The Serre-Grothendieck Finiteness Theorem for the cohomology of coherent sheaves

Abstract

We prove the converse of the Serre-Grothendieck finiteness theorem for the cohomology of coherent sheaves, originally due to Lipman. This result characterizes proper morphisms as those morphisms whose derived pushforward preserves pseudo-coherent complexes. Although the derived pushforward does not reflect pseudo-coherence, we prove that it does after twisting by all perfect complexes. This provides a new characterization of properness.

arXiv comment15 pages