INVARIANTS OF REAL CURVES ^{∗}

Dedicated to Paolo Salmon on his 60^{th} birthday

by

Claudio Pedrini^{1} and Charles Weibel^{2}

Abstract. In this paper we study several algebraic invariants of a real curve X: the Picard group Pic(X), the Brauer group Br(X), the K-theory matrix invariant SK_{1}(X), the étale cohomology groups H^{n}(X, Z/2), and the sheaves H^{n} associated to these cohomology groups. We relate these to two topological invariants of the space X(R) of real points of X: the number c of components and the number λ of loops. For singular X we compute PicX/2PicX and construct a Mayer-Vietoris sequence relating Br(X) to SK_{1}(X)/2SK_{1}(X) and use it to show that Br(X) = (Z/2)^{c} and SK_{1}(X)/2SK_{1}(X) = (Z/2)^{λ}.

§0. Introduction.

In this paper we study several algebraic invariants of a real curve X. By “real curve”we mean a quasiprojective 1-dimensional reduced scheme defined over R. The invariants are: the Picard group Pic(X), the Brauer group Br(X), the Ktheory matrix invariant SK_{1}(X), the étale cohomology groups H_{et}^{n}(X, Z/2), and the sheaves H^{n} associated to these cohomology groups. We relate these to two topological invariants of the space X(R) of real points (with the euclidean topology): the number c of connected components and the number λ of loops. We show that:

(0.1) Br(X) ≅ (Z/2)^{c}. This was previously known in the smooth projective case by [Witt] and in the affine case by [DK]. Since Br(X) = H^{0}(X, H^{2}), the smooth case also appears as a particular case of a general result in [CTP]: if X is a smooth quasi-projective variety over R then (Z/2)^{c} = H^{0}(X, H^{n}) for any n >dim X.

(0.2) There is a natural surjection from Pic(X)⊗Z/2 to (Z/2)^{λ}. If X is smooth this is an isomorphism except when X is projective and X(R) = ∅,when λ=0 but Pic(X) ⊗ Z/2 = Z/2. For general X we have Pic(X) ⊗ Z/2 ≅ (Z/2)^{λ+E}, where E is the number of components of X which are projective and have either no real points or have at most a finite set of isolated real points (Th.1.10).

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For smooth projective curves the calculation of Pic(X) goes back to Weichold’s 1882 thesis [Whd]. This includes some subtleties when X(R) = ∅ (prop. 1.1). For affine curves our calculation corrects an example of DeMeyer and Knus [DK, 3.5], where inequality is claimed.

(0.3) SK_{1}(X) ≅ (Z/2)^{λ} ⊕ D. Here D is a divisible abelian group, which is uniquely divisible if X is affine. In particular, SK_{1}(X) ⊗ Z/2 ≅ Z/2^{λ}. This seems to be a new result. The interest in SK_{1}(X) lies in the fact that for an affine curve X = Spec(A), SK_{1}(X) is the quotient SL_{3}(A)/E_{3}(A) of the 3 × 3 special linear group SL_{3}(A) (matrices of determinant 1) by E_{3}(A), the subgroup generated by the elementary matrices. Therefore computation of SK_{1}(A) ⊗ Z/2 reduces to a computation in a group of matrices. Our key technical result for this is the isomorphism SK_{1}(X) ⊗ Z/2 ≅ H^{1}(X, H^{2}), which in the smooth case is implicit in [MS].

(0.4) Let X̃ be the normalization of X, Y = Sing(X) and Ỹ = Y ×_{X} X̃. Then there is a natural exact sequence:

0 ⟶ Br(X) ⟶ Br(X̃)⊕Br(Y) ⟶ Br(Ỹ) ⟶ SK_{1}(X)⊗Z/2 ⟶ SK_{1}(X̃)⊗Z/2 ⟶ 0.

The first three terms of this sequence were found by DeMeyer and Knus when X is affine [DK]. The surprise is the continuation of the DeMeyer-Knus sequence to part of the Mayer-Vietoris sequence in algebraic K-theory.

(0.5) We can also compute all of the étale cohomology groups H_{et}^{i}(X, Z/2) for a real curve (Cor. 1.11). For i≥3 the isomorphism H_{et}^{i}(X, Z/2) ≅ (Z/2)^{c+λ} can be easily deduced from [Cox,1;2]. For i=2 we show that H_{et}^{i}(X, Z/2) ≅ (Z/2)^{c+λ+E}. In the case where X is either affine or smooth projective with X(R) ≠0, this appeared in [CT] as conseguence of Cox’s results.

We have laid out this paper as follows. §1 contains results and computations on the Picard group of real curves, as well as the calculations (0.5) on étale cohomology. In §2 we consider a Mayer-Vietoris sequence which relates the sheaves H^{q} with π_{∗}H^{q}, where π : X̃ ⟶ X is a finite birational map. Then we apply this to the case of a curve X and its normalization. In §3 we consider the Brauer group, proving (0.1). In §4 we give some technical results about the sheaf K_{2}/ℓ on a curve, using Chern classes and K-theory mod ℓ. These results are used in §5, where proofs of (0.3), and (0.4) are given. We give examples (see 5.7.1) computing SK_{1}(X) of a singular curve X obtained from a smooth affine curve by glueing points.

We want to thank L.Barbieri-Viale, J.L. Colliot-Thélène, J.F. Jardine and L. Mahe for useful discussions on some of the matters contained in this paper.

Notation. If A is an abelian group, we shall write _{ℓ}A for the ℓ-torsion subgroup {a ∈ A : ℓa = 0}. G_{m} and μ_{ℓ} are the étale sheaves of units and ℓ^{th} roots of unity, respectively. A “real curve”is a reduced quasiprojective 1-dimensional scheme defined over the real numbers R; if X is a real curve we write X(R) for the associated topological space of real points of X, topologized using the Euclidean topology.

§1. Picard groups of real curves.

We first turn our attention to the Picard group Pic(X) of an irreducible smooth projective real curve X. If X happens to be defined over C then of course X(R) = ∅ and Pic(X) ≅ Z ⊕ (R/Z)^{2g}. Otherwise X is geometrically irreducible and we set X_{C} = X ×_{R} C.

Proposition 1.1. Let X be geometrically irreducible, smooth, projective curve over R with c real components and genus g. Then Pic(X) is a subgroup of Pic(X_{C}) and as an abelian group:

Pic(X) ≅ Z × (R/Z)^{g} × \{ \begin{matrix}(Z/2)^{c−1} & if c ≠ 0 \\ 0 & if c = 0 \end{matrix}\} .

Thus Pic(X) ⊗ Z/2 ≃ (Z/2)^{c} if c ≠ 0, while Pic(X) ⊗ Z/2 ≃ Z/2 if c = 0.

Proof. Let A = Pic^{0}(X_{C}) be the 2g-dimensional Picard variety of X_{C}, so that as abelian groups Pic(X_{C}) ≅ Z × A and A ≅ (R/Z)^{2g}. The Galois group G = Gal(C/R) acts naturally on A and on Pic(X_{C}), and there is an exact sequence of G-modules: 0 ⟶ A ⟶ PicX_{C} ⟶ Z ⟶ 0. (The map from PicX_{C} to Z is the degree map.)

By using the calculations of Weichold [Whd] (and Klein [K,§7] when c = 0) on periods of real integrals, Comessatti [C1] and [C2] and Witt [Witt] showed that the invariant subgroup of A is:

A^{G} ≅ (R/Z)^{g} × \{ \begin{matrix}(Z/2)^{c−1} & if c ≠ 0 \\ 0 & if c = 0 and g is even \\ Z/2 & if c = 0 and g is odd \end{matrix}\} .

If c ≠ 0 it is well-known that Pic(X) = (PicX_{C})^{G} = Z × A^{G}, and we are done.

To analyze the case c=0, consider the Leray spectral sequence associated to the morphism f : X ⟶ Spec(R) and the sheaf G_{m} on X_{et}. This yields an exact sequence

0 ⟶ PicX ⟶ (PicX_{C})^{G} ⟶ BrR ⟶ H^{2}(X, G_{m}),

where BrR = Z/2 and H^{2}(X, G_{m}) = BrX because X is a real curve. Because X(R) = ∅, we have BrX = 0 by [Witt] (or 3.6 below), yielding the exact sequence:

(1.1.1)  0 ⟶ PicX ⟶ (PicX_{C})^{G} ⟶ Z/2 ⟶ 0.

So if c=0, PicX has index 2 in (PicX_{C})^{G}. The degree map PicX ⟶ PicX_{C} ⟶ Z has image 2Z and its kernel Pic^{0}X embeds into A^{G} via the following map of exact sequences:

0 ⟶ Pic^{0}X ⟶ PicX ⟶ 2Z ⟶  0
↓  ↓  ↓
0 ⟶ A^{G} ⟶ (PicX_{C})^{G} ⟶ Z ⟶ H^{1}(G, A)

Let σ denote complex conjugation. Since X(R) = ∅ every divisor on X is of the form x + σ(x) for some divisor x on X_{C}. Hence the usual transfer map from A = Pic^{0}X_{C} to Pic^{0}X̄ which sends x to x + σ(x) is onto. Therefore we may identify Pic^{0}X in A = Pic^{0}X_{C} with the image of the continuous map τ : A ⟶ A sending a to a + σ(a). A is a connected Lie group, so the image of τ is a connected Lie subgroup of A^{G} containing 2A^{G}. If g is even, this gives Pic^{0}X = A^{G}; if g is odd, τ cannot hit the subgroup Z/2 of A^{G}, and we are done.

Remark 1.1.2: If X(R) = ∅, the G-module structure of Pic(X_{C}) depends upon the parity of the genus g. Using Commessatti’s calculations as in [Sil, Prop.3], we find that:

H^{1}(G, A) ≅ \{ \begin{matrix}0 & if g is even \\ Z/2 & if g is odd. \end{matrix}

Therefore as a subgroup of Pic(X_{C}) we have (cfr. [Sil,Prop.10])

Pic(X_{C})^{G} ≅ \{ \begin{matrix}Z ⊕ (R/Z)^{g} ⊕ (Z/2)^{c−1} & if c ≠ 0 \\ Z ⊕ (R/Z)^{g} & if c = 0 and g is even \\ 2Z ⊕ (R/Z)^{g} ⊕ (Z/2) & if c = 0 and g is odd. \end{matrix}

If g is even then Pic(X_{C}) ≅ A × Z as a G-module, but if g is odd then PicX_{C} is not split as a G-module. In contast, the sequence (1.1.1) is not split when g is even, but is split when g is odd. In fact, when g is even, the map (PicX_{C})^{G} ⟶ Z/2 in (1.1.1) is just the degree map modulo 2.

