Say $K_\bullet$ is a bounded complex of vector bundles. I seem to want the determinant of $K_\bullet$ to be the alternating tensor product of the terms of the complex:

$\det(K) = \bigotimes_n \det(K_n)^{(-1)^n}$.

Is there a reason why this is the right definition (or the wrong definition)? Is there a better definition?

Say $K_\bullet$ is a bounded complex of vector bundles. I seem to want the determinant of $K_\bullet$ to be the alternating tensor product of the terms of the complex:

$$\det(K) = \bigotimes_n \det(K_n)^{(-1)^n}.$$

Is there a reason why this is the right definition (or the wrong definition)? Is there a better definition?