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Applications of perverse sheaves in commutative algebra

Bhargav Bhatt, Manuel Blickle, Gennady Lyubeznik, Anurag K. Singh, Wenliang Zhang

TL;DR

The paper develops a unified, topological approach to singularity invariants in commutative algebra by leveraging the Riemann–Hilbert correspondence and perverse sheaves in both characteristic 0 and characteristic $p$. In characteristic 0, it reinterprets local cohomology invariants (e.g., lcd, Bass numbers, Lyubeznik numbers) via perverse truncations and de Rham data, yielding topological proofs and embedding-independence results; it also connects Hodge-to-de Rham degeneracies to topology. In characteristic $p$, the authors construct and exploit perverse $\mathbf{F}_p$-sheaves through the Bhatt–Lurie RH framework, establishing a strong perverse Artin vanishing theorem and translating $F$-singularity phenomena (intersection cohomology, $F$-rationality, CM) into perverse-coherent/Frobenius-structure statements. They prove embedding-independence results for Lyubeznik complexes in characteristic $p$, express Bass numbers and gradings in topological terms, and derive Kodaira vanishing up to finite covers within this framework. Overall, the work provides soft, topological proofs and structural insights into singularity invariants, with broad implications for local cohomology, intersection cohomology, and $F$-singularity theory across characteristics.

Abstract

The goal of this paper is to explain how basic properties of perverse sheaves sometimes translate via Riemann-Hilbert correspondences (in both characteristic $0$ and characteristic $p$) to highly non-trivial properties of singularities, especially their local cohomology. Along the way, we develop a theory of perverse $\mathbf{F}_p$-sheaves on varieties in characteristic $p$, expanding on previous work by various authors, and including a strong version of the Artin vanishing theorem.

Applications of perverse sheaves in commutative algebra

TL;DR

The paper develops a unified, topological approach to singularity invariants in commutative algebra by leveraging the Riemann–Hilbert correspondence and perverse sheaves in both characteristic 0 and characteristic . In characteristic 0, it reinterprets local cohomology invariants (e.g., lcd, Bass numbers, Lyubeznik numbers) via perverse truncations and de Rham data, yielding topological proofs and embedding-independence results; it also connects Hodge-to-de Rham degeneracies to topology. In characteristic , the authors construct and exploit perverse -sheaves through the Bhatt–Lurie RH framework, establishing a strong perverse Artin vanishing theorem and translating -singularity phenomena (intersection cohomology, -rationality, CM) into perverse-coherent/Frobenius-structure statements. They prove embedding-independence results for Lyubeznik complexes in characteristic , express Bass numbers and gradings in topological terms, and derive Kodaira vanishing up to finite covers within this framework. Overall, the work provides soft, topological proofs and structural insights into singularity invariants, with broad implications for local cohomology, intersection cohomology, and -singularity theory across characteristics.

Abstract

The goal of this paper is to explain how basic properties of perverse sheaves sometimes translate via Riemann-Hilbert correspondences (in both characteristic and characteristic ) to highly non-trivial properties of singularities, especially their local cohomology. Along the way, we develop a theory of perverse -sheaves on varieties in characteristic , expanding on previous work by various authors, and including a strong version of the Artin vanishing theorem.
Paper Structure (23 sections, 56 theorems, 188 equations)

This paper contains 23 sections, 56 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Recollections on the Riemann-Hilbert correspondence in characteristic $0$
  3. Applications of the Riemann-Hilbert correspondence in characteristic $0$
  4. Recovering results of Ogus
  5. Gradings
  6. Bass numbers
  7. Hodge-to-de-Rham spectral sequence for de Rham homology of affine schemes
  8. Embedding independence for Lyubeznik numbers
  9. The Riemann-Hilbert functor in characteristic $p$ and perverse $\mathbf{F}_{\!p}$-sheaves
  10. Perverse coherent sheaves
  11. Recollection on the Riemann-Hilbert correspondence from BhattLurieRH
  12. A nilpotency result for Frobenius modules
  13. Perverse $\mathbf{F}_{\!p}$-sheaves
  14. Relation to Gabber's perverse $t$-structure
  15. Reminders on compactly supported cohomology
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $f\colon X \longrightarrow Y$ be an affine map of varieties over a field of characteristic $p$. Then is $t$-exact for the perverse $t$-structure on the source and the target. In particular, for any affine variety $Z$, we have $H^i_c(Z,F) = 0$ for $i \neq 0$, and $F$ a perverse $\mathbf{F}_{\!p}$-sheaf.

Theorems & Definitions (142)

  • Theorem 1.1: The perverse Artin vanishing theorem for $\mathbf{F}_{\!p}$-coefficients in characteristic $p$
  • Theorem 2.1
  • proof
  • Example 2.2: Local cohomology under Riemann-Hilbert
  • Theorem 3.1: Ogus
  • proof
  • Theorem 3.2: Ogus
  • proof
  • Proposition 3.3
  • proof
  • ...and 132 more