Historical Remark 1.1.3 In his 1882 Leipzig thesis [Whd], Weichold computed the period matrix Ω which gives the pairing

H_{1}(X_{C}, Z) × H^{0}(X_{C}, Ω^{1}) ⟶ C.

The columns of Ω generate the lattice in C^{g} giving the Jacobian variety of X_{C}. Weichold’s calculations were later refined by Klein [K]. Using the “pseudonormal”matrix of periods of real integrals Comessatti [C1] [C2] defined the so-called “real character”of a real abelian variety in the 1920’s; by applying this to the case of the Jacobian A of a real curve of genus g he computed A^{G}. In the case X(R) ≠ ∅ this also yields a computation of PicX.

A modern translation of these results, using the action of the Galois group and cohomology may be found in [Sil]. The case X(R) = ∅ has been considered by Gross and Harris in [GH, prop.2.2], where they also establish (1.1.1) and compute PicX ⊗ Z/2 [GH, prop.4.2]. We warn the reader that the calculation of PicX/2PicX attributed to Witt in [DK,1.7] is false when X(R) = ∅.

Example 1.2. In order to motivate the definition of loops below, consider the following classes of smooth affine curves X, each obtained from a smooth irreducible projective curve X̄ by removing points. Let c be the number of connected components of X(R); each component is diffeomorphic to a circle.

Suppose first that we remove a single point p. Then there is an exact sequence

Z ⟶^{i_{p}} PicX̄ ⟶ PicX ⟶ 0.

If p is a real point then geometrically X(R) = X̄(R) − {p} consists of λ = c − 1 smooth circles and one unbounded component, which is diffeomorphic to the real line. In this case the image of i_{p} is Z, and PicX ≅ (R/Z)^{g} ⊕ (R/Z)^{c−1}. If p is a “complex point”(i.e., a scheme p = SpecC), then geometrically X(R) = X̄(R) consists of λ = c smooth circles. If X is defined over C, then the image of i_{p} is Z and PicX = (R/Z)^{g}. Otherwise, X is geometrically irreducible and the image of i_{p} is 2Z. In this case PicX ≃ (R/Z)^{g} ⊕ (Z/2)^{c}.

Now suppose that X is obtained from X̄ by removing a finite set {p_{1}, ⋯, p_{r}} of more than one point. The subgroup Γ of Pic(X̄) generated by the classes of the p_{i}’s can be fairly complicated, but as an abelian group the quotient Pic(X) = Pic(X̄)/Γ will be the sum of a divisible abelian group and a finite number λ of copies of Z/2. Geometrically, it is not hard to show that X(R) consists of λ smooth circles and c − λ unbounded components (each diffeomorphic to the real line).

Definition 1.3. Let X be a curve defined over R, possibly singular. The \underline{number of loops} λ(X) of X(R) is the dimension of H^{1}(X(R), Z/2).

Topologically, each connected component of X(R) is either contractible (no loops), a circle (one loop) or (when X is singular) homotopy equivalent to a bouquet of a finite number λ_{i} of circles (λ_{i} loops); λ(X) is the sum over the set of components of X(R) of the number of the loops on each component. If X is smooth and projective then λ = c is just the number of connected components of X(R). A singular example is the node y^{2} = x^{2} + x^{3}; the affine node has λ = 1 loop but the projective node has λ = 2 loops.

Combining Prop. 1.1 with the above definition and with (1.2) we get

Corollary 1.4. Let X be a irreducible smooth real curve with λ loops. Then with one exception:

Pic(X) ⊗ Z/2 ≅ (Z/2)^{λ}.

The exception is an irreducible projective curve with X(R) = ∅ , when

Pic(X) ⊗ Z/2 ≅ Z/2.

More generally, if X is any smooth real curve with λ loops, let E denote the number of irreducible algebraic components X_{i} of X which are projective and for which X_{i}(R) is empty. Then:

Pic(X) ⊗ Z/2 ≅ (Z/2)^{λ+E}.

In the remainder of this section we are going to extend the computation of Pic(X) ⊗ Z/2 to any real curve.

Let π : X′ ⟶ X be a finite morphism of curves, Y = SingX, Y′ = π^{−1}(Y). The cartesian square

Y′ ⟶ X′
↓  ↓^{π}
Y ⟶ X

yields the following Mayer-Vietoris sequence

0 ⟶ U(X) ⟶ U(X′) ⊕ U(Y) ⟶^{Φ} U(Y′) ⟶ PicX ⟶ PicX′ ⟶ 0

where U(X) = H^{0}(X, O_{X}^{∗}) is multiplicative group of global units on X etc. For example U(Y′) consists of one copy of R^{∗} for every real point of Y′ and one copy of C^{∗} for every complex point. Tensoring with Z/2, the Mayer-Vietoris exact sequence yields the following one

(1.5) H ⊕ (U(Y) ⊗ Z/2) ⟶ U(Y′) ⊗ Z/2 ⟶ PicX ⊗ Z/2 ⟶ PicX′ ⊗ Z/2 ⟶ 0

where H = (U(X′) ⊗ Z/2) ⊕ _{2}PicX.

If U(Y) ⊗ Z/2 maps onto U(Y′) ⊗ Z/2 then PicX ⊗ Z/2 ≅ PicX′ ⊗ Z/2. This is the case if Y′(R) = ∅, or more generally when the map Y′(R) ⟶ Y (R) is 1-1. This observation motivates the following

Definition 1.6. Let X be a real curve with normalization X̃and singular locus Y. We say that an intermediate curve X̃ ⟶ X′ ⟶ X has property (P) if no two real points of Y′ = Y ×_{X} X′ lie over the same (real) point of Y. If X′_{1} and X′_{2} have this property then so does X′_{1} ×_{X} X′_{2}. Therefore there is a largest intermediate curve X′ having property (P); we call X′ the \underline{partial normalization} of X. It may be obtained from X̃ by glueing together the real points lying over each point of Y. From (1.5) it follows that PicX ⊗ Z/2 ≅ PicX′ ⊗ Z/2 if X′ is the partial normalization of X.

Example 1.7. Let X be the affine real curve defined by y^{2} = x^{2}(x + 1)(x + 2)^{2}. If t = \frac{y}{x+2} then the partial normalization X′ is the affine node t^{2} = x^{2}(x + 1) and the normalization is the affine line X̃ = Spec(R[\frac{t}{x}]). In this case Y consists of two real points, Y′ consists of one real point and one complex point, and X′ is obtained from X̃ by glueing the two real points \frac{t}{x} = ±1 together into the real point of Y′.

Proposition 1.8. (Cox, cf.[CT]). If X is any real curve then

H^{i}(X, Z/2) ≅ (Z/2)^{c+λ} for all i ≥ 3.

Proof. The result may be deduced from [Cox, 1.2] as follows. Let X be a real variety, G the Galois group of C over R, EG the infinite sphere S^{∞}. EG is a principal G-space with quotient BG = RP^{∞}, where G acts on EG by the antipodal map. Let X(C) ×_{G} EG be the quotient of X(C) × EG by the diagonal action of G. By [Cox,1.1] there is a isomorphism for any q ≥ 0 :

H_{et}^{q}(X, Z/2) ≅ H^{q}(X(C) ×_{G} EG, Z/2).

Applying the Künneth formula, we also have isomorphisms for any q ≥ 1:

H^{q}(X(R) × BG, Z/2) ≅ H^{0}(X(R), Z/2) ⊕ H^{1}(X(R), Z/2) ≅ (Z/2)^{c} ⊕ (Z/2)^{λ}.

To compare these isomorphisms, we introduce the quotient space X_{∞} = X(C)/G. X_{∞} inherits the Euclidean topology from X(C) and X(R) is a closed subset both of X_{∞} and X(C). By [Bred, VII.1.1] the relative cohomology groups H^{∗}(X_{∞}, X(R), Z/2) and H^{∗}(X(C) ×_{G} EG, X(R) × BG, Z/2) are isomorphic. Using this and the exact sequence of the pair (X(C)×_{G} EG, X(R)×BG) one gets a long cohomology exact sequence [Cox, 1.2]

⋯ H^{q}(X_{∞}, X(R), Z/2) ⟶^{α^{q}} H^{q}(X(C)×_{G}EG, Z/2) ⟶ H^{q}(X(R)×BG, Z/2) ⟶ ⋯

Since X_{∞} is a 2-dimensional complex, H^{q}(X_{∞}, X(R), Z/2) = 0 for q ≥ 3 and the proposition follows.

Lemma 1.9. Let X be a real curve and let X_{i} be its irreducible algebraic components. Assume that for each i either X_{i} is affine or X_{i}(R) ≠ ∅. Then

H_{et}^{2}(X, Z/2) ≅ (Z/2)^{c+λ}.

Proof. We use the notation of the proof of Cox’s result. The map α^{q} factors through H^{q}(X_{∞}, Z/2) via the commutative diagram:

H^{q}(X_{∞}, X(R), Z/2) ⟶^{≅} H^{q}(X(C) ×_{G} EG, X(R) × BG, Z/2)
↓  ↓_{α^{q}}
H^{q}(X_{∞}, Z/2) ⟶^{β^{q}}  H^{q}(X(C) ×_{G} EG, Z/2)
↓  ↓
H^{q}(X(R), Z/2) ⟶  H^{q}(X(R) × BG, Z/2)

If X is a curve then as cellular complexes we have dim(X_{∞})=2 and dim(X(R))=1. Therefore we get an exact sequence:

H^{2}(X_{∞}, Z/2) ⟶^{β^{q}} H_{et}^{2}(X, Z/2) ⟶ H^{2}(X(R) × BG, Z/2) ⟶ 0.

