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!