Fixed-Point Tambara Functors

• Fixed-point Tambara functors are defined by assigning to each orbit G/H the H-fixed subring of a G-ring, encapsulating equivariant data.
• They systematically integrate restriction, transfer, and norm maps, enabling explicit computations in cyclic and Galois settings.
• Their construction provides concrete examples for studying étale extensions in equivariant algebra, field theory, and Kummer constructions.

A fixed-point Tambara functor is a Tambara functor obtained from a $G$-ring by assigning to each orbit $G/H$ an $H$-fixed subring and equipping these orbitwise rings with the restriction, transfer, and norm maps required by Tambara theory. In the cyclic examples most often used in the recent literature, this construction appears as the “fix-Tambara functor” $T^{\mathrm{fix}}$, while more general treatments write it as $\underline{R}$ or $\mathrm{FP}(R)$. The construction is important for two distinct reasons: it gives explicit Tambara functors directly from equivariant algebra, and it supplies a large class of examples in which norms, transfers, fixed points, coinduction, and étaleness can all be computed concretely (Lindenstrauss et al., 2023, Wisdom, 2024).

1. Tambara-theoretic setting

A Tambara functor for a finite group $G$ is the multiplicative refinement of a Mackey functor. In Strickland’s formulation, one works with the category $U_G$ of bispans

$$X \xleftarrow{p} A \xrightarrow{q} B \xrightarrow{r} Y,$$

and a Tambara functor is a product-preserving functor

$S:U_G\to \mathbf{Sets}.$

From special bispans one recovers the three structure maps attached to an equivariant map $G/H$0: restriction $G/H$1, additive transfer $G/H$2, and multiplicative norm $G/H$3. These satisfy functoriality, pullback compatibilities, and the distributive law

$G/H$4

where $G/H$5 is the distributor diagram associated to $G/H$6 and $G/H$7 (Strickland, 2012).

This formalism makes fixed-point constructions especially natural. Values on orbits $G/H$8 are subgroup-indexed, and many standard examples behave as “$G/H$9-components.” Strickland also emphasizes invariant-style constructions such as

$H$0

together with the fact that evaluation on orbits often reduces Tambara data to subgroup-fixed information (Strickland, 2012).

The semiring structure at each $H$1 is built from transfer and norm along fold maps. Thus Tambara functors are not merely orbitwise rings; they are orbitwise semirings or rings tied together by contravariant and covariant operations whose interaction encodes equivariant multiplicative algebra. Fixed-point Tambara functors occupy a distinguished place in this landscape because their orbitwise values are literally fixed subrings of a single $H$2-ring.

2. Definition of fixed-point Tambara functors

For a $H$3-ring $H$4, the fixed-point Tambara functor is defined in the cyclic $H$5-group classification by

$H$6

Restriction maps are inclusions

$H$7

while transfers and norms are the usual additive orbit-sums and multiplicative orbit-products dictated by the Tambara axioms (Wisdom, 2024).

In the cyclic prime-order setting, the étale-extension paper writes this construction more explicitly. For a commutative $H$8-ring $H$9,

$T^{\mathrm{fix}}$0

with restriction given by inclusion,

$T^{\mathrm{fix}}$1

transfer

$T^{\mathrm{fix}}$2

and norm

$T^{\mathrm{fix}}$3

where $T^{\mathrm{fix}}$4 is a generator of $T^{\mathrm{fix}}$5 (Lindenstrauss et al., 2023).

A basic structural fact is that this construction is adjoint to evaluation at the free orbit. In the $T^{\mathrm{fix}}$6 setting, the assignment $T^{\mathrm{fix}}$7 is right adjoint to evaluation at $T^{\mathrm{fix}}$8. In the later coinduction paper, the fixed-point functor $T^{\mathrm{fix}}$9 is characterized by the adjunction

$\underline{R}$0

so the bottom level $\underline{R}$1 determines maps into fixed-point Tambara functors (Wisdom, 12 May 2025).