We claim that H_{2}(X_{∞}, Z/2) = 0, and therefore that H^{2}(X_{∞}, Z/2) = 0. As in the proof of 1.8, this will yield our statement:

H_{et}^{2}(X, Z/2) ≅ H^{2}(X(R) × BG, Z/2) = (Z/2)^{c+λ}

To establish the claim, we may assume that X is irreducible. Indeed, X_{∞} = X(C)/G differs from the disjoint of the X_{i}(C)/G by a finite set of points, so H_{2}(X_{∞}, Z/2) is the direct sum of the H_{2}(X_{i}(C)/G, Z/2).

We now assume X is irreducible and write M for X_{∞} considered as a 2dimensional simplicial complex. Because of our assumptions, M is a pseudomanifold with boundary X(R) in the sense of [Spa,p.150]. That is, every edge (1-simplex) is the boundary of at most two faces (2-simplices) of M and every pair of faces may be connected by a sequence of faces and edges.

The following counting argument shows that H_{2}(M, Z/2) = 0. Because M is a pseudomanifold, the only possible simplicial 2-cycle (modulo 2) is the sum σ of the faces of M. If X is affine then M is not compact, so there are infinitely many faces and such a 2-cycle cannot exist. If M is compact, the boundary of σ is the sum of the edges in the boundary X(R) of M. If X(R) ≠ ∅ then the 2-chain σ cannot be a 2-cycle, so H_{2}(M, Z/2) = 0.

Theorem 1.10. Let X be any real curve. Then

PicX ⊗ Z/2 ≅ (Z/2)^{λ+E}.

Where λ is the number of loops in X(R) and E is the number of irreducible algebraic components X_{i} of X which are proper and for which X_{i}(R) is rither empty or finite.

Proof. If X is smooth then the result follows from 1.4.

Assume X is singular. The numbers λ and E do not change if we pass from X to the partial normalization X′. Moreover PicX ⊗ Z/2 ≅ PicX′ ⊗ Z/2. By this process we may assume that there are no complex points in the normalization X̃ lying over the singular points of X. Hence we may assume that X is a disjoint union of E components X_{i} which are smooth, proper and have X_{i}(R) = ∅, together with a curve for which E = 0. Using 1.4 we are now left to show that if E=0 then PicX ⊗ Z/2 ≅ (Z/2)^{λ}. From Lemma 1.9 we get the isomorphism

H_{et}^{2}(X, Z/2) ≅ (Z/2)^{c+λ}.

We conclude using the exact Kummer sequence

0 ⟶ PicX ⊗ Z/2 ⟶ H_{et}^{2}(X, Z/2) ⟶ _{2}BrX ⟶ 0

and the isomorphism BrX ≅ (Z/2)^{c}; see §3 for the proof of this isomorphism.

Example 1.10.1 If X is the projective curve defined by XY (X^{2} + Y^{2}) = 0, then λ = 2 and E = 1. Thus Pic(X) ⊗ Z/2 ≅ (Z/2)^{3}.

Corollary 1.11. Let X be a real curve. Then:

a) H_{et}^{0}(X, Z/2) ≅ (Z/2)^{a} where a is the number of connected components of X

b) H_{et}^{1}(X, Z/2) ≅ H^{0}(X, O_{X}^{∗})/2 ⊕ _{2}PicX.

c) H_{et}^{2}(X, Z/2) ≅ (Z/2)^{c+λ+E}

d) H_{et}^{i}(X, Z/2) ≅ (Z/2)^{c+λ} for every i ≥ 3.

Proof. a) is obvious and b) comes from Kummer theory. d) is 1.8. c) follows from 1.10 and the isomorphism BrX ≃ (Z/2)^{c} of 3.6 below.

Remark 1.12 Let X be a irreducible affine real curve such that X(R) ≠ ∅ and X has no isolated singularities. Then 1.10 yields PicX ⊗ Z/2 ≃ (Z/2)^{λ}. This seems to contradict the example in [DK,3.5], where X is the curve obtained from the curve X̃ : x^{4} = y^{2} − 1 by glueing together the points A = (1, 0) and B = (−1, 0). X has no loops and E=0, yet in [DK] it is claimed that PicX ⊗ Z/2 ≅ Z/2. In fact their claim is false: the following correct computation yields PicX ⊗ Z/2=0.

The Mayer-Vietoris sequence for units and Pic yields the exact sequence

0 ⟶ U(X) ⟶ U(X̃) × R^{∗} ⟶ R^{∗} × R^{∗} ⟶ PicX ⟶ PicX̃ ⟶ 0

and the image of the map U(X̃) ⟶ R^{∗} × R^{∗} is the diagonal copy of R^{∗}. Now PicX̃ ≅ R/Z, so PicX̃ ⊗ Z/2=0 and _{2}PicX̃ ≅ Z/2. Tensoring with Z/2 we get the following exact sequence:

0 ⟶ Z/2 ⟶ _{2}PicX ⟶^{φ} _{2}PicX̃ ⟶^{ψ} R^{∗}/(R^{∗})^{2} ⟶ PicX ⊗ Z/2 ⟶ 0

(where Z/2 is the 2-torsion in R^{∗}). To conclude it is enough to show that φ is the zero map and ψ is surjective. Let R[x, y] be the coordinate ring of X̃: we have (x − 1) = 2A, (x + 1) = 2B as divisors on X̃. Hence 2A = 2B = 0 in PicX̃. Let t = x^{2} − y: t is a unit in R[x, y] and the divisor of \frac{y}{t+1} is A + B, hence A = B in PicX̃. Let L be the line through A: a(X − 1) + bY =0 where a ≠0 b ≠0. Then (L) = A + D in PicX̃ where D is a divisor not supported on A nor on B. Therefore A represents an element D in PicX too. Let f = \frac{L^{2}}{x−1} then (f) = 2D so that 2D=0 in PicX̃. Moreover f(A) > 0 and f(B) < 0. Therefore the image of f ∈ U(X̃) represents the class of {−1} in R^{∗}/(R^{∗})^{2}, hence ψ is surjective. Finally, 2D ≠ 0 in _{2}PicX and 2D = 0 in _{2}PicX̃; therefore φ is the zero map.

§2. Cohomological interlude. Let π : X̃ ⟶ X be a proper birational map of schemes which are essentially of finite type over a field k. Then there is a reduced closed subscheme Y of X with inverse image Ỹ = π^{−1}(Y) such that U = X − Y and Ũ = X̃ − Ỹ are isomorphic. If we write i and j for the inclusions Y ⊂ X and Ỹ ⊂ X̃, we have a square

Ỹ ⟶^{j} X̃
↓^{πỸ}  ↓^{π}
Y ⟶^{i} X.

Proposition 2.1. For every torsion abelian sheaf T on the big étale site of X there is a Mayer-Vietoris sequence in étale cohomology:

⋯ H_{et}^{q}(X, T) ⟶ H_{et}^{q}(X̃, T) ⊕ H_{et}^{q}(Y, T) ⟶ H_{et}^{q}(Ỹ , T) ⟶^{δ} H_{et}^{q+1}(X, T) ⋯

Proof. Writing η : U ⊂ X and γ : Ũ ⊂ X̃ for the inclusions of U = X − Y and Ũ = X̃ − Ỹ, we have the canonical exact sequences of étale sheaves on X and X̃:

0 ⟶ η_{!}(T |U) ⟶ T_{X} ⟶ i_{∗}(T |Y) ⟶ 0
0 ⟶ γ_{!}(T |Ũ) ⟶ TX̃ ⟶ j_{∗}(T |Ỹ) ⟶ 0.

There is also a map of the cohomology long exact sequences:

⋯H^{q+1}(Y, T) ⟶H^{q}(X, η_{!}(T |U)) ⟶H^{q}(X, T) ⟶ H^{q}(Y, T)⋯
↓  ↓  ↓  ↓
H^{q+1}(Ỹ , T) ⟶H^{q}(X̃, γ_{!}(T |Ũ)) ⟶H^{q}(X̃, T) ⟶H^{q}(Ỹ , T)⋯

where H^{∗}(Y, T) = H^{∗}(X, i_{∗}(T |Y)), H^{∗}(X̃, j_{∗}(T |Ỹ)) = H^{∗}(Ỹ , T). By a standard diagram chase it suffices to show that when Ũ ≅ U the maps from H^{∗}(X, η_{!}(T |U)) to H^{∗}(X̃, γ_{!}(T |Ũ)) are isomorphisms. If X is complete, this is just the fact that the cohomology groups with compact support H_{c}^{q}(U, T) are independent of the embedding U ⊂ X [M, V.3.1]. The proof in general is similar. First one observes that the sheaves π_{∗}γ_{!}(T |Ũ) and η_{!}(T |U) are isomorphic because they are both supported on U and they have the same stalks at every point of U. Hence H^{∗}(X, η_{!}(T |U)) ≅ H^{∗}(X, π_{∗}γ_{!}(T |Ũ)).

Then one considers the Leray spectral sequence

H^{p}(X, R^{q}π_{∗}γ_{!}(T |Ũ)) ⇒ H^{p+q}(X̃, γ_{!}(T |Ũ)).

According to the proper base change theorem [M, VI 2.5] the stalk of the sheaf R^{q}π_{∗}γ_{!}(T |Ũ) at a geometric point x̄ of X may be computed on the geometric fibres of π. Since the fiber consists of one point if x ∈ U and is 0 if x ∉ U, we get R^{q}π_{∗}(γ_{!}(T |Ũ))=0 for q >0. Thus the spectral sequence degenerates yielding the isomorphism

H^{p}(X, π_{∗}γ_{!}(T |Ũ)) ≅ H^{p}(X̃, γ_{!}(T |Ũ)).

The sequence of 2.1 is natural, so we may sheafify in the Zariski topology of X.The resulting sequence of sheaves on X is

(2.2)  ⋯ H_{X}^{q} ⟶ R^{q}π_{∗}(T) ⊕ i_{∗}H_{Y}^{q} ⟶ R^{q}(πj)_{∗}(T) ⟶ H_{X}^{q+1} · · · .

Here H_{X}^{q} and H_{Y}^{q} are the sheaves on X_{Zar} and Y_{Zar} associated to the presheaf U ↦ H_{et}^{q}(U, T) and R^{q}π_{∗}(T) is the sheaf associated to the presheaf U ↦ H_{et}^{q}(π^{−1}(U), T). By inspection, R^{0}π_{∗}(T) ≅ π_{∗}H0 ; X̃. Thus H_{X}^{1} ≅ O_{X}^{∗} ⊗ Z/ℓ and the sequence (2.2) begins with the sequence

0 ⟶ H_{X}^{0} ⟶ π_{∗}H0 ; X̃ ⊕ i_{∗}H_{Y}^{0} ⟶ (πj)_{∗}H0 ; Ỹ ⟶^{δ} H_{X}^{1}.

