I think I am confused about some terminology in algebraic geometry, specifically the meaning of the term "torsor". Suppose that I fix a scheme S. I want to work with torsors over S. Let $\mu$ be a sheaf of abelian groups over S. Then my understanding is that a $\mu$-torsor, what ever that is, should be classified by the cohomology gorup $H^1(X; \mu) \cong \check H^1(X; \mu)$.

Now suppose that $\mu$ is representable in the category of schemes over S, i.e. there is a group object $$\mathbb{G} \to S$$ in the category of schemes over $S$, and maps (over S) to $\mathbb{G}$ is the same as $\mu$. Lots of interesting example arise this way.

I also thought that in this case a torsor over S can be defined as a scheme $P \to S$ over S with an action of the group $\mathbb{G}$ such that locally in S it is trivial. I.e. there exists a cover $U \to S$ such that $$ P \times_S U \cong \mathbb{G} \times_S U $$ as spaces over S with a $\mathbb{G}$-action (or rather as spaces over U with a $\mathbb{G} \times_S U$-action).

The part that confuses me is that these two notions don't seem to agree. Here is an example that I think shows the difference. Let $S= \mathbb{A}^1$ be the affine line (over a field k) and let $x_1$ and $x_2$ by two distinct points in $S$. Consider the subscheme $Y = x_1 \cup x_2$, and let $C_Y$ be the complement of Y in S. Let $A$ be your favorite finite abelian group which we consider as a constant sheaf over S. Then we have an exact sequence of sheaves over S, $$0 \to A_{C_Y} \to A \to i_*A \to 0$$ Where $i_*A(U) = A(U \cap Y)$. I believe the first two are representable by schemes over S, namely $$C_Y \times A \cup S \times 0$$ and $S \times A$, where we are viewing the finite set $A$ as a scheme over $k$ (and these products are scheme-theoretic products of schemes over $spec \; k$).

In any event, the long exact sequence in sheaf cohomology shows that $$H^1(S; A_{C_Y}) \cong \check H^1(S; A_{C_Y}) \cong A$$ and it is easy to build an explicit C$\check{\text{e}}$ch cocycle using the covering given by the two opens consisting of the subschemes $U_i = S \setminus x_i$, for $i = 1,2$.

Now the problem comes when I try to glue these together to get a representable object over S, i.e. a torsor in the second sense. Then I am looking at the coequalizer of $$C_Y \times A \rightrightarrows (C_Y\cup C_Y) \times A$$ where the first map is the inclusion and the second is the usual inclusion together with addition by a given fixed element $a \in A$. This seems to just gives back the trivial "torsor" $C_Y \times A$.

Am I doing this calculation wrong, or is there really a difference between these two notions of torsor?

`$U_1 \times A$`

to`$U_2 \times A$`

along`$C_Y \times A$`

to get the correct torsor over S, no? The restriction of your torsor to`$U_i$`

should be trivial. $\endgroup$ – Tyler Lawson Mar 25 '10 at 18:46`$C_Y \times A$`

, when in fact that was your goal. The torsor's only nontrivial when considered over the whole base scheme. Your torsor should be the coequalizer`$$ C_Y \times A \rightrightarrows (U_1 \cup U_2) \times A $$`

$\endgroup$ – Tyler Lawson Mar 25 '10 at 19:00`$C_Y \times A$`

, with set of sections isomorphic to $A$. There is preciselyoneof these sections that has an extension to`$U_1$`

, and another that has an extension to`$U_2$`

, and the difference between those two sections is the element of $A$ that you're looking for. $\endgroup$ – Tyler Lawson Mar 25 '10 at 19:19