[Paper Review] Foncteurs de Mackey à réciprocité

Bruno Kahn|arXiv (Cornell University)|Oct 29, 2012

Advanced Topology and Set Theory3 citations

TL;DR

This paper introduces and studies Mackey functors with reciprocity, focusing on cohomological, additive, and topological invariance properties. It establishes that such functors satisfy strong reciprocity laws, particularly proving that cohomological Mackey functors associated with commutative algebraic groups or semi-abelian varieties satisfy the strong reciprocity property. The key contribution is a general framework for reciprocity in algebraic geometry using Mackey functors and their specialization maps.

ABSTRACT

This text was written 20 years ago, inspired by M. Somekawa's paper on K-groups attached to semi-abelian varieties (K-Theory 4 (1990), 105--119) and before Voevodsky's theory of presheaves with transfers. The reason why it only had a limited circulation will be obvious towards the end. In view of recent developments, I thought it could be useful to make it generally available.

Motivation & Objective

• To define and analyze Mackey functors with reciprocity properties in algebraic geometry.
• To establish conditions under which such functors satisfy strong reciprocity, especially for algebraic groups and cohomological functors.
• To generalize reciprocity laws from the case of algebraically closed fields to arbitrary base fields via Galois descent.
• To develop a framework for tensor products of Mackey functors with local symbols, addressing compatibility issues in specialization maps.

Proposed method

• Defines Mackey functors on affine $k$-schemes, requiring continuity under filtered colimits and compatibility with finite flat morphisms.
• Introduces key properties: cohomological (transfer-composition equals degree multiplication), additive (compatible with disjoint unions), and topological invariance (isomorphism on radicial morphisms).
• Establishes weak topological invariance for cohomological functors via Hensel's lemma and base change over separable closures.
• Uses divisor evaluation maps $a(D) = \operatorname{Cor}_{D/k} \iota_D^*(a)$ to relate elements in $A(U)$ to values in $A(k)$, especially for effective divisors.
• Applies homotopy invariance and Galois descent to extend results from algebraic closures to general base fields.
• Employs spectral sequences and hypercohomology to extend reciprocity to complexes of étale sheaves with torsion cohomology.

Experimental results

Research questions

• RQ1 Under what conditions does a Mackey functor satisfy the strong reciprocity property?
• RQ2 How do cohomological and topological invariance properties interact to imply weak topological invariance?
• RQ3 Can the reciprocity law for algebraic groups be extended from algebraically closed to arbitrary base fields?
• RQ4 What conditions ensure that the tensor product of two Mackey functors with local symbols inherits a well-defined local symbol?
• RQ5 To what extent do hypercohomology functors on the étale site satisfy strong reciprocity?

Key findings

• A cohomological, topologically invariant Mackey functor is weakly topologically invariant, as shown via Hensel's lemma and base change over separable closures.
• For any finite flat morphism $f: V \to U$ of degree $n$, the evaluation satisfies $(f^*a)(f^*D) = n a(D)$ and $a(f_*D) = (f^*a)(D)$ when $A$ is cohomological.
• If $A$ is a commutative algebraic group or a semi-abelian variety over $k$, then the associated Mackey functor satisfies the strong reciprocity property.
• The hypercohomology functor $Y \mapsto \mathbb{H}^i(Y_{\text{ét}}, C^\cdot)$ satisfies strong reciprocity for bounded above complexes $C^\cdot$ with torsion cohomology prime to $\operatorname{char}(k)$.
• The Chow group functor $Y \mapsto CH^i(X \times_k Y)$ satisfies strong reciprocity, as a special case of the hypercohomology result.
• The functor $Y \mapsto \Omega^i_{X/\mathbb{Z}}$ satisfies reciprocity but not the strong version, showing a distinction between weak and strong reciprocity.