Example 2.3. Let T be the étale sheaf μ_{ℓ} of ℓ^{th} roots of unity, where char(k) ≠ ℓ. Then H_{X}^{0} is the Zariski sheaf μ_{ℓ}. Since Pic(O_{X,x}) = 0, Kummer theory quickly yields isomorphisms for every point x in X:

(H_{X}^{1})_{x} ≅ O_{X,x}^{∗} ⊗ Z/ℓ  and  (H_{X}^{2})_{x} ≅ _{ℓ}Br(O_{X,x}).

Here _{ℓ}Br(O_{X,x}) is the ℓ-torsion in the Brauer group, which equals the ℓ-torsion in H_{et}^{2}(SpecO_{X,x}, G_{m}). Thus H_{X}^{1} ≅ O_{X}^{∗} ⊗ Z/ℓ and the sequence (2.2) begins

0 ⟶ μ_{ℓ}(X) ⟶ μ_{ℓ}(X̃) ⊕ μ_{ℓ}(Y) ⟶ μ_{ℓ}(Ỹ) ⟶^{δ} O_{X}^{∗} ⊗ Z/ℓ.

The boundary map δ can be nonzero, since Ỹ may have more roots of unity than either X̃ or Y.

Returning to (2.2), we remark that in general the sheaves R^{q}π_{∗}(T) can be fairly complicated. The stalk of R^{q}π_{∗}(T) at x ∈ X is H_{et}^{q}(V, T), where V = π^{−1}(Spec(O_{X,x})). Since the stalk of the direct image sheaf π_{∗}(Hq ; X̃) at x ∈ X is H_{Zar}^{0}(V, H_{V}^{q}), there is a canonical map

η^{q} : R^{q}π_{∗}(T) ⟶ π_{∗}Hq ; X̃.

This is not an isomorphism in general, being merely an edge map in the Leray spectral sequence for X_{et} ⟶ X_{Zar}:

E_{2}^{pq} = R^{p}π_{∗}(Hq ; X̃) ⇒ R^{p+q}π_{∗}(T),

which stalkwise is the Leray spectral sequence

E_{2}^{pq} = H_{Zar}^{p}(V, H^{q}) ⇒ H_{et}^{p+q}(V, T).

Example 2.4. This spectral sequence does not always degenerate, even if k is algebraically closed and T = μ_{ℓ}.

(a) Let π : X̃ ⟶ X be the blow up of the affine cone X over a smooth projective curve C, with Y = {y} the vertex, and Ỹ ≅ C. For p ≠ 0, R^{p}π_{∗}(H^{q}) is the skyscraper sheaf i_{∗}H^{p}(V, H^{q}) where V = π^{−1}(SpecO_{X,y}). For p = q = 1, we know from [BO,7.7] that

H^{1}(V, H^{1}) ≅ Pic(V) ⊗ Z/ℓ ≅ Pic(C) ⊗ Z/ℓ ≅ Z/ℓ.

On the other hand, [BO, (0.3)] states that H^{p}(V, H^{q}) = 0 for p > q, so R^{1}π_{∗}μ_{ℓ} ≅ π_{∗}H^{1} = π_{∗}OX̃ ⊗ Z/ℓ and we have the exact sequence

0 ⟶ i_{∗}(Z/ℓ) ⟶ R^{2}π_{∗}(μ_{ℓ}) ⟶ π_{∗}H^{2} ⟶ 0.

(b) Let X̃ be the triangle xy(x + y − 1) = 0 in the affine plane, and X the simple node defined by t^{2} = s^{2} + s^{3}. There is a finite cover π : X̃ ⟶ X ramified at the node point Y = {y}. In this case it is easy to see that H_{Zar}^{1}(V, μ_{ℓ}) ≅ Z/ℓ, so this time there is an exact sequence

0 ⟶ i_{∗}(Z/ℓ) ⟶ R^{1}π_{∗}(μ_{ℓ}) ⟶ π_{∗}H^{1} ⟶ 0.

In order to proceed further we need to identify the sheaf R^{q}π_{∗}(T) with π_{∗}Hq ; X̃. Example 2.4 (a) shows this need not be the case when π is not finite; example 2.4 (b) shows that even assuming π finite and T ≅ μ_{ℓ}^{⊗n} is not enough. However for q=1,2 we have the following two results.

Lemma 2.5. If π is finite and T = μ_{ℓ}, where char k ≠ ℓ, then the map of sheaves

R^{1}π_{∗}(μ_{ℓ}) ⟶ R^{1}(πj)_{∗}(μ_{ℓ}) is onto.

Proof. We show that the map R^{1}π_{∗}(μ_{ℓ}) ⟶ R^{1}(πj)_{∗}(μ_{ℓ}) is stalkwise onto, hence onto. For x ∈ Y, V = π^{−1}(O_{X,x}) is Spec(A) for some semilocal ring A. Set V |Y = V ×_{X} Y = π^{−1}(O_{Y,x}). Then V |Y is Spec(A/I) for some ideal I of A. The map of stalks at x ∈ Y is H_{et}^{1}(V, μ_{ℓ}) ⟶ H^{1}(V |Y, μ_{ℓ}). Since A is semilocal, Pic(A) = Pic(A/I) = 0 and A^{∗} ⟶ (A/I)^{∗} is onto. By Kummer theory, H_{et}^{1}(V, μ_{ℓ}) = A^{∗} ⊗ Z/ℓ and H_{et}^{1}(V |Y, μ_{ℓ}) = (A/I)^{∗} ⊗ Z/ℓ.

Lemma 2.6. If π is finite, dim(Y) = 0 and T = μ_{ℓ}^{⊗2} (char k ≠ ℓ) then the map R^{2}π_{∗}(μ_{ℓ}^{⊗2}) ⟶ R^{2}(πj)_{∗}(μ_{ℓ}^{⊗2}) is onto.

Proof. Keeping the notation of the previous lemma, the map of stalks at x ∈ Y is H_{et}^{2}(V, μ_{ℓ}^{⊗2}) ⟶ H^{2}(V |Y, μ_{ℓ}^{⊗2}), the bottom arrow in the commutative diagram:

H^{1}(V, μ_{ℓ}) ⊗ H^{1}(V, μ_{ℓ}) ⟶ H^{1}(V |Y, μ_{ℓ}) ⊗ H^{1}(V |Y, μ_{ℓ})
↓^{∪}  ↓^{∪}
H^{2}(V, μ_{ℓ}^{⊗2})  ⟶  H^{2}(V |Y, μ_{ℓ}^{⊗2}).

The vertical arrows are given by cup-product. The top horizontal arrow is the 2fold tensor of the surjection of 2.5, so it is onto. Since the reduced scheme of V |Y is a finite disjoint union of spectra of fields k(y), y ∈ Ỹ, H^{1}(V |Y, μ_{ℓ}) is the direct sum ⊕(k(y)^{∗}⊗Z/ℓ) and the right vertical arrow is the sum of the homomorphisms:

k(y)^{∗} ⊗ k(y)^{∗} ⊗ Z/ℓ ⟶ K_{2}(k(y)) ⊗ Z/ℓ ⟶ H^{2}(k(y), μ_{ℓ}^{⊗2}).

Now the left hand map is surjective because K_{2} of a field is generated by symbols and the right hand map is surjective by Merkurjev-Suslin [MS]. Hence the bottom arrow is also a surjection.

The next proposition gives a sufficient condition for the map η^{q} : R^{q}π_{∗}(T) ⟶ π_{∗}Hq ; X̃ to be an isomorphism.

Proposition 2.7. If X̃ is regular, π is finite and T = μ_{ℓ}^{⊗n} (n ≥ 0) then

R^{q}π_{∗}(μ_{ℓ}^{⊗n}) ≅ π_{∗}Hq ; X̃ for all q.

Proof. It is enough to show that R^{p}π_{∗}(Hq ; X̃) = 0 for p ≠ 0. The stalk of this sheaf at x ∈ X is H^{p}(V, H^{q}) = lim_{→} H^{p}(U, H^{q}), where V = π^{−1}(Spec(O_{X,x})) and

the direct limit runs over all open U in X̃ containing V. By [BO,4.2.2 and 4.7], H^{∗}(V, H^{q}) is the cohomology of the complex:

(2.7.1) 0 ⟶ ⊕_{V^{0}}H_{et}^{q}(v, μ_{ℓ}^{⊗n}) ⟶ ⊕_{V^{1}}H_{et}^{q−1}(v, μ_{ℓ}^{⊗n−1}) ⟶ ⋯ ⟶ ⊕_{V^{q}}H_{et}^{0}(v, μ_{ℓ}^{⊗n−q}) ⟶ 0.

Here ⊕_{V^{i}} means the direct sum is taken over the points v ∈ V of codimension i.

We have to show that the complex (2.7.1) is exact (except at the left, where the kernel is H^{0}(V, H^{q})). Since π is finite, S = π^{−1}(x) is a finite set of points in X̃. We proceed as on pp.190-191 of [BO]. Let H_{Z^{p}}^{q}(V, n) denote the direct limit over all open U and closed Z in X̃ such that V ⊆ U and codim(Z) ≥ p

H_{Z^{p}}^{q}(V, n) = lim_{→_{U,Z}}H_{Z∩U}^{q}(U, μ_{ℓ}^{⊗n})

Then there is an exact sequence (analogous to [BO,4.3]):

⋯ ⊕_{V^{p}} H^{q−p}(v, μ^{⊗n−p}) ⟶^{δ} H_{Z^{p+1}}^{q+1}(V, n) ⟶^{i} H_{Z^{p}}^{q+1}(V, n) ⟶^{j} ⊕_{V^{p}}H^{q−p+1}(v, μ^{⊗n−p}) ⋯

such that the differential in (2.7.1) is obtained by composing δ with j. Therefore it suffices to show that the maps i are zero. Using the direct limit definition of H_{Z^{p}}^{q}(V, n) and Poincaré Duality [BO, 1.3.5], the problem reduces to proving the following:

CLAIM. Given Z′ ⊆ X̃ of codimension p + 1 and a finite subset S of X̃ there is a Z ⊆ X̃ of codimension p containing Z′ and an affine neighborhood U of S in X̃ such that the map

H_{i}(Z′, μ_{ℓ}^{⊗n}) ⟶ H_{i}(Z, μ_{ℓ}^{⊗n}) ⟶ H_{i}(Z ∩ U, μ_{ℓ}^{⊗n}) is zero.

