Sunday, 2 July 2006

ag.algebraic geometry - Modules and Square Zero Extensions

If X is a scheme and mathcalM is a quasi-coherent sheaf on X then we can
form a sheaf of rings mathcalA:=mathcalOXoplusmathcalM, on which multiplication
of sections is given just by the same formula as for RoplusM.



The pair (X,mathcalA) is then a scheme which is an infinitesimal thickening of
X, and this is precisely how you pass from a quasi-coherent sheaf to the corresponding
thickening; it is just a sheafified version of the construction in your posting.



(Regarding cohomology, in your question you seemed most interested in the case when
X= Spec R is affine, in which case quasi-coherent sheaves have vanishing higher cohomology, so I'm not sure there is much to say about this.)



Added in response to comment below: To see how these come up geometrically,
consider for example a k-scheme X embedded diagonally into XtimesX.
(Here k is a field, and everything is happening over Spec k.)



Let mathcalIX be the ideal sheaf on XtimesX cutting out the diagonal,
and consider the square-zero thickening
mathcalOXtimesX/mathcalIX2 of X.



This sits in the short exact sequence
0toOmegaX1=mathcalIX/mathcalIX2tomathcalOXtimesX/mathcalIX2tomathcalOXtimesX/IX=mathcalOXto0. The projection p1:XtimesXtoX gives a spliting of this short exact
sequence, and so we find that mathcalOXtimesX/mathcalIX2=mathcalOXoplusOmegaX1.



Recapitulating, we see that in the special case mathcalM=OmegaX1, then
(X,mathcalOXoplusOmegaX1) is equal to the first order infinitesimal neighbourhood of X in XtimesX.



Suppose for example that X is a smooth curve, so that OmegaX1 is a line-bundle.
Then (X,mathcalOXoplusOmegaX1) is locally like the dual numbers (as you observe
in your comment) but is globally twisted (unless X is an elliptic curve, i.e. the genus is 1, which is the one case when OmegaX1 is actually trivial).



This should give you some sense of how these kinds of objects arise geometrically (and
why one would consider other examples rather than just the dual numbers).

No comments:

Post a Comment