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I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known.

Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the support of $F$ has codimension 1. The support of $F$ can be given a natural scheme structure, corresponding to the Fitting ideal sheaf $Fitt_0(F)$ locally generated by maximal minors of the matrix in a finite presentation of $F$ by trivial bundles.

In my present circumstances I know a bit more about $F$: it locally admits a resolution of the form

$0\to A\to B\to F \to 0$

where $A,B$ are vector bundles of the same rank. Under this hypothesis, I believe it is relatively straightforward to show that

$Fitt_0(F)^\ast \cong \det F$,

where the determinant is defined in terms of a resolution by vector bundles by the usual trick.

Does anyone know a reference where things like this are considered? And is my extra hypothesis needed? (I suspect that perhaps without it the Fitting ideal sheaf may fail to be a line bundle, but I am not sure).

Thanks!

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  • $\begingroup$ Assuming that your variety is projective, you may assume that $A$ and $B$ are globally defined. $\endgroup$ Dec 14, 2013 at 18:48
  • $\begingroup$ @LevBorisov: why is that? $\endgroup$ Dec 15, 2013 at 0:50
  • $\begingroup$ Take $B$ to be a sum of line bundles $\mathcal O(-d_i)$ that surjects onto $F$. Then the kernel is a vector bundle by projective dimension considerations. $\endgroup$ Dec 15, 2013 at 12:29
  • $\begingroup$ Have you looked at matrix factorizations? $\endgroup$
    – Youngsu
    Dec 16, 2013 at 2:35
  • $\begingroup$ @LevBorisov: thank you. I see now that this is also a consequence of Ex. III.6.5 in Hartshorne, and weaker hypotheses than projective are necessary. (Presumably the proof is similar to what you suggest, but the goal of this was to cite a reference and avoid a proof in the paper--the result is quite orthogonal to what I am actually doing.) $\endgroup$ Dec 16, 2013 at 3:14

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