To obtain a proof of our Claim, simply copy the proof of the Claim on p.191 of [BO], replacing all occurrences of x by S. This completes the proof of Proposition 2.7.

Remark 2.7.2 If Ỹ is also regular, then 2.7 also gives R^{q}(πj)_{∗}(μ_{ℓ}^{⊗n}) ≅ (πj)_{∗}Hq ; Ỹ . However even if X̃ is regular and π is finite, the map R^{q}(πj)_{∗}(μ_{l}^{⊗n}) ⟶ (πj)_{∗}H^{q} need not be an isomorphism in general. This follows from Example 2.4 (b), letting X be obtained by glueing the triangle Ỹ: XY (X + Y − 1) = 0 in the affine plane X̃ = A_{k}^{2} to the simple node Y by the map π : Ỹ ⟶ Y considered in 2.4 (b).

§3. Brauer groups of curves.

Let X be a curve over a field k, with normalization π : X̃ ⟶ X. To study the Brauer group Br(X) of X, we will apply the machinery of the last section to the sheaf T = μ_{ℓ}. The Leray spectral sequence for X_{et} ⟶ X_{Zar} degenerates to yield the exact sequence:

0 ⟶ H_{Zar}^{1}(X, H^{1}) ⟶ H_{et}^{2}(X, μ_{ℓ}) ⟶ H_{Zar}^{0}(X, H^{2}) ⟶ 0.

Since H_{Zar}^{1}(X, −) is a right exact functor, and H^{1}(μ_{ℓ}) ≅ O_{X}^{∗} ⊗Z/ℓ by 2.3, it follows that H^{1}(X, H^{1}(μ_{ℓ})) ≅ Pic(X) ⊗ Z/ℓ. Since dim X=1 there is an isomorphism H_{et}^{2}(X, G_{m}) ≅ Br(X) ([M,IV.2]). On the other hand, Kummer theory yields the exact sequence:

0 ⟶ Pic(X) ⊗ Z/ℓ ⟶ H_{et}^{2}(X, μ_{ℓ}) ⟶ _{ℓ}Br(X) ⟶ 0.

Comparing these sequences as in [CTR, lemma 1.1] we obtain:

Lemma 3.1. ( [BV, 3.5]) For every curve X and ℓ prime to char(k),

H_{Zar}^{0}(X, H^{2}(μ_{ℓ})) ≅ _{ℓ}Br(X).

Remark 3.1.1 By [BO,7.7] and [CTR,1.1] we also have an isomorphism

H^{0}(Z, H^{2}) ≅ _{ℓ}Br(Z)

for every regular scheme Z which is quasiprojective over k.

As X is a curve, X̃ is regular and π is finite with dim(Y) = 0. Hence much of the last section may be simplified. For example, R^{q}(πj)_{∗}(μ_{ℓ}) ≅ (πj)_{∗}Hq ; Ỹ by 2.7.2. This is a skyscraper sheaf on Y whose stalk at y ∈ Y is the sum (over all ỹ ∈ Ỹ lying over y) of the groups H^{q}(ỹ, μ_{ℓ}). For q = 2 we also have

H^{0}(X, R^{2}(πj)_{∗}μ_{ℓ}) = H^{0}(Ỹ , H^{2}) = _{ℓ}Br(Ỹ) = ⊕_{ỹ∈Ỹ}_{ℓ}Br(k(ỹ)).

From (2.2), 2.5 and 2.7 we have an exact sequence of sheaves on X_{Zar}:

(3.2) 0 ⟶ H_{X}^{2}(μ_{ℓ}) ⟶ π_{∗}H2 ; X̃(μ_{ℓ}) ⊕ i_{∗}H_{Y}^{2} (μ_{ℓ}) ⟶ (πj)_{∗}H2 ; Ỹ (μ_{ℓ}).

Now apply the left exact functor H^{0}(X, −) to (3.2). Combining with 3.1 we obtain:

Proposition 3.3. If X is any curve over k and char(k) ≠ ℓ, there is a natural exact sequence

0 ⟶ _{ℓ}Br(X) ⟶ _{ℓ}Br(X̃) ⊕ _{ℓ}Br(Y) ⟶ _{ℓ}Br(Ỹ).

Here π : X̃ ⟶ X is the normalization, Y = Sing(X) and Ỹ = Y ×_{X} X̃.

Application 3.4. Suppose that X is a real curve, i.e., a curve defined over R. We know that Br(X) = _{2}Br(X) because Br(X_{C}) = 0 [GB III,(1.2)]. Therefore (3.3) yields the DeMeyer-Knus exact sequence of elementary abelian 2-groups

0 ⟶ Br(X) ⟶ Br(X̃) ⊕ Br(Y) ⟶ Br(Ỹ).

This sequence was originally obtained for affine curves by DeMeyer and Knus in [DK]. The following improvement on (3.2) will be used in §5 to get a long exact sequence in cohomology, whose H^{0}-terms are just the DeMeyer-Knus sequence.

Proposition 3.5. Let X be any curve over k = R. Then the map

(3.5.1)  π_{∗}H2 ; X̃(μ_{ℓ}) ⟶ (πj)_{∗}H2 ; Ỹ (μ_{ℓ})

is onto. Therefore the following sequence is exact:

(3.5.2)  0 ⟶ H_{X}^{2}(μ_{ℓ}) ⟶ π_{∗}H2 ; X̃(μ_{ℓ}) ⊕ i_{∗}H_{Y}^{2} (μ_{ℓ}) ⟶ (πj)_{∗}H2 ; Ỹ (μ_{ℓ}) ⟶ 0.

Proof. We will show that the map (3.5.1) is onto stalkwise; (3.5.2) is a consequence of this and (3.2). Let y be any point in Y and B the semilocal ring of X̃ at the set of points ỹ ∈ Ỹ lying over y. Then the stalk of (3.5.1) at y is the map

H^{0}(SpecB, H^{2}(μ_{ℓ})) ⟶ ⊕_{ỹ} _{ℓ}Br(k(ỹ)).

If ℓ is odd, then _{ℓ}Br(k(ỹ)) = 0 for any ỹ because BrR = Z/2 and BrC = 0. Hence the map (3.5.1) is onto.

Now suppose that ℓ = 2. Since −1 ∈ R the sheaves μ_{2} and μ_{2}^{⊗2} are isomorphic (non-canonically) to Z/2. By 2.6 and 2.7 the map (3.5.1) is onto.

Finally, suppose that ℓ is even. Then ⊕_{ỹ} _{2}Br k(ỹ) ≅ ⊕_{ỹ} _{ℓ}Br k(ỹ). We claim that

the inclusion of μ_{2} into μ_{ℓ} induces an isomorphism between H^{2}(μ_{2}) and H^{2}(μ_{ℓ}), and therefore that the map (3.5.1) is onto by the case ℓ = 2. To see this, we compare the Bloch-Ogus resolutions [BO, (0.1)] of the sheaves. Because BrR(X) is 2-torsion, it is isomorphic to both H^{2}(R(X), μ_{2}) and H^{2}(R(X), μ_{ℓ}). Similarly, the inclusion induces an isomorphism between H^{1}(ỹ, Z/2) = Z/2 and

H^{1}(ỹ, μ_{ℓ}) = Hom(Z/2, Z/ℓ) = Z/2

for any real closed point ỹ in SpecB. Therefore the Bloch-Ogus resolution for the sheaves H^{2}(μ_{2}) and H^{2}(μ_{ℓ}) on SpecB are isomorphic, as claimed.

The following result extends the main theorem of [DK] to all algebraic curves defined over R. For smooth real curves, it is essentially Satz III′ of Witt’s 1934 paper [Witt]. We write X(R) for the set of real points of X with the Euclidean topology. The number c of connected components of X(R) is a topological invariant; in terms of cohomology we have c = dim H^{0}(X(R), Z/2).

Proposition 3.6. If X is a curve over R, and X(R) has c connected components, then Br(X) ≅ (Z/2)^{c}.

Proof. Topologically, the scheme X is obtained from X̃ by glueing Ỹ along the map Ỹ ⟶ Y (this may be checked locally). Hence X(R) is obtained from X̃(R) and Y (R) by identifying along Ỹ (R).

Suppose first that every component of X̃(R) is contractible, and form a bipartite graph Γ as follows. The m + n vertices of Γ are the m connected components of X̃(R), together with the n points of Y (R). There is one edge for each of the q points ỹ ∈ Ỹ (R), connecting π(ỹ) ∈ Y (R) to the appropriate component of X̃(R). By construction, Γ is homotopy equivalent to X(R). On the other hand, the simplicial cohomology of Γ is computed by the exact sequence:

0 ⟶ H^{0}(Γ, Z/2) ⟶ (Z/2)^{m} ⊕ (Z/2)^{n} ⟶^{δ} (Z/2)^{q} ⟶ H^{1}(Γ, Z/2) ⟶ 0,

where δ is an incidence matrix. But Br(X̃) ≅ (Z/2)^{m} by [W] or [CT] and it is well-known that Br(Y) = (Z/2)^{n}, Br(Ỹ) = (Z/2)^{q}. Since the incidence matrix δ is the natural map Br(X̃) ⊕ Br(Y) ⟶ Br(Ỹ), we obtain the desired isomorphism

Br(X) ≅ H^{0}(Γ, Z/2) ≅ H^{0}(X(R), Z/2).

In the general case, suppose that s of the connected components of X̃(R) are homeomorphic to a circle. (The other components are homeomorphic to a line.)

