Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Six-Functor Formalisms

Peter Scholze

Abstract

These are lecture notes for a course in Winter 2022/23, updated and completed in October 2025. The goal of the lectures is to present some recent developments around six-functor formalisms, in particular: the abstract theory of 6-functor formalisms; the 2-category of cohomological correspondences, and resulting simplifications in the proofs of Poincaré--Verdier duality results; the relation between 6-functor formalisms and ``geometric rings''; many examples of 6-functor formalisms, both old and new.

Six-Functor Formalisms

Abstract

These are lecture notes for a course in Winter 2022/23, updated and completed in October 2025. The goal of the lectures is to present some recent developments around six-functor formalisms, in particular: the abstract theory of 6-functor formalisms; the 2-category of cohomological correspondences, and resulting simplifications in the proofs of Poincaré--Verdier duality results; the relation between 6-functor formalisms and ``geometric rings''; many examples of 6-functor formalisms, both old and new.

Paper Structure

This paper contains 24 sections, 86 theorems, 340 equations.

Table of Contents

  1. Lecture I: Introduction
  2. Lecture II: Six-Functor Formalisms
  3. Lecture III: Symmetric monoidal $\infty$-categories
  4. Lecture IV: Construction of Six-Functor Formalisms
  5. Lecture V: Poincaré Duality
  6. Lecture VI: Complements on the abstract formalism
  7. Lecture VII: Topological spaces
  8. Lecture VIII: Coherent sheaves
  9. Reminders on derived schemes
  10. Right and left adjoints to pullback
  11. Option 1: All maps are proper.
  12. Option 2: $I$ consists of the proper maps of finite Tor-dimension, $P$ consists of the open immersions.
  13. Option 3: Replace $\mathrm{Ind}(\mathrm{Perf})$ by $\mathrm{Pro}(\mathrm{Perf})$
  14. Option 4: Replace $\mathrm{Pro}(\mathrm{Perf})$ by $\mathrm{Pro}(\mathrm{Coh})$
  15. Option 5: Replace $\mathrm{Pro}(\mathrm{Coh})$ by $\mathrm{Ind}(\mathrm{Coh})$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $f: X\to Y$ be a proper map in $C$ and let $y\in Y$ with fibre $X_y=X\times_Y \{y\}$, with inclusions $i_X: X_y\to X$ and $i: \{y\}\to Y$. Take any $A\in D(X,\mathbb Z)$. Then the natural map is an isomorphism, where $(f_\ast A)_y := i^\ast f_\ast A\in D(\{y\},\mathbb Z)=D(\mathbb Z)$ is the stalk of $f_\ast A$ at $y$. More generally, for any other map $g: Y'\to Y$ with base change $f':X'=X\t

Theorems & Definitions (221)

  • Theorem 1.1: Proper Base Change
  • Theorem 1.2: Künneth Formula
  • Theorem 1.3: Projection Formula
  • Theorem 1.4: Poincaré Duality
  • Theorem 1.5: Verdier duality
  • Theorem 1.6: Verdier duality
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 211 more