Étale cohomology of curves

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Algebraic Geometry

A computation of the étale cohomology of the multiplicative sheaf on curves

Published

June 4, 2026

Introduction

This post grew out of a theorem I encountered last semester in a course on étale cohomology. I wanted to collect in one place the ingredients entering the proof, mostly for my own understanding.

The theorem we prove is the following.

Theorem 1 Let $C$ be an algebraic curve (of dimension 1, finite type, and separated over a field) over a separably closed field $k$. Then \[ \mathrm{H}^i(C, \mathcal{O}^\times_{C,\mathrm{ét}})= \begin{cases} \mathcal{O}_C(C)^\times &i=0\\ \mathrm{Pic}(C) &i=1 \\ 0 &i>1\text{ and $k$ is algebraically closed}\\ p^\infty \text{-torsion group } &i>1 \text{ and $k$ is non-perfect with char($k$)=p} \end{cases} \]

One of the crucial points of the proof is to reduce the computation of higher cohomology of $\mathcal{O}_{C,\mathrm{ét}}^\times$ to the generic point of the curve. Once this is done, the problem becomes a statement in Galois cohomology.

Preliminaries

In this section we collect the auxiliary facts used in the proof: vanishing results for finite schemes over a field, finite sums of skyscraper sheaves, and torsion sheaves.

Lemma 1 Let $X$ be a finite scheme over a separably closed field $k$. Then $\mathrm{H}^i(X,\mathcal{F})=0$ for $i>0$ and for all $\mathcal{F}\in \mathrm{Sh}(X_{\mathrm{ét}})$.

Proof. Since the structure map $f:X\to \mathrm{Spec}(k)$ is finite, $f_*:\mathrm{Sh}(X _{\mathrm{ét}})\to\mathrm{Sh}(\mathrm{Spec}(k)_{\mathrm{ét}})$ is exact. Therefore, the derived functor $\mathrm{R}^qf_*$ vanishes for every $q>0$.

Consider the Grothendieck spectral sequence \[ \mathrm{E}_2 ^{p,q}=\mathrm{H}^p(\mathrm{Spec}(k) _{\mathrm{ét}}, \mathrm{R}^qf_* \mathcal{F})\Longrightarrow \mathrm{H} ^{p+q}(X _{\mathrm{ét}}, \mathcal{F}) \] By the vanishing of $R^qf_*$ we get that $\mathrm{H}^p(\mathrm{Spec}(k) _{\mathrm{ét}}, f_* \mathcal{F})\cong \mathrm{H}^p(X _{\mathrm{ét}}, \mathcal{F})$. Since $k$ is separably closed, it is a strictly henselian ring, so global sections are isomorphic to stalks. Hence global sections are exact, and the cohomology vanishes.

Lemma 2 Let $X$ be a scheme, $i: Z\to X$ a closed immersion, and $\mathcal{F}\in \mathrm{Sh}(X_{\mathrm{ét}})$ be supported on $Z$. Then the canonical morphism $\eta: \mathcal{F}\to i_*i^* \mathcal{F}$ is an isomorphism.

Proof. We check that $\eta$ is an isomorphism on stalks. Let $\bar x$ be a geometric point of $X$, and let $x$ be the underlying point. Since $i$ is a closed immersion, it is finite. Hence, by Stacks project Tag 0DK3, we have \[ (i_*i^* \mathcal{F}) _{\bar x}\cong \prod_{\bar z}(i^*\mathcal{F}) _{\bar z}, \] where the product is taken over the geometric points $\bar z$ of $Z$ lying over $\bar x$.

If $x\notin Z$, this product is empty, hence \[ (i_*i^* \mathcal{F}) _{\bar x}=0. \] Since $\mathcal{F}$ is supported on $Z$, also $\mathcal{F}_{\bar x}=0$, so $\eta_{\bar x}$ is an isomorphism.

