TRANSFER FUNCTORS ON k-ALGEBRAS

Charles Weibel

Dedicated to Wolmer Vasconcelos on his 65^{th} birthday

Abstract. We introduce and study the notion of a transfer functor for normal algebras over a field, i.e., a functor equipped with transfer maps for finite integral extensions. Any Hecke functor (such as a Galois module) induces a transfer functor, and conversely.

In this paper, we introduce the notion of a transfer functor F for normal algebras over a field k. It is a modification of Voevodsky’s notion of a presheaf with transfers, taking advantage of the fact (0.2) that every correspondence may be normalized, and is inspired by the work of Suslin and Voevodsky [SV] [SV1]. The restriction of transfer functors to smooth algebras forms a special case of Voevodsky’s theory of presheaves with transfer, which is developed in [SV1], [V] and [MVW].

If k has characteristic zero, a transfer functor consists of a covariant functor F, from the category Norm_{k} of normal k-algebras of finite type to Abelian groups, together with contravariant “transfer” maps j^{T} : F (A′) → F (A) associated to finite extensions j : A′ ⊇ A, subject to certain axioms which go back to Mackey’s study [Ma] of group representations. The axioms allow us to compose the covariant and contravariant maps.

The forgetful functor F (A) = A and the units functor U(A) = A^{×} are the prototype transfer functors, upon which other examples are based. Their respective transfer maps are the classical trace and norm (see 6.1 and 6.2). Hecke functors for the absolute Galois group, such as Galois modules, form another family of examples (see 6.4). Transfer functors F satisfying the “homotopy invariance” condition F (A) = F (A[t]) form the building blocks of Voevodsky’s triangulated category of motives [V].

An equivalent way to define transfer functors is to make them additive functors from a certain additive category Cor_{k} to abelian groups, where Cor_{k} is concocted so as to contain both Norm_{k} and (in characteristic zero) the dual of the category of finite extensions B ⊃ A of normal domains. This is the approach we shall follow. Thus for us a transfer functor will be an additive functor from Cor_{k} to abelian groups.

Except for the composition law, it is easy to define the category Cor_{k}. The objects of Cor_{k} are the normal k-algebras of finite type, and the algebra B_{1} ×B_{2} is the categorical product of B_{1} and B_{2}. Morphisms are integer combinations of elementary correspondences; we write Cor_{k}(A, B) for the group of morphisms from A to B in Cor_{k}. When A and B are smooth, our Cor_{k}(A, B) is the group Cor_{k}(Spec B, Spec A) of [MVW].

1991 Mathematics Subject Classification. Primary 19E15; Secondary 14F42.

Weibel was partially supported by NSF and NSA grants.

Definition 0.1. If B is a normal domain, an elementary correspondence from A to B is a prime ideal P in A ⊗_{k} B such that P ∩ B = 0 and the inclusion B ⊆ (A ⊗_{k} B)/P is finite. If B = ∏ B_{i} is a finite product of normal domains, an elementary correspondence from A to B is a prime ideal P in A ⊗_{k} B such that, for some i, P ∩ B_{i} = 0 and the inclusion B_{i} ⊆ A ⊗_{k} B/P is finite. When P is understood, we will say that the quotient B′ = A ⊗_{k} B/P is the elementary correspondence. A finite correspondence is a ℤ-linear sum of elementary correspondences.

If char(k) = 0, Cor_{k}(A, B) is the group of all finite correspondences from A to B. Note that Cor_{k}(A, ∏ B_{i}) = ⊕Cor_{k}(A, B_{i}) by construction.

If char(k) = p > 0, Cor_{k}(A, B) is defined to be the subgroup of all finite correspondences which are universally integral. Being universally integral is a mild technical restriction; if [P] is an elementary correspondence then p^{n}[P] is universally integral for some n. (See 5.6 below.)

There are two distinguished kinds of elementary correspondences. To a ring homomorphism f : A → B (with B a domain) we associate the kernel P of A ⊗_{k} B → B, and sometimes write [f] for its class in Cor_{k}(A, B); this association is compatible with composition ([gf] = [g] ∘ [f]) and induces the embedding of the category Norm_{k} into Cor_{k} that we mentioned above. To a finite inclusion j : B′ ⊇ B of domains, we associate the kernel of B′ ⊗_{k} B → B′. This will induce the embedding of the dual of the category of finite extensions into Cor_{k}. Therefore if F is a transfer functor and char(k) = 0, both f : F (A) → F (B) and j^{T} : F (B′) → F (B) are defined.

If P is any elementary correspondence from A to B, and we write B_{1} for the integral closure of the domain A⊗_{k}B/P, then P will be the composition in Cor_{k} of the canonical homomorphism f : A → B_{1} and j : B_{1} ⊇ B (see 3.2.1). Thus, for any transfer functor F, the map F (A) → F (B) induced by P is just the composition:

(0.2)  F (A) ⟶^{f} F (B_{1}) ⟶^{j^{T}} F (B).

Theorem 0.3. There is a unique associative composition Cor(A, B) ⊗ Cor(B, C) → Cor(A, C) satisfying the following axioms.

(1) If j : B ⊂ B_{1} is a finite inclusion of normal domains, then the composition j^{T} ∘ j : B → B_{1} → B is d times the identity of B, where d is the degree of j.

(2) Let B′ be an elementary correspondence from A to B such that B ⊆ B′ is flat, and let f : B → C be a homomorphism of normal domains. Then [f] ∘ [B′] is the sum ∑_{i} λ_{i}[P_{i}]. Here P_{1}, ..., P_{r} denote the inverse images in A ⊗_{k} C of the minimal primes of B′ ⊗_{B} C and λ_{i} denotes the length of (B′ ⊗_{B} C)_{P_{i}}.

(3) If S ⊂ B is multiplicatively closed, then Cor_{k}(A, B) → Cor_{k}(A, S^{−1}B) is the canonical injection sending [P] to [S^{−1}P].

Sections 1 and 2 describe the restriction of a transfer functor to finite field extensions of k. This recovers a well-known construction, that of Hecke functors for the Galois group G = Gal(k̄/k). In this setting, finite correspondences are called Hecke operators, or cohomological Mackey functors in the topological literature. I am grateful to Gaunce Lewis for pointing me in this expositional direction, and for sharing his notes [Lew].

For expositional purposes, we defer the description of composition to §3 and the proof of theorem 0.3 to §4 when char(k) = 0; the modification needed in finite characteristic is given in section 5. Examples are given in section 6.

§1. G-correspondences and Hecke functors

Let G be a group. Our first goal is to construct an additive self-dual category Cor_{G} containing the category G−sets_{f} of finite G-sets. This material is scattered through the literature of analytic number theory and topology, and is not new.

By a (formal) multi-valued function f from a set X to a set Y we mean a function from X to ℤ[Y], the free abelian group on the elements of Y. Multi-valued functions are composed in the usual way: if f(x) = ∑ n_{i}[y_{i}] then (g∘f)(x) = ∑ n_{i}g(y_{i}). Composition is associative, with identity functions 1_{X}(x) = [x].

If X and Y are G-sets, f is called equivariant if f(γ · x) = γ · f(x) for all γ ∈ G. Let Cor_{G}(X, Y) denote the abelian group of equivariant multi-valued functions from X to Y. Since the composition of equivariant multi-valued functions is easily seen to be equivariant, we get a category.

Definition 1.1. Let Cor_{G} denote the category of finite G-sets and equivariant multivalued functions. It is easy to see that this is an additive category, with disjoint union of G-sets as the direct sum operation.

A Hecke functor M is a contravariant additive functor from Cor_{G} to abelian groups, or more generally to any additive category A.

Any G-map f: X → Y is an equivariant single-valued function. In this way, we may regard Hom_{G}(X, Y) as a subset of Cor_{G}(X, Y). Note that if f and g are single-valued G-maps then g ∘ f is the usual composition of G-maps. In this way, we get a natural embedding of the category of finite G-sets into Cor_{G}.

Correspondences 1.2. Here is an equivalent description of Cor_{G}. A G-correspondence from X to Y (both G-sets) is an element of the free abelian group on the set of finite G-orbits Γ in X × Y. The group of all G-correspondences from X to Y is isomorphic to the group Cor_{G}(X, Y) by the following rule. To each G-correspondence ∑ n_{i}Γ_{i}, we associate the equivariant multi-valued function f(x) = ∑ n_{i}[y_{ij}], where the sum is over all y_{ij} such that (x, y_{ij}) ∈ Γ_{i}.

If f and g are G-correspondences from X to Y, and from Y to Z, respectively, we define their composition g ∘ f to be the G-correspondence associated to the composition of multi-valued functions. This definition ensures that the category Cor_{G} is equivalent to the category of finite G-sets and G-correspondences.

It will be useful to have an orbit-theoretic construction of composition. Given a Gmap p : T → X ×Y, we write [T] = [T]_{p} for the sum ∑ n_{i}[Γ_{i}] over all orbits Γ_{i} ∈ X ×Y, where n_{i} is the cardinality of p^{−1}(γ), γ ∈ Γ_{i}. Since n_{i} = |p^{−1}Γ_{i}|/|Γ_{i}|, the element [T] is independent of the choice of γ.

We say that a G-correspondence ∑ n_{i}Γ_{i} is effective if all the n_{i} are non-negative. The composition of two effective G-correspondences Γ_{1} and Γ_{2} is:

(1.2.1)  Γ_{2} ∘ Γ_{1} = [Γ_{12}], Γ_{12} = (Γ_{1} × Z) ∩ (X × Γ_{2}).

By additivity, a Hecke functor is uniquely determined by its restriction to the full subcategory of Cor_{G} on the orbits G/H. This subcategory is called the Hecke category by Yoshida in [Y], following its popularization in [Sh]. Correspondences between orbits have the following interpretation.

Example 1.3. Let H_{1}, H_{2} ⊆ G be subgroups of finite index. Then Cor_{G}(G/H_{1}, G/H_{2}) is the free abelian group on the set of all double cosets H_{1}aH_{2} in G. This is because the set of G-orbits in G/H_{1} × G/H_{2} is in 1–1 correspondence with the set of double cosets H_{1}aH_{2} in G, by the rule that sends the orbit of (x_{1}H_{1}, x_{2}H_{2}) to H_{1}(x_{1}^{−1}x_{2})H_{2}.

Historical Remark 1.3.1. In the 1959 paper [Sh1] (see [Sh, 3.1]), Shimura defined the Hecke ring with respect to H, using a partial multiplication · on double cosets H\G/H. If G/H is infinite, there is a commensurability restriction on the double cosets. If G/H is finite, the Hecke ring with respect to H is just the endomorphism ring Cor_{G}(G/H, G/H), because it is easy to see from (3.1.1) of [Sh] that Shimura’s product ‘·’ is the composition law in Cor_{G}.

Shimura’s definition was abstracted from a construction of Hecke [H], in which G = SL_{2}(ℝ) and Hecke only considered subgroups commensurate with SL_{2}(ℤ). In fact, one of the main thrusts of [Sh] is to connect the action of the Hecke rings with the zeta-functions associated with modular forms.

Independently in 1971, Green introduced the notion of a (cohomological) G-functor in [Gr]. Yoshida proved in [Y] that cohomological G-functors are the same as Hecke functors, i.e., additive functors on Cor_{G}. Central to Green’s definition was the Mackey axiom for subgroups H, K ⊂ L ⊂ G that in our language describes the composition G/H → G/L → G/K as the sum over double cosets HγK of composites G/H ≅ G/H^{γ} → G/(H^{γ} ∩ K) → G/K. This formula is a special case of (1.2.1).

Example 1.4. Let M be a left G-module. It is easy to see from the definitions that M(G/H) = M^{H} is a Hecke functor; if H ⊂ K, the “transfer” map M^{H} → M^{K} associated to G/H → G/K is m ↦ ∑_{γ∈K/H} γ(m).

