Derived Moduli Stacks, Perfect Stacks, and the \(D\)-Module / Betti Realization¶
Job: develop the derived moduli \(\mathrm{Perf}_X\) (cotangent / Ext / VFC / shifted symplectic), the perfect stacks / self-duality, the Betti / de Rham / étale realization / six-functor formalism, and the derived mapping stack / loop / Hochschild — with the derived moduli \(\mathrm{Perf}_X\) on a curve / the VFC as the worked example.
Core¶
The derived moduli stack of sheaves / perfect complexes¶
The derived moduli stack \(\mathrm{Perf}_X\) (the moduli of perfect complexes on a derived stack \(X\), the derived enhancement of the classical moduli). The cotangent complex \(\mathbb{L}_{\mathrm{Perf}_X}\) at \([E]\) (the tangent \(T_{[E]}\mathrm{Perf}=R\mathrm{Hom}(E,E)[1]\), the \(\mathrm{Ext}\) groups, recalled from SAG / derived moduli).
| Derived moduli \(\mathrm{Perf}_X\) | Description |
|---|---|
| \(\mathrm{Perf}_X\) | the moduli of perfect complexes (the derived enhancement of the classical moduli) |
| Cotangent at \([E]\) | \(\mathbb{L}_{[E]}=R\mathrm{Hom}(E,E)^\vee[-1]\) (the tangent \(T=R\mathrm{Hom}(E,E)[1]\), the \(\mathrm{Ext}\)) |
| VFC | the \((-1)\)-shifted symplectic / the obstruction theory (the Behrend–Fantechi / the Pantev–Toën–Vaquié–Vezzosi) |
| Shifted symplectic | the derived moduli the \((-1)\)-shifted symplectic (the \(\mathrm{Ext}^1\) the symplectic, the PTVV) |
Perfect stacks and the self-duality¶
A derived stack \(X\) is perfect (the Toën–Vezzosi / the Ben-Zvi–Francis–Nadler) if \(\mathrm{QCoh}(X)\) is compactly generated by the perfect complexes (\(\mathrm{QCoh}(X)=\mathrm{Ind}(\mathrm{Perf}(X))\)).
| Perfect stacks / self-duality | Description |
|---|---|
| Perfect | \(\mathrm{QCoh}(X)=\mathrm{Ind}(\mathrm{Perf}(X))\) (the QCoh compactly generated by the Perf) |
| Self-duality | \(D_X(\mathrm{QCoh}(X))\simeq\mathrm{QCoh}(X)\) (the QCoh self-dual for the perfect stack, the Grothendieck–Verdier duality trivial) |
| Smooth | the \(\mathrm{Perf}\) of compact amplitude (the smooth = the compact / the finite amplitude) |
| Gorenstein | the dualizing \(\omega_X\) invertible (the Gorenstein = the dualizing, the \(\mathrm{IndCoh}\simeq\mathrm{QCoh}\)) |
The Betti / de Rham / étale realization and the six-functor formalism¶
| Realization / six functors | Description |
|---|---|
| Betti realization | \(\mathrm{QCoh}(X_{\mathrm{Betti}})\cong\mathrm{Shv}_{\mathrm{conv}}(X(\mathbb{C}))\) (the Betti stack, the constructible sheaves, the Ben-Zvi–Francis–Nadler) |
| De Rham | the \(D\)-modules, the \(\mathrm{IndCoh}(X_{\mathrm{dR}})\) (the de Rham stack, the ind-coherent) |
| Étale | the \(\ell\)-adic sheaves, the \(\mathrm{QCoh}(X_{\mathrm{\acute{e}t}})\) (the étale stack) |
| Six functors on derived stacks | the \(f_*,f^*,f_!,f^!,\otimes,\mathrm{RHom}\) on the \(\mathrm{QCoh}\) / \(\mathrm{IndCoh}\) (the Gaitsgory–Rozenblyum, the analog of the motivic six functors) |
The derived mapping stack and the loop / the Hochschild¶
| Derived mapping / loop / Hochschild | Description |
|---|---|
| Derived mapping stack \(\mathrm{Map}(Y,X)\) | the derived stack of maps \(Y\to X\) (the \(\infty\)-internal Hom, the derived enhancement of the classical Hom) |
| Loop space \(LX=\mathrm{Map}(S^1,X)\) | the free loop space (the derived stack, the \(S^1\)-equivariant the cyclic / the Hochschild) |
| Hochschild homology \(\mathrm{HH}_*(X)\) | \(\mathrm{HH}_*(X)=\mathrm{QCoh}(LX)\) (the Hochschild as the QCoh of the loop, the Ben-Zvi–Nadler) |
| \(S^1\) / cyclic / NCG | the \(S^1\)-equivariant the cyclic homology, the NCG / cyclic cohomology connection |
See it¶
Derived moduli of perfect complexes on a curve and the virtual fundamental class. For a smooth projective curve \(X\) (genus \(g\)), the derived moduli \(\mathrm{Perf}_X\) at \([E]\) (a perfect complex / a vector bundle \(E\)): the tangent \(T_{[E]}\mathrm{Perf}=R\mathrm{Hom}(E,E)[1]=R\Gamma(\mathrm{End}\,E)[1]\) (the derived \(\mathrm{Ext}\), the \(H^1\) deformations / \(H^0\) automorphisms / \(H^2=0\) unobstructed, recalled from SAG / derived moduli). The derived moduli is the smooth derived Artin stack (the cotangent \(\mathbb{L}_{[E]}=R\Gamma(\mathrm{End}\,E)^\vee[-1]\) the perfect amplitude \([-1,0]\)). The VFC: the derived moduli has the \((-1)\)-shifted symplectic structure (the Pantev–Toën–Vaquié–Vezzosi, the \(\mathrm{Ext}^1\) the symplectic / the \(\mathrm{Ext}^0\) the Poisson), the VFC the derived / shifted symplectic structure (the smooth derived stack, the symplectic the shifted). The key point: the derived moduli is the smooth derived Artin stack with the shifted symplectic, the VFC the derived structure — the derived AG as the natural home for the VFC / the shifted symplectic.
