The Stacks project

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54.28 Kummer theory

Let $n \in \mathbf{N}$ and consider the functor $\mu _ n$ defined by

\[ \begin{matrix} \mathit{Sch}^{opp} & \longrightarrow & \textit{Ab} \\ S & \longmapsto & \mu _ n(S) = \{ t \in \Gamma (S, \mathcal{O}_ S^*) \mid t^ n = 1 \} . \end{matrix} \]

By Groupoids, Example 38.5.2 this is a representable functor, and the scheme representing it is denoted $\mu _ n$ also. By Lemma 54.15.8 this functor satisfies the sheaf condition for the fpqc topology (in particular, it also satisfies the sheaf condition for the étale, Zariski, etc topology).

Lemma 54.28.1. If $n\in \mathcal{O}_ S^*$ then

\[ 0 \to \mu _{n, S} \to \mathbf{G}_{m, S} \xrightarrow {(\cdot )^ n} \mathbf{G}_{m, S} \to 0 \]

is a short exact sequence of sheaves on both the small and big étale site of $S$.

Proof. By definition the sheaf $\mu _{n, S}$ is the kernel of the map $(\cdot )^ n$. Hence it suffices to show that the last map is surjective. Let $U$ be a scheme over $S$. Let $f \in \mathbf{G}_ m(U) = \Gamma (U, \mathcal{O}_ U^*)$. We need to show that we can find an étale cover of $U$ over the members of which the restriction of $f$ is an $n$th power. Set

\[ U' = \underline{\mathop{\mathrm{Spec}}}_ U(\mathcal{O}_ U[T]/(T^ n-f)) \xrightarrow {\pi } U. \]

(See Constructions, Section 26.3 or 26.4 for a discussion of the relative spectrum.) Let $\mathop{\mathrm{Spec}}(A) \subset U$ be an affine open, and say $f|_{\mathop{\mathrm{Spec}}(A)}$ corresponds to the unit $a \in A^*$. Then $\pi ^{-1}(\mathop{\mathrm{Spec}}(A)) = \mathop{\mathrm{Spec}}(B)$ with $B = A[T]/(T^ n - a)$. The ring map $A \to B$ is finite free of rank $n$, hence it is faithfully flat, and hence we conclude that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is surjective. Since this holds for every affine open in $U$ we conclude that $\pi $ is surjective. In addition, $n$ and $T^{n - 1}$ are invertible in $B$, so $nT^{n-1} \in B^*$ and the ring map $A \to B$ is standard étale, in particular étale. Since this holds for every affine open of $U$ we conclude that $\pi $ is étale. Hence $\mathcal{U} = \{ \pi : U' \to U\} $ is an étale covering. Moreover, $f|_{U'} = (f')^ n$ where $f'$ is the class of $T$ in $\Gamma (U', \mathcal{O}_{U'}^*)$, so $\mathcal{U}$ has the desired property. $\square$

Remark 54.28.2. Lemma 54.28.1 is false when “étale” is replaced with “Zariski”. Since the étale topology is coarser than the smooth topology, see Topologies, Lemma 33.5.2 it follows that the sequence is also exact in the smooth topology.

By Theorem 54.24.1 and Lemma 54.28.1 and general properties of cohomology we obtain the long exact cohomology sequence

\[ \xymatrix{ 0 \ar[r] & H_{\acute{e}tale}^0(S, \mu _{n, S}) \ar[r] & \Gamma (S, \mathcal{O}_ S^*) \ar[r]^{(\cdot )^ n} & \Gamma (S, \mathcal{O}_ S^*) \ar@(rd, ul)[rdllllr] \\ & H_{\acute{e}tale}^1(S, \mu _{n, S}) \ar[r] & \mathop{\mathrm{Pic}}\nolimits (S) \ar[r]^{(\cdot )^ n} & \mathop{\mathrm{Pic}}\nolimits (S) \ar@(rd, ul)[rdllllr] \\ & H_{\acute{e}tale}^2(S, \mu _{n, S}) \ar[r] & \ldots } \]

at least if $n$ is invertible on $S$. When $n$ is not invertible on $S$ we can apply the following lemma.

