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Recently I read in Milne's book "etale cohomology" that the set $H^1(X,\mu_n)$ ($X$ a scheme, $n$ a nature number, the cohomology is flat cohomology) can be described as the set of pairs $(L,\phi)$, where $L$ is a line bundle on $X$, $\phi$ is a trivialization $O_X\to L^{\otimes n}$.

Given such a pair $(L,\phi)$ one can easily construct a torsor (as was explained in Milne's book page 125): Zariski locally (say Spec$(A)=U\subseteq X$) one gets a trivialization $\psi: O_U\to L$, using this one gets a global section $e\in \Gamma(U,L)$, so $e^{\otimes n}$ is a base of $L^{\otimes n}$. Using the trivialization $\phi: O_U\to L^{\otimes n}$, one gets an element $a\in$$A$ such that $e^{\otimes n}=a\phi(1)$. Then one gets a $\mu_n$ torsor Spec$A[T]/(T^n-a)$, we can patch all these torsors which are constructed Zariski locally together to get a $\mu_n$-torsor on $X$.

How about the converse, i.e given a $\mu_n$-torsor $\pi: P\to X$ how can we get such a pair?

I figured that since $P\times_X P\cong P\times_X\mu_n=$Spec($O_P[T]/(T^n-1)$), one gets a $n$-th roots of unity $a\in \Gamma(P\times_X P,O_{P\times_X P})$ by taking the image of $T$. Thus $a$ defines an automorphism of $O_{P\times_X P}$. If this automorphism satisfies cocycle condition $$p_{12}^*(a)p_{23}^*(a)=p_{13}^*(a)\in \Gamma(P\times_XP\times_XP,O_{P\times_XP\times_XP})$$ then one can get a line bundle on $X$ using fpqc descent. I think it is true that $a$ satisfies cocycle condition, but I have difficulty in verifying this, can anyone help me out? Thanks.

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  • $\begingroup$ A $\mu_n$-torsor isa special kind of $\mathbb G_m$ torsor, which is the same thing as a line bundle. Use that map. $\endgroup$
    – Will Sawin
    Mar 20, 2012 at 3:53
  • $\begingroup$ (If unknown's name were Luke, Will could have written «Use that map, Luke»...) $\endgroup$
    – Mariano Suárez-Álvarez
    Mar 20, 2012 at 4:59
  • $\begingroup$ @Will. Your assertion might be misleading. E.g. if $X$ is $\mathbb A^1_{\mathbb C}\setminus\{0\}$, there is no non-trivial $\mathbb G_m$ torsor and there are of course non-trivial $\mu_n$-torsors. More generally, the Kummer exact sequence tells us that $H^1(X,_\mu_n)$ is an extension of the $n$-torsion of $H^1(X,\mathcal O_X^\times)$ by the $n$-cotorsion of $H^0(X,\mathcal O_X^\times)$. The point in Milne's assertion is to interpret this extension as classifying isomorphism classes of "suitably enriched" $\mathbb G_m$-torsors. $\endgroup$
    – Jef
    Mar 20, 2012 at 8:50
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    $\begingroup$ @unknown. To expand on Will's first comment, start with P as in your question and choose a trivialisation $\mu_n\simeq P^{\otimes n}$. Then put $L:=P\times^{\mu_n}\mathbb G_m$ the induced $\mathbb G_m$-torsor. Your trivialision of $P^{\otimes n}$ will induce a trivialisation $\phi$ of $L^{\otimes n}\simeq (P^{\otimes n})\times^{\mu_n} \mathbb G_m$. $\endgroup$
    – Jef
    Mar 20, 2012 at 9:05

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