If $x\in Z$, then $\bar x$ factors uniquely through $Z$, say as a geometric point $\bar z$ of $Z$. Hence the product has exactly one factor, and \[ (i_*i^* \mathcal{F}) _{\bar x}\cong (i^*\mathcal{F}) _{\bar z}\cong \mathcal{F}_{\bar x} \]

Corollary 1 (of Lemma 2) Let $X$ be a scheme and $\mathcal{F}\in \mathrm{Sh}(X_{\mathrm{ét}})$ supported on finitely many closed points of $X$. Then \[ \mathcal{F}\cong\bigoplus_{x\in \mathrm{supp}(\mathcal{F})}x_* \mathcal{F} _{x} \]

Proof. We apply Lemma 2 with $Z:=\{x\to X: x \text{ is a geometric point}\}$ and $i: Z\to X$. We then have that \[ \mathcal{F}\cong i_*i^* \mathcal{F} \] and \[ i_*i^* \mathcal{F}\cong i_*\Big(\bigoplus_{x\in \mathrm{supp}(X)} \mathcal{F}_x\Big) \cong \bigoplus_{x\in \mathrm{supp}(X)} x_*\mathcal{F}_x \] where the latter isomorphism holds because the points are finitely many.

Lemma 3 Let $X$ be a scheme, $x$ be a geometric point of $X$, and $\mathcal{F}\in \mathrm{Sh}(x_{\mathrm{ét}})$. Then for every $i>0$ \[ \mathrm{R}^ix_* \mathcal{F}=0. \]

In particular, \[ \mathrm{H}^i(X_{\mathrm{ét}}, x_*\mathcal{F})=0 \] for all $i>0$.

Proof. Consider an exact sequence \[ 0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0 \] of étale sheaves over $x_{\mathrm{ét}}$. Since $x$ is a geometric point, the functor $x_*$ is exact. Indeed, sections of $x_*\mathcal{F}$ over an étale $U\to X$ are products of stalks over the geometric fibre of $U$ at $x$, and products of abelian groups are exact. Therefore \[ \mathrm{R}^ix_* \mathcal{F}=0 \] for every $i>0$.

As a consequence, with the same reasoning used in the proof of Lemma 1 using the Grothendieck spectral sequence, we obtain \[ \mathrm{H}^i(X_{\mathrm{ét}}, x_*\mathcal{F})=0 \] for every $i>0$.

Lemma 4 Let $X$ be a quasi-compact quasi-separated (qcqs) scheme, and $\mathcal{F}\in \mathrm{Sh}(X_{\mathrm{ét}})$ be a torsion sheaf. Then $\mathrm{H}^i(X, \mathcal{F})$ is a torsion group for all $i\geq 0$.

Proof. Let $\mathcal{F}[n]$ be the kernel of the multiplication by $n$ map $\mathcal{F}\xrightarrow{\cdot n} \mathcal{F}$. We have an isomorphism \[ \varinjlim (\mathcal{F}[n])\to \mathcal{F} \] because it is an isomorphism on stalks: \[ (\varinjlim \mathcal{F}[n])_x\cong\varinjlim((\mathcal{F}[n])_x)\cong \varinjlim((\mathcal{F}_x)[n])\cong \mathcal{F}_x \]

Since $X$ is qcqs, the cohomology commutes with colimits: \[ \mathrm{H}^i(X _{\mathrm{ét}}, \mathcal{F})\cong \mathrm{H}^i(X _{\mathrm{ét}}, \varinjlim(\mathcal{F}[n]))\cong \varinjlim(\mathrm{H}^i(X _{\mathrm{ét}}, \mathcal{F}[n])) \]

Since the multiplication by $n$ map is the zero map on $\mathcal{F}[n]$, in cohomology it is again the zero map (since the cohomology functor is additive). Therefore, the cohomology is a colimit of torsion groups, and therefore it is a torsion group.

Lemma 5 Let $k$ be a separably closed field, $K/k$ a field extension of transcendence degree $1$, and $L/K$ be an algebraic and separable field extension. Then $L':=(L\otimes_k \bar k)_{\mathrm{red}}$, where $\bar k$ denotes an algebraic closure, is a field with $C_1$ property and $L'/L$ is purely inseparable.

