# A generalized Contou-Carrère symbol and its reciprocity laws in higher dimensions

## Abstract

We generalize Contou-Carrère symbols to higher dimensions. To an -tuple where , denotes an algebra over a field we associate an element , extending the higher tame symbol for , and earlier constructions for , by Contou-Carrère, and by Osipov–Zhu. It is based on the concept of *higher commutators* for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols. -theory,

## 1. Introduction

This article concerns a higher-dimensional generalization of the Contou-Carrère symbol Reference CC94. The original symbol plays a key role in the local theory of generalized Jacobians for a relative curve, as developed by Contou-Carrère Reference CC79, Reference CC90. This theory was inspired by a conjectural picture due to Grothendieck Reference Gro01. If the relative curve is just a plain curve over a field, the symbol specializes to the tame symbol. We review this in detail along with an explicit definition below in §1.5. But in general the Contou-Carrère symbol is far richer. For example, one recovers the residue symbol in its tangent space. This aspect cannot be seen in the tame symbol.

If is a group functor, one defines its *(formal) loop group* as the group functor

The classical Contou-Carrère symbol is a non-degenerate pairing of loop groups

which can also be seen as the statement that is self-dual under Cartier duality. Our generalized symbol will be on -multilinear loops -fold

for any

### 1.1. The origins

Let us first review the classical story before Contou-Carrère’s theory. Suppose (for simplicity) that

sending a closed point ^{1} line bundle

^{1}

Other people prefer to fix an auxiliary point

Every morphism

where we consider *fppf* cohomology on the right side. This is essentially characterizing ^{2} Moreover, extensions of

This property provides a link to class field theory: As a special case of it, one obtains that every abelian finite étale covering of

There is a more precise formulation, where one replaces ^{3}

^{3}

It is standard to call this a *modulus* in this setting, but in this context it is the same thing as an effective Weil divisor. The Jacobian

Background can be found in Reference Ser88, but our exposition here follows Reference AGV71, Tome 3, Exposé XVIII and Reference BE01, Appendix, Deligne’s letter, (e).Footnote^{4}

^{4}

Recently, it has become more popular to re-interpret geometric class field theory as rank one local systems arising as pullbacks from the Jacobian. We refrain from using this slight shift of perspective in this text.

### 1.2. The relative situation

Contou-Carrère generalized this story to the situation of relative curves, i.e. the compactified curve

such that the fibers are geometrically integral of dimension one and locally projective over the base and

The present paper also concerns the relative situation, but we should first explain a few more concepts in a simpler setting.Footnote^{5}

^{5}

For the sake of completeness, we mention that Deligne Reference Del91 has also found the Contou-Carrère symbol, albeit in an analytic setting. This extends the overall picture in a different direction and would lead us too far here.

### 1.3. Local symbols and the Contou-Carrère symbol

Returning to the original formulation of class field theory for curves, i.e. back in the situation ^{6}

^{6}

That is: Approaches to the global class field theory of curves which do not rely on the Jacobian (there are several ways to do this).

In terms of the idèle class group, the choice of a modulus *local symbols*. The formalism of local symbols extends beyond the mere application in class field theory to all commutative algebraic ^{7}

^{7}

This theory has since found a new formulation in terms of reciprocity sheaves Reference IR17, Reference KSY16 or more broadly motives with modulus.

For example, the tame symbol is a local contribution which arises in the context of Kummer cyclic coverings. These abelian extensions arise as the pullback along an isogeny

This suggests the existence of a local analogue of the entire story, where the roles of

so that one can think of

Of course, one can choose a local coordinate and obtain (non-canonical) isomorphisms

The analogy to the loop group construction in Equation Equation 1 is apparent.

Before we continue, let us recall that these (formal) local contributions admit a class field theory in their own right, known as local class field theory.

### 1.4. Duality formulation of local class field theory

Let us first look at the original local theory originating from arithmetic. Suppose

is non-degenerate for any

is the

The same is true if ^{8}

^{8}

The story is entirely analogous to what happens in geometric class field theory, where

Let us now discuss a generalization of this which is vital for understanding the deeper motivations for the present paper.

