Determinant of a projective module

Summary

The determinant of a finitely generated projective module $P$ of constant rank $n$ is its top exterior power $\Lambda^n P$. It is an invertible module, the algebraic analogue of the determinant line bundle of a vector bundle. It turns direct sums into tensor products. When the spectrum of $R$ is connected, it gives a group homomorphism from $K_0(R)$ to the Picard group of $R$. For a Dedekind domain this homomorphism detects the ideal class of a projective module.