The multiplicative group on a curve

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0.

On a smooth curve, a rational function fails to be a unit at finitely many points, and the failure is measured by an integer at each point. This gives an exact sequence of étale sheaves. Tsen's theorem makes the rational-function sheaf acyclic; the point sheaves are acyclic for a different reason. Together these facts compute the higher cohomology of the multiplicative group and, through Kummer, the cohomology with roots of unity as coefficients.

Throughout, \(k\) is algebraically closed, \(X\) is a nonempty connected smooth separated curve of finite type over \(k\), \(K=k(X)\), and \(n\geq1\) is invertible in \(k\). All cohomology is on the small étale site. Smoothness makes irreducible components disjoint: a regular local ring is a domain, so distinct components cannot meet. Connectedness therefore makes \(X\) integral. For a disconnected smooth curve, apply the results component by component; there are finitely many components.

Write \(j:\eta=\operatorname{Spec}K\to X\) and \(i_x:\operatorname{Spec}k\to X\) for the inclusion of a closed point. We use Pushforward, pullback and finite morphisms, especially its precise continuity theorem for the strict-local limit, and Brauer groups and Tsen's theorem, Corollary 6.1. The identification \(H^1(X,\mathbf G_m)=\operatorname{Pic}(X)\) and the normalized-line-bundle description of Kummer classes were proved in Topologies on schemes and The étale site and its points. We supply the divisor arguments here.

1. Local rings and rational sections

We use two elementary geometric foundations with precise locators. A local ring at a closed point of a smooth curve is a discrete valuation ring, by Stacks, Tag 0B8Y and the regular-local-ring characterization Tag 00PD. A smooth curve over a perfect field has a smooth projective completion with finite complement, by Tag 0H1F. The latter follows by taking a projective completion and its normalization: normalization is finite for curves of finite type over a field, the normal one-dimensional local rings are regular, and over a perfect field the resulting regular curve is smooth. These are scheme-theoretic foundations; the étale cohomology results below will be proved.

A discrete valuation ring \(A\) has a uniformizer \(\pi\), and every nonzero element of its fraction field is uniquely of the form \(u\pi^m\), with \(u\in A^\times\) and \(m\in\mathbf Z\). Its valuation is \(v(u\pi^m)=m\).

Lemma 1.1. At a geometric point over a closed point \(x\), the strict henselization \(R=\mathcal O_{X,x}^{\mathrm{sh}}\) is itself a discrete valuation ring. Its fraction field \(F\) is algebraic and separable over \(K\), and has transcendence degree one over \(k\).

Proof. Compute \(R\) as the filtered colimit of pointed local étale neighborhoods \(B\) of \(A=\mathcal O_{X,x}\). Each \(B\) is a regular local ring of dimension one and hence a discrete valuation ring. Its residue field is \(k\), since the specified residue field is a finite separable extension of the algebraically closed field \(k\). The étale fiber has maximal ideal zero, so the maximal ideal of \(B\) is \(\pi B\). Every stage thus has the same uniformizer \(\pi\).

Transition maps between these pointed local rings are local and flat. They are faithfully flat and injective; equivalently, a nonzero \(\pi^m u\) remains nonzero, and its unit remains a unit. Their filtered colimit is a domain. Every nonzero element of it already occurs at one stage, where it is \(\pi^m u\). Thus every nonzero ideal of \(R\) is generated by an element of least valuation: all its other elements are multiples of that element. The ring \(R\) is a discrete valuation ring with uniformizer \(\pi\), including the assertion of Noetherianity. Its fraction field is the union of the fraction fields of the stages. Their generic fibers are étale and of relative dimension zero, so these fields are finite separable extensions of \(K\). The assertions about \(F\) follow. \(\square\)

For a quasi-compact étale \(U\to X\), \(U\) is a finite disjoint union of smooth integral curves, and

This is the group of invertible rational functions on its components. A rational function on a Noetherian curve has only finitely many zeros and poles: on a finite affine cover write it as a quotient of regular functions; the zero loci of the nonzero numerator and denominator are zero-dimensional closed subsets, hence finite. A rational function belongs to the local ring precisely when its valuation is nonnegative, and is a local unit precisely when its valuation is zero.