Proof. We first show that $L'/L$ is purely inseparable. If $\operatorname{char}(k)=0$, then $k$ is already algebraically closed, so $\bar k=k$, hence $L'=L$, and there is nothing to prove. If $\mathrm{char}(k)=p>0$, it is easy to verify that for every $x \in L\otimes _k \bar k$ we get for a suitable $n$ that $x^{p^n}\in L$ (write $x$ as a sum of elementary tensor). Thus, passing to the reduction gives the purely inseparable extension.

It remains to prove that $L'$ is $C_1$. Since $L/k(t)$ is algebraic, $A:=L\otimes_k \bar k$ is integral over $k(t)\otimes_k \bar k$. Now $t$ is transcendental over $k$, so \[ k(t)\otimes_k \bar k \cong \bar k(t). \] Hence $A$ is integral over the field $\bar k(t)$, and therefore $L'/\bar k(t)$ is algebraic (being $L'$ a quotient of $A$).

Now $\bar k$ is algebraically closed, so by Tsen’s theorem the field $\bar k(t)$ is $C_1$. Since algebraic extensions of $C_1$-fields are again $C_1$, it follows that $L'$ is $C_1$.

Proof of the theorem

We are now ready to prove that

\[ \mathrm{H}^i(C, \mathcal{O}^\times_{C,\mathrm{ét}})= \begin{cases} \mathcal{O}_C(C)^\times &i=0\\ \mathrm{Pic}(C) &i=1 \\ 0 &i>1 \text{ and $k$ is algebraically closed}\\ p^\infty \text{-torsion group } &i>1 \text{ and $k$ is non-perfect with char($k$)=p} \end{cases} \]

Proof. The case $i=0$ is clear. We will not prove the case $i=1$, since it follows from the interpretation of first cohomology in terms of torsors. More precisely, $H^1(C_{\mathrm{ét}},\mathcal{O}_{C,\mathrm{ét}}^\times)$ classifies $\mathcal{O}_{C,\mathrm{ét}}^\times$-torsors on $C_{\mathrm{ét}}$, see Tag 03AJ. These torsors are equivalent to line bundles on $C$, thus we obtain \[ H^1(C_{\mathrm{ét}}, \mathcal{O}_{C,\mathrm{ét}}^\times)\cong \mathrm{Pic}(C), \] see Tag 040E for the proof.

We now study the case of $i>1$.

Step 1: suppose the curve is regular and connected.

In particular, the curve is integral, since every stalk is a noetherian regular local ring (hence reduced) and since every point belongs to exactly one irreducible component. Since irreducible components are closed (because $C$ is a noetherian topological space), the curve is indeed integral.

Let $\eta$ be the generic point. Consider the following exact sequence: \[ 0\to \mathcal{O}_{C, \mathrm{ét}}^\times\to \eta_* \mathcal{O}_{\eta, \mathrm{ét}}^\times \to \bigoplus _{c\in C \text{ closed }} c_* \mathbb{Z}\to 0 \tag{1}\]

where the second map is the divisor map. This sequence is exact since on the stalk at a geometric point $\bar x$ over a closed point it is given by \[ 0\to (\mathcal{O}_{C, \bar x} ^{\mathrm{sh}})^\times \to (\mathrm{Frac}(\mathcal{O}_{C, \bar x} ^{\mathrm{sh}}))^\times\to \mathbb{Z}\to 0 \] and the right-hand side map is the valuation map.

