Residues, Duality, and the Fundamental Class of a Scheme-map

The duality theory of coherent sheaves on algebraic varieties goes back to Roch’s half of the Riemann-Roch theorem for Riemann surfaces (1870s). In the 1950s, it grew into Serre duality on normal projective varieties; and shortly thereafter, into Grothendieck duality for arbitrary varieties and more generally, maps of noetherian schemes. This theory has found many applications in geometry and commutative algebra. We will sketch the theory in the reasonably accessible context of a variety V over a perfect field k, emphasizing the role of differential forms, as expressed locally via residues and globally via the fundamental class of V/k. (These notions will be explained.) As time permits, we will indicate some connections with Hochschild homology, and generalizations to maps of noetherian (formal) schemes. Even 50 years after the inception of Grothendieck’s theory, some of these generalizations remain to be worked out.