2. The valuation sequence

Let

The sum is a sum of sheaves, or equivalently the filtered colimit of its finite partial sums. On a quasi-compact étale \(U\), the summand at \(x\) gives one integer on each point of \(U\times_Xx\). Taking the orders of a rational function at all the closed points of \(U\) defines a section of \(\mathcal D_X\). Its support is finite by the preceding paragraph. Étale restriction preserves these orders because a local étale map of the relevant discrete valuation rings has ramification index one. This constructs a sheaf map \(\operatorname{ord}\).

Theorem 2.1 (fundamental sequence). There is a short exact sequence

Here the zero notation refers to abelian sheaves; the first two groups are written multiplicatively.

Proof. It suffices to check geometric stalks. At a geometric generic point, the first two stalks are both \((K^s)^\times\), and the last is zero. At a geometric point over a closed point \(x\), they are respectively

where \(R,F\) are as in Lemma 1.1. The middle identification follows by passing through (1.1) and the pointed-neighborhood colimit; a rational element and its inverse occur at a common stage. Finite-pushforward base change identifies the stalk of \(i_{x*}\mathbf Z\) with \(\mathbf Z\) and all other point summands with zero. Stalks commute with the filtered colimit of finite sums, so there is no additional contribution from infinitely many points. The stalk map is \(v:F^\times\to\mathbf Z\). It is surjective because \(v(\pi)=1\), and its kernel is exactly \(R^\times\). The étale topos has enough geometric points, so the sequence is exact. \(\square\)

The proof is stronger than a calculation on rational functions over \(X\): it verifies local surjectivity in the étale topology, including all the strict-local rings.

3. Why the infinite divisor sheaf is acyclic

Passing from the vanishing for a single point to vanishing for an infinite sum requires an argument. We give it using the full hypercover comparison proved in Hypercoverings.

Lemma 3.1. On the small étale site of a Noetherian quasi-compact separated scheme \(S\), cohomology commutes with filtered colimits of abelian sheaves:

Proof. Use the basis of quasi-compact separated étale schemes of finite presentation over \(S\). It is stable under finite fiber products and finite limits. Every cover of a basis object has a finite subcover after refining its members by basis objects.

On this basis, a filtered presheaf colimit of sheaves is already a sheaf. Indeed, the matching condition for a finite covering family is the equalizer of two maps between finite products of section groups. Filtered colimits commute with these finite limits. A matching family and its compatibility therefore occur at one sufficiently late stage, where the sheaf axiom gives its unique glue. The finite-cover topology generates the topology on this basis, so this proves the assertion for all covers. Consequently sections on every basis object commute with filtered colimits.

There is a cofinal system of hypercovers of \(S\) having finitely many basis objects in each degree. To see this, refine any hypercover inductively. Having chosen finitely many objects in the lower degrees, their matching object in the next degree is a finite limit of quasi-compact étale objects, and is quasi-compact. Pull back the given hypercover's next covering map to that matching object and choose a finite basis cover. Include the finitely many forced degenerate components. The split-hypercover construction from the previous lesson then extends the simplicial refinement. Induction constructs all the degrees. The same construction refines a common refinement, so this system suffices for the colimit in hypercover comparison, with morphisms taken through simplicial homotopy.

For a fixed such hypercover \(U_\bullet\), degree \(p\) of its cochain complex is a finite product of groups \(\mathcal F(U_{p,a})\). It therefore commutes with filtered coefficient colimits. Filtered colimits of abelian groups are exact, so the cohomology of that complex commutes with them as well. Finally hypercover comparison identifies sheaf cohomology with the colimit of these cohomology groups over refinements. Interchanging the two colimits gives (3.1). \(\square\)

Lemma 3.2. For every closed point \(x\), and every \(q\geq1\),

Moreover \(H^0(X,\mathcal D_X)=\bigoplus_x\mathbf Z\).

Proof. The closed immersion \(i_x\) is finite. Its étale direct image is exact, and the finite-morphism comparison from the higher-direct-image lesson gives

The small étale topos of an algebraically closed field is the topos of sets; its abelian global-section functor is the identity, hence exact. This gives the first vanishing and gives \(H^0=\mathbf Z\). Finite partial sums of the point sheaves have the same vanishing by additivity. Apply Lemma 3.1 to their filtered union to obtain both assertions for \(\mathcal D_X\). \(\square\)

4. Higher direct images of rational units

Lemma 4.1. One has \(R^qj_*\mathbf G_{m,\eta}=0\) for every \(q\geq1\). Consequently

Proof. The generic inclusion is affine: on an affine open \(\operatorname{Spec}A\subset X\) containing the generic point its inverse image is \(\operatorname{Spec}\operatorname{Frac}(A)\). In particular it is quasi-compact and quasi-separated. The higher-direct-image stalk formula and the precise strict-local continuity theorem proved earlier apply.

