This is a sub-entry of
see there for background and context.
This entry contains a basic introduction to derived group schemes and their orientations.
Previous:
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 - derived group schemes and (pre-)orientations
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
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
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.
Recall that a commutative group scheme over a scheme $X$ is a functor
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 schemes. There are two problems
Because of the higher categorical nature of derived schemes Hom sets are spaces.
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
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? over $X$.
Examples
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 schemes over $\overline{X}$ and commutative group schemes which are flat over $X$.
For $X$ a derived scheme we have a map from commutative derived group schemes over $X$ to commutative group schemes which are flat over $\pi_0 X$.
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
Definition(Preliminary) A preorientation of $G$ is a morphism of commutative derived group schemes over $X$
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
is an isomorphism.
Suppose that $G$ is affine, then a map
corresponds to a map $A_{S^1} \to A^{\mathbb{C}P^\infty}$ which is the same as a map
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
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
that is an element of $\pi_2 (G(X))$.
Hence, we always have at least one preorientation: the trivial one which corresponds to $0 \in \pi_2 (G(X))$.
As motivation recall that a map $s: A \to B$ of 1-dim formal groups 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
$\pi_0 G \to \mathrm{Spec} \pi_0 A$ is smooth of relative dimension 1, and
The map induced by $\beta : \omega \to \pi_2 A$
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 schemes 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
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
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
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
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
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
$G_m$ 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.
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 $\mathrm{Spec} A \to \mathrm{Spec} \mathbf{S} [ \mathbb{C} P^\infty ]$.
We consider the map $\beta : \omega \to \pi_2 A$ where
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.
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 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 $A$
where as if it were flat we would have
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
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.