D-Modules, Crystals, and the Prismatic / Crystalline via DAG¶
Job: develop the \(D\)-modules via the de Rham stack / \(\mathrm{IndCoh}(X_{\mathrm{dR}})\) (Gaitsgory–Rozenblyum), the Riemann–Hilbert / six functors for \(D\)-modules, the crystals / crystalline cohomology / prismatic, and the prismatic \(D\)-modules / Fargues–Fontaine arithmetic \(D\)-modules — with the de Rham stack / \(D\)-modules / Riemann–Hilbert / prismatic as the worked example.
Core¶
The \(D\)-module functor and the de Rham stack¶
The \(\mathcal{D}\)-modules \(D\text{-}\mathrm{mod}(X)\) on a smooth variety \(X\) (the sheaves with the flat connection, the modules over the ring of differential operators \(\mathcal{D}_X\)). The de Rham stack \(X_{\mathrm{dR}}\) (the quotient \(X/X_{\mathrm{inf}}\) by the infinitesimal groupoid, the crystalline / the de Rham, the \(X_{\mathrm{dR}}(S)=X(S_{\mathrm{red}})\) the reduced points).
Theorem (Gaitsgory–Rozenblyum). \(\mathrm{IndCoh}(X_{\mathrm{dR}})\cong D\text{-}\mathrm{mod}(X)\) (the \(D\)-modules as the ind-coherent sheaves on the de Rham stack).
| \(D\)-modules / de Rham stack | Description |
|---|---|
| \(\mathcal{D}\)-modules \(D\text{-}\mathrm{mod}(X)\) | the sheaves with the flat connection (modules over \(\mathcal{D}_X\)) |
| De Rham stack \(X_{\mathrm{dR}}\) | the quotient \(X/X_{\mathrm{inf}}\) by the infinitesimal groupoid (\(X_{\mathrm{dR}}(S)=X(S_{\mathrm{red}})\)) |
| \(\mathrm{IndCoh}(X_{\mathrm{dR}})\cong D\text{-}\mathrm{mod}(X)\) | the Gaitsgory–Rozenblyum: the \(D\)-modules as the \(\mathrm{IndCoh}\) on the de Rham stack |
The Riemann–Hilbert correspondence and the six functors for \(D\)-modules¶
Theorem (Riemann–Hilbert, Kashiwara). The regular holonomic \(D\)-modules \(\leftrightarrow\) the constructible sheaves: \(\mathrm{DR}:D\text{-}\mathrm{mod}_{\mathrm{rh}}(X)\xrightarrow{\sim}\mathrm{Shv}_{\mathrm{conv}}(X(\mathbb{C}))\) (the de Rham functor).
| Riemann–Hilbert / six functors | Description |
|---|---|
| Riemann–Hilbert (Kashiwara) | \(\mathrm{DR}:D\text{-}\mathrm{mod}_{\mathrm{rh}}(X)\xrightarrow{\sim}\mathrm{Shv}_{\mathrm{conv}}(X(\mathbb{C}))\) (the de Rham / the constructible) |
| Six functors for \(D\)-modules | the \(f_*,f^*,f_!,f^!,\otimes,\mathrm{RHom}\) on the \(D\)-modules (the Gaitsgory–Rozenblyum, the analog of the motivic six functors) |
| \(!\) vs \(*\) | the \(D\)-module the two pullbacks / two pushforwards (the \(!\) and the \(*\), the distinguished vs the sheaves the one each) |
Crystals and the crystalline cohomology¶
| Crystals / crystalline / prismatic | Description |
|---|---|
| Crystals | the sheaves with the flat connection / the stratification (the \(\mathcal{D}\)-modules the de Rham crystals) |
| Crystalline cohomology (Berthelot) | the crystals on the crystalline site (the \(p\)-adic cohomology, the connection to the prismatic cohomology) |
| Prismatic (Bhatt–Scholze) | the prismatic the generalization of the crystalline (the Bhatt–Scholze, the prismatic cohomology) |
| Gauss–Manin | the relative / the family, the flat connection on the cohomology of the family (the Gauss–Manin the crystal) |
The prismatic \(D\)-modules and the Bhatt–Scholze¶
| Prismatic \(D\)-modules / arithmetic | Description |
|---|---|
| Prismatic site \((A,I)\) | the prism (the Bhatt–Scholze, the prismatic cohomology) |