At a geometric closed point they identify the stalk of \(R^qj_*\mathbf G_{m,\eta}\) with

More explicitly the stalk is the colimit of the cohomology of the generic fibers of pointed étale neighborhoods. Their inverse limit has coordinate field \(F\). These generic transition maps are étale; for this inverse system the inverse-image units sheaf is the units sheaf on the limit. Thus the continuity theorem is being used under its actual hypotheses, rather than asserting that the inverse image of a units sheaf agrees with the units sheaf for an arbitrary ring map.

Lemma 1.1 makes \(F/k\) an extension of transcendence degree one. Tsen and the all-degree units-cohomology consequence proved in the preceding lesson apply even though \(F\) need not be finitely generated. They give zero in (4.2) for all \(q\geq1\). At a geometric generic point the corresponding field is \(K^s\), whose étale topos has no positive units cohomology. All geometric stalks therefore vanish.

The Leray spectral sequence now has only its row \(R^0j_*\), giving the first equality in (4.1). Its final vanishing is the same Tsen consequence for \(K\). \(\square\)

Theorem 4.2. For the curve \(X\),

Proof. The first equality is the canonical torsor-to-line-bundle identification already proved by descent. For \(q\geq2\), the long exact sequence of (2.2) has outer groups

Both outer groups vanish by Lemmas 3.2 and 4.1. \(\square\)

5. Divisors, line bundles and the connecting sign

A divisor on \(X\) is a finite sum \(D=\sum_xm_x[x]\), with \(m_x\in\mathbf Z\). For \(f\in K^\times\), write \(\operatorname{div}(f)=\sum_xv_x(f)[x]\). Define \(\mathcal O_X(D)\) inside the constant rational-function sheaf by

It is invertible: near a point with coefficient \(m\), a uniformizer gives the frame \(\pi^{-m}\); shrink the neighborhood to remove the finitely many other zeros and poles involved. Outside the support it is \(\mathcal O_X\).

Proposition 5.1. Every line bundle on \(X\) is \(\mathcal O_X(D)\) for a divisor \(D\), and

Proof. Choose a nonzero vector \(s\) in the one-dimensional generic fiber of a line bundle \(L\). In local frames \(e_i\), write \(s=f_i e_i\). The orders of \(f_i\) agree on overlaps because the transition functions are units. A finite trivializing cover shows that only finitely many orders are nonzero. They define a divisor \(D(s)\). Multiplication \(f\mapsto fs\) gives \(\mathcal O_X(D(s))\simeq L\): at a point its frame \(\pi^{-v(f_i)}\) maps to a unit times \(e_i\).

Replacing \(s\) by \(a s\) replaces \(D(s)\) by \(D(s)+\operatorname{div}(a)\). Conversely \(\mathcal O_X(\operatorname{div}(a))=a^{-1}\mathcal O_X\), so principal divisors give trivial bundles. If \(\mathcal O_X(D)\) is trivial, a rational generator \(a\) of this subsheaf must have order \(-m_x\) at every point, so \(D=-\operatorname{div}(a)\). These statements prove both surjectivity and the exact kernel in (5.2). \(\square\)

Taking the beginning of the long exact sequence of (2.2) yields

There is a sign to remember. Our torsor convention is the one fixed in the earlier lessons: local sections satisfy \(t_j=t_i g_{ij}\), and the associated line frames satisfy \(e_j=e_i g_{ij}\). If local rational lifts \(f_i\) of \(D\) satisfy \(\operatorname{div}(f_i)=D\), their cocycle is \(f_j/f_i\). These \(f_i\) are the frames of \(\mathcal O_X(-D)\). Therefore

The familiar isomorphism (5.2) using \(D\mapsto\mathcal O_X(D)\) is the negative of this connecting map. Exactness and all degree computations are unaffected, but the two conventions must not be silently mixed.