More generally, the Hecke algebras act on the group cohomology of M, i.e., that each H^{n}(G/H) ↦ H^{n}(H; M) is a Hecke functor. This was first shown by Rhie and Whaples, in a modular setting. (See [RW] and [Sh, §8].) This result has been independently discovered by many people, including Cline, Parshall and Scott [CPS], Green [Gr] and Dress [D].

Returning to the general theory, note that the transpose τ: X × Y → Y × X gives a natural isomorphism between Cor_{G}(X, Y) and Cor_{G}(Y, X). By inspection of (1.2.1), the transpose commutes with the composition pairing: τ([Γ_{2}]∘[Γ_{1}]) = τ[Γ_{12}] = [τΓ_{1}]∘[τΓ_{2}]. Thus:

Corollary 1.5. The transpose τ provides an isomorphism between Cor_{G} and its opposite category.

Lemma 1.6. For every effective G-correspondence Γ from X to Y there is a G-set T and G-maps X ⟵^{p} T ⟶^{f} Y such that Γ factors in Cor_{G} as the composition

X ⟶^{[τp]} T ⟶^{f} Y.

Proof. If Γ ⊂ X × Y is an orbit, we may take T = Γ and let p and f be the projections. Since τp ∈ Cor_{G}(X, Γ) is the multi-valued function x ↦ ∑_{(x,y)∈Γ}[(x, y)], it is clear that f ∘ τp = g. The general case follows by taking disjoint unions of orbits.

Corollary 1.7. Given G-maps f_{i}: Y_{i} → X (i = 1, 2), let p_{i}: Y_{1} ×_{X} Y_{2} → Y_{i} be the projections from the pullback. Then the following diagram commutes in Cor_{G}:

Y_{1} ×_{X} Y_{2} ⟶^{[p_{2}]} Y_{2}
[τp_{1}]↑  ↑^{[τf_{2}]}
Y_{1}  ⟶^{[f_{1}]} X.

Proof. The pullback T = Y_{1}×_{X} Y_{2} is naturally a G-subset of Y_{1}×Y_{2}. Its correspondence [T] ∈ Cor_{G}(Y_{1}, Y_{2}) is effective, and equals the composition Y_{1} → T → Y_{2} by the lemma. For the composition [τf_{2}] ∘ [f_{1}] we obtain the set

Γ_{12} = {(y_{1}, x, y_{2}) ∈ Y_{1} × X × Y_{2}|f_{1}(y_{1}) = x = f_{2}(y_{2})}.

Clearly, Γ_{12} is isomorphic to the pullback T, so this composition is also [T].

We are now ready to connect the above construction with Mackey functors on finite G-sets. The following definition is due to Andreas Dress [D].

Definition 1.8. A Mackey functor M = (M_{∗}, M^{∗}) from G−sets_{f} to a category A consists of a rule which associates to every finite G-set X an object M(X) of A such that: (a) M sends finite coproducts to products; (b) M extends to both a covariant functor M_{∗} and a contravariant functor M^{∗}, and (c) for every pair of G-maps f_{i}: Y_{i} → X the following diagram commutes in A:

M(Y_{1} ×_{X} Y_{2}) ⟶^{pr_{2∗}} M(Y_{2})
p_{1}^{∗}↑  ↑^{f_{2}^{∗}}
M(Y_{1})  ⟶^{f_{1∗}} M(X).

If A is an additive category, a Mackey functor M: G−sets_{f} → A is additive if for every pair of G-maps f, g: X → Y the fold map ∇: X ∐ X → X satisfies:

(1) f_{∗} + g_{∗} is the composite M(X) ⟶^{∇^{∗}} M(X ∐ X) \underrightarrow{(f ∐ g)_{∗}} M(Y);

(2) f^{∗} + g^{∗} is the composite M(Y) \underrightarrow{(f ∐ g)^{∗}} M(X ∐ X) ⟶^{∇_{∗}} M(X).

Note that composition with an additive functor F : A → A′ sends additive Mackey functors to additive Mackey functors. Combining the above results, we see that we have proven:

Proposition 1.9. The embedding G−sets_{f} ⊂ Cor_{G} and its transpose G−sets_{f}^{op} ⊂ Cor_{G} form an additive Mackey functor.

Moreover, there is a 1–1 correspondence between additive Mackey functors and Hecke functors M : Cor_{G} → A.

Remark 1.9.1. Lindner [Li] has constructed a category Sp(G−sets_{f}) of “spans” in G−sets_{f}, and proved that it is universal in the sense that the category of Mackey functors from G−sets_{f} to A is isomorphic to the category of all functors from Sp(G−sets_{f}) to A which preserve finite products. Lewis has observed in [Lew] that the category Cor_{G} is a quotient category of Sp(G−sets_{f}); this gives another way to see that every Hecke functor is a Mackey functor.

§2. Galois theory and Hecke operators

When the group G is the absolute Galois group of k, we may translate the language into field theory. The category Cor_{G} translates into the zero-dimensional subcategory of the category Cor_{k} of 0.1, and composition takes a particularly nice form here.

If E′ and E′′ are finite-dimensional algebras over a field k, then E′ ⊗_{k} E′′ is a finite-dimensional E′′-algebra. A well-known problem of the 1950’s was to describe the decomposition E′ ⊗_{k} E′′ = ∏ E_{i} when E′ and E′′ are fields. This was essentially solved by Grothendieck’s Galois theory [Mi, p.43], which states that there is an equivalence between the dual category of finite separable k-algebras and discrete finite G-sets, where G is the Galois group of k̄/k. Under this correspondence, the finite separable field extension E = k̄^{H} corresponds to the G-set G/H.

Lemma 2.1. Let K be a Galois extension of k with Galois group G = Gal(K/k). If E′ and E′′ are subfields of K, finite over k, the decomposition E′ ⊗_{k} E′′ ≅ ∏ E_{i} is determined as follows.

Let H′ and H′′ be the subgroups of G corresponding to E′ and E′′, and write the finite G-set X = G/H′ × G/H′′ as a union of orbits, X ≅ ∐ G/H_{i}. Then E_{i} = k̄^{H_{i}}.

Proof. The product of separable extensions is separable, so E′ ⊗_{k} E′′ is finite separable, and hence a finite product of fields. To determine the decomposition, note that the coproduct E′ ⊗_{k} E′′ corresponds to the product G-set X under the Galois correspondence.

Example 2.1.1. Let K be the splitting field of x^{3} −2 over k = ℚ, and set E = ℚ(\sqrt[3]{2}). Then E ⊗_{ℚ} E ≅ E × K. To see this, note that Gal(K/k) is the dihedral group D_{3}, and E = K^{H} for a subgroup H of order 2. Since G/H × G/H ≅ G/H ∐ G as a G-set, the result follows from lemma 2.1.

If E′ and E′′ are finite over k, we define the group of correspondence Cor_{k}(E′, E′′) to be the free abelian group on the finite set of quotient fields (or equivalently, prime ideals) of the finite-dimensional algebra E′ ⊗_{k} E′′. This definition also makes sense if E′ and E′′ are finite-dimensional algebras. We shall sometimes write 1_{E} for the canonical element [E] of Cor_{k}(E, E), because it is the identity for the composition 2.2 below.

If both E′ and E′′ are separable over k, then Cor_{k}(E′, E′′) is the free abelian group on the E_{i} described in lemma 2.1. In the inseparable case, let i′ : E′_{s} ⊆ E and i′′ : E′′_{s} ⊆ E′′ denote the respective maximal separable subfields; then Cor_{k}(E′, E′′) ≅ Cor_{k}(E′_{s}, E′′_{s}), and 2.1 describes the generators of the right hand side.

More generally, suppose that E′ and E′′ are finitely generated field extensions of k. In this case, we define Cor_{k}(E′, E′′) to be the free abelian group on the set of quotient fields of E′ ⊗_{k} E′′ which are finite over E′′.

To define composition of correspondences, it is useful to introduce the notation [M] for the correspondence corresponding to an E′ ⊗_{k} E′′-module M of finite length. By definition, [M] is the sum ∑_{i} λ_{i}[E_{i}], where λ_{i} is E_{i}-primary length of M, i.e., the number of factors isomorphic to E_{i} in a composition series for M. If either E′ or E′′ is separable over k then E′ ⊗_{k} E′′ = ∏ E_{i} and M = ⊕M_{i}, so λ_{i} = dim_{E_{i}}(M_{i}).

Definition 2.2 (Composition). Suppose that E′, E′′ and E′′′ are finite-dimensional over k. Let K_{1} be a quotient field of E′ ⊗_{k} E′′, and K_{2} a quotient field of E′′ ⊗_{k} E′′′. Then K_{1} ⊗_{E′′} K_{2} is an E′ ⊗_{k} E′′′ module, and we define the composition K_{2} ∘ K_{1} of K_{1} and K_{2} to be the class [K_{1} ⊗_{E′′} K_{2}] in Cor_{k}(E′, E′′′).

The composition of arbitrary correspondences is defined so as to be bilinear. That is, (∑ m_{i}K_{1i}) ∘ (∑ n_{j}K_{2j}) = ∑ m_{i}n_{j}K_{2j} ∘ K_{1i}.

If E′, E′′ and E′′ are arbitrary field extensions, and we assume that K_{1} = E′⊗_{k}E′′/P_{1} and K_{2} = E′′ ⊗_{k} E′′′/P_{2} are finite field extensions of E′′ and E′′′, respectively, then K_{1} ⊗_{E′′} K_{2} is finite over E′′′ and again we define [K′] ∘ [K′′] to be the class [K_{1} ⊗_{E′′} K_{2}] in Cor_{k}(E′, E′′′).

It is clear from the definition that if K is a correspondence from E to F then we have 1_{F} ∘ K = K ∘ 1_{E} = K.

Example 2.2.1. Any field extension i : E → F determines an element [i] in Cor_{k}(E, F) corresponding to the quotient E⊗_{k}F → F sending x⊗y to i(x)·y. Since [i_{2}i_{1}] = [i_{2}]∘[i_{1}], the composition of field extensions agrees with the composition in 2.2. If [F : E] is finite, the analogous map F ⊗_{k} E → F (x ⊗ y ↦ x · i(y)) determines an element [i^{T}] of Cor_{k}(F, E). Again, [(i_{2}i_{1})^{T}] = [i_{2}^{T}] ∘ [i_{1}^{T}], so the composition of finite extensions agree with composition under the contravariant operation j ↦ j^{T}.

If j : F_{0} ⊂ F is finite, the image F′ of E ⊗_{k} F_{0} in F is a subfield. If d = [F : F′] then [j^{T}]∘[i] = d[F′] in Cor_{k}(E, F_{0}). Setting F_{0} = i(E), we obtain the formula [i^{T}]∘[i] = d·1_{E} in Cor_{k}(E, E), d = [F : E].

Example 2.2.2. Let ℓ be a finite Galois extension of k. Then the endomorphism ring Cor_{k}(ℓ, ℓ) is just the group ring ℤ[G] of G = Gal(ℓ/k). This follows from 2.2.1 since the automorphisms g : ℓ → ℓ in G from a ℤ-basis of Cor_{k}(ℓ, ℓ) by 2.1.

The following result verifying axiom 0.3(2) is elementary, and immediate from 2.2.

Lemma 2.3. If F′ is a quotient field of E ⊗_{k} F, finite over F, then [F′] ∈ Cor_{k}(E, F) is the composite [j^{T}] ∘ [i], where i : E ⊆ F′ and j : F ⊆ F′ are the evident inclusions.

