Ind-Coherent Sheaves and the Gaitsgory–Rozenblyum Framework¶
Job: develop the ind-coherent sheaves \(\mathrm{IndCoh}(X)\) (vs \(\mathrm{QCoh}\), the proper base change / duality), the Grothendieck–Serre duality / dualizability / Serre functor, the crystals / \(D\)-modules via \(\mathrm{IndCoh}(X_{\mathrm{dR}})\), and the Orlov singularity category / matrix factorizations — with the singular \(\mathrm{IndCoh}\) vs \(\mathrm{QCoh}\) / Grothendieck–Serre as the worked example.
Core¶
Ind-coherent sheaves \(\mathrm{IndCoh}(X)\)¶
The ind-coherent sheaves \(\mathrm{IndCoh}(X)\) on a derived stack \(X\) (the Gaitsgory–Rozenblyum category, the ind-completion of the coherent sheaves \(\mathrm{Coh}(X)\)). The \(\mathrm{IndCoh}(X)\) is the "correct" category for the derived AG (vs the \(\mathrm{QCoh}(X)\) the "naive"):
| \(\mathrm{IndCoh}\) vs \(\mathrm{QCoh}\) | \(\mathrm{IndCoh}(X)\) | \(\mathrm{QCoh}(X)\) |
|---|---|---|
| Base change | \(f_*\) commutes with base change for proper qcqs \(f\) | fails for singular / non-Gorenstein |
| Duality | the Grothendieck–Serre duality (the dualizable category) | the duality may fail |
| \(t\)-structure | the coherent / the standard \(t\)-structure | the quasi-coherent \(t\)-structure |
| Dualizability | \(\mathrm{IndCoh}(X)\) dualizable as a category | not always |
| Relation | \(\mathrm{IndCoh}(X)\simeq\mathrm{QCoh}(X)\) for Gorenstein / smooth \(X\) | the general / the naive |
The Grothendieck–Serre duality and the dualizability¶
Theorem (Grothendieck–Serre duality in the derived setting). For a proper \(f:X\to Y\) of derived stacks, \(f_*^{\mathrm{IndCoh}}\) has the right adjoint \(f^!\) (the exceptional pullback) with \(f^!=f^*\otimes\omega_{X/Y}\) (the relative dualizing \(\omega_{X/Y}\)).
| Duality / Serre | Description |
|---|---|
| Grothendieck–Serre | \(f_*^{\mathrm{IndCoh}}\dashv f^!\) with \(f^!=f^*\otimes\omega_{X/Y}\) (the relative dualizing) |
| Dualizability | \(\mathrm{IndCoh}(X)\) dualizable as a category (the \(1\)-morphism has the adjoints / the dual) |
| Serre functor | \(S_X(K)=K\otimes\omega_X[\dim X]\) (the Serre duality / the CY the trivial Serre) |
| Calabi–Yau | \(\omega_X\simeq\mathcal{O}_X[\dim X]\) (the CY, the trivial Serre, the Serre = the shift) |
Crystals and the \(D\)-modules via \(\mathrm{IndCoh}\)¶
| Crystals / \(D\)-modules via \(\mathrm{IndCoh}\) | Description |
|---|---|
| Crystals | the sheaves with the flat connection / the stratification (the \(\mathcal{D}\)-modules) |
| \(D\)-modules via \(\mathrm{IndCoh}\) | \(\mathrm{IndCoh}(X_{\mathrm{dR}})\cong D\text{-}\mathrm{mod}(X)\) (the Gaitsgory–Rozenblyum: the \(\mathcal{D}\)-modules as the de Rham crystals) |
| De Rham stack \(X_{\mathrm{dR}}\) | the quotient \(X/X_{\mathrm{inf}}\) by the infinitesimal groupoid (the crystalline / the de Rham) |
| Prismatic crystals | the connection to the prismatic cohomology: the prismatic crystals / the prismatic \(D\)-modules (the Bhatt–Scholze) |
The functor of points and the category of singularities¶
| Singularity category / MF | Description |
|---|---|
| Singularity category \(D_{\mathrm{sg}}(X)\) | \(D_{\mathrm{sg}}(X)=\mathrm{Coh}(X)/\mathrm{Perf}(X)\) (the Orlov singularity category, the quotient of the coherent by the perfect — the singularity / the non-smooth) |
| Matrix factorizations | the \(D_{\mathrm{sg}}\) of the LG model / the hypersurface (the Orlov / the Knörrer periodicity) |
| Orlov / Knörrer | the Orlov theorem (the singularity category the MF), the Knörrer periodicity (the MF the periodic) |
| Mirror / HMS | the matrix factorizations the B-side of the Fano / the LG HMS (mirror symmetry / Fukaya, the \(\mathrm{Perf}\) / the \(\mathrm{MF}\) the B-model) |
See it¶
\(\mathrm{IndCoh}\) vs \(\mathrm{QCoh}\) for a singular derived scheme and the Grothendieck–Serre duality. For a smooth \(X\), \(\mathrm{IndCoh}(X)\simeq\mathrm{QCoh}(X)\) (the smooth / Gorenstein case, the two coincide). For a singular \(X\) (e.g. the nodal curve / the local complete intersection singularity), the \(\mathrm{IndCoh}(X)\ne\mathrm{QCoh}(X)\): the \(\mathrm{IndCoh}\) has the proper base change / the duality, the \(\mathrm{QCoh}\) does not (the \(\mathrm{QCoh}\) fails the base change for the singular / the non-Gorenstein). The Grothendieck–Serre: for the nodal curve \(X\) (the proper, the Gorenstein), \(f_*^{\mathrm{IndCoh}}\) has the right adjoint \(f^!\) with \(f^!=f^*\otimes\omega_X\) (the relative dualizing \(\omega_X\) the canonical, the Gorenstein). The \(D\)-modules: the \(\mathrm{IndCoh}(X_{\mathrm{dR}})\cong D\text{-}\mathrm{mod}(X)\) (the de Rham stack, the \(D\)-modules as the crystals in \(\mathrm{IndCoh}\)). The key point: the \(\mathrm{IndCoh}\) is the "correct" category for the singular derived AG (the base change / the duality), the \(\mathrm{QCoh}\) the "naive" — the Gaitsgory–Rozenblyum framework.