Lemma 54.28.3. For any $n \in \mathbf{N}$ the sequence

\[ 0 \to \mu _{n, S} \to \mathbf{G}_{m, S} \xrightarrow {(\cdot )^ n} \mathbf{G}_{m, S} \to 0 \]

is a short exact sequence of sheaves on the site $(\mathit{Sch}/S)_{fppf}$ and $(\mathit{Sch}/S)_{syntomic}$.

Proof. By definition the sheaf $\mu _{n, S}$ is the kernel of the map $(\cdot )^ n$. Hence it suffices to show that the last map is surjective. Since the syntomic topology is weaker than the fppf topology, see Topologies, Lemma 33.7.2, it suffices to prove this for the syntomic topology. Let $U$ be a scheme over $S$. Let $f \in \mathbf{G}_ m(U) = \Gamma (U, \mathcal{O}_ U^*)$. We need to show that we can find a syntomic cover of $U$ over the members of which the restriction of $f$ is an $n$th power. Set

\[ U' = \underline{\mathop{\mathrm{Spec}}}_ U(\mathcal{O}_ U[T]/(T^ n-f)) \xrightarrow {\pi } U. \]

(See Constructions, Section 26.3 or 26.4 for a discussion of the relative spectrum.) Let $\mathop{\mathrm{Spec}}(A) \subset U$ be an affine open, and say $f|_{\mathop{\mathrm{Spec}}(A)}$ corresponds to the unit $a \in A^*$. Then $\pi ^{-1}(\mathop{\mathrm{Spec}}(A)) = \mathop{\mathrm{Spec}}(B)$ with $B = A[T]/(T^ n - a)$. The ring map $A \to B$ is finite free of rank $n$, hence it is faithfully flat, and hence we conclude that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is surjective. Since this holds for every affine open in $U$ we conclude that $\pi $ is surjective. In addition, $B$ is a global relative complete intersection over $A$, so the ring map $A \to B$ is standard syntomic, in particular syntomic. Since this holds for every affine open of $U$ we conclude that $\pi $ is syntomic. Hence $\mathcal{U} = \{ \pi : U' \to U\} $ is a syntomic covering. Moreover, $f|_{U'} = (f')^ n$ where $f'$ is the class of $T$ in $\Gamma (U', \mathcal{O}_{U'}^*)$, so $\mathcal{U}$ has the desired property. $\square$

By Theorem 54.24.1 and Lemma 54.28.3 and general properties of cohomology we obtain the long exact cohomology sequence

\[ \xymatrix{ 0 \ar[r] & H_{fppf}^0(S, \mu _{n, S}) \ar[r] & \Gamma (S, \mathcal{O}_ S^*) \ar[r]^{(\cdot )^ n} & \Gamma (S, \mathcal{O}_ S^*) \ar@(rd, ul)[rdllllr] \\ & H_{fppf}^1(S, \mu _{n, S}) \ar[r] & \mathop{\mathrm{Pic}}\nolimits (S) \ar[r]^{(\cdot )^ n} & \mathop{\mathrm{Pic}}\nolimits (S) \ar@(rd, ul)[rdllllr] \\ & H_{fppf}^2(S, \mu _{n, S}) \ar[r] & \ldots } \]

for any scheme $S$ and any integer $n$. Of course there is a similar sequence with syntomic cohomology.

Let $n \in \mathbf{N}$ and let $S$ be any scheme. There is another more direct way to describe the first cohomology group with values in $\mu _ n$. Consider pairs $(\mathcal{L}, \alpha )$ where $\mathcal{L}$ is an invertible sheaf on $S$ and $\alpha : \mathcal{L}^{\otimes n} \to \mathcal{O}_ S$ is a trivialization of the $n$th tensor power of $\mathcal{L}$. Let $(\mathcal{L}', \alpha ')$ be a second such pair. An isomorphism $\varphi : (\mathcal{L}, \alpha ) \to (\mathcal{L}', \alpha ')$ is an isomorphism $\varphi : \mathcal{L} \to \mathcal{L}'$ of invertible sheaves such that the diagram

\[ \xymatrix{ \mathcal{L}^{\otimes n} \ar[d]_{\varphi ^{\otimes n}} \ar[r]_\alpha & \mathcal{O}_ S \ar[d]^1 \\ (\mathcal{L}')^{\otimes n} \ar[r]^{\alpha '} & \mathcal{O}_ S \\ } \]

commutes. Thus we have

54.28.4.1
\begin{equation} \label{etale-cohomology-equation-isomorphisms-pairs} \mathit{Isom}_ S((\mathcal{L}, \alpha ), (\mathcal{L}', \alpha ')) = \left\{ \begin{matrix} \emptyset & \text{if} & \text{they are not isomorphic} \\ H^0(S, \mu _{n, S})\cdot \varphi & \text{if} & \varphi \text{ isomorphism of pairs} \end{matrix} \right. \end{equation}

