isomorphism

$$
  \mathcal{M}_{1,1} \simeq
  X//(S_3 \times C_2)
$$

where

$$
  X :=
\mathbb{P}^1 - \{0,1, \infty\}
$$

and $S_3$ acts on that by permuting $0,1, \infty$. (Think of $\mathbb{P}^1$ as the [[Riemann sphere]]: there is a unique holomorphic automorphism of that permuting these three points in a given fashion.) While $C_2$ acts trivially.

*proof**

$$
  \array{
   1 \to SL_2(\mathbb{Z})[2]
   &\to&
   SL_2(\mathbb{Z})
   &\to&
   S_3 \simeq SL_2(\mathbb{Z}/2\mathbb{Z})
   &\to&
   1
   \\
   &&& {}_{\phi}\searrow& \downarrow
   \\
  &&&& PSL_2(\mathbb{Z})
  }
$$

now $PSL_2(\mathbb{Z})[2]$ is known to be torsion free. It acts in a standard way on the [[upper half plane]] $\mathfrak{h}$.

A little discussion shows that

$$
  \mathfrak{h}/PSL_2(\mathb{Z})[2]
  \simeq
  X
$$

this implies that

$$
  PSL_2(\mathbb{Z})[2]
  \simeq
  F_2
$$

the [[free group]] on two generators.

Then the second but last map

$$
  1 \to C_2 \to SL_2(\mathbb{Z})[2]
  \to
  F_2
  \to 1
$$

has a [[section]], from which we get that

$$
  SL_2(\mathbb{Z})[2]
  \simeq
  F_2 \times C_2
$$

and so

$$
  X//(C_2 \times S_3)
  \simeq
  (X//C_2) // S_3
  \simeq
  ((\mathfrak{h}// SPL_2(\mathbb{Z})[2])//C_2)//S_3
  \simeq
  ((\mathfrak{h}//PSL_2(\mathbb{Z}[2]))//S_3)
  \simeq
  \mathfrak{h}/ SL_2(\mathbb{Z})
$$

which is the **end of the proof**.

**corollary** The [[Euler characteristic]] of the [[moduli stack]] of [[elliptic curve]]s is

$$
  \chi(\mathcal{M}_{1,1}) = \frac{-1}{12}
  \,.
$$

now consider the line bundle

$$
  \array{
    \mathbb{C} \times \mathfrak{h}
    \\
    \downarrow
    \\
     \mathfrak{h}//SL_2(\mathbb{Z})
  }
$$

with action on the total space for $k \in\mathbb{Z}$

$$
  \left(
    \array{
      a & b \\ c & d
    }
  \right)
  :
  (z, \tau)
  \mapsto
  (c \tau + d)^k z, \frac{a \tau + b}{c \tau + d}
$$

call this [[line bundle]] on the [[moduli stack]] $\mathcal{L}_k \to \mathcal{M}_{1,1}$. We will see that all line bundles are isomorphic to one of these.

**remark**

$$
  f : \mathfrak{h} \to \mathcal{C}
$$

is a [[section]] of $\mathcal{L}_k$ iff

$$
  f\left(
    \frac{a \tau + b}{c \tau + d}
  \right)
  =
  (c \tau + d)^k f(\tau)
$$

hence precisely if it defines a [[modular function]] of weight $k$! This gives a geometric interpretation of modular functions.

$$
  \array{
    \mathbb{C} \times \mathfrak{h}
    \\
    \downarrow
     \\
     \mathfrak{h}
  }
$$

and define an action of $G := SL_2(\mathbb{Z}) \letimes \mathbb{Z}^2$

where $\mathbb{Z}^2$ acts on $SL_2(\mathbb{Z})$ by

$$
  \left(
    \array{
       a & b \\ c & d
    }
  \right)
   : (m , n)
   \mapsto
   a m + b m, c m + d n
$$

and on $\mathbb{C} \times \mathfrak{h}$ by

$$
  (m, n) : (z,\tau) \mapsto (z + m \tau + n, \tau)
$$

the resulting bundle

$$
  \array{
    \mathbb{C} \times \mathfrak{h}//G
    \\
    \downarrow
     \\
     \mathcal{M}_{1,1}
  }
$$

we call

$$
  \mathcal{E} \to \mathcal{M}_{1,1}
$$

**theorem** for any [[complex manifold]] $T$ there is a bijection between families of elliptic curves over $T$ and [[orbifold]] maps $T \to \mathcal{M}_{1,1}$ classify them.

