Wednesday, November 21, 2012

Misconceptions about $K_X$

This is the title of a very short paper by Steven Kleiman published in L'enseignement mathématique, and which should be studied by every young student in scheme theory.

Here, $X$ is a scheme and $K_X$ is the sheaf of rational functions on $X$.

The misconceptions are the following, where we write $\mathop{Frac}(A)$ for the total ring of fractions of a ring $A$, namely the localized ring with respect to all element which are not zero divisors.

  1. $K_X$ is not the sheaf associated to the presheaf $U\mapsto \mathop{Frac}(\Gamma(U,O_X))$; indeed, that map may not be a presheaf.
  2. The germ $K_{X,x}$ of $K_X$ at a point $x$ may not be the total ring of fractions of the local ring $O_{X,x}$, it may be smaller.
  3. If $U=\mathop{Spec}(A)$ is an affine open subset of $X$, then $\Gamma(U,K_X)$ is not necessarily equal to $\mathop{Frac}(A)$.
These mistakes can be found in the writings of very good authors, even Grothendieck's EGA IV... 
By chance, the first one is corrected in a straightforward way, and the other two work when the scheme $X$ is locally noetherian.

Thanks to Antoine D. for indicating to me this mistake, and to Google for leading me to Kleiman's paper.

No comments :

Post a Comment