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I came across the notion as follows:

Let $X$ be a projective, smooth scheme. And let $$ 0\to M\to N\to \mathcal{O}_{X}\to0 $$ be an exact sequence of coherent $\mathcal{O}_X$-modules. What is meant by “the above exact seqence defines an $M$-torsor on $X$”? I think it may be a standard use of terminology. I just lack knowledge. Thank you!

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up vote 10 down vote accepted

From the associated long exact sequence you obtain a map $H^0(X,\mathcal O_X) \to H^1(X, M)$. $H^1(X,M)$ is just the group paramaterizing $M$-torsors. Since $H^0(X,\mathcal O_X)$ is free of rank one as a module over itself, this map is equivalent to a single element of $H^1(X,M)$, the image of $1$ - in other words, a single $M$-torsor.

Explicitly one is just viewing the inverse image of $1$ in $N$ as an $M$-torsor.

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It helps . Thank you very much! –  TOM Jun 10 '13 at 3:22
The torsor is the inverse image of 1 in $N$. –  Angelo Jun 10 '13 at 5:08

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