If j_{E} : E ⊂ E′ is a finite inclusion and i : E → F a field extension, then E′ ⊗_{E} F is a finite product ∏ A_{α} of Artin local rings. If A_{α} has residue field F_{α} and length λ_{α}, then in Cor_{k}(E′, F) we have [i] ∘ [j^{T}] = ∑ λ_{α}[F_{α}].

E′ ⟶^{(λ_{1},...)} ∏ F_{α}
j_{E}^{T} ↓  ↓^{j_{F}^{T}}
E ⟶_{i} F.

Example 2.3.1. If E′ is separable over E then λ_{α} = 1 for all α and we obtain the analogue of the Mackey condition 1.7. If F = E′ is a Galois extension of E with group G, the formula simplifies to [i] ∘ [i^{T}] = ∑_{g∈G}[g] in Cor_{k}(F, F). At the other extreme, if F is purely inseparable over E of degree p^{e} then [i] ∘ [i^{T}] = p^{e} · 1_{F} in Cor_{k}(F, F).

Lemma 2.4. Composition is associative, and the (commutative) finite reduced k-algebras form the objects of an additive category Cor_{k}^{0}, whose Hom-groups are given by Cor_{k}.

If k is a perfect field, Cor_{k}^{0} is equivalent to the opposite category of Cor_{G}, and a covariant additive functor on Cor_{k}^{0} is the same thing as a Hecke functor on the Galois group G = Gal(k̄/k).

Proof. The verification of associativity and additivity is routine, using 2.2 and 2.3, and is left to the reader. Now assume that k is perfect. The Galois correspondence of 2.1 gives a natural isomorphism between Cor_{k}(E′, E′′) and Cor_{G}(G/H′, G/H′′). Therefore we may transport the composition of G-correspondences to this setting, and formula (1.2.1) translates exactly into formula 2.2. This reproves associativity, and shows that the Cor_{k}(E′, E′′) form the Hom-sets of an additive category which is dual to Cor_{G}. □

Example 2.4.1. Suppose that k is perfect, and that M is a Galois module, i.e., a discrete module for Gal(k̄/k). Translating 1.4 into the current setting, we see that the étale cohomology groups E ↦ H_{et}^{n}(E, M) are Hecke functors, i.e., additive covariant functors from Cor_{k}^{0} to abelian groups.

If k is not perfect, purely inseparable extensions complicate composition in the category Cor_{k}^{0} of finite reduced algebras. If i : E_{s} ⊆ E and j : F_{s} ⊆ F are the maximal separable subfields, then i^{T} and j make Cor(E_{s}, F) and Cor(E, F_{s}) into subgroups of finite (p-primary) index in Cor(E, F).

Imperfect fields 2.5. Suppose that k is not perfect, and let G denote the absolute Galois group of k. Then the inclusion Cor_{G} ⊂ Cor_{k}^{0} has a right adjoint, sending a field E to E_{s}, the separable closure of k in E. That is, Cor_{k}(ℓ, E_{s}) ≅ Cor_{k}(ℓ, E) for all E. Hence any Hecke functor M has a canonical Kan extension to an additive covariant functor on Cor_{k}^{0}, determined by the formula M(E) = M(E_{s}).

Example 2.5.1. If k is not perfect, and M is a Galois module, the canonical maps H_{et}^{n}(E_{s}, M) → H_{et}^{n}(E, M) are isomorphisms. Hence E ↦ H_{et}^{n}(E, M) is the Kan extension of a Hecke functor to Cor_{k}^{0}.

Example 2.5.2. The underlying abelian group is a covariant additive functor on Cor_{k}^{0}, where the transfer map j^{T} : E′ → E is the trace map. This follows from standard properties of the trace, and is also immediate from 1.4 and 2.4 when k is perfect. If k is not perfect, it is not the Kan extension of a Hecke functor in the sense of 2.5.

Proposition 2.6. The composition ∘ of 2.2 is the unique associative composition satisfying axioms (1) and (2) of 0.3.

Proof. The composition ∘ is associative and satisfies the axioms by 2.4, 2.2.1 and 2.3.

Conversely, suppose given a composition □ satisfying the axioms. By axiom (2) and 2.3, we have [j]□ c = [j] ∘ c for every correspondence c and every homomorphism j.

We claim that i^{T} □[K′] must also equal [i^{T}]∘[K′] for every finite i : E′ ⊃ E and every [K′] in Cor_{k}(K, E′). To see this, choose a splitting field j : E ⊂ F of E′ over E. Axiom (2) shows that [j]□[i^{T}] = [j] ∘ [i^{T}] has the form ∑[f_{α}] for various homomorphisms f_{α} : E′ → F. By associativity,

[j] ∘ ([i^{T}]□[K′]) = [j]□[i^{T}]□[K′] = ∑[f_{α}]□[K′] = ∑[f_{α}] ∘ [K′] = [j] ∘ [i^{T}] ∘ [K′].

Since [j]∘ is a monomorphism in Cor_{k} by axiom (1), we must have [i^{T}]□[K′] = [i^{T}]∘[K′], as claimed.

By 2.3, every elementary correspondence F′ in Cor_{k}(E, F) factors as [j^{T}] ∘ [i], where i : E → F′ and j : F → F′ are the inclusions. The proposition now follows from the observation that [F′]□[E′] must equal [j^{T}]□[i]□[E′] = [j^{T}]□([i] ∘ [E′]), which we have seen is [j^{T}] ∘ [i] ∘ [E′] = [F′] ∘ [E′]. □

Proposition 2.7. If i : K ⊂ L is a Galois extension with group G, and E is any field over k, then G acts on Cor_{k}(E, L) and:

(1) Cor_{k}(E, K) ≅ Cor_{k}(E, L)^{G};

(2) i ∘ i^{T} = ∑_{g∈G} g as endomorphisms of Cor_{k}(E, L).

Proof. From 2.2.1 we have a homomorphism G → Hom_{k}(L, L) → Cor_{k}(L, L). Composing with Cor_{k}(E, L) is associative by 2.4, so G acts on Cor_{k}(E, L). If c ∈ Cor_{k}(E, L) then by 2.3.1 and 2.4 we have [i] ∘ [i^{T}] ∘ c = ∑[g] ∘ c.

Because [i^{T}] ∘ [i] is multiplication by [L : K], the canonical map i : Cor_{k}(E, K) → Cor_{k}(E, L)^{G} is an injection. To show that it is an isomorphism, it suffices to show that if L′ is a quotient field of E ⊗_{k} L, finite over L, and {L_{α}} is its G-orbit in Cor_{k}(E, L) then ∑[L_{α}] is in the image of Cor_{k}(E, L). Let K′ be the image of E ⊗_{k} K in L′. Since it is a Galois extension of K′, K′ ⊗_{K} L = ∏ L_{α} is a product of fields isomorphic to L′; by definition, i : Cor_{k}(E, K) → Cor_{k}(E, L) sends [K′] to ∑[L_{α}]. □

Example 2.8. Let ℓ be a finite Galois extension of k and consider the representable functor M(E) = Cor_{k}(ℓ, E) on Cor_{k}^{0}. From 2.2.2 it is clear that M(E) = ℤ[G/N], where N is the normal subgroup of G = Gal(k̄/k) corresponding to ℓ. If H is the subgroup of G fixing E elementwise, it follows from 2.7 that M(E) ≅ ℤ[G/N]^{H}, which is free abelian on the double cosets H\G/N. Thus M may be identified via 2.4 (and 2.5) with the Hecke functor associated to the Galois module M = ℤ[G/N] in 1.4.

Example 2.9. Both the Milnor K-groups K_{n}^{M}(F) and Quillen’s K-groups K_{n}(F) are not only Hecke functors but are also covariant additive functors on Cor_{k}^{0}. This is perhaps the motivating example for the entire subject of transfer functors.

For Quillen’s groups, this is easiest seen by observing that tensoring with a E′-E′′ bimodule M is an exact functor from E′-modules to E′′-modules, and so induces a homomorphism K_{n}(E′) → K_{n}(E′′). The composition of ⊗_{E′}M_{1} and ⊗_{E′′}M_{2} is induced by the exact functor V ↦ V ⊗_{E′} (M_{1} ⊗_{E′′} M_{2}), and since M_{1} ⊗_{E′′} M_{2} ≅ ⊕A_{i} as an E′ ⊗_{k} E′′′-module, and A_{i} ≅ E_{i}^{λ_{i}} as an E′′′-module, it follows that the functor K_{n} preserves composition of correspondences. The verification for Milnor K-groups is more tedious, but straightforward given 2.3.

Example 2.10. We claim that the E-module Ω_{E} of absolute Kähler differentials of E is a covariant additive functor on Cor_{k}^{0}. Indeed, the restriction to finite separable algebras (i.e., to Cor_{G}) is a Hecke functor by 2.5.2, because if E/k is separable then Ω_{E} ≅ Ω_{k} ⊗_{k} E. If k is not perfect, then the claim is a simple exercise in algebra, since for any p-basis {x_{i}}_{i∈I} of E the elements {dx_{i}}_{i∈I} form a basis of Ω_{E}. (See [Mat, 26.5].)

§3. Composition of correspondences

In this section, we describe the composition law for correspondences:

Cor_{k}(A, B) ⊗ Cor_{k}(B, C) ⟶^{∘} Cor_{k}(A, C),

and prove that it makes Cor_{k} into a category, when char(k) = 0. It suffices to describe the composition of elementary correspondences, represented by a prime ideal P of A⊗_{k}B and a prime ideal Q of B ⊗_{k} C. It will be an integral combination of the prime ideals Q_{i} described in the following lemma.

Lemma 3.1. If f : B → C is a ring homomorphism, with C a normal domain, and P ∈ Spec(A ⊗_{k} B) is an elementary correspondence from A to B, then each minimal prime Q_{i} ⊂ A ⊗_{k} C over P ⊗_{B} C is an elementary correspondence from A to C.

Proof. By assumption, B′ = A ⊗_{k} B/P contains and is finite over B. By “going up,” B′ ⊗_{B} f(B) contains and is finite over f(B). Since the quotient field E of C is flat over f(B), it follows that B′ ⊗_{B} E contains both C and E. Hence C′ = B′ ⊗_{B} C is finite over C, and contains C. Since the reduced algebra of C′ is the product of the C′_{i} = A ⊗ C/Q_{i}, they too are finite over C, and contain C. □

In order to describe the transfer map j^{T} associated to a finite inclusion j, it is useful to introduce some notation. If B′ ⊂ B′′ is any finite inclusion of domains, it will be convenient to write [B′′ : B′] for the degree [K′′ : K′], where K′′ and K′ are the quotient fields of B′′ and B′, respectively. If B′ is an elementary correspondence from A to B, we will write [B′′] for the element [B′′ : B′] · [B′] of Cor_{k}(A, B).

Definition 3.2. Let j : B ⊂ B_{1} be a finite inclusion of normal domains. Then j^{T} ∘ : Cor_{k}(A, B_{1}) → Cor_{k}(A, B) sends the class [B′′] of an elementary correspondence (from A to B_{1}) to the class [B′′] in Cor_{k}(A, B).

That is, if B′ is the image of A ⊗_{k} B in B′′, then j^{T} ∘ [B′′] = [B′′ : B′] · [B′].

We will define composition with the element [j^{T}] of Cor_{k}(B_{1}, B) to be the map j^{T} ∘. When A = B_{1} the map j^{T} ∘ : Cor_{k}(B_{1}, B_{1}) → Cor_{k}(B_{1}, B) sends 1_{B_{1}} to [j^{T}]. If j_{1} : B_{1} ⊂ B_{2} is another finite inclusion, it is easy to see that j^{T} ∘(j_{1}^{T} ∘[B′′]) = (j_{1}j)^{T} ∘[B′′] for all [B′′] in Cor_{k}(A, B_{2}). In particular, j^{T} ∘ [j_{1}^{T}] = [(j_{1}j)^{T}] in Cor_{k}(B_{2}, B).