6. Kummer and projective curves

For every curve under discussion, Kummer and (4.3) give the complete answer

The first equality follows because a function whose \(n\)-th power is one on an integral curve is one of the constant roots of \(T^n-1\). For the other equalities take the long exact sequence of \(0\to\mu_n\to\mathbf G_m\xrightarrow{n}\mathbf G_m\to0\), and use both vanishings adjacent to \(H^q(\mu_n)\) in degrees at least three. The middle identification is induced by the Kummer connecting homomorphism

In particular \(H^1=\operatorname{Pic}[n]\) is a projective-curve simplification. It is false for general affine curves, whose global units can contribute.

Let now \(C\) be smooth, connected and projective, of genus \(g=\dim_kH^1(C,\mathcal O_C)\). A global regular function on \(C\) is constant. Indeed it gives a morphism to \(\mathbf A^1\), and its composite with \(\mathbf A^1\subset\mathbf P^1\) has closed image because \(C\) is proper. An irreducible closed subset of \(\mathbf P^1\) avoiding infinity is a point. Thus \(\Gamma(C,\mathcal O_C)^\times=k^\times\), on which the \(n\)-th-power map is surjective.

The following input on Picard varieties is proved in the programme; its proof chain is recorded after the statement.

Picard input 6.1. The subgroup \(\operatorname{Pic}^0(C)\) of degree-zero line bundles is the group of \(k\)-points of an abelian variety of dimension \(g\). Multiplication by every positive integer on that group is surjective. If \(n\) is invertible in \(k\), its \(n\)-torsion is noncanonically \((\mathbf Z/n)^{2g}\).

Proof chain. Since \(k\) is algebraically closed, the Picard functor of \(C\) is representable [Stacks, Tag 0B9Z], and the representing scheme is a disjoint union of smooth proper varieties \(\operatorname{Pic}^d\) of dimension \(g\), whose \(k\)-points are the invertible modules of degree \(d\), with \(\operatorname{Pic}^0\) an open and closed subgroup scheme [Stacks, Tag 0BA0]. Thus \(\operatorname{Pic}^0\) is a proper, irreducible and smooth group scheme over \(k\), hence an abelian variety of dimension \(g\) by Abelian varieties, Proposition 10.1. For a positive integer \(m\), multiplication by \(m\) is finite and surjective by Theorem 6.1 of that lesson. It is therefore surjective on \(k\)-points: the fibre over a \(k\)-point is a nonempty scheme of finite type over \(k\), and any closed point of it has residue field \(k\). If \(n\) is invertible in \(k\), Theorem 7.1 there, with \(k_s=k\), gives the \(n\)-torsion (section 7). The three assertions are also collected, with proofs, in [Stacks, Tag 03RP], parts (3), (4) and (7).

Degree descends from divisors to line bundles. For completeness, a nonconstant \(a\in k(C)\) defines a morphism \(C\to\mathbf P^1\): at each discrete valuation ring either \(a\) or \(a^{-1}\) is regular. It is proper and has finite fibers, hence finite. The local length argument in Proposition 8.1 below shows that its zero and pole fibers have the same degree. Thus \(\deg\operatorname{div}(a)=0\); constants give zero as well. Since every closed point has residue field \(k\), \(\deg(\sum m_x[x])=\sum m_x\). There is a degree-one line bundle \(\mathcal O_C(x)\), so

Theorem 6.2. The projective-curve cohomology is

The degree-two identification is canonical: it sends \(c_1^{(n)}(L)\) to \(\deg L\bmod n\).

Proof. The unit quotient in (6.1) is zero. Torsion line bundles have degree zero by (6.3), giving the stated \(H^1\). To compute the quotient in degree two, if \(\deg L=na\), choose a bundle \(M\) of degree \(a\). The bundle \(L\otimes M^{-n}\) has degree zero, and is an \(n\)-th power by Picard input 6.1. Hence \(L\) itself is an \(n\)-th power. Conversely an \(n\)-th power has degree divisible by \(n\), and every degree occurs. This proves \(\operatorname{Pic}(C)/n\simeq\mathbf Z/n\) without choosing a splitting of (6.3). The other degrees follow from (6.1). \(\square\)

Only after choosing a primitive root \(\zeta\in\mu_n(k)\) do we identify the constant sheaf \(\mathbf Z/n\) with \(\mu_n\) by \(a\mapsto\zeta^a\). Thus the identifications

include a choice of root, and the first also a choice of basis. The canonical degree statement uses \(\mu_n\). This distinction is the Tate twist that becomes essential over non-algebraically-closed fields.

