Galois and separable extensions of Tambara functors

Abstract: For a map of Tambara functors $R \rightarrow T$ we define when $T$ is separable over $R$ and if $T$ carries an action by some finite group $H$ that fixes $R$ we also define when $T$ is an $H$-Galois extension of Tambara functors. We show that flat and separable extensions are (formally) étale in the sense of Hill and we prove that Galois extensions are separable. We express Nullstellensatzian Tambara functors in the context of Galois theory.

• The paper develops an enriched categorical framework extending Galois theory to Tambara functors, linking formal étaleness with equivariant homology.
• It rigorously proves fixed point and box product isomorphisms under invertibility conditions, advancing the treatment of equivariant Hochschild homology.
• The work establishes explicit Galois and separability criteria for Tambara functors, opening new avenues in the classification of field-like structures.

Galois and Separable Extensions in the Theory of Tambara Functors

Overview

This paper develops an enriched framework for Galois and separable extensions within the category of Tambara functors, extending key structural motifs from classical and equivariant algebra to an equivariant categorical context. The theory integrates concepts such as formal étaleness, genuine Kähler differentials, and equivariant Hochschild homology, and provides new characterizations and examples of Galois extensions in this setting. Additionally, it relates these extensions to deeper classification theorems in the theory of field-like and Nullstellensatzian Tambara functors.

Fixed Point Tambara Functors and Box Products

The paper begins by extending technical tools for manipulating fixed point Tambara functors and their box products. The central result establishes that if $G$ is a finite group and $|G|$ is invertible in the coefficient rings, then for abelian groups with $G$-actions $M$ and $N$,

$$(M)^{fix} \Box (N)^{fix} \cong (M \otimes N)^{fix},$$

as $G$-Mackey functors and $G$-Tambara functors. The proofs systematically generalize prior work for cyclic groups to arbitrary finite $G$, highlighting obstructions when $|G|$ is not invertible. This structural insight is critical for later development of separability and Galois concepts in the functorial context.

Equivariant Hochschild Homology and Kähler Differentials

A comprehensive account is given of equivariant Hochschild homology for Tambara functors, paralleling the non-equivariant context. The module of genuine Kähler differentials, as introduced by Hill, is precisely related to the first Hochschild homology group. The identification

$|G|$0

persists in the equivariant setting, where $|G|$1 is the canonical augmentation ideal, and this relation is leveraged throughout the development of formal étaleness, separability, and their applications. Importantly, the twisted Leibniz rule governing derivations in the Tambara context is shown to reduce to the classical Leibniz rule on fixed-point subcategories, simplifying many computations for field-like functors with injective restrictions.

Étale and Separable Extensions

Étaleness and separability in the category of Tambara functors are defined following categorical and algebraic analogues. The main structural results include:

• If $|G|$2 is separable and flat as an $|G|$3-algebra in Tambara functors, then $|G|$4 is formally étale, i.e., the module of genuine differentials vanishes.
• If $|G|$5 is the Burnside Tambara functor, any separable commutative $|G|$6-algebra $|G|$7 must possess non-trivial idempotents, mirroring separable closure phenomena for the integers.
• Restriction and norm functors preserve separability, and the box product computations for constant and fixed-point Tambara functors are fully explicit in the presence of invertibility conditions.

A notable application is the vanishing of higher homotopy groups for equivariant Loday constructions (and thus Hochschild-like theories) when the extension is separable and flat, with precise calculations at both the free and fixed-point levels.

Galois Extensions of Tambara Functors

The concept of $|G|$8-Galois extensions is defined for $|G|$9-Tambara functors, extending the classical requirements for (unramified) Galois ring extensions. The main axioms encode:

1. An $G$0-action on the target functor via $G$1-algebra automorphisms.
2. The fixed points under $G$2 coincide with $G$3, levelwise.
3. A canonical isomorphism between $G$4 and a product of $G$5 indexed by $G$6, reflecting Galois vanishing of differentials and separation of tensor coordinates.

All $G$7-Galois extensions are shown to be separable in this setting, with explicit construction of a section of the multiplication map using the combinatorial structure of $G$8 and direct analogues of the trace and norm from classic descent theory. However, the surjectivity of the trace map can fail at non-free levels, especially in positive characteristic.

Examples and Counterexamples

• The fixed-point construction $G$9 from a $M$0-Galois field extension induces a $M$1-Galois extension of Tambara functors, with detailed analysis both in characteristic not 2 and characteristic 2.
• The paper demonstrates explicit failure to construct naive Tambara functor analogues of ramified extensions such as the Gaussian integers $M$2 via free functor constructions and box products, proving a general no-go theorem based on degree and fixed-point analysis at both levels.
• Faithfulness and flatness of $M$3 as an $M$4-module in the Galois context is established under invertibility assumptions on $M$5, employing module splitting via trace.

Nullstellensatzian and Field-Like Tambara Functors

The theoretical implications are further explored via the classification of field-like and Nullstellensatzian Tambara functors, following foundational work by Nakaoka, Wisdom, and Schuchardt-Spitz-Wisdom. The following points are established:

• Every field-like $M$6-Tambara functor is a coinduction from an algebraically closed field, and any such coinduction from a Galois extension of fields produces a Galois extension of Tambara functors.
• Nullstellensatzian functors—Tambara analogues of algebraically closed fields—arise precisely as filtered colimits of coinduced finite Galois extensions, generalizing the classical notions to the equivariant categorical situation.

Implications and Future Directions

The extension of Galois theoretic, étale, and separability concepts to the categorical setting of Tambara functors yields a robust framework bridging equivariant algebra, Mackey functor theory, and algebraic topology. The structural results suggest further exploration in several directions:

• The lack of strong local-to-global and flatness criteria highlights a need for new descent and patching techniques in the Tambara context.
• The tight interplay between coinduction, field-likeness, and algebraic closure motivates further investigation into the role of Tambara functors in equivariant stable homotopy, computational aspects of homological algebra, and algebraic geometry with symmetry.
• Additional classification efforts, particularly in torsion settings and for more general (non-abelian) groups, may clarify the boundaries between classical and equivariant phenomena.

Conclusion

By formalizing and relating Galois, separable, and étale extensions in the world of Tambara functors, this work positions equivariant algebra squarely within the reach of modern homotopy-theoretic and categorical techniques. The results open new avenues for exploiting the equivariant nature of algebraic structures, and connect seemingly disparate classification results via general functorial principles and categorical constructions (2607.04411).