Any algebra map f : A → B induces a surjection A ⊗_{k} B → B and hence a canonical elementary correspondence [f] from A to B.

Lemma 3.2.1. Every elementary correspondence from A to B has a canonical factorization j^{T} ∘ [f], where f : A → B_{1} is homomorphism of normal k-algebras and j : B ⊆ B_{1} is finite.

Proof. Let B′ = A ⊗_{k} B/P be an elementary correspondence from A to B, and let B_{1} be the normalization of B′. There is a canonical algebra map f : A → B′ ⊂ B_{1}, and the map j^{T} sends the class of the elementary correspondence [f] to [B_{1}]. But since B′ and B_{1} have the same quotient field we have [B_{1} : B′] = 1, and therefore j^{T} ∘ [f] = [B′] in Cor_{k}(A, B). □

Let C′ = B ⊗_{k} C/Q be an elementary correspondence from B to C. In order to define the composition with [C′] from Cor_{k}(A, B) to Cor_{k}(A, C), it suffices by linearity to define [C′] ∘ [B′] for all B′. Write f : B → C′ for the algebra map implicit in the definition of C′.

Definition 3.3. Given elementary correspondences B′ and C′, from A to B and from B to C, respectively. Let {Q_{i}} denote the minimal primes of B′ ⊗_{B} C′ and set C′_{i} = B′ ⊗_{B} C′/Q_{i}. We define [C′] ∘ [B′] = ∑ n_{i}[C′_{i}] in Cor_{k}(A, C) ⊗ ℚ, where the coefficients n_{i} are defined as follows.

Let K and K′ denote the quotient fields of B and B′, respectively. Choose a finite normal field extension K′′ of K′ and let B′′ denote the integral closure of B′ in K′′. For each i, let l(i) denote the number of minimal primes {Q_{ij}} of B′′ ⊗_{B} C′ over Q_{i}, and write l = ∑ l(i) for the total number of minimal prime ideals of the finite C-algebra B′′ ⊗_{B} C′. Then

n_{i} = \frac{[B′ : B] · l(i)}{[C′_{i} : C′] · l}.

The number n_{i} is independent of the choice of K′′ by [SV, 5.13]. If char(k) = 0 then the n_{i} are always integers, and the composition [C′] ∘ [B′] belongs to Cor_{k}(A, C) by [SV, 5.11]. If char(k) = p > 0, then the n_{i} belong to Cor_{k}(A, C) ⊗ ℤ[1/p] by loc. cit.

Example 3.3.1. When B = C′, i.e., j : C ⊆ B is a finite inclusion, we recover the transfer formula 3.2: j^{T} ∘ [B′] = [B′]. Indeed, we have [C′_{1} : C′] = [B′ : B] and l = l(1) = 1 in the formula 3.3.

More generally, if we factor [C′] = j^{T} ∘ [f] as in 3.2.1, where f : B → C_{1} is a homomorphism and j : C ⊆ C_{1} is finite, then it is clear from the construction in 3.3 that [C′] ∘ [B′] = (j^{T} ∘ [f]) ∘ [B′] = j^{T} ∘ ([f] ∘ [B′]).

Paradigm 3.3.2. This is the special case where A = B′ and C = C′. Using 3.2.1, the correspondence [C′_{i}] factors as [j_{i}^{T}] ∘ [f_{i}], where f_{i} : B′ → C_{i} is the composition of B′ → C′_{i} with the normalization C′_{i} ⊆ C_{i}, and j_{i} : C ⊆ C_{i}. Definition 3.3 yields the formula [f] ∘ [j^{T}] = ∑ n_{i}[j_{i}^{T}] ∘ [f_{i}]. We may interpret this formula via the diagram:

B′ ⟶^{(n_{1}f_{1},...)} ∏ C_{i}
j^{T} ↓  ↓^{∏ j_{i}^{T}}
B ⟶^{f} C.

Example 3.3.3. If f : A → B and g : B → C are algebra maps, then [g] ∘ [f] = [gf]. More generally, for every correspondence C′ from B to C we have [C′] ∘ [f] = [C′] = d[C′_{A}], where C′_{A} is the image of A ⊗_{k} C in C′ (an elementary correspondence from A to C) and d = [C′ : C′_{A}].

These formulas come from the fact that (since B = B′) there is only one minimal prime of B′ ⊗_{B} C = C, so l(1) = l = 1 and C′_{i} = C in definition 3.3.

This formula is even associative: given an elementary correspondence [A′] from A_{0} to A, then [gf] ∘ [A′] = [g] ∘ ([f] ∘ [A′]) in Cor_{k}(A_{0}, C). This is proven in [SV, 5.15], by showing that the coefficient m_{j} of [C′′_{j}] in [gf] ∘ [A′] equals the sum of the products n_{i}n_{j}^{i}, where [f] ∘ [A′] = ∑ n_{i}[B′_{i}] and [g] ∘ [B′_{i}] = ∑ n_{j}^{i}[C′′_{ij}].

Example 3.3.4. Axiom (3) of 0.3 holds. That is, if S ⊂ B is multiplicatively closed, the composition 3.3 with B → S^{−1}B induces a canonical injection of Cor_{k}(A, B) into Cor_{k}(A, S^{−1}B), sending [B′] to [S^{−1}B′]. This is a degenerate case of 3.3 because S^{−1}B′ and S^{−1}B′′ are domains.

Passing to the limit over all S, we may also make sense of Cor_{k}(A, E) when E is the quotient field of B. If k(P) is the quotient field of A/P, then evidently Cor_{k}(A, E) ≅ ⊕_{P} Cor_{k}(k(P), E), the sum being taken over all primes P of A.

The following definition (taken from [SV, 5.6]) will allow us to avoid excess use of the adjective “normal” in describing the extension B′′ of B and B′ in definition 3.3.

Definition 3.4. Let B ⊂ B_{1} be a finite extension, with quotient fields K and K_{1}, respectively. We will say that B_{1} is a pseudo-Galois extension of B if K_{1} is a normal field extension of K, and if B_{1} is the integral closure of B in K_{1}. If K_{1}/K is separable and B_{1} is flat over B, we will say that B_{1} is Galois over B.

Example 3.4.1. Suppose that K is the quotient field of B, K ⊂ K_{1} is a purely inseparable extension of degree p^{r}, and B_{1} is the integral closure of B in K_{1}. Then j : B ⊂ B_{1} is pseudo-Galois and [j] ∘ [j^{T}] = p^{r} · 1_{B_{1}}, [j^{T}] ∘ [j] = p^{r} · 1_{B}. This follows directly from 3.3, since B → (B_{1} ⊗_{B} B_{1})_{red} is an isomorphism. It may also be deduced from 2.2.1 and 3.6 below.

Lemma 3.4.2. Given a finite inclusion j : B ⊂ B_{1} and f : B → C, then for every [B′_{1}] ∈ Cor_{k}(A, B_{1}) we have:

[f] ∘ (j^{T} ∘ [B′_{1}]) = ([f] ∘ [j^{T}]) ∘ [B′_{1}].

Proof. Choose a pseudo-Galois extension B′′ of B′_{1}, and let B′ denote the image of A ⊗_{k} B in B′_{1}. Let {Q_{i}}, {Q_{α}}, {Q_{β}} and {Q_{γ}} denote the minimal primes of B′ ⊗_{B} C, B_{1} ⊗_{B} C, B′_{1} ⊗_{B} C and B′′ ⊗_{B} C, respectively, and set C_{i} = B′ ⊗_{B} C/Q_{i}, C_{α} = B_{1} ⊗_{B} C/Q_{α}, C_{β} = B_{1} ⊗_{B} C/Q_{β} and C_{γ} = B′′ ⊗_{B} C/Q_{γ}. If there are l minimal primes in B′′ ⊗_{B} C, then the coefficient of [C_{i}] in the left-hand side is l(i) · [B′_{1} : B]/l · [C_{i} : C].

On the other hand, since [C_{β}] = [C_{β} : C_{i}][C_{i}] in Cor_{k}(A, C) for Q_{β} lying over Q_{i}, and l(i) = ∑_{β/i} l(β), the right side is

∑_{α,β/α} \frac{l(α)·[B_{1} : B]}{l·[C_{α} : C]} \frac{l(β)·[B′_{1} : B_{1}]}{l(α)·[C_{β} : C_{α}]}[C_{β}] = ∑_{β} \frac{l(β)·[B′_{1} : B]}{l·[C_{β} : C]}[C_{β}] = ∑_{i} \frac{l(i)·[B′_{1} : B]}{l·[C_{i} : C]}[C_{i}].

Now compare coefficients. □

Consider the category Norm_{k} of normal k-algebras of finite type and their algebra homomorphisms. Example 3.3.3 shows that the rule f ↦ [f] preserves composition. The following result shows that this rule defines a faithful functor, embedding Norm_{k} as a subcategory of Cor_{k} if char(k) = 0, and of Cor_{k} ⊗ ℤ[1/p] if char(k) = p > 0.

Theorem 3.5. If char(k) = 0, the normal k-algebras of finite type form the objects of an additive category Cor_{k}, whose morphisms are the groups Cor_{k}(A, B).

If char(k) = p > 0, they form the objects of an additive category Cor_{k} ⊗ℤ[1/p], whose morphisms are the groups Cor_{k}(A, B) ⊗ ℤ[1/p].

Proof. The only non-trivial point is to show that composition is associative. Let B′, C′ and D′ be elementary correspondences in Cor_{k}(A, B), Cor_{k}(B, C), and Cor_{k}(C, D), respectively. Let C′_{1} and D′_{1} denote the normalizations of C′ and D′, respectively, with f : B → C′_{1}, g : C → D′_{1}, j_{C} : C ⊂ C′_{1} and j_{D} : D ⊆ D′_{1} the natural maps. With this notation, 3.2.1 allows us to factor [C′] as j_{C}^{T} ∘ f and [D′] as j_{D}^{T} ∘ g. See (3.5.1).

(3.5.1)  \begin{aligned}B & ⟶^{f} & C′_{1} & ⟶^{(g_{α})} & ∏ D_{α} \\ & ^{j_{C}^{T}} ↓ & & ↓^{(j_{α}^{T})} \\ C & ⟶^{g} & D′_{1} & ⟶^{j_{D}^{T}} & D \end{aligned}

Using 3.3 and 3.2.1, choose j_{α} : D′_{1} ⊆ D_{α}, λ_{α} and g_{α} : C′_{1} → D_{α} so that [g] ∘ j_{C}^{T} = ∑ λ_{α}j_{α}^{T} ∘ [g_{α}] in Cor_{k}(C′_{1}, D′_{1}). By the construction in 3.3, we have

[D′] ∘ [C′] = j_{D}^{T} ∘ ∑ λ_{α}j_{α}^{T} ∘ [g_{α}f] = ∑ λ_{α}(j_{α}j_{D})^{T} ∘ [g_{α}f].

Using 3.3.1, 3.3.3 and 3.4.2, we have the desired result, viz.:

\begin{matrix}([D′] ∘ [C′]) ∘ [B′] = & (∑ λ_{α}(j_{α}j_{D})^{T} [g_{α}f]) ∘ [B′] \\ = & ∑ λ_{α}(j_{α}j_{D})^{T} ([g_{α}f] ∘ [B′]) \\ = & j_{D}^{T} ∘ \{∑ λ_{α}(j_{α}^{T} ∘ [g_{α}]) ∘ ([f] ∘ [B′])\} \\ = & j_{D}^{T} ∘ ([g] ∘ j_{C}^{T}) ∘ ([f] ∘ [B′]) \\ = & j_{D}^{T} ∘ [g] ∘ (j_{C}^{T} ∘ ([f] ∘ [B′]) \\ = & [D′] ∘ ([C′] ∘ [B′]). □ \end{matrix}

Lemma 3.6. When E and F are field extensions of k, the formula 3.3 for the composition Cor_{k}(A, E) ⊗ Cor_{k}(E, F) → Cor_{k}(A, F) agrees with the formula given in 2.2.