7. Affine curves and their boundary

Let \(X\) be affine and let \(C\) be its smooth projective completion. Its complement is a nonempty finite set \(S=\{s_1,\ldots,s_r\}\). If it were empty, the positive-dimensional affine scheme \(X=C\) would have only constant global functions, contrary to its affine coordinate ring having transcendence degree one.

Proposition 7.1. Restriction gives

The group \(\operatorname{Pic}(X)\) is divisible, including by the characteristic. In particular \(H^q(X,\mu_n)=0\) for every \(q\geq2\).

Proof. Extend a divisor representing any line bundle on \(X\) to the same finite divisor on \(C\). This proves surjectivity. If a bundle on \(C\) is trivial on \(X\), choose its rational section given by a trivializing frame there. Its divisor is supported on \(S\), proving the kernel assertion.

For any bundle \(L\) on \(C\), the twist \(L(-(\deg L)s_1)\) has degree zero and restricts to the same bundle on \(X\). Thus \(\operatorname{Pic}^0(C)\to\operatorname{Pic}(X)\) is surjective. A quotient of a divisible group is divisible: lift a class, divide the lift, and project. Apply Picard input 6.1 for every positive integer. The degree-two conclusion follows from (6.1), and the higher degrees already vanish there. The Kummer sequence itself is used only for \(n\) invertible in \(k\). \(\square\)

There is also a useful full description of the degree-one group. Write

Proposition 7.2. There is an exact sequence

Consequently \(H^1(X,\mu_n)\simeq(\mathbf Z/n)^{2g+r-1}\) noncanonically.

Proof. Recall the Kummer description by pairs \((L,\alpha:L^{\otimes n}\simeq\mathcal O_X)\), with isomorphisms preserving \(\alpha\). Extend \(L\) to a degree-zero bundle \(\bar L\) on \(C\), using a twist at \(s_1\). Extend \(\alpha\) as a rational map. There is a unique divisor \(D=\sum a_i[s_i]\) for which this rational map is an isomorphism

Indeed the target coefficient at each boundary point is the negative of the order of the rational map in local frames. Since \(\deg\bar L=0\), one has \(\sum a_i=0\).

Two degree-zero extensions of \(L\), with their identification over \(X\), differ by \(\mathcal O_C(D')\) for a degree-zero divisor \(D'\) supported on \(S\), by the kernel proof of (7.1). Their corresponding \(D\)'s differ by \(nD'\). An isomorphism of the original pairs gives the same comparison. Therefore \((a_i\bmod n)\) is well defined, lies in \(V_S\), and is additive under tensor product. This defines \(\mathrm{res}_S\).

Restriction from \(C\) is injective. If a normalized pair on \(C\) becomes trivial on \(X\), it has a rational section \(s\) with \(\alpha(s^{\otimes n})=1\) on \(X\), hence in the function field. It follows that \(n\operatorname{div}(s)=0\) on \(C\). Thus \(s\) has order zero everywhere and is a global normalized frame; the pair was trivial on \(C\).

If the residue vector is zero, write \(D=nD'\) with \(D'\) supported on \(S\) and of degree zero. Twisting \(\bar L\) by \(\mathcal O_C(-D')\) turns (7.3) into a global normalization to \(\mathcal O_C\), so its restriction comes from \(H^1(C,\mu_n)\). This proves exactness in the middle.

Finally lift any vector in \(V_S\) to integers \(a_i\). Their sum is divisible by \(n\); adjusting one lift by that multiple of \(n\) makes the sum zero. Picard input 6.1 gives an \(n\)-th root \(\bar L\) of \(\mathcal O_C(\sum a_i[s_i])\) in \(\operatorname{Pic}^0(C)\). Choose the isomorphism (7.3), and restrict it to \(X\). This proves surjectivity.

All groups in (7.2) are \(\mathbf Z/n\)-modules. Projection to the first \(r-1\) coordinates identifies \(V_S\) with the free module \((\mathbf Z/n)^{r-1}\). Lift its standard basis; the lifts define a module section because the middle group is killed by \(n\). Hence (7.2) splits as modules, though without a preferred splitting. Theorem 6.2 now gives the asserted rank. \(\square\)

The residue construction is compatible with the earlier normalization sign. For a unit \(a\) on \(X\), its Kummer class is the pair \((\mathcal O_X,a^{-1})\). In (7.3) this has \(D=\operatorname{div}_C(a)\), so its residues are the orders of \(a\) modulo \(n\).

