Sunday, 2 July 2006

ag.algebraic geometry - Modules and Square Zero Extensions

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



The pair (X,mathcalA)(X,mathcalA) is then a scheme which is an infinitesimal thickening of
XX, 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=X= Spec RR 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 kk-scheme XX embedded diagonally into XtimesXXtimesX.
(Here kk is a field, and everything is happening over Spec kk.)



Let mathcalIXmathcalIX be the ideal sheaf on XtimesXXtimesX cutting out the diagonal,
and consider the square-zero thickening
mathcalOXtimesX/mathcalI2XmathcalOXtimesX/mathcalI2X of XX.



This sits in the short exact sequence
0toOmega1X=mathcalIX/mathcalI2XtomathcalOXtimesX/mathcalI2XtomathcalOXtimesX/IX=mathcalOXto0.0toOmega1X=mathcalIX/mathcalI2XtomathcalOXtimesX/mathcalI2XtomathcalOXtimesX/IX=mathcalOXto0. The projection p1:XtimesXtoX gives a spliting of this short exact
sequence, and so we find that mathcalOXtimesX/mathcalI2X=mathcalOXoplusOmega1X.



Recapitulating, we see that in the special case mathcalM=Omega1X, then
(X,mathcalOXoplusOmega1X) is equal to the first order infinitesimal neighbourhood of X in XtimesX.



Suppose for example that X is a smooth curve, so that Omega1X is a line-bundle.
Then (X,mathcalOXoplusOmega1X) 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 Omega1X 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).

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