Check¶
Show that for a smooth \(X\), \(\mathrm{IndCoh}(X)\simeq\mathrm{QCoh}(X)\), and explain why the two differ for singular \(X\).
For a smooth \(X\) (the Gorenstein, the dualizing \(\omega_X\) invertible), the \(\mathrm{IndCoh}(X)\simeq\mathrm{QCoh}(X)\) (the Gorenstein / smooth case, the two categories coincide: the \(\mathrm{IndCoh}\) the ind-completion of \(\mathrm{Coh}\), the \(\mathrm{QCoh}\) the ind-completion of \(\mathrm{Perf}\), and for the smooth / Gorenstein the \(\mathrm{Coh}=\mathrm{Perf}\) (the perfect = the coherent), so the ind-completions coincide). For a singular \(X\) (the non-Gorenstein), the \(\mathrm{Coh}\ne\mathrm{Perf}\) (the coherent has the non-perfect / the singular sheaves), so the \(\mathrm{IndCoh}=\mathrm{Ind}(\mathrm{Coh})\ne\mathrm{Ind}(\mathrm{Perf})=\mathrm{QCoh}\). The \(\mathrm{IndCoh}\) has the proper base change / the duality (the Grothendieck–Serre, the \(f^!\)), the \(\mathrm{QCoh}\) does not (the base change fails for the singular / the non-Gorenstein, the \(f^!\) not defined / not the duality). So the smooth / Gorenstein the two coincide, the singular the two differ — the \(\mathrm{IndCoh}\) the "correct" (the duality), the \(\mathrm{QCoh}\) the "naive".
Explain the Orlov singularity category \(D_{\mathrm{sg}}=\mathrm{Coh}/\mathrm{Perf}\) and its relation to the matrix factorizations.
The Orlov singularity category \(D_{\mathrm{sg}}(X)=\mathrm{Coh}(X)/\mathrm{Perf}(X)\) (the quotient of the coherent by the perfect — the Verdier quotient, the stable category of the singular / the non-smooth sheaves). For a hypersurface \(X=\mathrm{Spec}\,A/(f)\) (the LG model / the zero locus of \(f\)), the Orlov theorem: \(D_{\mathrm{sg}}(X)\simeq\mathrm{MF}(A,f)\) (the matrix factorizations of \(f\), the \(\mathbb{Z}/2\)-graded complexes \(d_0,d_1\) with \(d_0d_1=d_1d_0=f\cdot\mathrm{id}\), the MF the stable category of the LG model). The Knörrer periodicity: the \(\mathrm{MF}(A,f)\simeq\mathrm{MF}(A[x],f+x^2)\) (the periodicity, the MF the same after adding a quadratic variable). The relation to the mirror / the HMS: the \(\mathrm{MF}\) the B-side of the Fano / the LG HMS (the mirror symmetry / Fukaya the \(\mathrm{Perf}\) / the \(\mathrm{MF}\) the B-model, the Fano the LG / the MF, the HMS the \(\mathrm{Fuk}(X)\simeq\mathrm{MF}(\check Y,W)\)).
Pitfalls¶
- The \(\mathrm{IndCoh}\simeq\mathrm{QCoh}\) for the smooth / Gorenstein (the singular / the non-Gorenstein the two differ) — the \(\mathrm{IndCoh}\) the "correct" (the base change / the duality), the \(\mathrm{QCoh}\) the "naive". The Gaitsgory–Rozenblyum the framework, not the equivalence (the two differ for the singular).
- 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 singularity category \(D_{\mathrm{sg}}=\mathrm{Coh}/\mathrm{Perf}\) is the quotient (the Verdier quotient, the stable category), not the subcategory — the \(\mathrm{Perf}\subset\mathrm{Coh}\), the quotient the singularity / the non-perfect.
- The matrix factorizations are the \(D_{\mathrm{sg}}\) of the LG model / the hypersurface (the Orlov), the Knörrer the periodicity — the MF the stable / the periodic, not the full \(\mathrm{Coh}\).
- The Grothendieck–Serre duality requires the proper \(f\) (the \(f^!\) the exceptional pullback, the right adjoint to \(f_*\)) — for non-proper \(f\) the duality is more subtle (the Borel–Moore / the \(f^!\) not the simple tensor with \(\omega\)).
Refs¶
- Gaitsgory–Rozenblyum, A Study in Derived Algebraic Geometry (AMS 2017) Vol. I (IndCoh) / Vol. II (D-modules)
- Grothendieck–Hartshorne, Residues and Duality (the classical Serre duality); the derived / Gaitsgory–Rozenblyum version
- Orlov, "Triangulated categories of singularities and D-branes" (2004); the Knörrer periodicity
- Knörrer, "Cohen–Macaulay modules on hypersurface singularities I" (1987); the matrix factorizations
Next¶
DAG index · Foundations · Derived Moduli & Perfect · Shifted Symplectic & DT · D-Modules & Crystals · SAG / derived moduli · Geometric Langlands / derived