8. Finite pullback, including inseparability

Proposition 8.1. Let \(f:C\to C'\) be a finite morphism between smooth connected projective curves, and let \(d=[k(C):k(C')]\). For every line bundle \(L\) on \(C'\),

Under the canonical degree identifications of Theorem 6.2, pullback on \(H^2(-,\mu_n)\) is multiplication by \(d\).

Proof. A finite morphism between these positive-dimensional integral curves is nonconstant and dominant: a map through a point would make the positive-dimensional source finite over that point. Fix a closed point \(y\in C'\), let \(A=\mathcal O_{C',y}\), and let \(B\) be the finite semilocal algebra obtained from \(f_*\mathcal O_C\) over \(A\). It is torsion free as an \(A\)-module, since multiplication by a nonzero element of the target function field is injective in the source field. A finite torsion-free module over a discrete valuation ring is free, and its rank is the generic dimension \(d\). Consequently

The Artinian algebra \(B/\pi_yB\) is the product of its local factors at the points \(x\) above \(y\). In the discrete valuation ring \(\mathcal O_{C,x}\) write \(\pi_y=u_x\pi_x^{e_x}\). Its local factor has length \(e_x\): the powers of \(\pi_x\) give \(e_x\) successive quotients, each the residue field \(k\). Therefore

This proves the degree formula for every divisor, including negative coefficients. Proposition 5.1 represents every line bundle by a divisor, and Cartier pullback satisfies \(f^*\mathcal O_{C'}(D)=\mathcal O_C(f^*D)\), giving (8.1).

Kummer is natural under pullback. In particular \(f^*c_1^{(n)}(L)=c_1^{(n)}(f^*L)\). These classes exhaust \(H^2(C',\mu_n)\) by (6.1). Their degrees modulo \(n\) change by multiplication by \(d\), which proves the cohomological assertion. No separability was used: the fiber lengths include all inseparable multiplicities. \(\square\)

For example the map \(\mathbf P^1\to\mathbf P^1\) given by \(t\mapsto t^d\) acts by \(d\bmod n\). In characteristic \(p\), the finite \(k\)-morphism \(t\mapsto t^p\) has degree \(p\) and acts by \(p\bmod n\), even though it has no separable degree.

9. Three examples

The projective line. Every finite point \(a\in k\) satisfies \(\operatorname{div}(t-a)=[a]-[\infty]\). Thus every divisor is equivalent to its degree times \([\infty]\). Degree-zero principal divisors and the degree-one bundle \(\mathcal O(\infty)\) give \(\operatorname{Pic}(\mathbf P^1)=\mathbf Z\), and (6.1) gives

The multiplicative curve. For \(X=\mathbf G_m=\operatorname{Spec}k[t,t^{-1}]\), every closed-point divisor \([a]\), \(a\ne0\), is the divisor of \(t-a\) on \(X\). Hence \(\operatorname{Pic}(X)=0\). An invertible Laurent polynomial is \(ct^m\): the difference between its largest and smallest exponents is additive under multiplication, so an inverse forces that difference to be zero. Consequently

The generator is the Kummer class of \(t\), represented by the right \(\mu_n\)-torsor \(z^n=t\), or the normalized pair \((\mathcal O_X,t^{-1})\). Its residue at \(0\) in the completion \(\mathbf P^1\) is \(1\), and at infinity is \(-1\). All its degree-at-least-two cohomology is zero. This example exhibits the unit contribution missing from the abbreviated \(H^1=\operatorname{Pic}[n]\) formula.

An elliptic curve. Let \(E\) be a smooth projective genus-one curve with an origin \(o\). Its usual Abel identification sends \(x\) to \(\mathcal O_E(x-o)\) and identifies \(E(k)\) with \(\operatorname{Pic}^0(E)\); equivalently this is the divisor description of the elliptic group law. To check bijectivity, the classical curve Riemann–Roch and duality formula Stacks, Tag 0BS6 says that a degree-one bundle on a genus-one curve has exactly one independent section: its dual tensored with the degree-zero canonical bundle has negative degree and no section. That section has a unique zero of degree one, so the bundle is uniquely \(\mathcal O_E(x)\). Tensoring by \(\mathcal O_E(-o)\) proves the claimed identification. Here a negative-degree bundle has no section because the divisor of a nonzero regular section would be effective of negative degree. The elliptic group law is the one transported by this identification. Thus

The Riemann–Roch formula used here is proved in the AI Integrated Stacks Project at Tag 0BS6; it is a geometric input to this example, not an étale-cohomology theorem.