Taking cohomology of sequence 1 gives that \[ \mathrm{H}^i(C, \mathcal{O}_{C, \mathrm{ét}}^\times)\cong \mathrm{H}^i(C, \eta_* \mathcal{O}_{\eta, \mathrm{ét}}^\times) \tag{2}\] for all $i>1$. Indeed, the sheaf $\bigoplus _{c\in C \text{ closed }} c_* \mathbb{Z}$ is the filtered colimit of the finite sums $\bigoplus _{c\in S} c_* \mathbb{Z}$, where $S$ ranges over the finite subsets of closed points of $C$. By Corollary 1 and Lemma 3, each such finite sum has vanishing higher cohomology. Since $C$ is qcqs, filtered colimits of abelian sheaves on $C_{\mathrm{ét}}$ commute with cohomology by the Stacks Project Tag 073D. Therefore \[ \mathrm{H}^i\left(C_{\mathrm{ét}},\bigoplus _{c\in C \text{ closed }} c_* \mathbb{Z}\right)=0 \] for every $i>0$, and the claimed isomorphism follows from the long exact sequence in cohomology.

By Tag 03Q9, we have for every geometric point $c$ of $C$ \[ (\mathrm{R}^q\eta _* \mathcal{O}_{\eta, \mathrm{ét}} ^\times)_c\cong \mathrm{H}^q((\eta\times_C \mathrm{Spec}(\mathcal{O}_{C,c} ^{\mathrm{sh}}))_{ét}, (\mathcal{O}^\times _{\eta,\mathrm{ét}})_{|\eta\times_C \mathrm{Spec}(\mathcal{O} _{C,c} ^{\mathrm{sh}})}) \tag{3}\]

As $\eta \times_C \mathrm{Spec}(\mathcal{O} _{C,c}^\mathrm{sh})=\mathrm{Spec}(\mathrm{Frac}(\mathcal{O}_{C,c}^\mathrm{sh}))$, write $L$ for $\mathrm{Frac}(\mathcal{O}_{C,c} ^{\mathrm{sh}})$, so that the restriction of $\mathcal{O}_{\eta ,\mathrm{ét}}^\times$ appearing in the right-hand side of 3 is just $\mathcal{O}_{\mathrm{Spec}(L),\mathrm{ét}}^\times$. Observe that $L$ is an algebraic separable extension of the function field $\kappa(C)$. Indeed, $\mathcal{O}_{C,c}^ \mathrm{sh}$ is the filtered colimit of local étale $\mathcal{O}_{C,c}$-algebras $B$, and for each such $B$ the field $\mathrm{Frac}(B)=B\otimes _{\mathcal{O}_{C,c}}\kappa(C)$ is a finite separable extension of $\kappa(C)$, thus every element of $L$ lies in some finite separable extension of $\kappa(C)$.

By the relation between Galois and étale cohomology, we have \[ \mathrm{H}^q(\mathrm{Spec}(L), \mathcal{O}_{\mathrm{Spec}(L), \mathrm{ét}}^\times)\cong \mathrm{H}^q(\mathrm{Gal}(L^\mathrm{sep}/L), (L^{\mathrm{sep}})^\times) \tag{4}\]

By Hilbert 90, this vanishes for $q=1$.

Claim. The group \[ \mathrm{H}^q(\mathrm{Gal}(L^\mathrm{sep}/L), (L^{\mathrm{sep}})^\times) \] is a $p^\infty$-torsion group for every $q\ge 2$.

Proof. By Lemma 5, $L':=(L\otimes_k \bar k)_{\mathrm{red}}$ is a $C_1$-field with $L'/L$ purely inseparable. It is a well-known Galois-cohomology fact (Stacks project Tag 03R8 and Tag 03RC) that for a $C_1$-field \[ \mathrm{H}^i(\mathrm{Gal}({L'}^{\mathrm{sep}}/L'), ({L'} ^{\mathrm{sep}})^\times)=0 \] for every $i>0$. Therefore, by pure inseparability of $L'/L$, we get, by the isomorphism on Galois groups, \[ \mathrm{H}^i(\mathrm{Gal}(L ^{\mathrm{sep}}/L), ({L'} ^{\mathrm{sep}})^\times)\cong \mathrm{H}^i(\mathrm{Gal}({L'} ^{\mathrm{sep}}/L'), ({L'} ^{\mathrm{sep}})^\times)=0 \] for every $i>0$.

