# determinant of a coherent sheaf, locally free on a big open set

Let $f:X\rightarrow S$ be a smooth projective morphisms of noetherian schemes and let $\mathcal{O}(1)$ be a relatively ample line bundle. Let $\mathcal{E}$ be a coherent sheaf over $X$ and flat over $S$. Suppose there exists an open subscheme $U\subset X$ such that $\mathcal{E}|_U$ is locally free and complement of $U$ has codimension atleast $2$ in $X$. Can we define determinant of $\mathcal{E}$? i.e., is there a line bundle on $X$ which is the "determinant" of $\mathcal{E}$?

Note that $\mathcal{E}$ is not torsion free. But has the extra assumption that there exists an open subscheme $U\subset X$ such that $\mathcal{E}|_U$ is locally free and complement of $U$ has codimension atleast $2$ in $X$.

## 1 Answer

For any sheaf which admits a finite locally free resolution its determinant is defined as the alternating tensor product of the determinants of its terms. Under your assumptions $\mathcal{E}$ admits such a resolution, so this definition applies.

• Where does the OP write that $\mathcal{E}$ has a finite locally free resolution? The morphism $f$ is smooth, but neither $S$ nor $X$ need be smooth. – Jason Starr Mar 10 '18 at 21:06
• Actually, existence of a finite locally free resolution does follow from the hypotheses (use $S$-flatness and Hilbert's syzygy theorem). – Jason Starr Mar 10 '18 at 21:25
• @JasonStarr: Yes, this is what I meant, smooth morphism and flatness of the sheaf over the base give finite locally free resolution. – Sasha Mar 11 '18 at 8:59