What is the nature of the zero locus of a section of a coherent sheaf? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T06:56:23Z http://mathoverflow.net/feeds/question/53097 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53097/what-is-the-nature-of-the-zero-locus-of-a-section-of-a-coherent-sheaf What is the nature of the zero locus of a section of a coherent sheaf? Ramsey 2011-01-24T18:07:58Z 2011-01-24T18:07:58Z <p>Suppose that $X$ is a reduced rigid space and $\scr{F}$ is a coherent sheaf on $X$. For a section $f\in {\scr F}(X)$, the <i>zero locus</i> of $f$ is the set of points $x\in X$ at which $f$ vanishes in the <i>fiber</i> ${\scr F}(x) = {\scr F}\otimes_{{\scr O}_X}k(x)$.</p> <p>If ${\scr F}$ is locally free, then the zero locus of a section is an analytic set (by which I mean the zero locus of a coherent ideal sheaf). In general, this is quite false (consider a sky-scraper sheaf at a point). </p> <p>Here a probably too general question:</p> <blockquote> <p>Which sets are the zero loci of such sections?</p> </blockquote> <p>Let's call ${\scr F}$ <i>torsion-free</i> if it is without torsion by non-zero-divisors in the structure sheaf. Equivalently, if the natural map ${\scr F}\to {\scr F}\otimes_{{\scr O}_X} {\scr M}_X$ is injective, where ${\scr M}_X$ is the sheaf of meromorphic functions on $X$.</p> <blockquote> <p>What if ${\scr F}$ is torsion-free? Are the zero loci analytic in this generality?</p> </blockquote> <p>Though I've phrased the problem in the context of rigid spaces, there are obvious analogues for schemes and complex analytic spaces. I'd welcome comments in any of these contexts.</p> <blockquote> <p></p> </blockquote>