The above duality formulation of local class field theory can be generalized to

There are more

A duality formulation of class field theory as in Equation Equation 6 remains intact also in this broader setting, but the cohomological dimension increases from

Letting

where the map is the norm residue isomorphism. We observe two key facts: (1) as the cohomological dimension increases, the duality moves to higher homological degrees, and (2) the role of

### 1.5. Back to the Contou-Carrère symbol

The duality considerations in §1.4 were only on the level of Galois cohomology, or the étale topos if you will. They are not geometric. Despite the formal similarity to Poincáre duality, the underlying scheme is just

We return to the situation of a relative curve. The *Contou–Carrère symbol* is a non-degenerate pairing on the loop group of

It can be given by an explicit formula. Using a presentation

for suitable

We can directly connect this to the local class field theory story of §1.4. If we evaluate the Contou–Carrère symbol on a field

sending

Here we exploit that since the fraction in the big brackets has degree zero, its evaluation at zero is possible and non-zero. This expression is known as the *tame symbol*. Its relation to local class field theory is as follows: Taking

can, through the norm residue isomorphism (as in Equation Equation 9) be realized as a quotient of the natural pairing in Milnor

and along with the boundary map

the composition of maps in the top row is given by the same formula as in Equation Equation 13. This shows that the duality maps which occur in local class field theory are at least close to the ones realized by the tame symbol; and thus are reasonable to generalize in some way to the Contou–Carrère symbol. We also get a strong hint of the relevance of

The boundary map

Here

In these low degrees there is no difference between Quillen

It turns out that this description generalizes without any problem to the *higher tame symbol*. Its role in higher-dimensional class field theory of schemes (as provided by Parshin Reference Par78, Reference Par84, Reference PF99 and Kato Reference Kat79, Reference Kat83, Kato–Saito Reference KS86) is analogous to the classical tame symbol. Its reciprocity laws have the same formal shape as reciprocity lawsFootnote^{9} for rational

For the higher tame symbol, one obtains the same object irrespective of whether one uses Milnor

which is just the

### 1.6. Central extensions

On the other hand, this approach also has a drawback: Going from Equation Equation 4 to Equation Equation 5 we chose a local coordinate. In other words, we were using Cohen’s Structure Theorem, telling us that an equicharacteristic discrete valuation field is always isomorphic to a Laurent series field,

where ^{11} Translated to the Contou–Carrère symbol, i.e. to abstract loop group functors

this suggests that our constructions should really be invariant under all ring automorphisms of ^{12} This property indeed holds for the original Contou–Carrère symbol, but note that it is not at all obvious from the complicated formula in Equation Equation 11. This suggests to look for a definition of the Contou-Carrère symbol (as well as its higher analogues) where this invariance is automatic by construction.

Tate in his famous paper Reference Tat68 had asked a related question: Suppose

is a clear candidate for a definition, it suffers from the same problem of depending on the isomorphism

holds for the choice

Next, let us explain how Tate’s solution works since this is also the foundation for our second construction of the higher symbol.

### 1.7. Tate spaces

Let us briefly recall Tate’s idea in modern terms: The ingredients for our local symbols can always be written as an ind-pro limit of finite-dimensional

(for any uniformizer ^{13} He manages to express the residue as a certain commutator of endomorphisms of these ind-pro objects. Since the ind-pro structure on

^{13}

These have since become known as *Tate vector spaces*. Alternatively (but equivalently), one can work in the setting of locally linearly compact topological

The papers Reference ADCK89, Reference APR04 now recover the tame symbol by studying the corresponding central extension of groups, i.e. a group

In fact, the Lie algebra

A key point of the present paper will be to explain how this approach is compatible with the ideas about

In order to treat such objects “by induction” in the number of loops *Tate categories*, Reference Pre11, Reference BGW16c. One then finds that the correct analogue of the group ^{14}

^{14}

This also works on the Lie algebra level. A Lie algebra

The two-dimensional tame symbol and its reciprocity law were set up by Osipov and Osipov–Zhu Reference Osi05, Reference OZ11. Osipov–Zhu also constructed a two-dimensional Contou-Carrère symbol using this method Reference OZ16 and showed its reciprocity law on surfaces. They also showed how the residue symbol for

This will also work and we pursue this in §5. In some sense it is more general since it really only relies on the iterated ind-pro structures.

### 1.8. Our approach through homotopy theory

A central part of this paper is devoted to establishing a clear connection between these two ideas. To this end, we need to work with Quillen

The boundary maps *Idea 1* really come from maps between spectra,Footnote^{15} e.g., using the boundary map of the localization sequence on the level of spectra,

taking

The dotted arrow does not quite exist because we ignored

Next, in Reference BGW18b, Theorem 1.4 (2) we showed that

where the plus superscript refers to the plus construction. This is an analogue of Quillen’s construction of