Proof. It suffices (by 3.2.1) to consider a finite inclusion j : E ⊆ E′ and a homomorphism i : E → F of fields over k. We may further assume that the extension E′/E is either purely inseparable or separable.

If E′ is purely inseparable over E, then E′ ⊗_{E} F is Artin local. If it has length λ and residue field F′, then [E′ : E] = λ[F′ : F] and l = l(1) = 1, so we have [i] ∘ [E′] = λ[F′], agreeing with 2.3.

Now suppose that E′ is a separable field extension of E. Being a separable algebra over F, E′ ⊗_{E} F is a product of separable field extensions F′_{α} of F. Embedding E′ in a Galois extension E′′ of E, the Galois group G = Gal(E′′/E) acts transitively on the set of minimal primes (or factors) of both E′ ⊗_{E} F and E′′ ⊗_{E} F. Fix a prime in each ring, and let G_{1} and G_{11} be the corresponding stabilizer subgroups. If H is the subgroup of G corresponding to E′, then it is easy to see that [G : H] = [E′ : E], [G_{1} : H] = [F′_{1} : F], l = [G : G_{11}], and l(i) = [G_{1} : G_{11}] for all i. From this we deduce that n_{i} = 1 for all i, in agreement with 2.3.1. □

Corollary 3.7. Suppose that f : B → C is an injection of normal domains, and C′_{i} = B′ ⊗_{B} C/Q_{i} for some minimal prime Q_{i}. Then the coefficient n_{i} of [C′_{i}] in [f] ∘ [B′] equals the length of the Artinian algebra (B′ ⊗_{B} C)_{Q_{i}}.

In particular, [f] ∘ [B′] = ∑[C′_{i}] if char(k) = 0, or more generally if the quotient field of either B′ or C is separable over the quotient field of B.

Proof. Let E, F and F_{i} denote the quotient fields of B, C and C′_{i}, respectively. Embedding Cor_{k}(A, B) in Cor_{k}(A, E) and Cor_{k}(A, C) in Cor_{k}(A, F), we see from 3.5, 3.6 and 2.2 that n_{i} equals the F_{i}-primary length of B′ ⊗_{E} F = (B′ ⊗_{B} C) ⊗_{C} F. □

Proposition 3.8. Suppose that f : B ⊂ C is a Galois extension with Galois group G. Then G acts on Cor_{k}(A, C) via G → Hom(C, C) → Cor_{k}(C, C) and:

(1) f induces an isomorphism Cor(A, B) ≅ Cor(A, C)^{G}.

(2) f ∘ f^{T} = ∑_{g∈G} g as an endomorphism of Cor(A, C).

(3) f : Cor(A, B) → Cor(A, C) sends [B′] to ∑[C′_{i}], using the notation of 3.3.

Proof. Item (3) implies (1) and (2) by the following argument. Given an elementary correspondence C′_{1} = A ⊗ C/P_{1} from A to C, let P be the restriction of P_{1} to A ⊗ B and set B′ = A ⊗ B/P. The group G acts transitively on the primes P_{i} over P and acts on Cor(A, C), with g sending [P_{i}] to [gP_{i}]. If G_{1} is the stabilizer of P_{1} then f^{T} ∘ [C′_{1}] = [G : G_{1}] · [B′] by 3.2. Items (1) and (2) follow since [f] ∘ [B′] = ∑ A ⊗ C/P_{i} by (3).

It remains to show that (3) holds i.e., that all coefficients n_{i} = 1. (Cf. [SV, 6.5(3)]). In the definition 3.3 of [f] ∘ [B′], note that G permutes the Q′_{i} and the Q′′_{ij}. If G_{1} and G_{11} are the stabilizer subgroups of Q′_{1} and Q′′_{11} then it is easy to see that |G| = [B′ : B], |G_{1}| = [C_{1} : C], l = [G : G_{11}], l(i) = [G_{1} : G_{11}] for all i. From this we deduce that n_{i} = 1 for all i, as claimed. □

§4. Uniqueness of composition

We now turn to the verification of theorem 0.3, stated in the introduction.

Lemma 4.1. If j : B ⊂ C is a finite inclusion of degree d = [C : B] then for every A: (1) [j]∘ : Cor_{k}(A, B) → Cor_{k}(A, C) is injective, and

(2) j^{T} ∘ [j]∘ : Cor_{k}(A, B) → Cor_{k}(A, B) is multiplication by d.

Proof. Since (2) implies (1) by 3.3.1, it suffices by associativity 3.5 to observe that j^{T} ∘ [j] = d · 1_{B}, by 3.2. □

We now have to show that axiom 0.3(2) holds.

Lemma 4.2. Suppose that j : B′ ⊃ B is a flat pseudo-Galois extension, and that f : B → C is a homomorphism of normal domains. Let {Q_{1}, …, Q_{l}} denote the minimal primes of C′ = B′ ⊗_{B} C and set C′_{i} = C′/Q_{i}. Then the length λ of the Artinian ring C′_{Q_{i}} is independent of i, and [f] ∘ [j^{T}] = λ ∑[C′_{i}] in Cor(B′, C).

Proof. The Galois group G acts transitively on the set of minimal prime ideals Q_{i} of C′ = B′ ⊗_{B} C, and induces isomorphisms between all the C′_{i}; see [Bour, 5.2.3]. This proves that λ and d = [C′_{i} : C] are independent of i. Since B′ is flat over B, C′ is flat over C and C′_{E} = C′ ⊗_{C} E is flat over the quotient field E of C. Since there are no embedded primes, C′_{E} = ∏ C′_{Q_{i}}. Thus [B′ : B] = [C′_{E} : E] = ∑[C′_{Q_{i}} : C] = dlλ. It follows from 3.3 that the coefficient of [C′_{i}] in [f] ∘ [j^{T}] is n_{i} = [B′ : B]/dl = λ. □

Recall that if B ⊂ B′ is a finite inclusion of Dedekind domains, and 𝔫 is a prime of B′ lying over a prime 𝔪 of B, the ramification index at 𝔫 is that integer e such that 𝔪B′ = 𝔫^{e}, i.e., e is the length of (B′/𝔪B′)_{𝔫}.

Corollary 4.3. Let j : B ⊂ B′ be a finite extension of normal domains and π : B → B/P the quotient by a height one prime ideal P of B such that B/P is integrally closed. If {P_{i}} are the primes of B′ lying over P, and e_{i} is the ramification index of B′_{P_{i}} over B_{P}, then [π] ∘ [j^{T}] = ∑ e_{i}[B′/P_{i}] in Cor_{k}(B′, B/P).

Proof. If B′ is pseudo-Galois over B, this is just a restatement of 4.2. In the general case, choose a pseudo-Galois extension j′ : B′ ⊆ B′′, let {P_{ij}} denote the primes of B′′ over the prime P_{i} of B′, with e_{ij} the ramification index of B′′_{P_{ij}} over B′_{P_{i}}, and let f_{ij} denote the degree [B′′/P_{ij} : B′/P_{i}]. Since B′′_{P} is flat over the Dedekind domain B′_{P}, ∑_{j} e_{ij}f_{ij} = [B′′ : B′] for all i. Since P B′′_{P} = ∏_{i,j} P_{ij}^{e_{i}e_{ij}}B′′_{P}, we see by 4.2 that

Cor_{k}(B′′, B) ⟶^{π} Cor_{k}(B′′, B/P) ⟶^{j′} Cor_{k}(B′, B/P) sends [B′′] = [(j′j)^{T}] to

[π] ∘ [B′′] ∘ [j′] = ∑_{i,j} e_{i}e_{ij}[B′′/P_{ij}] ∘ [j′] = [B′′ : B′] ∑_{i} e_{i}[B′/P_{i}].

By 4.1, [B′′]∘[j′] = [j^{T}]∘[j′^{T}]∘[j′] = [B′′ : B′]·[j^{T}], so [π]∘[B′′]∘[j′] = [B′′ : B′][π]∘[j^{T}] in Cor_{k}(B′, B/P). Since this group is torsionfree, we may divide by [B′′ : B′] to get the result. □

Proposition 4.4. Let B be a normal domain and π : B → B/P the quotient at a height one prime ideal P such that B/P is integrally closed. If B′ is an elementary correspondence from A to B, let {P_{i}} be the primes of B′ over P, and let λ_{i} denote the length of the Artin local ring (B′/P B′)_{P_{i}}.

Then Cor_{k}(A, B) → Cor_{k}(A, B/P) sends [B′] to ∑ λ_{i}[B′/P_{i}].

Note that the Artin local ring is R/I, where R = B′_{P_{i}} and I = P B′_{P_{i}}. In fact, the length λ_{i} of R/I is also the multiplicity e(I) of I. This is because R is flat over the discrete valuation domain B_{P}, so that I^{r}/I^{r+1} = P^{r}/P^{r+1} ⊗_{B} R ≅ R/I for all r.

Proof. Let B′_{1} denote the integral closure of B′, with j : B ⊆ B′_{1}, and write f : A → B′_{1} for the induced map. Since [B′] = [j^{T}] ∘ [f] we must compute the coefficient n_{i} of [B′/P_{i}] in [π] ∘ [j^{T}] ∘ [f]. If the set of primes of B′_{1} over P_{i} is {P_{ij}}, we see from 4.3 that n_{i}[B′/P_{i}] = ∑_{j} e_{ij}[B′_{1}/P_{ij}] ∘ [f] in Cor_{k}(A, B/P). If f_{ij} denotes the index [B′/P_{i} : B′_{1}/P_{ij}], then [B′_{1}/P_{ij}] ∘ [f] = f_{ij}[B′/P_{i}] in Cor_{k}(A, B/P), i.e., n_{i} = ∑_{j} e_{ij}f_{ij}. But the multiplicity λ_{i} of I = P B′_{P_{i}} is e(I, B′′_{P_{i}}) = ∑_{j} e(I, B′′_{P_{i}j}) = ∑_{j} e_{ij}f_{ij}. □

Lemma 4.5. Let f : B → C be a homomorphism of normal domains, and let h : C ↪ D be an injection. Given an elementary correspondence B′ from A to B, write {P_{i}} for the minimal primes of C′ = B′ ⊗_{B} C, and write {Q_{α}} for the minimal primes of D′ = B′ ⊗_{B} D, respectively. Suppose that [f] ∘ [B′] = ∑ n_{i}[C′/P_{i}] in Cor_{k}(A, C) and that [hf] ∘ [B′] = ∑ m_{α}[D′/Q_{α}] in Cor_{k}(A, D).

Then n_{i} = length(C′_{P_{i}}) for all i if and only if m_{α} = length(D′_{Q_{α}}) for all α.

Proof. By 3.7, [h] ∘ [f] ∘ [B′] equals ∑_{α} n_{i(α)}λ_{α}[D′/Q_{α}], where P_{i} = P_{i(α)} is the restriction of Q_{α} to C′ and λ_{α} is the length of (C′/P_{i} ⊗_{C} D)_{Q_{α}}. Localizing D, which is harmless by 3.3.4, we may assume that D is flat over C. In this case, tensoring a filtration of C′_{P_{i}} with D yields a filtration of C′_{P_{i}} ⊗_{C} D. Counting, we see that the length of C′_{P_{i}} ⊗_{C} D is λ_{α} times the length of C′_{P_{i}}. Comparing coefficients of [D′/Q_{α}] yields the result. □

We can now prove that axiom (2) holds for the composition ∘ of 3.3.