Consider the short exact sequence \[ 0\to (L ^{\mathrm{sep}})^\times \to ({L'} ^{\mathrm{sep}})^\times\to Q \to 0 \] where $Q$ is the cokernel of the first map. By pure inseparability of ${L'} ^{\mathrm{sep}}/(L ^{\mathrm{sep}})$, we have that $Q$ is a $p^\infty$-torsion group. Passing to cohomology, we get \[ \mathrm{H}^{i+1}(\mathrm{Gal}(L ^{\mathrm{sep}}/L), (L ^{\mathrm{sep}})^\times)\cong \mathrm{H} ^{i}(\mathrm{Gal}(L ^{\mathrm{sep}}/L), Q) \] for every $i>0$. Therefore, by Lemma 4, we get the claim.

By isomorphisms 3, 4, and Claim, we get that \[ (\mathrm{R}^q\eta_* \mathcal{O}_{\eta, \mathrm{ét}}^\times)_c \] is a $p^\infty$-torsion group at every geometric point $c$, thus the sheaf $R^q \eta_* \mathcal{O}_{\eta , \mathrm{ét}}^\times$ is a $p^\infty$-torsion sheaf.

Consider the Grothendieck spectral sequence \[ \mathrm{E}_2^{i,j}:=\mathrm{H}^i(C_{\mathrm{\acute et}},\mathrm{R}^j\eta_*\mathcal O^\times_{\eta,\mathrm{\acute et}}) \Longrightarrow \mathrm{H}^{i+j}(\eta_{\mathrm{\acute et}},\mathcal O^\times_{\eta,\mathrm{\acute et}}). \] Since this is a first-quadrant spectral sequence, we also get \[ \mathrm{E}_2^{i,j}\otimes_{\mathbb{Z}}\mathbb{Z}[p^{-1}] \Longrightarrow \mathrm{H}^{i+j}(\eta_{\mathrm{\acute et}},\mathcal O^\times_{\eta,\mathrm{\acute et}})\otimes_{\mathbb{Z}}\mathbb{Z}[p^{-1}]. \]

For every $j\geq 1$, we proved that $\mathrm{R}^j\eta_*\mathcal O^\times_{\eta,\mathrm{\acute et}}$ is a $p^\infty$-torsion sheaf. Hence $\mathrm{E}_2^{i,j}=\mathrm{H}^i(C_{\mathrm{\acute et}},\mathrm{R}^j\eta_*\mathcal O^\times_{\eta,\mathrm{\acute et}})$ is a $p^\infty$-torsion group for every $i\geq 0$ and every $j\geq 1$, by Lemma 4. Therefore \[ \mathrm{E}_2^{i,j}\otimes_{\mathbb{Z}}\mathbb{Z}[p^{-1}]=0 \qquad (j\geq 1). \] Thus the localized spectral sequence has only the row $j=0$ possibly nonzero, and therefore it degenerates at the $\mathrm{E}_2$-page. Hence for every $n>0$ one gets an isomorphism \[ \mathrm{E}_2^{n,0}\otimes_{\mathbb{Z}}\mathbb{Z}[p^{-1}] \cong \mathrm{H}^n(\eta_{\mathrm{\acute et}},\mathcal O^\times_{\eta,\mathrm{\acute et}})\otimes_{\mathbb{Z}}\mathbb{Z}[p^{-1}]. \]

Now $\mathrm{H}^1(\eta_{\mathrm{\acute et}},\mathcal O^\times_{\eta,\mathrm{\acute et}})=0$ by Hilbert-90, and for every $n\geq 2$ the group $\mathrm{H}^n(\eta_{\mathrm{\acute et}},\mathcal O^\times_{\eta,\mathrm{\acute et}})$ is a $p^\infty$-torsion group by the same Galois-cohomological argument as above, applied to the field $\kappa(C)$. Therefore \[ \mathrm{H}^n(\eta_{\mathrm{\acute et}},\mathcal O^\times_{\eta,\mathrm{\acute et}})\otimes_{\mathbb{Z}}\mathbb{Z}[p^{-1}]=0 \qquad (n>0). \] It follows that \[ \mathrm{E}_2^{n,0}\otimes_{\mathbb{Z}}\mathbb{Z}[p^{-1}]=0 \qquad (n>1). \]

