Derived / Spectral Moduli and Deformation Theory¶

Job: develop the $\infty$-categorical / spectral moduli via the cotangent complex (Schlessinger / Illusie), the derived moduli of vector bundles and coherent sheaves, the spectral moduli of formal groups / the chromatic picture, and the derived Artin representability — with the derived moduli of vector bundles on a curve as the worked example.

Core¶

Derived moduli problems and the cotangent complex¶

A derived moduli problem is a functor $F:(\mathrm{CAlg}^{\mathrm{cn}})^{\mathrm{op}}\to\mathcal{S}$ (spaces) on connective $\mathbb{E}_\infty$-rings, and the derived moduli stack is the $\infty$-categorical / spectral version of the classical moduli stack (the functor represented by a spectral DM / Artin stack). The cotangent complex $\mathbb{L}_X$ (of a spectral DM stack $X$) controls the deformation theory:

Deformation datum	Cotangent complex
Tangent complex	$T_X=\mathbb{L}_X^\vee$ (the dual)
Automorphisms (at a point)	$\mathrm{Ext}^0(\mathbb{L}_X,M)=\mathrm{Hom}(\mathbb{L}_X,M)$
Deformations (first-order)	$\mathrm{Ext}^1(\mathbb{L}_X,M)=\mathrm{Ext}^1(T_X,M)$
Obstructions	$\mathrm{Ext}^2(\mathbb{L}_X,M)$ (the obstruction class)
Smoothness / étaleness	$\mathbb{L}_X$ perfect of amplitude $[0,0]$ / $\mathbb{L}_X\simeq0$

The Schlessinger / Illusie deformation theory: the deformation of a point $x\in X$ over a square-zero extension $R'\to R$ (with kernel $M$) is controlled by $\mathrm{Ext}^i(\mathbb{L}_{X,x},M)$ — the $i=0$ automorphisms, $i=1$ deformations, $i=2$ obstructions. The derived / $\infty$-categorical version upgrades the classical Schlessinger (the truncation to the connected components and the $\pi_0$ / $\pi_1$).

The spectral moduli of stacks / the derived moduli of vector bundles¶

The moduli stack of vector bundles / the spectral $\mathrm{Bun}_G$ (the moduli of $G$-bundles on a spectral scheme / a curve), the derived moduli of coherent sheaves (the $\infty$-categorical enhancement, the cotangent complex from the $\mathrm{Ext}$ groups $\mathrm{Ext}^i(E,E)$).

The deformation-theoretic interpretation of the $\mathrm{Ext}$ groups: at a point $[E]$ (a coherent sheaf / vector bundle $E$), $$ T_{[E]}\mathrm{Bun}_G=\mathrm{Ext}^1(E,E),\qquad \mathrm{obstruction}\in\mathrm{Ext}^2(E,E),\qquad \mathrm{Aut}(E)=\mathrm{Ext}^0(E,E)=\mathrm{End}(E). $$

The Hodge-to-de-Rham / the spectral Riemann–Roch in the spectral / derived setting (the relation of the cotangent complex to the Hodge cohomology $\bigoplus H^q(X,\Omega^p)$ and the de Rham cohomology $H^*_{\mathrm{dR}}(X)$ via the Hodge-to-de-Rham spectral sequence, the spectral Riemann–Roch theorem of Toën–Vezzosi / Lurie).

The spectral moduli of formal groups and the chromatic picture¶

The moduli stack of formal groups $\mathcal{M}_{\mathrm{FG}}$ (the moduli of 1-dimensional formal groups, the height stratification by the height $n=0,1,2,\dots$ — the height-$n$ locus where the $p$-series $[p]_F$ has leading term $v_n x^{p^n}$), and the Morava $E$-theory $E_n$ (the Lubin–Tate deformation of a height-$n$ formal group $\Gamma_n$ over $\mathbb{F}_{p^n}$, the universal deformation ring $R(\Gamma_n)\cong W(\mathbb{F}_{p^n})[[u_1,\dots,u_{n-1}]]$).

Chromatic / formal-group moduli	Description
$\mathcal{M}_{\mathrm{FG}}$	the moduli of 1-d formal groups (the height stratification $n=0,1,2,\dots$)
Height-$n$ stratum $\mathcal{M}_{\mathrm{FG}}^{\ge n}$	the locus where $[p]_F=v_n x^{p^n}+\dots$ (the $v_n$ nonzero)
Lubin–Tate deformation $\mathrm{Def}(\Gamma_n)$	the universal deformation ring $R_n=W(\mathbb{F}_{p^n})[[u_1,\dots,u_{n-1}]]$
Morava $E$-theory $E_n$	the even-periodic $\mathbb{E}_\infty$-ring with $\pi_0E_n=R_n$
Relation to $\mathrm{MGL}$	the chromatic picture of algebraic cobordism: the $v_n$-periodic layers

