We now come to the most intuitive definition of a motivic Tambara functor as an effective homotopy module with norms.
First recall the notion of an exponential diagram [8, Definition 7]: given finite étale morphisms \(A \xrightarrow {q} X \xrightarrow {f} Y\) in \({\mathrm {S}\mathrm {m}}_k\), the corresponding exponential diagram is
Here e is the X-morphism corresponding by adjunction to the identity \(R_f A \rightarrow R_f A\), and p is the canonical projection.
Definition 1
Let \(\mathcal C \subset _\mathrm {f\acute{e}t}{\mathrm {S}\mathrm {m}}_k\). A \(\mathcal C\)-Tambara functor of the first kind consists of an effective homotopy module \(M \in \mathbf {HI}_0(k)\), together with for each \(f: X \rightarrow Y \in \mathcal C\) finite étale a map of sets \(N_f: M(X) \rightarrow M(Y)\) such that:
-
(1)
For \(X \in \mathcal C\) we have \(N_{{\text {id}}_X} = {\text {id}}_{M(X)}\) and if \(X \xrightarrow {f} Y \xrightarrow {g} Z\) are finite étale morphisms in \(\mathcal C\), then \(N_{gf} = N_g \circ N_f\).
-
(2)
Given a cartesian square
with \(X, Y \in \mathcal C\) and p finite étale, the following diagram commutes
-
(3)
Given finite étale morphisms \(A \xrightarrow {q} X \xrightarrow {f} Y\) in \(\mathcal C\), the following diagram (induced by the corresponding exponential diagram) commutes
A morphism \(\phi : M_1 \rightarrow M_2\) of \(\mathcal C\)-Tambara functors of the first kind is a morphism of the underlying effective homotopy modules such that for every \(f: X \rightarrow Y\) finite étale, the following diagram commutes
We denote the category of \(\mathcal C\)-Tambara functors of the first kind by \(T_\mathcal {C}^1(k)\), and we write \(U_1: T_\mathcal {C}^1(k) \rightarrow \mathbf {HI}_0(k)\) for the evident forgetful functor.
Remark 2
If \(f: X \rightarrow Y \in \mathcal C\) is an isomorphism, then condition (2) with \(Y'=X'=X\), \(p=f\), \(p'=f'={\text {id}}\) implies that \(N_f = (f^*)^{-1}\). It follows thus from condition (1) that for \(f: X \rightarrow Y \in \mathcal C\) finite étale, the map \(N_f: M(X) \rightarrow M(Y)\) is invariant under automorphisms of X/Y.
Remark 3
If \(M \in T_\mathcal {C}^1(k)\) and \(X \in \mathcal C\), then the fold map \(\nabla : X \coprod X \rightarrow X\) induces a binary operation \(N_\nabla : M(X \coprod X) \simeq M(X) \times M(X) \rightarrow M(X)\) (the first isomorphism because M is a sheaf), called multiplication. This operation is commutative by Remark 2. If \(f: \emptyset \rightarrow X\) is the unique map, then \(M(\emptyset ) = *\) (since M is a sheaf) and \(N_f(*) \in M(X)\) is a unit of this multiplication on M(X) (this follows from condition (1)). Condition (3) implies that multiplication distributes over addition in M(X) (apply it to the sequence of étale maps \(X \coprod (X \coprod X) \xrightarrow {{\text {id}}\coprod \nabla } X \coprod X \xrightarrow {\nabla } X\)). For this reason we refer to condition (3) as the distributivity law. We thus see that M(X) is naturally a commutative ring.
Condition (2) implies that \(M(X \coprod X) \simeq M(X) \times M(X)\) as rings, and condition (1) then implies that for \(f: X \rightarrow Y\) finite étale, the map \(N_f: M(X) \rightarrow M(Y)\) is multiplicative.
Remark 4
If \(\mathcal C = \mathrm {FEt}{}_k\), then the above definition coincides with [8, Definition 8].
Here is a basic structural property of the category of \(\mathcal C\)-Tambara functors of the first kind.
Lemma 5
The category \(T_\mathcal {C}^1(k)\) is presentable and the forgetful functor \(U_1: T_\mathcal {C}^1(k) \rightarrow \mathbf {HI}_0(k)\) is a right adjoint.
