Foundations of grothendieck duality for diagrams of schemes

Joseph Lipman, Mitsuyasu Hashimoto

Research output: Chapter in Book/Report/Conference proceedingChapter

17 Citations (Scopus)


This is a polished version of notes begun in the late 1980s, largely available from my home page since then, meant to be accessible to mid-level graduate students. The first three chapters treat the basics of derived categories and functors, and of the rich formalism, over ringed spaces, of the derived functors, for unbounded complexes, of the sheaf functors ⊗, Hom, f* and f* (where f is a ringed-space map). Included are some enhancements, for concentrated (= quasi-compact and quasi-separated) schemes, of classical results such as the projection and Kudie;unneth isomorphisms. The fourth chapter presents the abstract foundations of Grothendieck Duality-existence and tor-independent base change for the right adjoint of the derived functor Rf* when f is a quasiproper map of concentrated schemes, the twisted inverse image pseudofunctor for separated finite-type maps of noetherian schemes, some refinements for maps of finite tor-dimension, and a brief discussion of dualizing complexes.

Original languageEnglish
Title of host publicationLecture Notes in Mathematics
Number of pages482
Publication statusPublished - 2009
Externally publishedYes

Publication series

NameLecture Notes in Mathematics
ISSN (Print)00758434


ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Lipman, J., & Hashimoto, M. (2009). Foundations of grothendieck duality for diagrams of schemes. In Lecture Notes in Mathematics (Vol. 1960, pp. 1-482). (Lecture Notes in Mathematics; Vol. 1960).