I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFDs.

I know the basics of local cohomology but I have not seen a theorem which shows the connection between UFDs and Local Cohomology.

My Question:

Could someone tell me about a result which shows the connection, in particular a theorem which detects whether a ring is a UFD using local cohomology?



1 Answer 1


What you mentioned is about the Grothendieck's proof of Samuel's Conjecture:

If $R$ is a local domain which is a complete intersection and such that $R_{\mathfrak p}$ is UFD for every prime ideal of height $\le 3$, then $R$ is UFD.

See SGA 2, Corollaire 3.14, page 132. (Maybe the keyword here is parafactorial). But probably the best algebraic approach to Grothendieck's method can be found in this paper.

  • $\begingroup$ Thanks for the reference. Somehow i still remain unsatisfied because i was hoping to find a result that goes as " some statement about $H^i_I(R)$ implies UFD'ness of $R$, but perhaps something like this is hidden in the paper you attached. Thanks again. $\endgroup$
    – messi
    May 5, 2013 at 11:11

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