Derived Artin representability and the spectral moduli¶

Theorem (derived Artin representability, Lurie / Toën–Vezzosi). A derived moduli problem $F:(\mathrm{CAlg}^{\mathrm{cn}})^{\mathrm{op}}\to\mathcal{S}$ is representable by a spectral Artin stack if: (i) $F$ is a sheaf for the étale topology; (ii) $F$ has an obstruction theory (the cotangent complex $\mathbb{L}_F$ controls the deformations); (iii) $F$ has a smooth atlas (a smooth surjection from a spectral affine scheme). The cotangent complex controls the representability and the smoothness.

The spectral moduli of elliptic curves $\mathcal{M}_{\mathrm{ell}}$ as the prototypical derived moduli (the Goerss–Hopkins–Miller / $\mathcal{O}^{\mathrm{top}}$ sheaf, the spectral DM stack of elliptic curves with the $\mathbb{E}_\infty$-structure sheaf), and the derived moduli of Calabi–Yau / the derived Donaldson–Thomas theory (the virtual fundamental class as a derived structure, the $(-1)$-shifted symplectic structure of Pantev–Toën–Vaquié–Vezzosi).

The spectral Gromov–Witten (the derived enhancement of the GW moduli, the virtual fundamental class as the derived / $(-1)$-shifted symplectic structure, the relation to mirror symmetry / GW — the derived GW theory as the natural home for the VFC).

See it¶

Derived moduli of vector bundles on a curve and the cotangent complex. For a smooth projective curve $X$ and a vector bundle $E$ of rank $n$, the derived moduli stack $\mathrm{Bun}_n^{\mathrm{der}}(X)$ has the tangent complex at $[E]$ equal to $$ T_{[E]}\mathrm{Bun}_n^{\mathrm{der}}=R\Gamma(X,\mathrm{End}(E))[1], $$ the derived global sections of the endomorphism sheaf, shifted by $[1]$ (the shift comes from the moduli / deformation interpretation: a deformation of $E$ over $R[\epsilon]/\epsilon^2$ is a class in $H^1(X,\mathrm{End}(E))$, the first-order deformation, hence the $[1]$ shift of the tangent complex). The cohomology: - $H^0(R\Gamma(\mathrm{End}(E))[1])=H^1(X,\mathrm{End}(E))=\mathrm{Ext}^1(E,E)$ — the deformations (the tangent space); - $H^{-1}(R\Gamma(\mathrm{End}(E))[1])=H^0(X,\mathrm{End}(E))=\mathrm{Ext}^0(E,E)=\mathrm{End}(E)$ — the automorphisms (the stabilizer, the negative degree, the "underlying" classical stack); - $H^2=0$ — the obstruction vanishes on a curve (the classical unobstructedness of $\mathrm{Bun}_G$ on a curve, the semisimplicity of the category of coherent sheaves on a curve, $\mathrm{Ext}^2(E,E)=0$ for dimension reasons).

The derived enhancement adds the $H^0$ automorphisms (the stabilizers) and the higher $\mathrm{Ext}$ groups (the higher derived structure), giving a smooth derived Artin stack (the cotangent complex is perfect of amplitude $[-1,0]$, the smoothness). This is the "correct" (derived) home for the Donaldson–Thomas / the virtual fundamental class constructions — the derived moduli is the smooth derived Artin stack, and the classical moduli is its truncation (the underived scheme / stack with the automorphisms).

Check¶

Show that the cotangent complex of $\mathrm{Bun}_n^{\mathrm{der}}(X)$ at $[E]$ is $\mathbb{L}_{[E]}=R\Gamma(X,\mathrm{End}(E))^\vee[-1]$, and verify the deformation / obstruction groups.

The moduli of vector bundles represents the functor $R\mapsto\{$vector bundles on $X_R$ deforming $E\}$, and a deformation over $R'=R\oplus M$ (square-zero) is a class in $\mathrm{Ext}^1(E,E\otimes M)$, so the tangent is $T_{[E]}=\mathrm{Ext}^1(E,E)\otimes M=R\Gamma(\mathrm{End}(E))[1]\otimes M$. Dualizing, $\mathbb{L}_{[E]}=T_{[E]}^\vee=R\Gamma(\mathrm{End}(E))^\vee[-1]$. The deformation group $\mathrm{Ext}^1(\mathbb{L}_{[E]},M)=H^1(X,\mathrm{End}(E))\otimes M$ (the first-order deformations), the automorphisms $\mathrm{Ext}^0(\mathbb{L}_{[E]},M)=H^0(X,\mathrm{End}(E))\otimes M$ (the endomorphisms), and the obstructions $\mathrm{Ext}^2(\mathbb{L}_{[E]},M)=H^2(X,\mathrm{End}(E))\otimes M=0$ on a curve (the dimension-1 vanishing, $\mathrm{Ext}^2=0$).