Theorem 4.6. Let B′ be an elementary correspondence from A to B such that B ⊆ B′ is flat, and let f : B → C be a homomorphism of normal domains. Then [f] ∘ [B′] = ∑_{i} λ_{i}[C′/P_{i}], where: C′ = B′ ⊗_{B} C; {P_{i}} are the minimal primes of C′; and λ_{i} denotes the length of C′_{P_{i}}.

Proof. This is clear if f is an injection, by 3.7. In the general case, let B̄ denote the integral closure of B/P, where P is the kernel of f. Applying 4.5 to B̄ ⊆ C, we may assume that C = B̄. Replacing B by B[1/b] for suitable b ∉ P, allowed by 3.3.4, we may also assume that B̄ = B/P.

If R is any normal domain between B and its quotient field, of finite type over k, then R′ = B′ ⊗_{B} R is an elementary correspondence from A to R (by flatness of B′), and axiom (3) implies that [R′] is the image of [B′] in Cor_{k}(A, R).

Pick x ∈ P nonzero, let R_{0} denote the integral closure of B[P/x], and pick a prime ideal Q_{0} ⊂ R_{0} minimal over xR_{0}; there is an s ∈ R_{0} − Q_{0} so that both R = R_{0}[1/s] and R/Q are normal, where Q = Q_{0}[1/s]. By 4.4, the image of [R′] in Cor_{k}(A, R/Q) is ∑ length(R′/QR′)_{Q_{i}}. The theorem now follows from 4.5. □

Example 4.6.1. Suppose that B/P is normal and consider the effect of the projection π : B → B/P on a flat elementary correspondence B′, as in 4.6. If {P_{i}} are the primes of B′ over P, then the coefficient of [B′/P_{i}] in [π] ∘ [B′] is also the ratio of multiplicities e(P B′_{P_{i}})/e(P B_{P}). Indeed, because B′/B is flat, this follows from 4.6 by the formula e(P B′_{P_{i}}) = e(P B_{P})length(B′/P B′)_{P_{i}} of [BH, 4.6.9]. (Cf. [SV1, 3.5.4].)

Corollary 4.7. Any transfer functor F satisfies the flat Mackey condition: if j : B′ ⊃ B is a finite flat extension of normal algebras and f : B → C is a homomorphism, then

F (B′) ⟶^{(λ_{1}f_{1},...)} ∏ F (C_{i})
↓  ↓
F (B) ⟶^{f} F (C).

commutes, where the product is over all minimal primes Q_{i} of C′ = B′ ⊗_{B} C, C_{i} is the normalization of C′/Q_{i}, f_{i} : B′ → C_{i} and λ_{i} = length(C′_{Q_{i}}).

Theorem 4.8. The composition ∘ of 3.3 is the unique associative composition satisfying axioms (1), (2) and (3) of 0.3.

Proof. Axioms (1) and (3) hold for ∘ by 3.3.4, 4.6 and 4.4. Conversely, suppose given a composition law □ satisfying the axioms; we must show □ agrees with ∘.

Given a finite extension j : B ⊂ B′ of degree d, with quotient fields i : B ⊂ E and i′ : B′ ⊂ E′, and a ∈ Cor_{k}(A, B′), axiom (3) implies that [i]□[j^{T}]□a = [j_{E}^{T}]□[i′]□a. We saw in 2.6 that [j_{E}^{T}]□ = [j_{E}^{T}]∘ is determined by axioms (1) and (2). It follows that [j^{T}]□a = [j^{T}] ∘ a.

Next, we claim that any elementary correspondence [B′] from A to B has a canonical factorization as the composition of f : A → B′_{1} and j^{T} ∈ Cor_{k}(B′_{1}, B), as in 3.2.1. To see this we map [B′] and the composition to Cor_{k}(A, E) where E is the quotient field of B. Here [B′] and [j^{T}]□[f] agree, by the argument of 2.6 which uses axioms (1) and (2). By (3) they must agree in Cor_{k}(A, B), as claimed.

We are left to show that the composition [f]□[B′] of [B′] with a homomorphism f : B → C equals [f] ∘ [B′]. Let P denote the kernel of f and suppose first that B′_{P} is flat over B_{P}. (This is the case for example when f is an injection.) Choose S ⊂ (B −P) so that S^{−1}B′ is flat over S^{−1}B; the composition of [f]□[B′] with i : C → S^{−1}C must equal [S^{−1}f] ∘ [S^{−1}B′] by axioms (2) and (3). That is, [i]□[f]□[B′] = [i] ∘ [f] ∘ [B′]. This implies by (3) that [f]□[B′] = [f] ∘ [B′].

In the general case, pick x ∈ P nonzero and let R denote the integral closure of B[P/x] in the quotient field E of B. For any height one prime Q of R minimal over xR there is an extension j : C ⊂ D so that jf factors as B → R → R/Q → D. Since B ⊂ R ⊂ E, the composition of [B′] with B → R is an elementary correspondence [R′], uniquely determined by axiom (3). Since R′_{Q} is flat over the discrete valuation domain R_{Q}, the composition of [R′] with R → D is uniquely determined by the flat case above. But the composition with C → D is an injection from Cor_{k}(A, C) to Cor_{k}(A, D), by the previous paragraph and 3.7. Hence the composition [f]□[B′] must equal [f] ∘ [B′], as required. □

§5. Correspondences in positive characteristic

If k is a field of characteristic p > 0, the composition of two elementary correspondences need not have integer coefficients, as the following example shows. (The case n = 1 is due to Merkurev [SV, p.77].)

Example 5.1. Let a_{1}, a_{2} ∈ k be linearly independent in k^{×}/k^{×p} and set q = p^{n}, so that the field ℓ = k(α_{1}, α_{2}) has degree q^{2} over k, where α_{i} = \sqrt[q]{a_{i}}. Set A = k[T_{0}, T_{1}, T_{2}]/(a_{1}T_{1}^{q} + a_{2}T_{2}^{q} = T_{0}^{q}) and B = ℓ[T_{1}, T_{2}]. Both A and B are integrally closed, and the map j : A → B sending T_{0} to α_{1}T_{1} + α_{2}T_{2} makes B a finite A-algebra. The field L = ℓ(T_{1}, T_{2}) of fractions of B is purely inseparable of degree q over the field K of fractions of A. At the maximal ideal 𝔪 = (T_{0}, T_{1}, T_{2})A, we have A/𝔪 = k and B/𝔪B = ℓ, and the extension j̄ : k ⊂ ℓ has degree q^{2}.

Consider the projections π : A → A/𝔪 = k and π̃ : B → B/𝔪B = ℓ. Since B is pseudo-Galois over A, definition 3.3 yields the formula [π] ∘ [j^{T}] = (1/p^{n})[j̄^{T}] ∘ [π̃] in Cor_{k}(B, k).

B ⟶^{(1/q)π̃} ℓ
j^{T} ↓  ↓ j̄^{T}
A ⟶^{π} k.

Remark 5.1.1. The value of the coefficient n in the formula [π] ∘ [j^{T}] = n[j̄^{T}] ∘ [π̃] is forced by the formulas [j] ∘ [j^{T}] = q · 1_{B} and [j̄] ∘ [j̄^{T}] = q^{2} · 1_{ℓ} of 3.4.1. Indeed, we have

q[π̃] = [π̃] ∘ ([j] ∘ [j^{T}]) = [π̃j] ∘ [j^{T}]) = [j̄] ∘ [π] ∘ [j^{T}] = n[j̄] ∘ [j̄^{T}] ∘ [π̃] = nq^{2}[π̃].

The following definition is taken from [SV1, 3.3.9] [MVW, 1A.9]. Recall that a finite correspondence is just a ℤ-linear combination of elementary correspondences. The composition of two correspondences is a ℤ[1/p]-linear combination of elementary correspondences by [SV, 5.11]. If P is a prime of B, we write k(P) for the field B_{P} /P B_{P}.

Definition 5.2. A finite correspondence a from A to B is called universally integral when the composition [π] ∘ a has integer coefficients for every map π : B → k(P), where P ranges over all prime ideals of B. We write Cor_{k}(A, B) for the set of all universally integral correspondences from A to B; since these correspondences are clearly closed under addition, Cor_{k}(A, B) is an abelian group.

Note that the condition for being universally integral could also be phrased more awkwardly in terms of the map from B to the integral closure of B/P.

Examples 5.2.1. a) Any homomorphism f : A → B is universally integral by 3.3.3.

b) If B′ is flat over B, [B′] is universally integral by axiom 0.3(2).

c) If B is a Dedekind domain, every correspondence is universally integral by (b).

d) An elementary correspondence [B′] is universally integral if the field extensions k(P) ⊆ k(P′_{i}) are separable for every prime ideal P of B and every prime P′_{i} of B′ lying over P; see [SV, 5.11].

Nonexample 5.2.2. In example 5.1, the correspondence [j^{T}] from B to A is not universally integral, but p^{n}[j^{T}] is. This example also shows that transfer maps need not preserve universally integral correspondences; j^{T} ∘ does not send Cor_{k}(B, B) into Cor_{k}(B, A).

Lemma 5.3. If R is regular, every finite correspondence is universally integral.

Proof. (Cf. [SV, 5.18]) We show that R → k(P) = R_{P} /P preserves finite correspondences, by induction of the height of the prime ideal P. If P =0 this is axiom (3). Otherwise, pick an x ∈ P − P^{2} and observe that R_{P} and R_{P} /xR_{P} are regular local rings. Localizing R, we may assume that xR is prime and hence R/xR is regular. Since R → R/xR preserves finite correspondences by 4.4, and R/xR → k(P) preserves them by induction, we are done. □

Lemma 5.4. The composition with any homomorphism f : B → C sends universally integral correspondences to universally integral correspondences, i.e., sends Cor_{k}(A, B) to Cor_{k}(A, C).

Proof. Fix a universally integral correspondence a ∈ Cor_{k}(A, B). If Q is a prime ideal of C and π : C → C_{Q}/Q is the projection, we must show that the composition [π]∘[f]∘a has integer coefficients. Let P denote the restriction of Q to B, and ā the image of a in Cor_{k}(A, B_{P} /P). Since ā has integer coefficients, and B_{P} /P → C_{Q}/Q is an injection, the image of ā (and hence a) in Cor_{k}(A, C_{Q}/Q) has integer coefficients by 3.7. □

Proposition 5.5. The normal k-algebras of finite type form the objects of an additive category Cor_{k}, whose morphisms are the groups Cor_{k}(A, B).

Proof. Given theorem 3.5, we need only show that the composition of universally integral correspondences is universally integral. That is, the composition b ∘ a of b ∈ Cor_{k}(B, C) and a ∈ Cor_{k}(A, B) belongs to Cor_{k}(A, C). Fix a prime ideal Q of C; localizing C, we may suppose that C/Q is normal and need to show that [π] ∘ b ∘ a has integer coefficients, where π : C → C/Q.

Because b is universally integral, [π] ∘ b = ∑ n_{i}[Q_{i}] for certain primes Q_{i} in B ⊗ k(Q) and integers n_{i}. Each [Q_{i}] ∘ a has integer coefficients by 3.2 and 5.4, since it is the composition of a with f_{i} : B → B⊗k(Q)/Q_{i}, followed by the transfer Cor_{k}(A, k(Q_{i})) → Cor_{k}(A, k(Q)). Substituting, we see that [π] ∘ b ∘ a has integer coefficients. □

Proposition 5.6. If B′ is any elementary correspondence between normal k-algebras, and char(k) = p > 0, then some multiple p^{n}[B′] is universally integral.