Moreover, given two pairs $(\mathcal{L}, \alpha )$, $(\mathcal{L}', \alpha ')$ the tensor product

\[ (\mathcal{L}, \alpha ) \otimes (\mathcal{L}', \alpha ') = (\mathcal{L} \otimes \mathcal{L}', \alpha \otimes \alpha ') \]

is another pair. The pair $(\mathcal{O}_ S, 1)$ is an identity for this tensor product operation, and an inverse is given by

\[ (\mathcal{L}, \alpha )^{-1} = (\mathcal{L}^{\otimes -1}, \alpha ^{\otimes -1}). \]

Hence the collection of isomorphism classes of pairs forms an abelian group. Note that

\[ (\mathcal{L}, \alpha )^{\otimes n} = (\mathcal{L}^{\otimes n}, \alpha ^{\otimes n}) \xrightarrow {\alpha } (\mathcal{O}_ S, 1) \]

is an isomorphism hence every element of this group has order dividing $n$. We warn the reader that this group is in general not the $n$-torsion in $\mathop{\mathrm{Pic}}\nolimits (S)$.

Lemma 54.28.5. Let $S$ be a scheme. There is a canonical identification

\[ H_{\acute{e}tale}^1(S, \mu _ n) = \text{group of pairs }(\mathcal{L}, \alpha )\text{ up to isomorphism as above} \]

if $n$ is invertible on $S$. In general we have

\[ H_{fppf}^1(S, \mu _ n) = \text{group of pairs }(\mathcal{L}, \alpha )\text{ up to isomorphism as above}. \]

The same result holds with fppf replaced by syntomic.

Proof. We first prove the second isomorphism. Let $(\mathcal{L}, \alpha )$ be a pair as above. Choose an affine open covering $S = \bigcup U_ i$ such that $\mathcal{L}|_{U_ i} \cong \mathcal{O}_{U_ i}$. Say $s_ i \in \mathcal{L}(U_ i)$ is a generator. Then $\alpha (s_ i^{\otimes n}) = f_ i \in \mathcal{O}_ S^*(U_ i)$. Writing $U_ i = \mathop{\mathrm{Spec}}(A_ i)$ we see there exists a global relative complete intersection $A_ i \to B_ i = A_ i[T]/(T^ n - f_ i)$ such that $f_ i$ maps to an $n$th power in $B_ i$. In other words, setting $V_ i = \mathop{\mathrm{Spec}}(B_ i)$ we obtain a syntomic covering $\mathcal{V} = \{ V_ i \to S\} _{i \in I}$ and trivializations $\varphi _ i : (\mathcal{L}, \alpha )|_{V_ i} \to (\mathcal{O}_{V_ i}, 1)$.

We will use this result (the existence of the covering $\mathcal{V}$) to associate to this pair a cohomology class in $H^1_{syntomic}(S, \mu _{n, S})$. We give two (equivalent) constructions.

First construction: using Čech cohomology. Over the double overlaps $V_ i \times _ S V_ j$ we have the isomorphism

\[ (\mathcal{O}_{V_ i \times _ S V_ j}, 1) \xrightarrow {\text{pr}_0^*\varphi _ i^{-1}} (\mathcal{L}|_{V_ i \times _ S V_ j}, \alpha |_{V_ i \times _ S V_ j}) \xrightarrow {\text{pr}_1^*\varphi _ j} (\mathcal{O}_{V_ i \times _ S V_ j}, 1) \]

of pairs. By (54.28.4.1) this is given by an element $\zeta _{ij} \in \mu _ n(V_ i \times _ S V_ j)$. We omit the verification that these $\zeta _{ij}$'s give a $1$-cocycle, i.e., give an element $(\zeta _{i_0i_1}) \in \check C(\mathcal{V}, \mu _ n)$ with $d(\zeta _{i_0i_1}) = 0$. Thus its class is an element in $\check H^1(\mathcal{V}, \mu _ n)$ and by Theorem 54.19.2 it maps to a cohomology class in $H^1_{syntomic}(S, \mu _{n, S})$.

