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Grothendieck Duality theories -- abstract and concrete, I: pseudo-coherent finite maps

Joseph Lipman

TL;DR

The paper investigates how concrete duality constructions for the twisted inverse image functor $(-)^!$ align with abstract Grothendieck/Verdier–Deligne duality, focusing on pseudo-coherent finite maps. It constructs a concrete right adjoint $f^{flat}$ to $Rf_*$ for pseudo-coherent finite maps and a counit $\bar t_f$, yielding a realizable model of the duality data $(f^!,\tau_f)$ and enabling base-change and tensor/Hom compatibilities in the derived category setting. It also provides explicit affine/commutative-algebra translations for finite maps (e.g., when $f$ is affine, or finite locally free) and frames these constructions within an abstract dualizing structure of monoidal pseudofunctors with oriented squares. The work positions the Ideal Theorem as extending to essentially-finite-type maps of noetherian schemes with bounded-below qc cohomology, and sets the stage for subsequent treatment of smooth maps and broader geometric contexts. Overall, Part I develops the concrete realization of duality for pseudo-coherent finite maps and connects it to the abstract framework, paving the way for broader applicability in algebraic geometry.

Abstract

Grothendieck Duality -- the theory of the twisted inverse image pseudofunctor (-)^! over a suitable category of scheme-maps -- can be developed concretely, with emphasis on explicit constructions, or abstractly, with emphasis on category-theoretic considerations. It is not obvious that the two resulting theories are essentially the same. This is a semi-expository account of the connection between these approaches, a nontrivial matter involving some alluring relations, for instance among differential forms, residues and duality. In particular, it emerges that the culminating Ideal Theorem in Hartshorne's "Residues and Duality" holds for arbitrary essentially-finite-type maps of noetherian schemes and bounded-below complexes with quasi-coherent cohomology. What appears in this first part mostly concerns pseudo-coherent finite maps. The rest is being prepared.

Grothendieck Duality theories -- abstract and concrete, I: pseudo-coherent finite maps

TL;DR

The paper investigates how concrete duality constructions for the twisted inverse image functor align with abstract Grothendieck/Verdier–Deligne duality, focusing on pseudo-coherent finite maps. It constructs a concrete right adjoint to for pseudo-coherent finite maps and a counit , yielding a realizable model of the duality data and enabling base-change and tensor/Hom compatibilities in the derived category setting. It also provides explicit affine/commutative-algebra translations for finite maps (e.g., when is affine, or finite locally free) and frames these constructions within an abstract dualizing structure of monoidal pseudofunctors with oriented squares. The work positions the Ideal Theorem as extending to essentially-finite-type maps of noetherian schemes with bounded-below qc cohomology, and sets the stage for subsequent treatment of smooth maps and broader geometric contexts. Overall, Part I develops the concrete realization of duality for pseudo-coherent finite maps and connects it to the abstract framework, paving the way for broader applicability in algebraic geometry.

Abstract

Grothendieck Duality -- the theory of the twisted inverse image pseudofunctor (-)^! over a suitable category of scheme-maps -- can be developed concretely, with emphasis on explicit constructions, or abstractly, with emphasis on category-theoretic considerations. It is not obvious that the two resulting theories are essentially the same. This is a semi-expository account of the connection between these approaches, a nontrivial matter involving some alluring relations, for instance among differential forms, residues and duality. In particular, it emerges that the culminating Ideal Theorem in Hartshorne's "Residues and Duality" holds for arbitrary essentially-finite-type maps of noetherian schemes and bounded-below complexes with quasi-coherent cohomology. What appears in this first part mostly concerns pseudo-coherent finite maps. The rest is being prepared.

Paper Structure

This paper contains 2 sections, 3 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Pseudo-coherent finite maps

Key Result

Proposition 2.1.6

The functor $\bar{f\:\mkern-1mu}^{\!\mkern-1mu*}\colon\boldsymbol{\mathsf{D}}(\mkern1mu\overline{Y\mkern-1mu}\mkern1mu)\to\boldsymbol{\mathsf{D}}(X)$ induces an equivalence from $\boldsymbol{\mathsf{D}}_{\mathsf{qc}}(\overline{Y\mkern-1mu}\mkern1mu)$ to $\boldsymbol{\mathsf{D}}_{\mathsf{qc}}(X)$$($r

Theorems & Definitions (6)

  • Example 2.1.1
  • Proposition 2.1.6
  • proof
  • Corollary 2.1.8: Sheafified duality for $\bar{f}\,$
  • Corollary 2.1.9
  • proof