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A Gersten complex on real schemes

Fangzhou Jin, Heng Xie

TL;DR

The paper develops a Gersten-type complex on real schemes to unify coherent duality with Verdier duality in a real-analytic context, producing a dualizing object in $D(X_r)$ that is compatible with the exceptional inverse image $f^!$. It constructs a Rost–Schmidt complex via twisted residues and transfers, and shows it coincides with the Gersten–Witt complex when $2$ is invertible, enabling a refined devissage for $ ext{I}^j$-cohomology and its support. Through six-functor formalism, absolute purity, and Cousin-complex refinements, the authors establish functoriality of the Gersten complex for closed immersions and smooth morphisms, and demonstrate that the Gersten complex provides biduality in $D(X_r)$. A key payoff is the identification of real Borel–Moore homology with twisted $ ext{I}^ullet$-cohomology via the signature map, unifying algebraic duality with semialgebraic topology and yielding a practical computational framework. These constructions pave the way for applications to real-analytic topology, including a robust signature isomorphism $H_n(X, ext{I}^ abla(L)) o H^{ ext{BM}}_n(X_r,L)$ and a realizable Borel–Moore theory compatible with the six-functor setup.

Abstract

We discuss a connection between coherent duality and Verdier duality via a Gersten-type complex of sheaves on real schemes, and show that this construction gives a dualizing object in the derived category, which is compatible with the exceptional inverse image functor $f^!$. The hypercohomology of this complex coincides with hypercohomology of the sheafified Gersten-Witt complex, which in some cases can be related to topological or semialgebraic Borel-Moore homology.

A Gersten complex on real schemes

TL;DR

The paper develops a Gersten-type complex on real schemes to unify coherent duality with Verdier duality in a real-analytic context, producing a dualizing object in that is compatible with the exceptional inverse image . It constructs a Rost–Schmidt complex via twisted residues and transfers, and shows it coincides with the Gersten–Witt complex when is invertible, enabling a refined devissage for -cohomology and its support. Through six-functor formalism, absolute purity, and Cousin-complex refinements, the authors establish functoriality of the Gersten complex for closed immersions and smooth morphisms, and demonstrate that the Gersten complex provides biduality in . A key payoff is the identification of real Borel–Moore homology with twisted -cohomology via the signature map, unifying algebraic duality with semialgebraic topology and yielding a practical computational framework. These constructions pave the way for applications to real-analytic topology, including a robust signature isomorphism and a realizable Borel–Moore theory compatible with the six-functor setup.

Abstract

We discuss a connection between coherent duality and Verdier duality via a Gersten-type complex of sheaves on real schemes, and show that this construction gives a dualizing object in the derived category, which is compatible with the exceptional inverse image functor . The hypercohomology of this complex coincides with hypercohomology of the sheafified Gersten-Witt complex, which in some cases can be related to topological or semialgebraic Borel-Moore homology.

Paper Structure

This paper contains 25 sections, 27 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Conventions and preliminaries
  3. On schemes and sheaves
  4. On dualizing complexes
  5. On real schemes
  6. Main results of the paper
  7. Borel-Moore Devissage
  8. The Gersten-Witt complex
  9. The Rost-Schmid complex
  10. $\mathrm{I}$-cohomology with dualizing complex
  11. $\mathrm{I}$-cohomology with support
  12. The (twisted) signature map
  13. Twisted residue
  14. Twisted transfer
  15. The Gersten complex of a minimal dualizing complex
  16. ...and 10 more sections

Key Result

Lemma 2.3.4

For any Zariski open subset $U\subset X$ and regular function $f\in{\mathcal{O}}_X^\times(U)$, the sign map $\mathrm{Sign}(f)\colon U_r \to \{\pm 1\}$ given by $(x,P)\mapsto \mathrm{Sign}_P(f_x)$ is locally constant.

Theorems & Definitions (52)

  • Lemma 2.3.4
  • proof
  • Definition 2.3.5
  • Theorem 3.0.1
  • Theorem 3.0.2
  • Theorem 3.0.3
  • Theorem 3.0.4
  • Remark 4.2.3
  • Theorem 4.2.5
  • proof
  • ...and 42 more