Choose a set S of smooth real points of X (not in Y), one on each circular component of X̃(R). Then X(R) and X(R) − S(R) both have c components, so it suffices to prove that Br(X) ≅ Br(X − S). Since X̃(R) and X̃(R) − S(R) have the same number of components, this follows from the diagram

0 ⟶ Br(X) ⟶ Br(X̃) ⊕ Br(Y) ⟶^{δ} Br(Ỹ)
↓  ↓≅  ↓^{=}
0 ⟶ Br(X − S) ⟶ Br(X̃ − S) ⊕ Br(Y) ⟶ Br(Ỹ)

which commutes and has exact rows by (3.5).

Remark 3.6.1 The cokernel of δ is H^{1}(Γ, Z/2), whose dimension is the number of loops in X(R) − S(R). Thus if X̃(R) has s circular components and X(R) has λ loops, the cokernel of δ is (Z/2)^{λ−s}. We will return to this point in §5.

Remark 3.6.2 By selecting a set S so that X − S is affine, we could reduce the argument to the result of [DK, pp.230-232].

§4. Chern classes and curves.

We now wish to relate algebraic K-theory to étale cohomology using the natural Chern class map ([Sh],[Sou],[Gi],[J])

c_{ij} : K_{n}(X) ⟶ H_{et}^{j}(X, μ_{ℓ}^{⊗i}), n + j = 2i.

Here X be a scheme essentially of finite type over a field k and ℓ be an integer (typically a prime power) such that \frac{1}{ℓ} ∈ k. Let K_{n} and K_{n}/ℓ denote the Zariski sheaves on X associated to the presheaves sending U to K_{n}(U) and K_{n}(U) ⊗ Z/ℓ. Note that K_{n}/ℓ = K_{n} ⊗ Z/ℓ. The Chern classes sheafify to yield maps

c_{ij} : K_{n} ⟶ K_{n}/ℓ ⟶ H^{j}(μ_{ℓ}^{⊗i}).

Example 4.1 The map c_{11} : K_{1}(X) ⟶ H^{1}(X, μ_{ℓ}) is the determinant map up to sign by [Sh], and it sheafifies to yield the canonical isomorphism c_{11} : K_{1}/ℓ ≅ O_{X}^{∗}/ℓ ≅ H^{1}(μ_{ℓ}). The stalk of K_{2} at x ∈ X is K_{2}(O_{X,x}) so again by [Sh] the map c_{22} : K_{2}/ℓ ⟶ H^{2}(μ_{ℓ}^{⊗2}) is the Galois symbol.

Lemma 4.2. If Y = SpecA where A is an Artinian k-algebra and \frac{1}{ℓ} ∈ k then

K_{∗}(Y) ⊗ Z/ℓ ≅ K_{∗}(Y_{red}) ⊗ Z/ℓ and H^{∗}(Y, μ_{ℓ}^{⊗i}) ≅ H^{∗}(Y_{red}, μ_{ℓ}^{⊗i}).

Consequently we may identity the Chern classes c_{ij} on Y and Y_{red}.

Proof. Let J be the Jacobson radical of A, so that Y_{red} = Spec(A/J). Because A contains a field, A ⟶ A/J splits and K_{∗}(A) ≅ K_{∗}(A/J) × K_{∗}(A, J). Since \frac{1}{ℓ} ∈ A, K_{∗}(A, J) is a Z[\frac{1}{ℓ}] -module and K_{∗}(A, J) ⊗ Z/ℓ = 0 by [W1,1.4]. This establishes the K-theory result K_{∗}(A) ⊗ Z/ℓ ≅ K_{∗}(A/J) ⊗ Z/ℓ. The cohomology result follows from the observation that the canonical isomorphism of sites Y_{et} ≅ (Y_{red})_{et} identifies the sheaves μ_{ℓ} on Y and on Y_{red}.

Lemma 4.3. Let X be a smooth variety over a field k, with \frac{1}{ℓ} ∈ k. Then c_{22} : K_{2}/ℓ ≅ H^{2}(μ_{ℓ}^{⊗2}).

Proof. By [Gray,Cor.6], the sheaf K_{2}/ℓ has the flasque resolution:

0 ⟶ K_{2}/ℓ ⟶ η_{∗}(K_{2}k(X)/ℓ) ⟶ ∐_{x∈X^{1}}i_{x∗}(k(X)^{∗}/ℓ) ⟶ ∐_{x∈X^{o}}i_{x∗}(Z/ℓ) ⟶ 0,

where η : {ξ} ⟶ X, is the inclusion of the generic point ξ of X. Given this, the proof on p.168 of [CTR] applies verbatim to yield the lemma, (even though k is supposed to be separably closed in op.cit.).

Remark 4.3.1 Lemma 4.3 is implicit in [MS,§18]. If we assume that k contains all ℓ^{th}-roots of unity, then [CTR,1.1] gives a natural isomorphism H^{0}(X, K_{2}/ℓ) ≅_{ℓ} Br(X) ⊗ μ_{ℓ}(k). By [S,4.4], [BO,7.7] there is an exact sequence

0 ⟶ H^{1}(X, K_{2}) ⊗ Z/ℓ ⟶ H^{1}(X, K_{2}/ℓ) ⟶ _{ℓ}CH^{2}(X) ⟶ 0

and an isomorphism H^{2}(X, K_{2}/ℓ) ≅ CH^{2}(X) ⊗ Z/ℓ for every smooth variety X.

Now let X be a curve with normalization π : X̃ ⟶ X. As in §2, there is a square with Y =Sing(X):

Ỹ ⟶^{j} X̃
↓  ↓^{π}
Y ⟶^{i} X.

Proposition 4.4. If \frac{1}{ℓ} ∈ k there is an exact sequence of Zariski sheaves on X, with F supported on Y :

0 ⟶ F ⟶ K_{2}/ℓ ⟶ π_{∗}K_{2}/ℓ × i_{∗}K_{2}/ℓ ⟶ (πj)_{∗}K_{2}/ℓ ⟶ 0.

Proof. Since the kernel and cokernel of K_{2}/ℓ ⟶ π_{∗}K_{2}/ℓ are supported on Y, it is enough to check exactness at a point x ∈ Y. Write A = O_{X,x} and let B be the (semilocal) coordinate ring of π^{−1}(SpecA). If I denotes the conductor ideal from B to A, A/I and B/I are Artinian rings. Moreover Y = Spec(A/I)_{red} and Ỹ = Spec(B/I)_{red}. By Lemma 4.2 the stalks of i_{∗}K_{2}/ℓ and (πj)_{∗}K_{2}/ℓ at x are K_{2}(A/I) ⊗ Z/ℓ and K_{2}(B/I) ⊗ Z/ℓ, respectively. Furthermore B is regular so it follows from Quillen’s proof of Gersten’s conjecture ([Q] or [Gray]) that the stalk of π_{∗}K_{n}/ℓ at x ∈ Y is:

H^{0}(SpecB, K_{n}/ℓ) = K_{n}(B) ⊗ Z/ℓ.

for any n. Therefore it suffices to show that the following sequence is exact.

(4.4.1) K_{2}(A) ⊗ Z/ℓ ⟶ K_{2}(B) ⊗ Z/ℓ × K_{2}(A/I) ⊗ Z/ℓ ⟶ K_{2}(B/I) ⊗ Z/ℓ ⟶ 0.

To prove this we use K-theory with coefficients mod ℓ. The groups K_{n}(A; Z/ℓ) fit into Universal Coefficients sequences [W1, 2.1]:

(4.4.2) 0 ⟶ K_{n}(A) ⊗ Z/ℓ ⟶ K_{n}(A; Z/ℓ) ⟶ T or(K_{n−1}(A), Z/ℓ) ⟶ 0.

Since \frac{1}{ℓ} ∈ k we know (from [W1, (1.3)]) that “excision holds”in the sense that associated to a cartesian square of rings

A ⟶ B
↓  ↓
A/I ⟶ B/I

there is a long exact sequence:

⋯ ⟶ K_{2}(A; Z/ℓ) ⟶ K_{2}(B; Z/ℓ) × K_{2}(A/I; Z/ℓ) ⟶ K_{2}(B/I; Z/ℓ) ⟶ ⋯

Now we consider the following diagram:

(4.4.3)

0  0  0
↓  ↓  ↓
K_{2}(A) ⊗ Z/ℓ ⟶(K_{2}(B) ⊗ Z/ℓ) × (K_{2}(A/I) ⊗ Z/ℓ) ⟶K_{2}(B/I) ⊗ Z/ℓ ⟶⋯
↓  ↓  ↓
K_{2}(A; Z/ℓ) ⟶ K_{2}(B; Z/ℓ) × K_{2}(A/I; Z/ℓ)  ⟶ K_{2}(B/I; Z/ℓ) ⟶⋯
↓  ↓  ↓
0 ⟶  _{ℓ}A^{∗}  ⟶  _{ℓ}B^{∗} ×_{ℓ}(A/I)^{∗}  ⟶ _{ℓ}(B/I)^{∗}  ⟶⋯
↓  ↓  ↓
0  0  0

where the columns are the exact Universal Coefficient sequences, so that the diagram commutes. The bottom sequence is the T or(−, Z/ℓ) sequence for

0 ⟶ A^{∗} ⟶ B^{∗} × (A/I)^{∗} ⟶ (B/I)^{∗} ⟶ 0

(where K_{1}(A) = A^{∗} etc.) so it is exact. A diagram chase shows that the top sequence

K_{2}(A) ⊗ Z/ℓ ⟶ (K_{2}(B) ⊗ Z/ℓ) × (K_{2}(A/I) ⊗ Z/ℓ) ⟶ K_{2}(B/I) ⊗ Z/ℓ

is exact. So to prove (4.4.1) we need only show that the map K_{2}(B) ⟶ K_{2}(B/I) is surjective. This follows from the fact that both B and B/I are semilocal, so that K_{2}(B) and K_{2}(B/I) are generated by Steinberg symbol {b, c} with b and c units. Since units of B/I lift to units of B the Steinberg symbols lift.

Corollary 4.5. Under the same assumption as in (4.4) assume furthermore that k contains a primitive ℓ^{th}-root of unity. Then the following sequence of Zariski sheaves on X is exact.

0 ⟶ K_{2}/ℓ ⟶ π_{∗}K_{2}/ℓ × i_{∗}K_{2}/ℓ ⟶ (πj)_{∗}K_{2}/ℓ ⟶ 0.