Proof. Let ν denote the number of generators of B′ as a B-module, and let B′′, l, etc. be as in definition 3.3. For every prime Q of B, the k(Q)-vector space B′ ⊗_{B} k(Q) has dimension at most ν. Hence the denominators of the coefficients arising in 3.3 for the image of B′ under B → k(Q) are powers of p bounded above by lν. If p^{n} is the largest power of p less than or equal to lν, then evidently p^{n}[B′] is universally integral. □

Proposition 5.7. The inclusion Cor_{k}^{0} ⊂ Cor_{k} has a right adjoint A ↦ ℓ_{A}, where ℓ_{A} is the integral closure of k in A.

Proof. First note that if A is a domain then ℓ_{A} is finite-dimensional over k, because it injects into A/𝔪 for any maximal ideal 𝔪 of A. A finite inclusion i : A ⊂ B induces i_{ℓ} : ℓ_{A} ⊆ ℓ_{B}; the adjoint sends the finite correspondence [i^{T}] to the element [B : Aℓ_{B}]·[i_{ℓ}^{T}] of Cor_{k}(ℓ_{B}, ℓ_{A}). [Since Cor_{k}(A, B) is a subgroup of all finite correspondences, the adjoint is defined.] Using the factorization 3.2.1 of elementary correspondences in Cor_{k}, it is easy to verify that ℓ_{A} depends functorially on A, and that the inclusion η_{A} : ℓ_{A} ⊂ A is a natural transformation.

We must show that the natural map [η_{A}]∘ : Cor_{k}(E, ℓ_{A}) → Cor_{k}(E, A) is an isomorphism for all E and A. Since it is an injection by 3.7, it suffices to prove that it is a surjection. Given an elementary correspondence A′ from E to a domain A, let ℓ′ denote the subfield Eℓ_{A} of A′ generated by the canonical subfields E and ℓ_{A}. Since ℓ′ and A are linearly disjoint over ℓ_{A}, ℓ′A = ℓ′ ⊗_{ℓ_{A}} A is a domain, finite over A. From 3.7 we see that [η_{A}] ∘ [ℓ′] = [ℓ′A]. Since the natural surjection E ⊗_{k} A → (E ⊗_{k} ℓ_{A}) ⊗_{ℓ_{A}} A → A′ factors through ℓ′A, we must have ℓ′A = A′, as desired. □

Base-change 5.8. If k ⊂ ℓ is a finite field extension, there is an obvious forgetful functor Cor_{ℓ} → Cor_{k}; an elementary correspondence of ℓ-algebras determines an elementary correspondence of the underlying k-algebras. Hence any transfer functor on normal k-algebras restricts to a transfer functor on normal ℓ-algebras.

Now suppose that ℓ is a separable field extension of k. If A is a k-algebra and B is an ℓ-algebra, then there is a canonical identification Cor_{k}(A, B) ≅ Cor_{ℓ}(A ⊗_{k} ℓ, B). Indeed, the separability of ℓ implies that A ⊗_{k} ℓ is normal; (see [BH, 2.2.23]), and a sum of primes in A ⊗_{k} B being universally integral is the same over k as it is over ℓ.

Given this, it is easy to see that the forgetful functor is right adjoint to the additive functor Cor_{k} → Cor_{ℓ} sending A to A ⊗_{k} ℓ. Hence transfer functors over ℓ extend to transfer functors over k.

§6. Examples of Transfer functors

By definition, a transfer functor F is an additive covariant functor from Cor_{k} to abelian groups. In order to specify F, we need only construct F as a functor from normal k-algebras to abelian groups, and describe the transfer maps (in characteristic p > 0 we need slightly more). It suffices to work with integral domains, because F must commute with the categorical product in Cor_{k}: F (A × B) = F (A) × F (B).

Trace maps 6.1. The underlying abelian group (A ↦ A) is a transfer functor. If A ⊂ B is a finite integral extension of normal domains, with fields of fractions K ⊂ L, then the trace map L → K induces a trace map B → A. Indeed, the trace of b ∈ B is integral over A and lies in K, so it is in A.

Norm maps 6.2. The group of units U(A) = A^{×} is a transfer functor. If A ⊂ B is a finite extension of normal domains, with fields of fractions K ⊂ L, then the norm map L^{×} → K^{×} induces a norm map B^{×} → A^{×}. Indeed, the norm of a unit b ∈ B^{×} and its inverse are both integral over A and lie in K, so they belong to A.

Constant transfer functors 6.3. The constant transfer functor associated to an abelian group M satisfies F_{M}(A) = M for every domain A, and if A → B is a map of domains then F_{M}(A) → F_{M}(B) is the identity. If B′ is an elementary correspondence from A to B then the map F_{M}(B) → F_{M}(A) is multiplication by the rank [B′ : B].

Representable transfer functors 6.4. Fix a k-algebra R. Then A ↦ Cor_{k}(R, A) is a transfer functor. The Yoneda Lemma, that F (R) = Hom(Cor_{k}(R, −), F) for every transfer functor F, implies that Cor_{k}(R, −) is a projective object in the abelian category of transfer functors, and that every transfer functor is a quotient of a direct sum of such representable transfer functors.

a) If R = k then Cor_{k}(k, A) = ℤ for every domain A. If A ⊂ B is a finite inclusion then the transfer map Cor_{k}(k, B) → Cor_{k}(k, A) is multiplication by the rank of B over A, [B : A]. In fact, this is just the constant transfer functor F_{ℤ} in the sense of 6.3.

b) If R = k[x] then Cor_{k}(k[x], A) is the free abelian group on the set of irreducible monic polynomials in A[t]. If A ⊂ B then we may identify the transfer map Cor_{k}(k[x], B) → Cor_{k}(k[x], A) with the norm map applied to the monic polynomials.

Flasque transfer functors 6.4.1. If H is a contravariant additive functor from abelian groups to itself, and S is a fixed algebra, then A ↦ H Cor_{k}(A, S) is a transfer functor. In particular, for S = k and H = Hom(−, M), we see that F (A) = ∏_{𝔪} M is a transfer functor, where the product ranges over all maximal ideals 𝔪 of A. (F (A) = ⊕_{𝔪}M is a transfer sub-functor.) If A ⊃ B is finite, the component of F (A) → F (B) corresponding to a maximal ideal 𝔫 of B is the sum of the components indexed by the finitely many maximal ideals 𝔪_{i} of A over 𝔫. This example is interesting because F is a flasque Zariski sheaf on each Spec(A).

There are many variations, among which is the following. Set E(A) = ∏_{𝔪} A_{𝔪}^{×}, where the product is again over all maximal ideals of A. If A ⊃ B is finite, the component of E(A) → E(B) corresponding to 𝔫 ⊂ B is the product of the norms A_{𝔪_{i}}^{×} → B_{𝔫}^{×} of 6.2, over the maximal ideals 𝔪_{i} of A over 𝔫. If f : B → C is a homomorphism, E(B) → E(C) is defined because Spec(C) → Spec(B) preserves maximal ideals. The verification that E is a transfer functor is easy, given 6.2.

Hecke Functors 6.5. Recall from 1.1 that a Hecke functor for the Galois group Gal(k̄/k) is a contravariant additive functor from Cor_{G} to abelian groups. Since Cor_{G} is a subcategory of Cor_{k} by 3.6, 2.4 and 2.5, the restriction of any transfer functor to Cor_{G} is a Hecke functor.

Conversely, any Hecke functor M induces a transfer functor by the formula M(A) = M(ℓ_{A}^{s}), where ℓ_{A}^{s} ⊆ ℓ_{A} is the separable closure of k in A. Indeed, this is just the composition of M with the adjunction Cor_{k} → Cor_{k}^{0} → Cor_{G} of 5.7.

A slightly more bizarre example occurs when k is not perfect. If ℓ_{A} is the integral closure of k in A, then F (A) = Ω_{ℓ_{A}/k} is a transfer functor by 2.10 and 5.7.

Galois modules 6.6. Let G denote the Galois group of k̄/k, and let M be a discrete G-module. If A is a domain, and ℓ_{A} is the integral closure of k in A, we set M(A) = M^{H}, where H = Gal(k̄/ℓ_{A}). This is a transfer functor, because M is a Hecke functor by 1.4 and the formula in 6.5 comes from 2.5.1.

Étale Sheaves 6.7. Regarding a transfer functor F as a presheaf on normal schemes over k, we may sheafify to get an étale sheaf F_{et}. The associated covariant functor on normal algebras is a transfer functor. This is proven in [MVW, 6.17] (where smoothness is not needed), or in [V, 3.1.12 and 3.3.1].

For example, if E and F are the flasque Zariski sheaves of 6.4.1, then the transfer functors associated to the flasque étale sheaves E_{et} and F_{et} may be described as follows. Fix an algebraic closure k̄ of k, and set F_{et}(A) = ∏_{x} M, the product being taken over the set of all k-algebra maps x : A → k̄. The transfer formula for F_{et}(A) → F_{et}(B) is similar to that of 6.4.1. Similarly, E_{et}(A) = ∏_{x}(A_{x}^{sh})^{×}, where A_{x}^{sh} denotes the strict henselization of A along x; again the transfer map is induced from that of 6.4.1.

Étale Cohomology 6.7.1. The examples in 6.7 are special cases of a construction due to Deligne: the terms in the canonical flasque resolution of an étale sheaf with transfers F are themselves étale sheaves with transfers. (See [MVW, 6.20].) It follows that the étale cohomology of F, A ↦ H_{et}^{∗}(Spec(A), F), is a transfer functor.

Picard groups 6.8. The Picard group is a well known functor from rings to abelian groups. By 6.2 and 6.7.1, it is also a transfer functor. This was first observed, in nascient form, by Roggenkamp and Scott [RS].

Here is a proof avoiding 6.7.1. Define C by the short exact sequence of transfer functors 0 → A^{×} → E_{et}(A) → C(A) → 0, where the middle term is the transfer functor described in 6.7. Let C_{et} denote the étale sheaf associated to C; it is a transfer functor by 6.7. The long exact cohomology sequence yields an exact sequence, proving that Pic is a transfer functor:

0 → A^{×} → ∏_{x}(A_{x}^{sh})^{×} → C_{et}(A) → Pic(A) → 0.

It is instructive to describe the transfer map i_{∗}: Pic(B) → Pic(A) associated to a finite integral extension i : B ⊃ A of normal domains. If we regard L ∈ Pic(B) as a locally free B-module, there are s_{1}, ..., s_{n} ∈ A and units b_{ij} ∈ B[1/s_{i}s_{j}]^{×} so that L is obtained by patching: there are isomorphisms L[1/s_{i}] ≅ B[1/s_{i}] so that the composite isomorphisms of B[1/s_{i}s_{j}] are multiplication by b_{ij}. We define i_{∗}(L) to be the A-module obtained by patching the modules A[1/s_{i}] using the norms of the b_{ij}. It is not hard to see that this is independent of the choices made.

Alternatively, note that B is flat over A (say of rank d) except on a set Y of codimension ≥ 2, because at every height one prime A_{P} is a discrete valuation ring. Hence ∧_{A}^{d}(L) is locally free of rank one except on Y, and its double dual ∧_{A}^{d}(L)^{∗∗} is locally free; we have i_{∗}(L) = ∧_{A}^{d}(L)^{∗∗}.

Homotopy invariant transfer functors 6.9. We say that F is homotopy invariant if k → k[t] induces an isomorphism F (A) ≅ F (A[t]) for all A. The units and constant transfer functors (6.2 and 6.3) are homotopy invariant but examples 6.1 and 6.4 are not.

For any transfer functor F and any n, C_{n}F (A) = F (A[T_{n}]) is a transfer functor, where k[T_{n}] denotes the polynomial ring k[t_{0}, …, t_{n}]/(∑ t_{i} = 1). Moreover, any algebra map k[T_{m}] → k[T_{n}] induces a morphism of transfer functors C_{m}F → C_{n}F. The coequalizer [F]A of F (A[t]) ⇉ F (A) (induced by t ↦ 0, 1) is homotopy invariant, and in fact F → [F] is universal with respect to this property.