10. Exercises and complete solutions

1. Compute every \(H^q(\mathbf P^1,\mu_n)\) directly from Kummer.
2. For \(X=\mathbf P^1-\{s_1,\ldots,s_r\}\), \(r\geq1\), compute \(H^1(X,\mu_n)\) using its units. Relate the generators to boundary orders.
3. Prove that \(\operatorname{Pic}(X)\) is divisible for every smooth connected affine curve over \(k\), and deduce the degree-two Kummer vanishing. Explain what happens for division by the characteristic.
4. Let \(A=\mathbf Z/\ell\), with \(\ell\) prime. Show \(H^1(X,A)=\operatorname{Hom}_{\mathrm{cont}}(\pi_1(X,\bar x),A)\) for a connected scheme \(X\). For a smooth projective genus-\(g\) curve over \(k\) and \(\ell\ne\operatorname{char}k\), deduce that its maximal abelian quotient of exponent \(\ell\) has order \(\ell^{2g}\).
5. Prove the finite-pullback formula on \(H^2(-,\mu_n)\), allowing inseparable maps. Determine when the pullback is an isomorphism.

Solution 1. The units of \(\mathbf P^1\) are \(k^\times\), which is \(n\)-divisible. The divisor computation preceding (9.1) gives \(\operatorname{Pic}=\mathbf Z\), with no \(n\)-torsion. Thus the first Kummer segment gives \(H^1=0\), while the second identifies \(H^2\) with \(\mathbf Z/n\), generated by \(c_1^{(n)}(\mathcal O(\infty))\). The degree-zero group is \(\mu_n(k)\). In each degree \(q\geq3\), the adjacent groups \(H^{q-1}(\mathbf G_m)\) and \(H^q(\mathbf G_m)\) are zero by Theorem 4.2, so \(H^q(\mu_n)=0\). This computes all degrees, including \(n=1\), when all groups displayed are zero.

Solution 2. Change the coordinate by an automorphism of \(\mathbf P^1\) so that \(s_r=\infty\), and write the other points as distinct \(a_1,\ldots,a_{r-1}\in k\). The coordinate ring is

Unique factorization in \(k[t]\) implies that its units are exactly \(c\prod_{i=1}^{r-1}(t-a_i)^{m_i}\), with \(c\in k^\times\) and \(m_i\in\mathbf Z\). Indeed both a unit and its inverse can have only the inverted irreducible factors; conversely all these products are invertible. Also \(\operatorname{Pic}(X)=0\): every divisor on this open curve is a sum of points \(b\) outside the removed set, and each \([b]\) is \(\operatorname{div}_X(t-b)\). Kummer therefore identifies \(H^1(X,\mu_n)\) with \((\mathbf Z/n)^{r-1}\), with basis the classes of \(t-a_i\).

Their boundary residue vectors are \(1\) at \(s_i\), \(-1\) at \(s_r\), and zero elsewhere. They form a basis of \(V_S\). For \(r=1\), the curve is \(\mathbf A^1\), its unit quotient is zero, and the stated rank is zero.

Solution 3. Complete \(X\) to \(C\) and choose a boundary point \(s\). Every line bundle on \(X\) extends by Proposition 5.1. Twisting an extension \(L\) by \(-(\deg L)s\) gives a degree-zero extension. Thus it lifts to \(\operatorname{Pic}^0(C)\). For any positive integer \(m\), surjectivity of multiplication by \(m\) on this group gives a bundle \(M\) with \(M^{\otimes m}\) equal to that extension. Its restriction is an \(m\)-th root on \(X\). Hence \(\operatorname{Pic}(X)\) is divisible.

If \(m\) is divisible by the characteristic, this group-divisibility assertion remains true by Picard input 6.1. The étale Kummer sequence used in this lesson still requires \(n\) invertible in \(k\); one cannot infer a characteristic-primary étale Kummer sequence from that divisibility. For the allowed \(n\), (6.1) gives \(H^2(X,\mu_n)=\operatorname{Pic}(X)/n=0\).