Proof. According to (4.4) it is enough to show F_{x} = 0 for any point x in Y. From the exactness of sequence (4.4.1) and commutativity of the diagram (4.4.3) we get

0  0  0
↓  ↓  ↓
0 ⟶  F_{x}  ⟶K_{2}(A) ⊗ Z/ℓ ⟶ (K_{2}(B) × K_{2}(A/I)) ⊗ Z/ℓ ⟶⋯
↓  ↓  ↓
H ⟶K_{3}(B/I; Z/ℓ) ⟶ K_{2}(A; Z/ℓ) ⟶K_{2}(B; Z/ℓ) × K_{2}(A/I; Z/ℓ) ⟶⋯
↓  ↓  ↓
0  ⟶  _{ℓ}A^{∗}  ⟶  _{ℓ}B^{∗} ×_{ℓ}(A/I)^{∗}  ⟶⋯
↓  ↓
0  0

where H = K_{3}(B; Z/ℓ) × K_{3}(A/I; Z/ℓ). So it is enough to show that the map

φ : K_{3}(B; Z/ℓ) ⟶ K_{3}(B/I; Z/ℓ)

is onto. By 4.2 and (4.4.2) K_{3}(B/I; Z/ℓ) = K_{3}((B/I)_{red}; Z/ℓ) so we may assume that B/I is reduced, i.e., that B/I = F is a finite product of fields. By [L,4.12] there is a short exact sequence

K_{3}^{M}(F) ⟶ K_{3}(F; Z/ℓ) ⟶^{c_{21}} H^{1}(F, μ_{ℓ}^{⊗2}) ⟶ 0,

where K_{3}^{M}(F) is the Milnor K-group generated by products {a, b, c} with a, b, c ∈ F^{∗}. Since B^{∗} maps onto F^{∗} = (B/I)^{∗}, every element of K_{3}^{M}(F) can be lifted to an element of K_{3}(B), hence to an element of K_{3}(B; Z/ℓ), by (4.4.2).Therefore it suffices to show that each element x ∈ K_{3}(F; Z/ℓ) lifts to an element y ∈ K_{3}(B; Z/ℓ) in the sense that c_{21}(φ(y)) = c_{21}(x), where φ is the map K_{3}(B; Z/ℓ) ⟶ K_{3}(F; Z/ℓ).

For this we use a simple method which is taken from [Sou, IV.1.5]: fix a primitive ℓ^{th}-rooth of unity ξ in B, and let β ∈ K_{2}(B; Z/ℓ) be an element mapping to ξ ∈ _{l}B^{∗} under the map defined in (4.4.2). Since H^{1}(F, μ_{ℓ}^{⊗2}) ≅ F^{∗} ⊗ μ_{ℓ} we can write c_{21}(x) = ā ⊗ ξ, for some a ∈ B^{∗}. Using the K-theory product (as in [S,p.13]) K_{1}(B)×K_{2}(B; Z/ℓ) ⟶ K_{3}(B; Z/ℓ), the elements a and β give an element y = {a, β} in K_{3}(B; Z/ℓ). Then the element y = {a, β} in K_{3}(B; Z/ℓ). The map φ maps the element y to {ā, β} in K_{3}(F; Z/ℓ) and c_{21}({ā, β}) = ā⊗ξ by the product formula [Sou,Th.1]. Hence c_{21}(φ(y)) = c_{21}(x).

Remark 4.5.1. We do not know if it is possible to remove the hypothesis that ξ ∈ B when B contains a field. An easy transfer argument shows that the result is still true if ℓ is prime and ξ ∉ B.

§5. SK_{1} of curves.

In section 3 we related the ℓ-torsion in the Brauer group Br(X) of a curve X to H^{0}(X, H^{2}(μ_{ℓ})). In this section we relate the K-theory group SK_{1}(X) to H^{1}(X, H^{2}(μ_{ℓ}^{⊗2})).

Recall that SK_{1}(X) is defined to be the kernel of the determinant map K_{1}(X) ⟶ H^{0}(X, O_{X}^{∗}). In fact K_{1}(X) is the direct sum of SK_{1}(X) and H^{0}(X, O_{X}^{∗}). Let K_{2} and K_{2}/ℓ be the sheaves defined in §4. There is a Chern class map (see [Gi]) from SK_{1}(X) to H^{1}(X, K_{2}). Our first result is:

Theorem 5.1. Let X be a curve over a field k with \frac{1}{ℓ} ∈ k. Then

SK_{1}(X) ⊗ Z/ℓ ≅ H^{1}(X, K_{2}/ℓ) ≅ H^{1}(X, H^{2}(μ_{ℓ}^{⊗2})).

The second isomorphism is induced from the Chern class c_{22} : K_{2}/ℓ ⟶ H^{2}(μ_{ℓ}^{⊗2}).

Proof. When X is a curve it is well known (see [W2,p.814]) that SK_{1}(X) ≅ H^{1}(X, K_{2}). Since dim X=1, H_{Zar}^{1}(X, −) is a right exact functor; hence the sequence

H^{1}(X, K_{2}) ⟶^{ℓ} H^{1}(X, K_{2}) ⟶ H^{1}(X, K_{2}/ℓ) ⟶ 0

is exact. This yields the first isomorphism. If X is a smooth curve then c_{22} : K_{2}/ℓ ≅ H^{2}(μ_{ℓ}^{⊗2}) by Lemma 4.3, yielding the second isomorphism in the smooth case. In general, we compare the Mayer-Vietoris sequences for K-theory and étale cohomology, associated to π : X̃ ⟶ X, where X̃ is the normalization (cfr. §2).

Let G be the image of the map (πj)_{∗}H1 ; Ỹ (μ_{ℓ}^{⊗2}) ⟶ H_{X}^{2}(μ_{ℓ}^{⊗2}) of (2.2). By naturality of c_{22}, 2.6, 2.7 and 4.4 we have a commutative diagram with exact rows:

(5.1.1)  \begin{aligned}0 & ⟶ & F & ⟶ & K_{2}/ℓ & ⟶ & π_{∗}K_{2}/ℓ × i_{∗}K_{2}/ℓ & ⟶ & (πj)_{∗}K_{2}/ℓ & ⟶ & 0 \\ & & ↓ & ^{c_{22}}↓ & & ≅↓ & & ≅↓ \\ 0 & ⟶ & G & ⟶ & H_{X}^{2} & ⟶ & π_{∗}H2 ; X̃ × i_{∗}H_{Y}^{2} & ⟶ & (πj)_{∗}H2 ; Ỹ & ⟶ & 0 \end{aligned}

It follows that the kernel and cokernel of K_{2}/ℓ ⟶ H_{X}^{2} are supported on Y, and therefore that H^{1}(X, K_{2}/ℓ) ≅ H^{1}(X, H_{X}^{2}), as claimed.

We can in fact make a slightly stronger assertion when k contains ℓ^{th}-roots of unity.

Corollary 5.2. Let X be a curve over a field k. Assume that \frac{1}{ℓ} ∈ k and that k contains a primitive ℓ^{th}-roots of unity. Then F = G = 0 in (5.1.1) and c_{22} is a isomorphism:

K_{2}/ℓ ≅ H_{X}^{2}(μ_{ℓ}^{⊗2})

Proof. It is enough to check that F_{x} = G_{x} = 0 for every x ∈ Y; from Corollary 4.5 we get F = 0. On the other hand, G_{x} is the cokernel of the map from

(π_{∗}H1 ; X̃)_{x} × (i_{∗}H_{Y}^{1} )_{x} = H_{et}^{1}(B, μ_{ℓ}^{⊗2}) × H_{et}^{1}(A/I, μ_{ℓ}^{⊗2}) ≅ (B^{∗} × (A/I)^{∗}) ⊗ μ_{ℓ}(k)

to (πj)_{∗}H^{1} = H^{1}(B/I, μ_{ℓ}^{⊗2}) ≅ (B/I)^{∗} ⊗ μ_{ℓ}(k). Since B^{∗} maps onto (B/I)^{∗}, this map is a surjection and G_{x} = 0.

Theorem 5.3. Let X be a curve over a field k with \frac{1}{ℓ} ∈ k. Assume k contains a primitive ℓ^{th}-root of unity. Then there is a natural exact sequence

0 ⟶ _{ℓ}Br(X) ⟶ _{ℓ}Br(X̃) ⊕ _{ℓ}Br(Y) ⟶ _{ℓ}Br(Ỹ) ⟶

⟶ SK_{1}(X) ⊗ Z/ℓ ⟶ SK_{1}(X̃) ⊗ Z/ℓ ⟶ 0

Proof. This is the cohomology sequence of the short exact sequence in (3.5.2) of Zariski sheaves on X. The first three terms are the exact sequence of Proposition 3.3, and we have used 5.1 to identify SK_{1}(X) ⊗ Z/ℓ with H^{1}(X, H^{2}(μ_{ℓ}^{⊗2})) = H^{1}(X, H^{2}(μ_{ℓ})).

Application 5.4 (Real Curves) Suppose that X is a curve over R. We first observe that if X(R) = ∅ then H^{1}(X, H^{2}) = SK_{1}(X)/2 = 0. If X is smooth the Bloch-Ogus resolution for H^{2} yields H^{2} ≅ i_{∗}(H_{et}^{2}(R(X), Z/2) = 0 by [CTP, 1.2.1]. If X is not smooth apply 5.3 to X and X̃ to get SK_{1}(X)/2 ≅ SK_{1}(X̃)/2 = 0.

If X(R) ≠ ∅ then the sequence (5.3) begins with Br(X) ≅ (Z/2)^{c} as described in 3.4 and 3.6. We showed in 3.6.1 that if X(R) has λ loops and X̃(R) has s loops (circular components) then the cokernel of the second map is (Z/2)^{λ−s}. Thus 5.3 yields the exact sequence

(5.4.1)  0 ⟶ (Z/2)^{λ−s} ⟶ SK_{1}(X) ⊗ Z/2 ⟶ SK_{1}(X̃) ⊗ Z/2 ⟶ 0.

We are going to show that in fact SK_{1}(X) ⊗ Z/2 ≅ (Z/2)^{λ}.

To do this, we use the topological K-theory KO^{∗} associated to real vector bundles [Hu]. For any quasiprojective scheme X over R there is a natural map from SK_{1}(X) to

SKO^{−1}(X(R)) = [X(R), SO],

the homotopy classes of maps from X(R) to the special orthogonal group SO. Since [S^{−1}, SO] = π_{1}(SO) = Z/2, it follows that SKO^{−1}(X(R)) ≅ (Z/2)^{λ}, where λ is the number of loops of X(R), defined in 1.3.