Applying this construction to the simplicial ring k[T_{•}] yields a chain complex C_{•}F in the abelian category of transfer functors. Hence the homology A ↦ H_{∗}C_{•}F (A) and the cohomology A ↦ H^{∗} Hom(C_{•}F (A), ℤ/m) are transfer functors. They are also homotopy invariant by [W81, 2.4].

Zariski sheaves 6.9.1. If F is homotopy invariant, then the Zariski sheaf F_{zar} is also a transfer functor by [V, 3.1.12] or [MVW, 21.1].

Betti Cohomology 6.10. If k = ℂ then the Betti cohomology H^{∗}(X, ℤ/m) of the topological space X of ℂ-points of Spec(A) are transfer functors. This is a special case of 6.7.1, since the Betti cohomology agrees with the étale cohomology groups H_{et}^{n}(Spec A, ℤ/m).

This is essentially the main result 7.8 of [SV], since the singular homology A ↦ H_{∗}^{sing}(Spec(A), ℤ/m) of [SV] is isomorphic to the cohomology of the cochain complex S^{∗}(A) = Hom(C_{•}Cor_{k}(A, k), ℤ/m) of 6.9 by [SV, 6.7–6.8]. Since S^{∗} is a complex of transfer functors by 6.4.1, its cohomology also consists of transfer functors.

Base change 6.11. If k ⊂ ℓ is a finite field extension, we saw in 5.8 that the restriction of any transfer functor on normal k-algebras is also a transfer functor on normal ℓalgebras. If ℓ is a separable field extension of k, the extension F̃(A) = F (A ⊗_{k} ℓ) of a transfer functor F over ℓ is also a transfer functor over k by 5.8. For Galois modules, considered as transfer functors by 6.6, these operations are just the classical restriction and extension of modules over a group.

The following lemma could have been used for most of the above examples.

Lemma 6.12. Let F be a functor from normal k-algebras to abelian groups which preserves products and injections. Suppose that F also has a functorial transfer for finite extensions which commutes with localization and satisfies the flat Mackey condition 4.7. If char(k) = p > 0, we assume that each F (A) is uniquely p-divisible. Then F extends to a transfer functor.

Proof. Any elementary correspondence has a canonical factorization A → B_{1} ⊇ B by 3.2.1; we define F (A) → F (B) by (0.2). To see that composition in Cor_{k} is preserved by F, it suffices to compare the composition of the transfer F (B_{1}) → F (B) with the map induced by f : B → C. (This uses 3.5.) Localizing C, we may suppose that each C_{i} = C′/Q_{i} is normal and flat over C. As in the proof of 4.6, there is a normal domain R between B and its quotient field, and an extension C ⊆ D, such that B → C → D factors through R, and R_{1} = B_{1} ⊗_{B} R is flat over R. Localizing D, we may assume that B_{1} ⊗_{B} D = ∏ D_{α} is normal, so we have a commutative diagram

F (B_{1}) ⟶ F (R_{1}) ⟶ ∏ F (D_{α}) ⟵ ∏ F (C_{i}) ⟵ F (B_{1})
↓  ↓  ↓  ↓  ↓
F (B) ⟶ F (R) ⟶ F (D) ⟵ F (C) ⟵ F (B).

The two bottom composites F (B) → F (D) are the same, and the map F (C) → F (D) is an injection because C → D is. The middle two squares commute by the flat Mackey condition. The upper two composites F (B_{1}) → ∏ F (D_{α}) are the same by 3.2.1 applied to the correspondence from B_{1} to D. The left square commutes because transfer commutes with localization (compare B and R to a common localization). The result now follows by a diagram chase. □

Following [SV, 6.1], a functor F from normal algebras to abelian groups is called a qfh-sheaf if it defines a Zariski sheaf on each normal affine scheme Spec(B) and if F (A) → F (B)^{G} is an isomorphism for every pseudo-Galois extension A ⊂ B with Galois group G. Since any constant sheaf is a qfh-sheaf, it is clear that qfh-sheaves need not be p-divisible in characteristic p.

Corollary 6.13. Let F be a qfh-sheaf. If char(k) = p > 0, assume that the groups F (A) are uniquely p-divisible. Then F has transfer maps, making it into to a transfer functor.

Proof. Given A′ ⊃ A, we define the transfer map F (A′) → F (A) by embedding A′ in a pseudo-Galois extension A′′ with group G, and sending a ∈ F (A′) to

[k(A′) : k(A)]_{insep} ∑_{h∈Hom_{A}(A′,A′′)} h_{∗}(a),

regarded as an element of F (A′′)^{G} ≅ F (A). Suslin and Voevodsky verify the (flat) Mackey condition 3.3.2 in [SV, 5.17]. □

Let τ(A) denote the torsion submodule of the absolute Kähler differentials Ω_{A} where A is a noetherian domain. If A is regular, then τ(A) = 0 because Ω_{A} is projective. If A is normal, then the annihilator of τ(A) has height at least two. If pd_{A}Ω_{A} < ∞ and A is separable over k, this is actually equivalent to the normality of A by [Vas]. The following lemma uses the trick introduced in the proof of theorem 4.6.

Lemma 6.14. Suppose that ℚ ⊂ A. If f : A → B is a homomorphism of domains, then Ω_{A} → Ω_{B} sends τ(A) into τ(B).

Proof. We may assume that B is a field E. In characteristic zero, any field extension E ⊂ F induces an injection of Ω_{E} into Ω_{F}, so we may pass to arbitrary extensions of E. Let P be the kernel of f and x nonzero in P; passing first to A[P/x], normalizing and localizing at a height one prime, we get a DVR R between A and its quotient field such that (after extending E) f factors through R. But τ(A) already vanishes in the torsionfree module Ω_{R}. □

Lemma 6.15. Let A be a finite-dimensional augmented local algebra over a field K, so that there is a canonical splitting Ω_{A} ≅ Ω_{K} ⊕ Ω_{A,K}. Then the transfer map Ω_{A} → Ω_{K} is zero on Ω_{A,K}, and is multiplication by dim_{K}(A) on the summand Ω_{K}.

Proof. Choose a K-basis of A compatible with the filtration A ⊃ I ⊇ I^{2} ⊇ ⋯ ⊇ I^{n} = 0, where I is the kernel of the augmentation A → K. For any a ∈ A, the matrix (a_{ij}) representing multiplication by a is lower triangular, and strictly lower triangular if a ∈ I. Since the trace of a db is ∑_{i,j} a_{ij} db_{ji} and Ω_{A,K} is generated over K by the terms a db with either a or b in I, the result is immediate. □

Corollary 6.16. Let A be a finite-dimensional local algebra over a field K, with residue field L separable over K. Then the transfer Ω_{A} → Ω_{K} is length(A) times the projection Ω_{A} → Ω_{L}, followed by the usual transfer Ω_{L} → Ω_{K}.

Theroem 6.17. In characteristic zero, the Kähler differentials induce a transfer functor A ↦ Ω_{A}/τ(A).

Proof. This is a functor on normal k-algebras by 6.14, and preserves products and injections in characteristic zero. If A ⊃ B is a finite inclusion, the extension of quotient fields E ⊃ F induces a transfer Ω_{E} → Ω_{F} as in 2.10, sending dx to dy, y = trace(x). It is easy to see that this transfer sends the image Ω_{A}/τ(A) of Ω_{A} into the image Ω_{B}/τ(B) of Ω_{B}, yielding a transfer maps constructed so as to commute with localization.

Finally, the flat Mackey condition is easy to check. Let A ⊃ B be finite and flat, with quotient field extension E ⊃ F. If B → F′ is an injection, with F′ a field, the diagram

Ω_{A}/τ(A) ⟶ Ω_{E} ⟶ Ω_{E⊗_{F} F′}
↓  ↓  ↓
Ω_{B}/τ(B) ⟶ Ω_{F} ⟶ Ω_{F′}

commutes, which by 3.7 is the Mackey condition. Given this, it suffices to consider the maps B → B/P, where P is a prime ideal of B. The transfer Ω_{A} → Ω_{B} sends P Ω_{A} to P Ω_{B} and d(P A) to d(P), so it induces a transfer from Ω_{A/P A}/τ(A) to Ω_{B/P} /τ(B). We may localize B (and A) at P to assume that A/P A is a finite-dimensional algebra over the field K = B/P. If {P_{i}} are the primes of A over P, then A/P A = ∏ A_{i}, where each A_{i} is a finite-dimensional nilpotent extension of the field K_{i} = A/P_{i}. By 6.16, the transfer on Ω_{A_{i}} is the projection onto the submodule Ω_{K_{i}}, followed by λ_{i} = length(A_{i}) times the usual transfer Ω_{K_{i}} → Ω_{K}. □

Ω_{A} ⟶ Ω_{A/P A} ⟶^{project} Ω_{K_{i}}
↓  ↓  ↓^{λ_{i}·trace}
Ω_{B}/τ(B) ⟶ Ω_{B/P} ⟶^{=} Ω_{K}

References

[BH] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Univ. Press, 1993.

[Bour] N. Bourbaki, Algèbre Commutative, Herman, Paris, 1964.

[CPS] E. Cline, B. Parshall and L. Scott, Cohomology of finite groups of Lie type I, Publ. IHES 45 (1975), 169–191.

[D] A. Dress, Contributions to the theory of induced representations, Lecture Notes in Math., vol. 342, Springer-Verlag, 1973, pp. 183–240.

[Gr] J. Green, Axiomatic representation theory for finite groups, J. Pure Appl. Alg. 1 (1970), 41–77.

[H] E. Hecke, Zur Theorie der elliptischen Modulfunktionen, Math. Ann. 97 (1926), 210–242; Math Werke, 428–460.

[Lew] L. G. Lewis, An introduction to Mackey functors, lecture notes in preparation.

[Li] H. Lindner, A remark on Mackey functors, Manuscr. Math. 18 (1976), 273–278.

[Ma] G. Mackey, On induced representations of groups, Amer. J. Math. 73 (1951), 576–592.

[Mat] H. Matsumura, Commutative ring theory, Cambridge Univ. Press, 1986.

[Mi] J. Milne, Étale Cohomology, Princeton Univ. Press, 1980.

[MVW] C. Mazza, V. Voevodsky and C. Weibel, Lectures on Motivic cohomology, preprint, 2002,

        available at http://math.rutgers.edu/∼weibel.

[RS] K. Roggenkamp and L. Scott, Hecke actions on Picard groups, J. Pure Appl. Alg. 26 (1982), 85–100.

[RW] Y. Rhie and G. Whaples, Hecke operators in cohomology of groups, J. Math. Soc. Japan 22 (1970), 431–442.

[Sh] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, Iwanami Shoten and Princeton Univ. Press, 1971.

[Sh1] G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 1959 291–311.

[SV] A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), 61–94.

[SV1] A. Suslin and V. Voevodsky, Relative Cycles and Chow Sheaves, Cycles, Transfers and Motivic Cohomology Theories, Princeton Univ. Press, 2000, pp. 10–86.

[V] V. Voevodsky, Triangulated Categories of Motives over a field, Cycles, Transfers and Motivic Cohomology Theories, Princeton Univ. Press, 2000, pp. 188–238.

[Vas] W. Vasconcelos, A note on normality and module of differentials, Math. Zeit. 105 (1968), 291–293.

[W81] C. Weibel, KV -theory of categories, Trans. AMS 267 (1981), 621–635.

[Y] T. Yoshida, On G-functors (II): Hecke operators and G-functors, J. Math. Soc. Japan 1 (1983), 179–190.

Rutgers University

   E-mail address: weibel@math.rutgers.edu