Proof
We first construct auxiliary categories \(\mathcal D\) and \(\mathcal D'\). The objects of both \(\mathcal D\) and \(\mathcal D'\) are objects of \(\mathcal C\). For \(X, Y \in \mathcal C\), the morphisms from \(X \rightarrow Y\) in \(\mathcal D'\) are given by equivalence classes spans, i.e. diagrams \(X \xleftarrow {f} T \rightarrow Y\), where f is required to be finite étale. In other words \(\mathcal D'\) is just the homotopy 1-category of the bicategory \(\mathrm {Span}(\mathcal C, \mathrm {f\acute{e}t}, \mathrm {all})\).
The morphisms from \(X \rightarrow Y\) in \(\mathcal D'\) are given by equivalence classes of bispans, i.e. diagrams \(X \xleftarrow {f} T_1 \xleftarrow {g} T_2 \xrightarrow {p} Y\), where f and g are required to be finite étale. We shall identify two bispans if they fit into a commutative diagram
with a, b isomorphisms. If \(f: X \rightarrow Y \in \mathcal C\) then we denote the bispan \(X \xleftarrow {{\text {id}}} X \xleftarrow {{\text {id}}} X \xrightarrow {f} Y\) by \(\rho _f\). If \(f: X \rightarrow Y \in \mathcal C\) is finite étale, we denote the bispan \(X \xleftarrow {f} Y \xleftarrow {{\text {id}}} Y \xrightarrow {{\text {id}}} Y\) by \(\tau _f\) and we denote the bispan \(X \xleftarrow {{\text {id}}} X \xleftarrow {f} Y \xrightarrow {{\text {id}}} Y\) by \(\nu _f\).
Before explaining composition in \(\mathcal D\), let us explain what the category \(\mathcal D\) is supposed to do. We will have a functor \(F: \mathcal D' \rightarrow \mathcal D\) which is the identity on objects and sends \(X \xleftarrow {g} T \xrightarrow {f} Y\) to \(\rho _f \circ \tau _g\). This induces \(F^*: PSh(\mathcal D) \rightarrow PSh(\mathcal D')\). The objects in \(PSh(\mathcal D)\) are going to be “presheaves with norm and transfer” in the following sense. Let \(G: \mathbf {HI}_0(k) \rightarrow PSh(\mathcal D)\) denote the forgetful functor. Then we have a cartesian square of 1-categories
The category \(\mathbf {HI}_0(k)\) is presentable, being an accessible localization of the presentable category \(\mathcal {SH}(k)^{\text {veff}}\). The categories \(PSh(\mathcal D)\) and \(PSh(\mathcal D')\) are of course presentable. The functor \(F^*\) has a left adjoint, given by left Kan extension. The functor G also has a left adjoint; indeed it is a functor between presentable categories which preserves limits and filtered colimits (see Lemma 11 below), so the claim follows from the adjoint functor theorem [20, Corollary 5.5.2.9(2)]. It follows that \(F^*\) and G are morphisms in \(Pr^R\). Thus the square is also a pullback in \(Pr^R\) [20, Theorem 5.5.3.18], and in particular \(T_\mathcal {C}^1(k)\) is presentable and \(U_1\) is a right adjoint.
It remains to finish the construction of \(\mathcal D\). The composition in \(\mathcal D\) is determined by the following properties: (1) if \(\alpha = (X \xleftarrow {f} T_1 \xleftarrow {g} T_2 \xrightarrow {p} Y)\) is a bispan, then \(\alpha = \rho _p \nu _g \tau _f\). (2) if \(X \xrightarrow {f} Y \xrightarrow {g} Z \in \mathcal C\), then \(\rho _{gf} = \rho _g \rho _f\). If f, g are finite étale then \(\tau _{gf} = \tau _f \tau _g\) and \(\nu _{gf} = \nu _f \nu _g\). (3) The \(\tau \) and \(\nu \) morphisms satisfy the basechange law with respect to the \(\rho \) morphisms. (4) the distributivity law holds. For a more detailed construction of similar categories, see [27, Section 5, p. 24 and Proposition 6.1]. \(\square \)
Remark 6
The cartesian square (1) can be used to elucidate the nature of motivic Tambara functors of the first kind: the category is equivalent to the category of triples \((T, M, \alpha )\) where T is a presheaf on a certain bispan category \(\mathcal D\), M is an effective homotopy module, and \(\alpha \) is an isomorphism between the presheaves with finite étale transfers underlying T and M. In fact, if \(char(k) = 0\) then one may show that the functor \(\mathbf {HI}_0(k) \rightarrow PSh(\mathcal D')\) is fully faithful (use [4, Corollary 5.17] and [8, paragraph before Proposition 22]), whence so is \(T_\mathcal {C}^1(k) \rightarrow PSh(\mathcal D)\). We deduce that in this situation the category \(T_\mathcal {C}^1(k)\) has a particularly simple description: it consists of presheaves on the bispan category \(\mathcal D\) such that the underlying presheaf with finite étale transfers extends to an effective homotopy module (in particular, is a strictly homotopy invariant sheaf).
