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This volume contains two related, though independently written, mo-
graphs. In Notes on Derived Functors and Grothendieck Duality the
?rst three chapters treat the basics of derived categories and
functors, and of the rich formalism, over ringed spaces, of the
derived functors, for unbounded com- ? plexes, ofthesheaffunctors?,
Hom, f andf wheref isaringed-spacemap. ? Included are some
enhancements, for concentrated (i.e., quasi-compact and
quasi-separated) schemes, of classical results such as the
projection and K] unneth isomorphisms. The fourth chapter presents
the abstract foun- tions of Grothendieck Duality-existence and
tor-independent base change for the right adjoint of the derived
functor Rf when f is a quasi-proper ? map of concentrated schemes,
the twisted inverse image pseudofunctor for separated ?nite-type
maps of noetherian schemes, re?nements for maps of ?nite
tor-dimension, and a brief discussion of dualizing complexes. In
Equivariant Twisted Inverses the theory is extended to the context
of diagrams of schemes, and in particular, to schemes with a
group-scheme action. An equivariant version of the twisted
inverse-image pseudofunctor is de?ned, and equivariant versions of
some of its important properties are proved, including Grothendieck
duality for proper morphisms, and ?at base change. Also,
equivariant dualizing complexes are dealt with. As an appli- tion,
ageneralizedversionofWatanabe'stheoremontheGorensteinproperty of
rings of invariants is proved. More detailed overviews are given in
the respective Introductions."
This book presents a new homological approximation theory in the category of equivariant modules, unifying the Cohen-Macaulay approximations in commutative ring theory and Ringel's theory of Delta-good approximations for quasi-hereditary algebras and reductive groups. The book provides a detailed introduction to homological algebra, commutative ring theory and homological theory of comodules of coalgebras over an arbitrary base. It aims to overcome the difficulty of generalizing known homological results in representation theory.
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