# classify \mu_n torsors

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$$A such that e^{\otimes n}=a\phi(1). Then one gets a \mu_n torsor SpecA[T]/(T^n-a), we can patch all these torsors which are constructed Zariski locally together to get a \mu_n-torsor on X. How about the converse, i.e given a \mu_n-torsor \pi: P\to X how can we get such a pair? I figured that since P\times_X P\cong P\times_X\mu_n=Spec(O_P[T]/(T^n-1)), one gets a n-th roots of unity a\in \Gamma(P\times_X P,O_{P\times_X P}) by taking the image of T. Thus a defines an automorphism of O_{P\times_X P}. If this automorphism satisfies cocycle condition$$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. - A$\mu_n$-torsor isa special kind of$\mathbb G_m$torsor, which is the same thing as a line bundle. Use that map. – Will Sawin Mar 20 '12 at 3:53 (If unknown's name were Luke, Will could have written «Use that map, Luke»...) – Mariano Suárez-Alvarez Mar 20 '12 at 4:59 @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. – Jef Mar 20 '12 at 8:50 @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\$. – Jef Mar 20 '12 at 9:05