We immediately deduce the following.
Corollary 7
The category \(T_\mathcal {C}^1(k)\) has all (small) limits and colimits.
Recall now that if F is a presheaf on a category \(\mathcal D\), then F extends uniquely to a continuous presheaf on \(Pro(\mathcal D)\), the category of pro-objects. Moreover, consider the subcategory \({\mathrm {S}\mathrm {m}}^{\text {ess}}_k \subset \mathrm {S}\mathrm {ch}{}_k\) on those schemes which can be obtained as cofiltered limits of smooth k-schemes along diagrams with affine transition morphisms. Then \({\mathrm {S}\mathrm {m}}^{\text {ess}}_k\) embeds into \(Pro({\mathrm {S}\mathrm {m}}_k)\) [17, Proposition 8.13.5], and consequently for \(X \in {\mathrm {S}\mathrm {m}}^{\text {ess}}_k\) the expression F(X) makes unambigious sense, functorially in X. It follows in particular that Definition 1 makes sense more generally for \(\mathcal C \subset _\mathrm {f\acute{e}t}{\mathrm {S}\mathrm {m}}^{\text {ess}}_k\).
Let \(\mathcal C \subset \mathrm {S}\mathrm {ch}{}\). Write \(\mathcal C^{sl}\) for the subcategory of \(\mathrm {S}\mathrm {ch}{}\) on those schemes obtained as semilocalizations of schemes in \(\mathcal C\) at finitely many points. We write \(\mathcal C \subset _{\mathrm {f\acute{e}t},\mathrm {op}} \mathrm {S}\mathrm {ch}{}\) to mean that \(\mathcal C \subset _\mathrm {f\acute{e}t}\mathrm {S}\mathrm {ch}{}\) and \(\mathcal C\) is closed under passage to open subschemes.
A convenient property of the category \(T_\mathcal {C}^1(k)\) is that, in reasonable cases, it is invariant under replacing \(\mathcal C\) by \(\mathcal C^{sl}\):
Proposition 8
Let \(\mathcal C \subset _{\mathrm {f\acute{e}t},\mathrm {op}} {\mathrm {S}\mathrm {m}}_k\). Then \(\mathcal C^{sl} \subset _\mathrm {f\acute{e}t}{\mathrm {S}\mathrm {m}}^{\text {ess}}_k\) and the canonical forgetful functor \(T_{\mathcal C}^1(k) \rightarrow T_{\mathcal C^{sl}}^1(k)\) is an equivalence of categories.
In the proof, we shall make use of the unramifiedness property of homotopy modules [24, Lemma 6.4.4]: if \(X \in {\mathrm {S}\mathrm {m}}_k\) is connected and \(\emptyset \ne U \subset X\), then \(M(X) \rightarrow M(U)\) is injective. In particular, if \(\eta \) is the generic point of X, then \(M(X) \hookrightarrow M(\eta )\).
Proof
Let \(X \in \mathcal C^{sl}\) and \(f: Y \rightarrow X\) finite étale. Then X is a cofiltered limit along open immersions, so there exists a cartesian square
with \(U \in \mathcal C\), i an open immersion and \(f'\) finite étale [17, Théorèmes 8.8.2(ii) and 8.10.5(x), and Proposition 17.7.8(ii)]. It follows that \(V \in \mathcal C\), and Y is a cofiltered limit (intersection) of open subschemes of V. Since \(Y \rightarrow X\) is finite (so in particular closed and quasi-finite [29, Tags 01WM and 02NU]), Y is semilocal, and so must be a semilocalization of V. This proves the first claim.
Note that \(T^1_{\mathcal C}(k) \rightarrow T^1_{\mathcal C^{sl}}(k)\) is full. Indeed if \(M_1, M_2 \in T^1_{\mathcal C}(k)\) and \(\alpha : U_1 M_1 \rightarrow U_1 M_2\) is a morphism of the underlying homotopy modules, compatible with the norms on semilocal schemes, then it is compatible with the norms on generic points, and hence it is compatible with all norms, by Lemma 10 below.