This point of view is compatible with the older “fixed point functor” notation

$\underline{R}$2

used in the localization paper. On an orbit $\underline{R}$3, such equivariant maps identify with $\underline{R}$4-fixed elements of $\underline{R}$5, so the construction is again orbitwise fixed-point data written in a different language (Nakaoka, 2011).

3. Constant versus fixed-point constructions

The literature repeatedly contrasts fixed-point Tambara functors with constant Tambara functors. For a ring $\underline{R}$6, the constant $\underline{R}$7-Tambara functor $\underline{R}$8 has

$\underline{R}$9

for all subgroups $\mathrm{FP}(R)$0, trivial Weyl action, identity restriction, transfer given by multiplication by the index, and norm $\mathrm{FP}(R)$1 (Lindenstrauss et al., 2023).

The contrast is conceptual as well as formal. Constant Tambara functors encode the same ring at every orbit with trivial equivariance, whereas fixed-point Tambara functors encode the actual $\mathrm{FP}(R)$2-action on a ring by varying the value from $\mathrm{FP}(R)$3 at the free orbit to $\mathrm{FP}(R)$4 at $\mathrm{FP}(R)$5. This distinction is central in the étale-extension examples: the base field is represented by a constant Tambara functor, while the extension field with its Galois action is represented by a fixed-point Tambara functor (Lindenstrauss et al., 2023).

For a $\mathrm{FP}(R)$6-Galois extension $\mathrm{FP}(R)$7, the $\mathrm{FP}(R)$8-equivariant inclusion at the free orbit induces the Tambara map

$\mathrm{FP}(R)$9

Thus a classical field extension is converted into an equivariant extension between a constant object and a genuinely fixed-point object. This is the basic mechanism behind the new étale examples constructed in the paper (Lindenstrauss et al., 2023).

A related but broader observation appears in the 2025 coinduction paper: the $G$0-level is often decisive. If the bottom ring already looks coinduced, then the whole Tambara functor is coinduced. Fixed-point Tambara functors are therefore part of a larger architecture in which bottom-level equivariant algebra controls global Tambara structure (Wisdom, 12 May 2025).

4. Étale extensions from Galois and Kummer theory

The principal new family of fixed-point Tambara functors in the 2023 paper arises from field extensions with cyclic Galois action. Its central theorem states that if $G$1 is a $G$2-Galois extension of fields, then

$G$3

is $G$4-Tambara étale (Lindenstrauss et al., 2023).

The proof proceeds through the underlying Green functor. Étaleness is taken in Hill’s sense: flatness together with vanishing of the equivariant Kähler differentials. In the examples treated, the decisive computation is that the ideal

$G$5

satisfies

$G$6

Consequently $G$7, so the equivariant Kähler differentials vanish already at the Green-functor level; this is why the paper emphasizes Green étaleness and then deduces Tambara étaleness (Lindenstrauss et al., 2023).

The $G$8-argument splits into characteristic $G$9 and characteristic $U_G$0. In the Artin–Schreier case, the kernel of multiplication at the fixed level is generated by

$U_G$1

and this generator is idempotent. In the Kummer case, the relevant generator is

$U_G$2

In both cases the explicit box-product calculation forces the differential module to vanish, while projectivity over $U_G$3 gives the required flatness (Lindenstrauss et al., 2023).

Section 4 of the same paper extends the Kummer argument from $U_G$4 to arbitrary cyclic $U_G$5. Under the hypotheses that $U_G$6 contains a primitive $U_G$7-th root of unity, $U_G$8 is invertible in $U_G$9, and $$X \xleftarrow{p} A \xrightarrow{q} B \xrightarrow{r} Y,$$0 is a $$X \xleftarrow{p} A \xrightarrow{q} B \xrightarrow{r} Y,$$1-Kummer extension, the induced map [ K^