001 What is a derived stack

Source: “What is…a derived stack?”, Gabriele Vezzosi

Higher Stacks

  • Motivation:
    • manifolds are modeled by Euclidean space,
    • schemes are modeled by commutative rings
    • stacks are modeled by quotients of schemes by group actions
    • derived schemes are modeled by derived commutative rings: \({ \mathsf{c}{\mathsf{dg \mathsf{Alg} } }}\) (commutative differential graded algebras) or \({\mathrm{ss}}(\mathsf{CAlg})\) (simplicial commutative algebras)
      • Idea for \({\mathsf{sSet}}\mathsf{CAlg}\): a simplicial set which at every level is an object of \(\mathsf{CAlg}\) whose face/degeneracy maps are algebra morpisms
  • Use: classification problems where we want to weaken the notion of isomorphism, e.g. chain complexes with quasi-isomorphisms.
  • Idea: replace a scheme \((X, {\mathcal{O}}_X)\in {\mathsf{Top}}\times {\mathsf{Sh}}({-}; {}_{k} \mathsf{Alg} )\) with its functor of points \(h_x: {\mathsf{Sch}}\to {\mathsf{Set}}\) . Regard a set as a discrete groupoid (no cross-morphisms), and replace the target category to get \(h_x: {\mathsf{Sch}}\to{\mathsf{Grpd}}\)
    • Why: don’t mod out by isomorphisms, keep witnesses to the isomorphisms to remember how things were isomorphic.
  • A word about higher equivalences: filling cells.
  • Motivating example: Eilenberg-MacLane stacks.
    • For $G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {k}} $, form \({{\mathbf{B}}G}= K(G, 1)\) the stack classifying principal \(G{\hbox{-}}\)bundles (or \(G{\hbox{-}}\)torsors).
    • Via the nerve construction \({ \mathcal{N}({{-}}) }: {\mathsf{Grpd}}\to{\mathsf{sSet}}\), view \(K(G, 1): \mathsf{C} \to \mathsf{sSet}\)
    • If $G\in {\mathsf{Ab}}{\mathsf{Grp}}{\mathsf{Sch}}_{/ {k}} $, can iterate this construction to get \({\mathbf{B}}^2 G = K(G, 2)\), the first higher stack.
    • Yields a notion of spheres: \(S^1 = K({\mathbf{Z}}, 1)\)
  • General definition: higher stacks form the category \([{\mathsf{C}} {}_{k} \mathsf{Alg} , \mathsf{sSet}]\)

Derived Stacks

  • Idea for derived stacks: enlarge source category
  • Motivations from intersection theory:

    • $X\in {\mathsf{sm}}\mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}} $ with \(Z, T\leq X\), possibly singular, with \(\dim Z + \dim T = \dim X\) with \(\dim(Z \cap T) = 0\)

    • Recall graded dimension as a Laurent series (motivated e.g. by the Jones polynomial as a way to categorify dimension): \begin{align*}{\mathsf{gr}\,}\dim_k V_\bullet = d_{V_{/ {k}} }(z) \coloneqq\sum_{n\in {\mathbf{Z}}} \dim_k V_n z^n \in {\mathbf{Z}}[z, z^{-1}]\end{align*}

    • Points \(p\in Z\cap T\) have an intersection multiplicity computed by Serre’s intersection formula using the local rings at \(p\): \begin{align*} \mu_p(X, Z, T) \coloneqq{\mathsf{gr}\,}\dim_C \qty{ {\mathcal{O}}_{Z, \, p} \overset{\mathbb{L}}{ \otimes} _{{\mathcal{O}}_{X. p}} {\mathcal{O}}_{T, \, p} } \Big|_{z=1} \end{align*}

    • Idea behind derived tensor product: \(A \overset{\mathbb{L}}{ \otimes} _R B\) is a space with \(\pi_0(A \overset{\mathbb{L}}{ \otimes} _R B) = A\otimes_R B\), but contains higher homotopical data in \(\pi_{\geq 1}\). Note that \(\pi_*\) reduces to \(H_*\) here. (For experts: this is realized by a smash product of spectra.)

    • Unpacking this: \begin{align*} \mu_{p}(X ; Z, T)\coloneqq\sum_{i\geq 0} \operatorname{dim}_{C} \operatorname{Tor}_{i}^{R}\left(\mathcal{O}_{Z, \,p}, \mathcal{O}_{T, \,p}\right) \qquad \sim {}_{R}{\mathsf{Mod}} , \quad R = O_{X, p} \end{align*}

    • This can be computed using standard homological algebra: take an acyclic (flat) resolution, tensor, etc.

    • For flat intersections, i.e. if either entry in the Tor is a flat \(R{\hbox{-}}\)module, this collapses: \begin{align*}\mu_p(X, Z, T) = \dim_C {\mathcal{O}}_{Z, p}\otimes_{{\mathcal{O}}_{X, p}} {\mathcal{O}}_{T, p}\end{align*}

  • Cotangent complex: goes but to Quillen, Grothendieck, Illusie
    • Idea: for \(X = \operatorname{Spec}A\) and $A\in\mathsf{CAlg}_{/ {k}} $ in characteristic zero, take a simplicial resolution of \(A\) by cdgas and apply \(\Omega_{{-}_{/ {k}} }\) (algebraic/Kahler differentials) to get \({\mathbb{L}}_X\)
    • Becomes the actual tangent space of a derived scheme
  • Definition of derived stacks: \begin{align*} {\mathsf{d}}{\mathsf{St}}\leq [{\mathsf{d}}\mathsf{CAlg}_{/ {k}} , {\mathsf{sSet}}] = [{\mathsf{sSet}}\mathsf{CAlg}_{/ {k}} , {\mathsf{sSet}}] \underset{\operatorname{ch}k = 0}= [{ \mathsf{c}{\mathsf{dg \mathsf{Alg} } }}_{/ {k}} , {\mathsf{sSet}}], \end{align*} where we take the functors that send quasi-isomorphisms to weak equivalences (preserving homotopy theories on both sides), plus descent (basically making the functors “etale sheaves”).
    • Idea for site structure on source: covering families are morpisms \(A\to B_i\) where \(\pi_0 A\to \pi_0 B_i\) is an etale covering family and \(\pi_i A \otimes_{\pi_0 A} \pi_0 B { \, \xrightarrow{\sim}\, }\pi_i B\)
    • Think of \(\pi_i = H^i\) here.
    • For any derived ring \(A\), get a derived spectrum \({\mathbf{R}}\operatorname{Spec}A\).
    • Derived stacks admit truncations \(\tau_0({\mathbf{R}}\operatorname{Spec}A) = \operatorname{Spec}\pi_0 A\) which are stacks.
  • Interesting application:
    • Write \({ \mathsf{Vect} }_n(X)\) as the classifying stack for rank \(n\) vector bundles over \(X\) a smooth proper scheme. Then \({\mathbf{R}}{ \mathsf{Vect} }_n({-}) = {\mathbf{R}}\mathop{\mathrm{Hom}}({-}, \mathbf{B}\mkern-3mu \operatorname{GL} _n)_\bullet\), which recovers \({ \mathsf{Vect} }_n(X)\) in degree zero.
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