Solution 4. We use the precise foundational meaning of the étale fundamental group: with a geometric point \(\bar x\), the fiber functor identifies finite étale covers of a connected scheme with finite continuous left \(\pi_1(X,\bar x)\)-sets. This is Stacks, Tag 0BND, part (1), proved from the finite-étale Galois category and its equivalence theorem Tag 0BN4. This supplies the definition-and-classification prerequisite for \(\pi_1\); the following torsor and quotient calculations are the exercise.

The degree-one torsor classification proved earlier identifies \(H^1(X,A)\) with right torsors under the constant finite group \(A\). Such a sheaf torsor is represented by a finite étale cover: locally it is the disjoint union of \(|A|\) copies of the base, and effective étale descent for these finite locally free algebras glues it. Its geometric fiber is a right \(A\)-torsor. Choose one point of that fiber to identify it with \(A\). The left monodromy action commutes with the right \(A\)-action, so it has the form

for a homomorphism \(\chi:\pi_1\to A\). It is continuous because the fiber action is continuous. Choosing another fiber point conjugates \(\chi\); since \(A\) is abelian this does not change it. Conversely a continuous \(\chi\) gives the finite set \(A\) with action (10.2) and commuting regular right \(A\)-action. Finite-étale classification produces the corresponding cover and action. The torsor map is an isomorphism on the geometric fiber, hence an isomorphism of finite étale covers. These constructions are inverse, and the contracted product of torsors adds \(\chi\), so they give the asserted group isomorphism.

For the projective curve and invertible \(\ell\), choose a primitive \(\ell\)-th root to compare \(A\) with \(\mu_\ell\). Theorem 6.2 makes the character space \(\operatorname{Hom}_{\mathrm{cont}}(\pi_1,\mathbf F_\ell)\) have dimension \(2g\). Choose a basis \(\chi_1,\ldots,\chi_{2g}\). The evaluation map \(\chi:\pi_1\to\mathbf F_\ell^{2g}\) is surjective: if its image were a proper subspace, a nonzero linear functional annihilating it would give a nontrivial linear relation among the basis characters.

Its kernel is the intersection of the kernels of all continuous characters. Every finite abelian quotient of exponent \(\ell\) is a vector space over \(\mathbf F_\ell\), so its coordinate characters force this kernel into its own kernel. The same holds for a profinite abelian quotient of exponent \(\ell\), since its finite quotients separate its points. It therefore factors through the evaluation quotient. This proves that the maximal such quotient is \(\mathbf F_\ell^{2g}\), of order \(\ell^{2g}\). The invertibility assumption is necessary; the argument does not give that order for the characteristic prime.

Solution 5. At a point \(y\) of the target, the finite torsion-free module \(B\) over its local discrete valuation ring is free of rank \(d\). Reduction modulo a uniformizer has dimension \(d\) over \(k\). Its local factors at \(x\mid y\) have dimensions \(e_x=v_x(\pi_y)\), so their sum is \(d\). Thus \(f^*[y]=\sum e_x[x]\) has degree \(d\). Additivity gives \(\deg f^*D=d\deg D\) for every divisor. Represent a line bundle by a divisor and apply naturality of the Kummer connecting map. Its class in the target \(\mathbf Z/n\) becomes \(d\) times that class in the source \(\mathbf Z/n\). This is an isomorphism exactly when \(d\) is a unit modulo \(n\), equivalently \(\gcd(d,n)=1\). For \(n=1\) both groups are zero and the unique map is an isomorphism, in agreement with this criterion.

11. Sources and the arithmetic continuation

The corresponding results of the Stacks project are Stacks, Tag 03R0 for the curve-cohomology setup; Tag 03RH for the valuation sequence and higher units cohomology; and Tag 03RN for the projective, affine and pullback computations. Lemma 3.1 supplies the filtered-colimit argument needed for the infinite point sum; Proposition 8.1 supplies the local-length proof of the pullback assertion whose details Tag 0AMB omits. Picard input 6.1 (proof chain in Section 6), Riemann–Roch for curves (Tag 0BS6) and the classification of finite étale covers (Tags 0BND and 0BN4) are not reproved here; their proofs are in the AI Integrated Stacks Project and in Abelian varieties of the group-schemes course, at the places cited.

Divisors, degree and the Picard group also carry the arithmetic of curves over finite fields, the subject of Weil's proof of the Riemann hypothesis for curves. Over a finite field the arithmetic Frobenius acts on geometric cohomology; keeping \(\mu_n\) distinct from an unchosen constant \(\mathbf Z/n\) is already preparing for that calculation.