Since $\mathrm{E}_2^{n,0}=\mathrm{H}^n(C_{\mathrm{\acute et}},\eta_*\mathcal O^\times_{\eta,\mathrm{\acute et}})$, we obtain \[ \mathrm{H}^n(C_{\mathrm{\acute et}},\eta_*\mathcal O^\times_{\eta,\mathrm{\acute et}})\otimes_{\mathbb{Z}}\mathbb{Z}[p^{-1}]=0 \qquad (n>1). \]

By the isomorphism 2, this gives the claim in the case of a regular connected curve.

If $k$ is algebraically closed, the previous argument gives a stronger conclusion. Indeed, in this case the field $L=\mathrm{Frac}(\mathcal{O}_{C,c}^{\mathrm{sh}})$ is a $C_1$-field, being an algebraic extension of the function field $\kappa(C)$. Hence, by the same Galois-cohomological vanishing used above, we have \[ \mathrm{H}^q(\mathrm{Gal}(L^{\mathrm{sep}}/L),(L^{\mathrm{sep}})^\times)=0 \] for every $q>0$. Therefore \[ (\mathrm{R}^q\eta_* \mathcal{O}_{\eta,\mathrm{ét}}^\times)_c=0 \] for every geometric point $c$ of $C$ and every $q>0$, so \[ \mathrm{R}^q\eta_* \mathcal{O}_{\eta,\mathrm{ét}}^\times=0 \] for every $q>0$.

Thus the Grothendieck spectral sequence degenerates and gives \[ \mathrm{H}^n(C_{\mathrm{ét}},\eta_*\mathcal{O}_{\eta,\mathrm{ét}}^\times) \cong \mathrm{H}^n(\eta_{\mathrm{ét}},\mathcal{O}_{\eta,\mathrm{ét}}^\times) \] for every $n\geq 0$. Since $\kappa(C)$ is again a $C_1$-field, the right-hand side vanishes for every $n>0$. Therefore \[ \mathrm{H}^n(C_{\mathrm{ét}},\eta_*\mathcal{O}_{\eta,\mathrm{ét}}^\times)=0 \] for every $n>0$. By 2, this proves the vanishing of \[ \mathrm{H}^n(C_{\mathrm{ét}},\mathcal{O}_{C,\mathrm{ét}}^\times) \] for every $n>1$ in the case where $k$ is algebraically closed.
Step 2: suppose $C$ is regular.

Apply Step 1 to every connected component.
Step 3: suppose $C$ is reduced.

Consider the normalization $\nu: C'\to C$ of the curve. We first prove that the natural morphism \[ \mathcal{O}_{C, \mathrm{ét}}^\times\to \nu_* \mathcal{O}_{C',\mathrm{ét}}^\times \] is injective.

Let $C _{\mathrm{reg}}$ be the regular locus of $C$. It is Zariski open in $C$, since $C$ is locally of finite type over a field. Moreover, since $C$ is reduced, each generic point of $C$ is regular, thus $C _{\mathrm{reg}}$ is dense. Since regular points are normal, the normalization is an isomorphism over $C _{\mathrm{reg}}$. Hence \[ (\mathcal{O}_{C, \mathrm{ét}}^\times)_{|C _{\mathrm{reg}}}\to (\nu_* \mathcal{O}_{C', \mathrm{ét}}^\times)_{|C _{\mathrm{reg}}} \] is an isomorphism.

Now let $V\to C$ be an affine étale morphism. Since étale morphisms are open, \[ V_{\mathrm{reg}}:=V\times_C C_{\mathrm{reg}} \] is a dense open subset of $V$. After base change to $V$, the normalization map is still an isomorphism over $V_{\mathrm{reg}}$. Therefore the map \[ \Gamma(V,\mathcal{O}_V^\times)\to \Gamma(V\times_C C',\mathcal{O}_{V\times_C C'}^\times) \] becomes an isomorphism after restricting to the dense open subset $V_{\mathrm{reg}}$.