Why is the derived moduli of coherent sheaves on a curve smooth (the cotangent complex of amplitude $[-1,0]$), while on a surface it is obstructed (amplitude $[-2,0]$)?

On a curve, $\mathrm{Ext}^i(E,E)=0$ for $i>1$ (the dimension-1 vanishing, the coherent sheaves on a curve have $\mathrm{Ext}^i=0$ for $i>1$), so the cotangent complex has cohomology only in degrees $-1, 0$ (the $\mathrm{End}$ and the $\mathrm{Ext}^1$), hence the amplitude $[-1,0]$ and the smoothness. On a surface, $\mathrm{Ext}^2(E,E)$ may be nonzero (the obstruction group), so the cotangent complex has cohomology in degrees $-2,-1,0$, the amplitude $[-2,0]$, and the moduli is obstructed (the virtual fundamental class is needed, the $(-1)$-shifted symplectic structure of the DT theory).

Pitfalls¶

The derived moduli / the cotangent complex is the $\infty$-categorical deformation theory (the classical Schlessinger is the truncation to $\pi_0$ / $\pi_1$) — do not conflate the derived / $\infty$-categorical version with the classical Schlessinger (the latter sees only the tangent and the obstruction, not the full derived structure).
The Morava $E$-theory $E_n$ is the Lubin–Tate deformation (the universal deformation of the height-$n$ FGL $\Gamma_n$, the Lubin–Tate ring $R_n$), not an arbitrary height-$n$ theory — the universality is the key (every height-$n$ deformation factors through $E_n$).
The derived Artin representability requires the full obstruction theory (the cotangent complex), the smooth atlas, and the sheaf condition — the representability is a theorem under these hypotheses, not automatic.
The spectral Gromov–Witten / the virtual fundamental class as a derived structure is the derived / $(-1)$-shifted symplectic framework (Pantev–Toën–Vaquié–Vezzosi), not the classical VFC of Behrend–Fantechi — the two are related but the derived version is the natural / intrinsic one.
The cotangent complex controls both smoothness and the deformation theory — the amplitude $[0,0]$ is smooth, $[0,1]$ is derived smooth, $[-1,0]$ is the stacky smooth, etc.; the amplitude encodes the geometric properties.
The moduli of formal groups $\mathcal{M}_{\mathrm{FG}}$ is a stack over $\mathrm{Spec}\,\mathbb{Z}_{(p)}$ (the height stratification is at each prime $p$); the Lubin–Tate deformation is the formal neighborhood of the height-$n$ point, not a global neighborhood.

Refs¶

Lurie, Spectral Algebraic Geometry (derived Artin representability, the cotangent complex); Lurie, DAG series
Toën–Vezzosi, "Algèbres de Simpson" / HAG II (the derived Artin representability, the spectral moduli)
Illusie, Complexe cotangent et déformations (LNM 1971); Schlessinger, "Functors of Artin rings" (1968)
Pantev–Toën–Vaquié–Vezzosi, "Shifted symplectic structures" (Publ. Math. IHÉS 2013)
Lubin–Tate, "Formal moduli for one-parameter formal Lie groups" (BAMS 1966); Hopkins–Miller (the Morava $E$-theory $\mathbb{E}_\infty$-structure)

Next¶

SAG index · Foundations · Elliptic Cohomology & TMF · Spectral GAGA & Descent · Chromatic & Morava E-theory · Derived AG · DAG / ind-coherent — the $\mathrm{IndCoh}$ / the Gaitsgory–Rozenblyum · DAG / shifted symplectic — the shifted symplectic / the DT · Mirror symmetry / GW

Chromatic / formal-group moduli	Description
\(\mathcal{M}_{\mathrm{FG}}\)	the moduli of 1-d formal groups (the height stratification \(n=0,1,2,\dots\))
Height-\(n\) stratum \(\mathcal{M}_{\mathrm{FG}}^{\ge n}\)	the locus where \([p]_F=v_n x^{p^n}+\dots\) (the \(v_n\) nonzero)
Lubin–Tate deformation \(\mathrm{Def}(\Gamma_n)\)	the universal deformation ring \(R_n=W(\mathbb{F}_{p^n})[[u_1,\dots,u_{n-1}]]\)
Morava \(E\)-theory \(E_n\)	the even-periodic \(\mathbb{E}_\infty\)-ring with \(\pi_0E_n=R_n\)
Relation to \(\mathrm{MGL}\)	the chromatic picture of algebraic cobordism: the \(v_n\)-periodic layers