| Prismatic crystals | the sheaves on the prismatic site (the \(A\)-modules with the Frobenius / the connection) |
| Prismatic \(D\)-modules | the Bhatt–Scholze / the Drinfeld: the prismatic \(D\)-modules as the crystals on the prismatic site (the \(p\)-adic / the prismatic \(D\)-module theory) |
| Arithmetic \(D\)-modules | the Fargues–Fontaine / the \(p\)-adic Riemann–Hilbert (the arithmetic / the \(p\)-adic the analog of the classical Riemann–Hilbert) |
| SAG / chromatic connection | the prismatic / the cyclotomic / the BMS, the \(p\)-adic the spectral / the prismatic (SAG / chromatic) |
See it¶
De Rham stack and the \(D\)-modules via \(\mathrm{IndCoh}(X_{\mathrm{dR}})\) and the Riemann–Hilbert. For a smooth complex variety \(X\), the de Rham stack \(X_{\mathrm{dR}}\) (the quotient by the infinitesimal groupoid \(X_{\mathrm{inf}}\), the \(X_{\mathrm{dR}}(S)=X(S_{\mathrm{red}})\) the reduced points). The \(\mathrm{IndCoh}(X_{\mathrm{dR}})\cong D\text{-}\mathrm{mod}(X)\) (the \(D\)-modules as the ind-coherent sheaves on the de Rham stack, the Gaitsgory–Rozenblyum — the \(\mathcal{D}_X\)-modules the \(\mathrm{IndCoh}\) on the de Rham). The Riemann–Hilbert: the \(\mathrm{DR}:D\text{-}\mathrm{mod}_{\mathrm{rh}}(X)\xrightarrow{\sim}\mathrm{Shv}_{\mathrm{conv}}(X(\mathbb{C}))\) (the de Rham functor, the regular holonomic \(D\)-modules to the constructible sheaves, the Kashiwara). The prismatic: the prismatic site \((A,I)\) the Bhatt–Scholze, the prismatic crystals the \(A\)-modules with the Frobenius, the prismatic \(D\)-modules the Bhatt–Scholze / the Drinfeld, the arithmetic \(D\)-module the \(p\)-adic Riemann–Hilbert (the Fargues–Fontaine). The key point: the \(D\)-modules are the \(\mathrm{IndCoh}\) on the de Rham stack (the Gaitsgory–Rozenblyum), the Riemann–Hilbert the de Rham / the constructible, the prismatic the \(p\)-adic / the arithmetic generalization (the Bhatt–Scholze / the Drinfeld) — the DAG / the derived stack the framework for the \(D\)-module / the crystal / the prismatic.
Check¶
Show that \(\mathrm{IndCoh}(X_{\mathrm{dR}})\cong D\text{-}\mathrm{mod}(X)\) for a smooth \(X\), and explain the de Rham stack.
The de Rham stack \(X_{\mathrm{dR}}\) is the quotient \(X/X_{\mathrm{inf}}\) by the infinitesimal groupoid \(X_{\mathrm{inf}}\) (the groupoid of the infinitesimal thickenings, the formal neighborhood of the diagonal). The \(X_{\mathrm{dR}}(S)=X(S_{\mathrm{red}})\) (the \(S\)-points of the de Rham stack are the \(S_{\mathrm{red}}\)-points of \(X\), the reduced / the classical points — the de Rham stack "forgets the nilpotents"). The \(\mathrm{IndCoh}(X_{\mathrm{dR}})\): the ind-coherent sheaves on the de Rham stack. By the Gaitsgory–Rozenblyum, \(\mathrm{IndCoh}(X_{\mathrm{dR}})\cong D\text{-}\mathrm{mod}(X)\): the \(D\)-modules (the modules over \(\mathcal{D}_X\) the ring of differential operators, the sheaves with the flat connection) are the ind-coherent sheaves on the de Rham stack. The mechanism: the de Rham stack the quotient by the infinitesimal groupoid, the \(\mathrm{IndCoh}\) on the quotient the \(\mathcal{D}\)-modules (the crystals / the stratification the descent along the infinitesimal groupoid, the \(\mathcal{D}_X\)-modules the crystals in \(\mathrm{IndCoh}\)). So the \(D\)-modules are the \(\mathrm{IndCoh}\) on the de Rham stack, the Gaitsgory–Rozenblyum the derived / the stacky framework.