Second construction: Using torsors. Consider the presheaf

\[ \mu _ n(\mathcal{L}, \alpha ) : U \longmapsto \mathit{Isom}_ U((\mathcal{O}_ U, 1), (\mathcal{L}, \alpha )|_ U) \]

on $(\mathit{Sch}/S)_{syntomic}$. We may view this as a subpresheaf of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}, \mathcal{L})$ (internal hom sheaf, see Modules on Sites, Section 18.27). Since the conditions defining this subpresheaf are local, we see that it is a sheaf. By (54.28.4.1) this sheaf has a free action of the sheaf $\mu _{n, S}$. Hence the only thing we have to check is that it locally has sections. This is true because of the existence of the trivializing cover $\mathcal{V}$. Hence $\mu _ n(\mathcal{L}, \alpha )$ is a $\mu _{n, S}$-torsor and by Cohomology on Sites, Lemma 21.5.3 we obtain a corresponding element of $H^1_{syntomic}(S, \mu _{n, S})$.

Ok, now we have to still show the following

  1. The two constructions give the same cohomology class.

  2. Isomorphic pairs give rise to the same cohomology class.

  3. The cohomology class of $(\mathcal{L}, \alpha ) \otimes (\mathcal{L}', \alpha ')$ is the sum of the cohomology classes of $(\mathcal{L}, \alpha )$ and $(\mathcal{L}', \alpha ')$.

  4. If the cohomology class is trivial, then the pair is trivial.

  5. Any element of $H^1_{syntomic}(S, \mu _{n, S})$ is the cohomology class of a pair.

We omit the proof of (1). Part (2) is clear from the second construction, since isomorphic torsors give the same cohomology classes. Part (3) is clear from the first construction, since the resulting Čech classes add up. Part (4) is clear from the second construction since a torsor is trivial if and only if it has a global section, see Cohomology on Sites, Lemma 21.5.2.

Part (5) can be seen as follows (although a direct proof would be preferable). Suppose $\xi \in H^1_{syntomic}(S, \mu _{n, S})$. Then $\xi $ maps to an element $\overline{\xi } \in H^1_{syntomic}(S, \mathbf{G}_{m, S})$ with $n \overline{\xi } = 0$. By Theorem 54.24.1 we see that $\overline{\xi }$ corresponds to an invertible sheaf $\mathcal{L}$ whose $n$th tensor power is isomorphic to $\mathcal{O}_ S$. Hence there exists a pair $(\mathcal{L}, \alpha ')$ whose cohomology class $\xi '$ has the same image $\overline{\xi '}$ in $H^1_{syntomic}(S, \mathbf{G}_{m, S})$. Thus it suffices to show that $\xi - \xi '$ is the class of a pair. By construction, and the long exact cohomology sequence above, we see that $\xi - \xi ' = \partial (f)$ for some $f \in H^0(S, \mathcal{O}_ S^*)$. Consider the pair $(\mathcal{O}_ S, f)$. We omit the verification that the cohomology class of this pair is $\partial (f)$, which finishes the proof of the first identification (with fppf replaced with syntomic).

To see the first, note that if $n$ is invertible on $S$, then the covering $\mathcal{V}$ constructed in the first part of the proof is actually an étale covering (compare with the proof of Lemma 54.28.1). The rest of the proof is independent of the topology, apart from the very last argument which uses that the Kummer sequence is exact, i.e., uses Lemma 54.28.1. $\square$


Comments (6)

Comment #220 by Matt on

In Lemma 41.28.1 (and then again in 41.28.3) there seems to be conflicting notation or maybe just a typo. Let and then there is , but is already a section over . Since the next sentence says it seems the idea was to work affine locally on the base and so there is a possible relabeling or those two places that is used should be changed to .

Comment #273 by Andrei Halanay on

Shouldn't be intead of ?

Comment #3236 by Dario Weißmann on

Typos: - In the introduction: it is also satisfies - In the discussion directly above Lemma 53.28.5: 'Note that ... hence every element of this group... ' Are we missing an 'is an isomorphism' here?


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