Suppose we have an "isotrivial family" (meaning all fibers are isomorphic elliptic curves, i.e. a fiber bundle of elliptic curves)

$$
  \array{

    \\
    \downarrow
    \\
    T &\stackrel{\phi}{\to}& \mathcal{M}_{1,1}
  }
$$

recall that the group that defines $T$ as an [[orbifold]] is the first [[homotopy group]] $\pi_1(T)$.

The only condition that we get from the definition of orbifold maps is that

$$
  \phi : \pi_1(T) \to SL_2(\mathbb{Z})
$$

factors through the [[stabilizer group]] $\simeq Aut(E_p)$ of our base point $p \in \mathcal{M}_{1,1}$

# compactified moduli stack #

one can see that over _compact_ $T$ with $\mathcal{M}_{1,1}$ we cannot have nontrivial famlies _without_ singular fibers.

To get around that we want a **compactification** $\bar \mathcal{M}_{1,1}$ of the [[moduli stack]].

also fur purposes of intersection theory, we need to further compactify.

recall the  description of $\mathcal{M}_{1,1}$ as a weak quotient of $\mathbb{P}^1$.  Then consider:

**definition**

Let

$$
  \bar \mathcal{M}_{1,1} := \mathbb{P}^1//(C_2 \times S_3)
$$

otice that this is now an [[orbifold]] which is no longer _basic_ by the above definition. In fact, we can cover it by charts of basic orbifolds as follows: consider

$$
  \array{
    && \mathfrak{h}//(C_2 \times \mathbb{Z})
    \\
    &  \swarrow && \searrow
    \\
    \mathfrak{h}//SL_2(\mathbb{Z})
    &&&&
    \mathbb{D}//C_2
  }
$$

>with the arrows being maps of orbifolds whose precise details I haven't typed.

then let $\mathbb{D}^*$ be the punctured disk and realize the diagram

$$
  \array{
    && \mathbb{D}^*//C_2
    \\
    &  \swarrow && \searrow
    \\
    \mathcal{M}_{1,1}
    &&&&
    \mathbb{D}//C_2
  }
$$

where the right morphism is just the inclusion

now we build a chart of $\bar \mathcal{M}_{1,1}$ consisting of the two patches $\mathcal{M}_{1,1}$ and $\mathbb{D}/C_2/$

from this we get the alternative

**definition**

$$
  \bar \mathcal{M}_{1,1}
  :=
  \mathcal{M}_{1,1} \coprod_{\mathbb{D}^*//C_2}
  \mathbb{D}//C_2
$$

the [[colimit]] on the right manifestly glues in the "point at infinity" that is not hit by the map $\mathbb{D}^*//C_2 \to \mathcal{M}_{1,1}$.

# Gromov-Witten invariants  #

**definition** A **stable curve** (over $\mathcal{C}$) of genus $g$ with $n$ marked points is a proper, connected curve with $n$ smooth marked points such that all singularities are nodes and such that the the [[automorphism]] group (of autos respecting the smooth marked points) is finite,

$$
  |Aut(C)| \lt + \infty
$$

and such that the [[arithmetic genus]] is $g$.

Now $\bat \mathcal{M}_{g,n}$ is the [[fine moduli space]] for smooth curves of genus $g$.

There is a [[line bundle]]

$$
  \mathcal{T}_i^* \to \bar \mathcal{M}_{g,n}
$$

built fiberwise from the cotangent spaces of the elliptic curves.

one of them is obtained from one of the $n$ sections $s_i$ of the universal family $\mathcal{F} \to \bar \mathcal{M}_{g,n}$. The fiber over a point is the cotangent space of the elliptic curve over that point at this section.