Check¶
Verify that the derived moduli \(\mathrm{Perf}_X\) on a smooth curve is the smooth derived Artin stack (the cotangent amplitude \([-1,0]\)), and identify the shifted symplectic.
The tangent \(T_{[E]}\mathrm{Perf}=R\mathrm{Hom}(E,E)[1]=R\Gamma(\mathrm{End}\,E)[1]\), the cohomology \(H^0=\mathrm{End}(E)\) (automorphisms, degree \(-1\)), \(H^1=\mathrm{Ext}^1(E,E)\) (deformations, degree \(0\)), \(H^2=0\) (unobstructed on a curve, the dimension-1 vanishing). The cotangent \(\mathbb{L}_{[E]}=R\Gamma(\mathrm{End}\,E)^\vee[-1]\) has the cohomology in degrees \(-1, 0\) (the dual, the amplitude \([-1,0]\)), so the derived moduli is the smooth derived Artin stack (the cotangent of perfect amplitude \([-1,0]\), the smoothness). The shifted symplectic: the Serre duality on the curve \(\mathrm{Ext}^i(E,E)\cong\mathrm{Ext}^{1-i}(E,E)^*\) (the genus \(g\), the Serre duality \(\mathrm{Ext}^i\cong\mathrm{Ext}^{1-i*}\) for the dimension 1) gives the symplectic pairing on \(\mathrm{Ext}^1\) (the \((-1)\)-shifted: the \(\mathrm{Ext}^1\) the degree \(0\) of the tangent, the symplectic the \((-1)\)-shifted from the \(\mathrm{Ext}^0\) the degree \(-1\)). So the \(\mathrm{Perf}_X\) the \((-1)\)-shifted symplectic (the PTVV, the curve the dimension 1, the shifted symplectic the \((-1)\)-shift).
Explain the Hochschild homology as the \(\mathrm{QCoh}\) of the loop space, and identify the \(S^1\)-equivariant / the cyclic.
The derived mapping stack \(\mathrm{Map}(S^1,X)=LX\) (the free loop space, the derived stack of maps from \(S^1\) to \(X\), the \(\infty\)-internal Hom). The Hochschild homology \(\mathrm{HH}_*(X)=\mathrm{QCoh}(LX)\) (the Ben-Zvi–Nadler: the Hochschild as the QCoh of the loop space, the derived / the noncommutative the Hochschild the \(\mathrm{QCoh}\) of the loop). The \(S^1\)-action on \(LX\) (the rotation of the loop) gives the \(S^1\)-equivariant \(\mathrm{QCoh}^{S^1}(LX)\) the cyclic homology \(\mathrm{HC}_*(X)\) (the \(S^1\)-equivariant the cyclic, the Connes the \(S^1\)-action, the NCG / cyclic cohomology connection). The Hochschild the \(\mathrm{QCoh}(LX)\), the cyclic the \(S^1\)-equivariant — the derived loop the framework for the Hochschild / the cyclic, the NCG the connection.
Pitfalls¶
- The perfect stacks (the QCoh compactly generated by Perf) — the smooth the Perf of compact amplitude, the Gorenstein the dualizing, the perfect the general / the compact generation. The perfect is the general (the compact generation), the smooth / the Gorenstein the special.
- The Betti / de Rham / étale realization (the Ben-Zvi–Francis–Nadler) is the derived stack perspective: the \(\mathrm{QCoh}\) of the Betti / de Rham / étale stack the constructible / the \(D\)-modules / the \(\ell\)-adic sheaves. The three realizations the three categories (the constructible / the \(D\)-modules / the \(\ell\)-adic), the derived stack the framework.
- The six functors on the derived stacks (the Gaitsgory–Rozenblyum) the \(f_*,f^*,f_!,f^!,\otimes,\mathrm{RHom}\) on the \(\mathrm{QCoh}\) / \(\mathrm{IndCoh}\), the analog of the motivic six functors — the derived the categorical, the motivic the homotopy-theoretic, the two related but distinct.
- The derived mapping stack \(\mathrm{Map}(Y,X)\) is the \(\infty\)-internal Hom (the derived enhancement of the classical Hom), not the classical Hom — the derived the higher / the derived structure.
- The Hochschild as the \(\mathrm{QCoh}(LX)\) (the Ben-Zvi–Nadler) is the derived / the geometric Hochschild (the loop space), the classical Hochschild the underived shadow (the \(HH_*(A)=\mathrm{Tor}_*^A(A,A)\) the classical, the derived the \(\mathrm{QCoh}\) of the loop).
Refs¶
- Toën–Vezzosi, HAG II (the perfect stacks, the self-duality); Ben-Zvi–Francis–Nadler, "Integral transforms and Drinfeld centers" (2010)
- Ben-Zvi–Nadler, "Loop spaces and connections" (the Hochschild as the \(\mathrm{QCoh}(LX)\)); Toën, "Derived algebraic geometry" (the mapping stack)
- Gaitsgory–Rozenblyum, A Study in DAG Vol. I–II (the six functors, the \(D\)-modules)
- Pantev–Toën–Vaquié–Vezzosi, "Shifted symplectic structures" (Publ. Math. IHÉS 2013) (the VFC / the shifted symplectic)
Next¶
DAG index · Foundations · Ind-Coherent Sheaves · Shifted Symplectic & DT · D-Modules & Crystals · SAG / derived moduli · Motivic / six functors · NCG / cyclic cohomology