Consider the map from K_{0}(X) = Z×Pic(X) to K_{1}(X) = H^{0}(X, O_{X}^{∗})×SK_{1}(X) obtained by multiplication by [−1] ∈ K_{1}(R). It is compatible with the map from KO(X) to KO^{−1}(X) obtained by multiplication by the canonical class η in

KO^{−1}(S^{0}) = [S^{0}, O] ≅ Z/2.

This yields a commutative diagram:

(5.4.2)

Pic(X) ⟶ Pic(X(R)) ≅ (Z/2)^{λ}
[−1]↓  ≅↓^{η}
SK_{1}(X) ⟶ SKO^{−1}(X(R)).

Since X(R) is a 1-dimensional complex, Bott periodicity implies that the right vertical map is a isomorphism. The top map is a surjection by Theorem 1.10. Hence the bottom map in (5.4.2) is a surjection. The following theorem says that it is an isomorphism modulo 2.

Theorem 5.5. Let X be any curve, and let λ be the number of loops in X(R). Then the natural map

SK_{1}(X) ⊗ Z/2 ⟶ SKO^{−1}(X(R)) ≅ (Z/2)^{λ}

is a isomorphism.

Proof. We have already seen that the map is surjective. From (5.4.1) we see that it suffices to prove the result for smooth curves. When X is a smooth curve over R, the localization sequence in K-theory is a sequence of K_{∗}(R)-modules, so there is a commutative diagram with exact rows:

(5.5.1)  \begin{matrix}R(X)^{∗} & ⟶ & ⨿_{x∈X^{1}}Z & ⟶ & Pic(X) & ⟶ & 0 \\ ↓^{[−1]} & & ↓^{[−1]} & & ↓^{[−1]} \\ K_{2}R(X) & ⟶ & ⨿_{x∈X^{1}}k(x)^{∗} & ⟶ & SK_{1}(X) & ⟶ & 0 \end{matrix}

Tensoring with Z/2 yields the exact diagram

(5.5.2)  \begin{matrix}⨿_{x}Z/2 & ⟶ & Pic(X) ⊗ Z/2 & ⟶ & 0 \\ ↓^{[−1]} & & ↓^{[−1]} \\ ⨿_{x}k(x)^{∗}/k(x)^{∗2} & ⟶ & SK_{1}(X) ⊗ Z/2 & ⟶ & 0 \end{matrix}

In the lower left corner, k(x)^{∗}/k(x)^{∗2} is R^{∗}/R^{∗2} ≅ Z/2 if x ∈ X(R) and C^{∗}/C^{∗2} = 0 otherwise. If X(R) = ∅ this implies that SK_{1}(X) ⊗ Z/2 = 0, the conclusion of 5.5 already made in 5.4. When X(R) ≠ ∅, the left vertical map of (5.5.2) is onto, proving that the map Pic(X) ⊗ Z/2 ⟶ SK_{1}(X) ⊗ Z/2 is onto. This establishes 5.5 because from (5.4.2) the composition

(Z/2)^{λ} ≅ Pic(X) ⊗ Z/2 ⟶ SK_{1}(X) ⊗ Z/2 ⟶ SKO^{−1}(X(R)) ≅ (Z/2)^{λ}

is a isomorphism.

Remark 5.5.3 Here is an alternative proof, using a recent result of Scheiderer ([Sch],[CT]). Let Spec_{r}(X) denote the real spectrum of X and φ : Spec_{r}(X) ⟶ X_{Zar} the support map. Scheiderer proves that if X is smooth then H^{n} ≅ φ_{∗}(Z/2) for n ≥ 2 and that R^{i}φ^{∗}(Z/2) = 0 for i ≠ 0. Consequently for all i ≥ 0 and n ≥ 2

H^{i}(X, H^{n}) ≅ H^{i}(Spec_{r}X, Z/2) ≅ H^{i}(X(R), Z/2).

In particular, using 5.1 we have the result for smooth X:

SK_{1}(X) ⊗ Z/2 ≅ H^{1}(X, H^{2}) ≅ H^{1}(X(R), Z/2) ≅ (Z/2)^{λ}.

Remark 5.5.4 If X is smooth projective and X(R) ≠ ∅, then the isomorphism SK_{1}(X) ⊗ Z/2 ≃ (Z/2)^{c} follows from [CTR,prop.4.9.b)]

Lemma 5.6. Let X be an irreducible smooth curve over C. If X is projective then SK_{1}(X) = C^{∗} × V (X) where V (X) is uniquely divisible. If X is affine then SK_{1}(X) is either 0 or it is uniquely divisible. In either case K_{2}(X) is a divisible abelian group.

Proof. The sequence (4.4) of [S] is (for ℓ ∈ Z):

0 ⟶ H^{0}(X, K_{2}) ⊗ Z/ℓ ⟶ H_{et}^{2}(X, μ_{ℓ}) ⊗ μ_{ℓ} ⟶ _{ℓ}SK_{1}(X) ⟶ 0.

Suppose first that X is projective. The assertion about SK_{1}(X) is proven in [R,1.1]. Since H_{et}^{2}(X, μ_{ℓ}) ≅ Z/ℓ and _{ℓ}SK_{1}(X) ≅ μ_{ℓ}, this shows that H^{0}(X, K_{2}) is divisible.

If X is affine then X has étale cohomological dimension 1. Therefore the above sequence immediately shows that SK_{1}(X) is torsionfree and that H^{0}(X, K_{2}) is divisible. Since SK_{1}(X) ⊗ Z/ℓ = 0 by 5.1, SK_{1}(X) is divisible. Finally, we use the localization sequence

∐_{x∈X^{1}}K_{2}(C) ⟶ K_{2}(X) ⟶ H^{0}(X, K_{2}) ⟶ 0

and divisibility of K_{2}(C) to deduce that K_{2}(X) is divisible.

Proposition 5.7. If X is a curve defined over R, then there is a uniquely divisible abelian group D = D(X) and a natural decomposition

SK_{1}(X) ≅ (Z/2)^{λ} ⊕ D ⊕ (C^{∗})^{r},

where r is the number of components of X which are projective and defined over C.

Proof. Let X̃ ⟶ X be the normalization of X and suppose that X̃ has r_{1} real and r_{2} complex projective components. Because Ỹ_{C} ⟶ Y_{C} is étale, excision holds for K_{1}; this follows from [GR] in the affine case, and the general case easily reduces to the affine case. Therefore there is an exact Mayer-Vietoris sequence

K_{2}(X̃_{C}) ⊕ K_{2}(Y_{C}) ⟶ K_{2}(Ỹ_{C}) ⟶ SK_{1}(X_{C}) ⟶ SK_{1}(X̃_{C}) ⟶ 0.

Since K_{2}(Ỹ_{C}) = ∐ K_{2}(C) and K_{2}(C) is uniquely divisible, lemma 5.6 shows that SK_{1}(X_{C}) is the direct sum of a uniquely divisible group and SK_{1}(X̃_{C}). Using lemma 5.6 again, we get that SK_{1}(X_{C}) ≅ V (X_{C}) ⊕ (C^{∗})^{h}, for some uniquely divisible group V (X_{C}), where h is the number of components of X̃_{C} which are projective. Now let D′ be the image of the transfer map from SK_{1}(X_{C}) to SK_{1}(X). Since SK_{1}(X) ⟶ SK_{1}(X_{C}) ⟶ SK_{1}(X) is multiplication by 2, we get 2D′ ⊆ 2SK_{1}(X) ⊆ D′. As D′ is divisible, 2SK_{1}(X) = D′. By 5.5, SK_{1}(X) ≅ (Z/2)^{λ} ⊕ D′. If X has a projective component which is defined over R then the transfer map sends C^{∗} to the group of positive reals which has no torsion. If X has a projective component defined over C then X_{C} has 2 components and the transfer map maps the corresponding summand C^{∗} ⊕ C^{∗} onto C^{∗} yielding torsion. Hence D′ is the sum of a uniquely divisible group D and possibly some C^{∗}’s, one for every projective component of X defined over C. Therefore D′ ≅ D ⊕ (C^{∗})^{r} and, by 5.5, SK_{1}(X) = (Z/2)^{λ} ⊕ D′ = (Z/2)^{λ} ⊕ D ⊕ (C^{∗})^{r} where D is uniquely divisible.

Example 5.8 a) Let X̃ = Spec(R[t, t^{−1}]) and let X be the curve obtained from X̃ by glueing the points t = 1 and t = α where α ≠ 0,1. Excision holds (by [GR,Thm.2]) so we get a mayer-Vietoris exact sequence

K_{2}(X̃) ⟶^{Φ} K_{2}(R) ⟶ SK_{1}(X) ⟶ 0.

Hence SK_{1}(X) ≅ K_{2}(R)/ImΦ. In [RW, prop.4.10] ImΦ has been computed for different values of α: if α=-1 then SK_{1}(X) = K_{2}(R)/{−1, −1} = K_{2}(C)^{+} where K_{2}(C) = K_{2}(C)^{+} ⊕ K_{2}(C)^{−}. Hence SK_{1}(X)/2=0, and X(R) has no loops. If α < 0, α ≠ −1 then SK_{1}(X) = K_{2}(R)/{R^{∗}, α}; in this case we also have SK_{1}(X)/2 = 0. If α > 0, α ≠ 1 then SK_{1}(X) = Z/2 ⊕ V, for some divisible group V. In the first two cases the real loop has been punctured, while in the last case the curve X has 1 loop.

b) Let X̃ = Spec(R[X, Y]/(X^{2}+Y^{2}−1). For each θ let X be the curve obtained by identifying the real points (1,0) and (cos θ,sin θ) of X̃. In this case X(R) has always two loops so that SK_{1}(X)/2 = (Z/2)^{2}. However the computation in [RW,Ex. 4.11] show that the divisible part of SK_{1}(X) depends on the choice of θ. We always have SK_{1}(X) = (Z/2)^{2} ⊕ V; if e^{iθ} is a root of unity then V = K_{2}(C)^{+}, otherwise V is just a quotient of K_{2}(C)^{+}.

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