The functor \(T^1_{\mathcal C}(k) \rightarrow T^1_{\mathcal C^{sl}}(k)\) is also faithful, since \(U_1: T^1_\mathcal {C}(k) \rightarrow \mathbf {HI}_0(k)\) and \(U_1^{sl}: T^1_{\mathcal C^{sl}}(k) \rightarrow \mathbf {HI}_0(k)\) are. It remains to show that it is essentially surjective.
Thus let \(M \in T^1_{\mathcal C^{sl}}(k)\). Let \(p: X \rightarrow Y \in \mathcal C\) be finite étale. We need to construct a norm \(N_p: M(X) \rightarrow M(Y)\). We may assume that Y is connected. Let \(\eta \) be the generic point of Y. We are given a norm map \(N_p^\eta : M(X_\eta ) \rightarrow M(\eta )\). By unramifiedness (and compatibility of norms with base change), there is at most one map \(N_p\) compatible with \(N_p^\eta \). What we need to show is that \(N_p^\eta (M(X)) \subset M(Y)\). In order to do this, by unramifiedness again, it suffices to prove this for Y replaced by the various localizations of Y at its points of codimension one. But then Y is semilocal, so the result holds by assumption.
It remains to show that these norms turn M into a \(\mathcal C\)-motivic Tambara functor of the first kind. The base change formula (i.e. condition (2) of Definition 1) is satisfied by construction. It implies using unramifiedness of M that it is enough to check conditions (1) and (3) when the base is a field (use that Weil restriction commutes with arbitrary base change [12, Proposition A.5.2(1)]), in which case they hold by assumption. \(\square \)
Remark 9
It follows that if \(\mathcal C_1, C_2 \subset _{\mathrm {f\acute{e}t},\mathrm {op}} {\mathrm {S}\mathrm {m}}_k\) such that \(\mathcal C_1^{sl} = \mathcal C_2^{sl}\), then \(T^1_{\mathcal C_1}(k) \simeq T^1_{\mathcal C_1}(k)\). This applies for example if \(\mathcal C_1 = {\mathrm {S}\mathrm {m}}_k\) and \(\mathcal C_2 = \mathrm {SmQP}{}_k\).
In the course of the proof of Proposition 8, we used the following lemma of independent interest. Let \(\mathcal C \subset \mathrm {S}\mathrm {ch}{}\). Write \(\mathcal C^{gen}\) for the full subcategory of \(\mathrm {S}\mathrm {ch}{}\) consisting of the subschemes of generic points of schemes in \(\mathcal C\).
Lemma 10
Let \(\mathcal C \subset _{\mathrm {f\acute{e}t},\mathrm {op}} {\mathrm {S}\mathrm {m}}_k\). Then \(\mathcal C^{gen} \subset _\mathrm {f\acute{e}t}{\mathrm {S}\mathrm {m}}^{\text {ess}}_k\).
Let \(F, G \in T^1_\mathcal {C}(k)\) and let \(\alpha \in {\text {Hom}}_{\mathbf {HI}_0(k)}(F,G)\) be a morphism of the underlying homotopy modules. If \(\alpha \in {\text {Hom}}_{T^1_{\mathcal {C}^{gen}}(k)}(F,G)\), then \(\alpha \in {\text {Hom}}_{T^1_{\mathcal {C}}(k)}(F,G)\). If \(U_1G \in \mathbf {HI}_0(k)[1/e]\) where e is the exponential characteristic of k, then the above criterion we may replace \(\mathcal C^{gen}\) by \(\mathcal C^{gen,perf}\), consisting of the perfect closures of objects in \(\mathcal C^{gen}\).
Proof
The first claim is proved exactly as in the proof of Lemma 8. Suppose given \(\alpha \) with the claimed properties. We need to show that, if \(p: X \rightarrow Y \in \mathcal C\) is finite étale, then the following diagram commutes
Let \(Y^{(0)}\) denote the set of generic points of Y. Then \(G(Y) \rightarrow G(Y^{(0)})\) is injective, by unramifiedness. The base change formula thus allows us to assume that \(Y \in \mathcal C^{gen}\). It follows that \(X \in \mathcal C^{gen}\), and so the diagram commutes by definition.