Explain the six functors for \(D\)-modules and the \(!\) vs \(*\), and contrast with the sheaf six functors.
The \(D\)-module six functors: the \(f_*,f^*,f_!,f^!,\otimes,\mathrm{RHom}\) on the \(D\text{-}\mathrm{mod}(X)\) (the Gaitsgory–Rozenblyum, the analog of the motivic six functors). The \(!\) vs \(*\): the \(D\)-modules have the two pullbacks (\(f^*\) the \(*\)-pullback, \(f^!\) the \(!\)-pullback) and the two pushforwards (\(f_*\) the \(*\)-pushforward, \(f_!\) the \(!\)-pushforward), the distinguished (the \(!\) the exceptional / the shriek, the \(*\) the ordinary). The contrast with the sheaves: the sheaves have the one pullback (\(f^*\)) and the one pushforward (\(f_*\), with the \(f_!\) for the proper / the compact support), the \(D\)-modules the two (the \(!\) and the \(*\), the distinguished). The reason: the \(D\)-modules the de Rham / the crystals, the \(!\) the de Rham / the \(*\) the direct (the two the de Rham / the direct, the Riemann–Hilbert the \(!\) the constructible the \(*\)). The six functors the \(!\) and the \(*\), the \(D\)-modules the two, the sheaves the one (the \(!\) the proper / the compact, the \(D\)-module the two always).
Pitfalls¶
- The \(D\)-modules the \(\mathrm{IndCoh}(X_{\mathrm{dR}})\) (the de Rham stack, the Gaitsgory–Rozenblyum) — the classical \(\mathcal{D}_X\)-modules the underived shadow (the derived the \(\mathrm{IndCoh}\) on the de Rham stack, the underived the \(\mathcal{D}_X\)-modules).
- The six functors for \(D\)-modules the \(!\) vs \(*\) (the two pullbacks / two pushforwards, the distinguished vs the sheaves the one each) — the \(D\)-modules the two, the sheaves the one, the \(!\) the de Rham / the \(*\) the direct.
- The crystalline cohomology (Berthelot) is the \(p\)-adic the crystals on the crystalline site, the connection to the prismatic cohomology the Bhatt–Scholze the generalization (the prismatic the general, the crystalline the special).
- The prismatic \(D\)-modules (the Bhatt–Scholze / the Drinfeld) the \(p\)-adic / the arithmetic, the Fargues–Fontaine the \(p\)-adic Riemann–Hilbert — the relation to the classical Riemann–Hilbert via the reduction / the \(p\)-adic (the \(p\)-adic the analog, not the direct).
- The Gauss–Manin connection is the crystal / the flat connection on the cohomology of the family (the relative / the family), not the absolute (the absolute the \(D\)-module, the relative the Gauss–Manin).
- The arithmetic \(D\)-modules / the Fargues–Fontaine are the \(p\)-adic the analog of the classical Riemann–Hilbert (the \(p\)-adic the constructible sheaves the \(p\)-adic the \(D\)-modules), the relation via the Fargues–Fontaine curve (the \(p\)-adic the geometric framework), not the direct.
Refs¶
- Gaitsgory–Rozenblyum, A Study in DAG Vol. II (the \(D\)-modules, the de Rham stack, the six functors)
- Kashiwara, "The Riemann–Hilbert problem for holonomic systems" (1984); the Riemann–Hilbert
- Berthelot, Cohomologie cristalline des schémas de caractéristique \(p>0\) (1974); the crystalline
- Bhatt–Scholze, "Prisms and prismatic cohomology" (Annals 2022); Drinfeld, "Prismatization" (the prismatic \(D\)-modules); Fargues–Fontaine, Courbes et fibrés vectoriels (the \(p\)-adic Riemann–Hilbert)
Next¶
DAG index · Foundations · Ind-Coherent Sheaves · Derived Moduli & Perfect · Shifted Symplectic & DT · Prismatic cohomology · SAG / chromatic · Geometric Langlands / derived