Write for the first [[Chern class]]

$$
  c_1(\mathcal{T}_i^*) = \Psi_i
$$

$$
  k_1, \cdots, k_n
  \in \mathbb{Z}_{\geq 0}
$$

such that

$$
  \sum_{i = 1}^n k_i = 3 g - 3 + n
$$

then we get numbers called the [[Gromov-Witten invariants]] ("of the point")

$$
  \langle
    \tau_{k_1}, \cdots, \tau_{k_n}
  \rangle_g
   :=
  \int_{\bar \mathcal{M}_{1,1}}
  \prod_{i = 1}^n
   \Psi_i^{k_i}
$$

## example: $\langle \tau_1\rangle_1$ ##

Let $x, y$ by affine coordinates on $\mathbb{P}^2$

Let $f(x,y)$ and $g(x,y)$ be two generic cubics, in particular there are nine joint zeros

$$
 |\langle (x,y)| f(x,y) = g(x,y) = 0\rangle| = 0
$$

called $p_1, \cdots, p_9$.

define then

$$
  F :=
  \left\lbrace
    (x,y,t)
    \in \mathbb{P}^2 \times \mathbb{P}^1
    :
    f(x,y) - t g(x,y) = 0
  \right\rbrace
$$

and consider

$$
  \array{
    F
    \\
    \downarrow^{pr_2}
    \\
    \mathbb{P}^1
    &\stackrel{\phi}{\to}&
    \bar \mathcal{M}_{1,1}
    \\
    &\searrow & \downarrow^{q}
    \\
    && \bar M_{1,1}
  }
$$

That map $q$ has degree $\frac{1}{2}$ (!) since $\mathbb{P}^1 \to \bar \mathcal{M}_{1,1}$ has degree 12

we also find that the diaginal map $\mathbb{P}^1 \to \bar M_{1,1}$ has degree 12. It follows that $\phi$ has degree 24:

$$
  deg(\phi) = 24
  \,.
$$

Now let $\mathcal{T}_i^* \to \bar \mathcal{M}_{g,n}$ be one of these line bundles. Consider the [[pullback]] $\phi^*(\mathcal{T}_1)$

then by some argument not reproduced here we find

$$
  \int_{\mathbb{P}^1} c_1(\phi^*(\mathcal{T}_1)^*)
  \,.
$$

Then since the order of $\phi$ is 24 we find that the first [[Gromov-Witten invariant]] is

$$
  \langle
    \tau_1
  \rangle_1
  =
  \frac{1}{24}
  \,.
$$

#extending structures to the compactified moduli space#

recall that the [[moduli stack]] of [[elliptic curve]] is, as a global [[orbifold]]

$$
  \mathcal{M}_{1,1} := \mathfrak{h}//SL_2(\mathbb{Z})
$$

and also

$$
  \cdots \simeq (\mathbb{P}^1 - \{0,1,\infty\})//(C_2 \times S_3)
$$

and there is a [[line bundle]] on this given by

$$
  \mathcal{L}_k :=
  (\mathbb{C} \times \mathfrak{h})//SL_2(\mathbb{Z})
$$

where the action is given by

$$
  \left(
    \array{
       a& b \\ c & d
    }
  \right)
  : (z, \tau)
   \mapsto
   (c\tau + d)^{k} z, \frac{a \tau + b}{c \tau + d}
$$

since $SL_2(\mathbb{Z})^{ab} = \mathbb{Z}/12\mathbb{Z}$
one finds the [[Picard group]]

$$
  Pic \mathcal{M}_{1,1} \simeq \mathbb{Z}/12\mathbb{Z}
$$

meromorphic (holomorphic) sections $f$ of $\mathcal{L}_k$ are [[modular function]]s of weight $k$, i.e. $f : \mathfrak{h} \to \mathbb{C}$ such that

$$
  \forall \gamma = (\frac{a b}{c d}) \in SL_2(\mathbb{Z})
  :
  f(\gamma \tau) = (c \tau + d)^k f(\tau)
$$

the **universal elliptic curve** over $\mathcal{M}_{1,1}$ is

$$
  \mathcal{E} := (\mathbb{C} \times \mathfrak{h})//(SL_2(\mathbb{Z}) \ltimes \mathbb{Z}^2)
$$