For the last claim, we use that if \(X \in \mathcal C^{gen}\) has perfect closure \(X'\), then \(G(X) \rightarrow G(X')\) is injective [7, Lemma 17]. \(\square \)
Corollary 7 assures us that \(T^1_\mathcal {C}(k)\) has all limits and colimits. In the final part of this section, we wish to investigate how these limits and colimits are computed. We begin with the case of homotopy modules:
Lemma 11
Let \(X \in {\mathrm {S}\mathrm {m}}^{\text {ess}}_k\). The functor \(ev_X: \mathbf {HI}_0(k) \rightarrow Ab, F \mapsto F(X)\) preserves finite limits and filtered colimits. If \(X \in {\mathrm {S}\mathrm {m}}_k^{sl}\), then the functor \(ev_X\) preserves arbitrary colimits as well, whereas if \(X \in {\mathrm {S}\mathrm {m}}_k\), it preserves arbirary limits.
Proof
By [5, Proposition 5(3)], the functor \(o: \mathbf {HI}_0(k) \rightarrow Shv_{Nis}({\mathrm {S}\mathrm {m}}_k)\) preserves limits and colimits. Taking global sections of sheaves preserves limits, so the claim about preservation of limits when \(X \in {\mathrm {S}\mathrm {m}}_k\) is clear. If \(X \in {\mathrm {S}\mathrm {m}}^{\text {ess}}_k\), say \(X = {{\,\mathrm{lim}\,}}_i X_i\) (the limit being cofiltered), then \(F(X) = {{\,\mathrm{colim}\,}}_i F(X_i)\). Since this colimit is filtered, and finite limits commute with filtered colimits, the claim about preservation of finite limits follows.
A filtered colimit of Nisnevich sheaves, computed in the category \(PSh({\mathrm {S}\mathrm {m}}_k)\), is still a Nisnevich sheaf, since Nisnevich sheaves are detected by the distinguished squares and filtered colimits commute with finite limits of sets. It follows that \(ev_X\) preserves filtered colimits (for any X). Since \(ev_X\) preserves finite limits and our categories are abelian, \(ev_X\) preserves finite sums. Now let X be semi-local. It remains to show that \(ev_X\) preserves cokernels. Let \(\alpha : F \rightarrow G \in \mathbf {HI}_0(k)\), let \(K = ker(\alpha ), C = cok(\alpha ), I = im(\alpha )\) and consider the short exact sequences \(0 \rightarrow K \rightarrow F \rightarrow I \rightarrow 0\) and \(0 \rightarrow I \rightarrow G \rightarrow C \rightarrow 0\). It is enough to show that \(ev_X\) preserves these exact sequences. Let \(0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0 \in \mathbf {HI}_0(k)\) be an exact sequence. Then \(0 \rightarrow o(F) \rightarrow o(G) \rightarrow o(H) \rightarrow 0\) is exact, and hence to show that \(0 \rightarrow F(X) \rightarrow G(X) \rightarrow H(X) \rightarrow 0\) is exact it suffices to show that \(H^1_{Nis}(X, o(F)) = 0\). This is proved in [3, last paragraph of Theorem 10.12] (if k is infinite, this follows directly from [2, Lemma 3.6], noting that o(F) has MW-transfers, e.g. by Theorem 31). \(\square \)
We can deduce the desired result.
Corollary 12
Let \(\mathcal C \subset _{\mathrm {f\acute{e}t},\mathrm {op}} {\mathrm {S}\mathrm {m}}_k\). Then \(U_1: T^1_\mathcal {C}(k) \rightarrow \mathbf {HI}_0(k)\) preserves sifted colimits.
Note that a functor between categories with small colimits (such as ours) preserves sifted colimits if and only if it preserves filtered colimits and geometric realizations [20, Corollary 5.5.8.17], which for 1-categories (such as ours) is the same as preserving filtered colimits and reflexive coequalizers. We will not use this observation.
Proof
By Lemma 8, we may replace \(\mathcal C\) by \(\mathcal C^{sl}\). Let \(F: \mathcal D \rightarrow T^1_{\mathcal C^{sl}}(k)\) be a sifted diagram, and let \(C = {{\,\mathrm{colim}\,}}_\mathcal {D} U_1F\). Note that the forgetful functor \(Ab \rightarrow Set\) preserves sifted colimits. Hence if \(X \in \mathcal C^{sl}\) then by Lemma 11 we find that \(C(X) = {{\,\mathrm{colim}\,}}_\mathcal {D} F(\bullet )(X)\), where the colimit is taken in the category of sets. In particular if \(f: X \rightarrow Y \in \mathcal C^{sl}\) is finite étale, then there is a canonical induced norm \(N_f: C(X) \rightarrow C(Y)\). It is easy to check that C, together with these norms, defines an object of \(T^1_{\mathcal C^{sl}}(k)\) which is a colimit of F. This concludes the proof. \(\square \)