We know a coherent sheaf $E$ over a smooth projective variety $X$ admits a finite locally free resolution. $0\longrightarrow E_n\longrightarrow E_{n-1}\longrightarrow\cdots\longrightarrow E_0\longrightarrow E\longrightarrow 0$.

So we define the determinant of $E$ to be $\textrm{det}(E)=\otimes\textrm{det}({E}_i)^{(-1)^i}$, this is a line bundle.

My doubt is as follows : How do we know that this is independent of the choice of locally free resolution? Any clarification or reference would really help.

Thanks in advance!

  • 1
    $\begingroup$ The right hand side of your formula for the determinant does not make sense. $\endgroup$ – Georges Elencwajg Feb 18 '15 at 7:33

The proof, that this definition is independent of the choice of a resolution, can be found in detail in S.Kobayashi - Differential Geometry of Complex Vector Bundles. It is in Chapter V, Paragraph 6, "Determinant bundles".


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