Then we ended last time with describing the compactified moduli space

$$
  \bar \mathcal{M}_{1,1} := \mathbb{P}^1//(C_2 \times S_3)
$$

## extending the line bundles ##

**proposition** $\mathbb{L}_k$ has a universal extension $\bar \mathcal{L}_k$ to $\bar \mathcal{M}_{1,1}$

**proof**

take

$$
  (\mathbb{C} \times \mathbb{D}//C_2) \to (\mathbb{D}//C_2)
$$

where $C_2$ acts by

$$
  \pm 1 \;\; (z,\tau) \mapsto (\pm^k z , \tau)
$$

note that since $\left(\array{1 & n \\ 0 1}\right) \in S_2(\mathbb{Z})$ for all $n \in \mathbb{Z}$ ay [[modular function]] $f : \mathfrak{h} \to \mathbb{C}$

$$
  f(\tau)
  =
  \sum{-\infty}^\infty
  a_n q^ n
$$

where $q := e^{2 \pi i \tau}$

is called a holomorphic **[[modular form]]** of weight $k$ if $f : \mathfrak{h} \to \mathbb{C}$ is holomorphic and $a_n = 0$ for all $n \lt 0$

**remark** modular forms of weight $k$ are in bijection with sections of the line bundle $\bar \mathcal{L}_k$.

**example** for any [[lattice]] $\Lambda$ in $\mathbb{C}$ and for any $k \gt 2$ we have

$$
  S_k(\Lambda) := \Sum_{0 \neq \lambda \in \Lambda}
    \frac{1}{\lambda^k}
$$

obviously for all $u \in \mathbb{C}^*$ $S_k(u \Lambda) = u^{-k} S_k(\Lambda)$

and for all $\tau \in \mathfrak{h}$ with $C_k(\tau) := S_k(\Lambda_\tau)$ it follows that $G_k : \mathfrak{h} \to \mathbb{C}$ is holomorphic

since $\Lambda_{\gamma \tau} = (c \tau + d)^{-1} \Lambda_\tau$ it follows that $G_k$ is a modular function of weight $k$

**fact** $G_{2k} = 2 \zeta(2 k) ü 2 \frac{(2 \pi i)^{2k}}{(2k-1)!} \sum_{n=1}^\infty b_{2k-1}(n)q^n$

with $b_k(n) := \sum_{d|n} d^k$ and where $\zeta$ is the [[zeta-function]]

it follows that

$G_{2k}$ is a modular form of weight $2k$ (which is not a cusp form).

an important cusp form is

setting $g_2 = 60 G_4$ and $g_3 = 140 G_6$

the [[modular form]]

$$
  \Delta := g_2(\tau)^3 - 27 g_3(\tau)^2
$$

is a cusp form of weight 12. $\Delta$ does not have any 0 in $\mathfrak{h}$ and it has a simple zero at $q = 0$.

we have an isomorphism

$$
  \bar \mathcal{L}_{12}
  \simeq
  \mathcal{O}_{\bar \mathcal{M}_{1,1}(\infty)}
$$

where on the right is the sheaf with at most a pole at $\infty$. This isomorphism going from right to left is induced by multiplication with $\Delta$.

we have an exact sequence

$$
  0 \to \mathbb{Z} \to Pic(\bar \mathcal{M}_{1,1})
  \to \mathbb{Z}/12\mathbb{Z} \to 0
$$

where the first nontrivial map sends 1 to $\bar \mathcal{L}_{12}$ and the second one $\bar \mathcal{L}_1$ to the generator.

set for all $k$

$$
  M_k := \{modular~forms~of~weight~k\}
$$

$$
  M_k^\circ := \{cusp~forms~of~weight~k\}
$$

**proposition** $M_\bullet$ is an even graded algebra freely generated by $G_4$ and $G_6$ and the ideal $M_\bullet^\circ$ is generated by $\Delta$.

the dimensions are

$$
  dim M_{2k}  =
  \left\{
     floor k/6 & for k = 1 mod 6
     \\
     1+ floor k/6 & otherwise
  \right.
$$

> ???????

## extending the universal family of elliptic curves  ##

Recall the three definition of [[elliptic curve]]s from above.

Now a fourth definition:

**definition** an elliptic curve is a smooth curve of degree 3 in $\mathbb{C}\mathbb{P}^2$ together with a point in it.

* that this equation implies the first one above follows from the genus formula, which says that a degree $n$ curve as in the definition has genus $g = \frac{(n-1)(n-2)}{2}$

* that the first def implies this one