Since $C$ is reduced and $V\to C$ is étale, also $V$ is reduced. Hence restriction to a dense open subset is injective, thus \[ \Gamma(V,\mathcal{O}_V^\times)\to \Gamma(V\times_C C',\mathcal{O}_{V\times_C C'}^\times) \] is injective. Since this holds for every affine étale $V\to C$, the morphism \[ \mathcal{O}_{C, \mathrm{ét}}^\times\to \nu_* \mathcal{O}_{C',\mathrm{ét}}^\times \] is injective.

Now define $Q$ as the cokernel of this injective morphism. We have an exact sequence \[ 0\to \mathcal{O}_{C, \mathrm{ét}}^\times\to \nu_* \mathcal{O}_{C',\mathrm{ét}}^\times\to Q\to 0 \tag{5}\]

Since the normalization is an isomorphism over $C _{\mathrm{reg}}$, the sheaf $Q$ vanishes on $C _{\mathrm{reg}}$. Hence $Q$ is supported on finitely many points of $C$. Therefore, by Corollary 1, we get \[ \mathrm{H}^i(C _{\mathrm{ét}}, \mathcal{O}_{C, \mathrm{ét}}^\times)\cong \mathrm{H}^i(C _{\mathrm{ét}}, \nu_* \mathcal{O}_{C', \mathrm{ét}}^\times) \] for every $i\geq 2$.

Moreover, by the Stacks Project Tag 035S and Tag 035B, $\nu$ is finite. Hence, by the exactness of the direct image of a finite morphism on étale sheaves, $\nu_*$ is exact, giving \[ \mathrm{H}^i(C _{\mathrm{ét}}, \nu_* \mathcal{O}^\times_{C', \mathrm{ét}})\cong \mathrm{H}^i(C', \mathcal{O}^\times_{C', \mathrm{ét}}) \]

Since $C'$ is normal of dimension $1$, it is regular, so we can apply Step 2.
Step 4: no additional hypotheses on $C$.

Consider $\mathcal{N}:=\mathrm{Nil}(\mathcal{O}_C)$. We proceed by induction on $n$ such that $\mathcal{N} ^{2^n}=0$. The case $n=0$ is given by Step 3. Assume now that $n>0$ and that the result is known whenever the nilradical is killed by $2^{n-1}$. Define $J:=\mathcal{N} ^{2^{n-1}}$, let $C_0$ be the closed subscheme defined by it, and let $\pi: C_0\to C$. Since $J^2=\mathcal{N}^{2^n}=0$, we get an exact sequence \[ 0\to J\to \mathcal{O}_{C, \mathrm{ét}}^\times\to \pi_*\mathcal{O}_{C_0, \mathrm{ét}}^\times\to 1 \] where the left map sends an element $a$ to $1+a$ on sections. Since $J$ is quasi-coherent, \[ \mathrm{H}^i(C _{\mathrm{ét}}, J)\cong \mathrm{H}^i(C _{\mathrm{Zar}}, J) \] for every $i\geq 0$. Thus, by dimension vanishing, we get an isomorphism \[ \mathrm{H}^i(C _{\mathrm{ét}}, \mathcal{O}_{C, \mathrm{ét}}^\times)\cong \mathrm{H}^i(C _{\mathrm{ét}}, \pi_* \mathcal{O}_{C_0, \mathrm{ét}}^\times) \] for every $i>1$.

Since $\pi$ is a closed immersion, it is finite, thus $\pi_*$ is exact, giving \[ \mathrm{H}^i(C_{\mathrm{ét}}, \pi_* \mathcal{O}_{C_0, \mathrm{ét}}^\times)\cong \mathrm{H}^i((C_0)_{\mathrm{ét}}, \mathcal{O}_{C_0, \mathrm{ét}}^\times) \] for every $i\geq 0$. Moreover, the nilradical of $\mathcal{O}_{C_0}$ is killed by $2^{n-1}$. Applying the induction hypothesis, we get the claim.