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In September 1997, the Working Week on Resolution of Singularities
was held at Obergurgl in the Tyrolean Alps. Its objective was to
manifest the state of the art in the field and to formulate major
questions for future research. The four courses given during this
week were written up by the speakers and make up part I of this
volume. They are complemented in part II by fifteen selected
contributions on specific topics and resolution theories. The
volume is intended to provide a broad and accessible introduction
to resolution of singularities leading the reader directly to
concrete research problems.
In September 1997, the Working Week on Resolution of Singularities
was held at Obergurgl in the Tyrolean Alps. Its objective was to
manifest the state of the art in the field and to formulate major
questions for future research. The four courses given during this
week were written up by the speakers and make up part I of this
volume. They are complemented in part II by fifteen selected
contributions on specific topics and resolution theories. The
volume is intended to provide a broad and accessible introduction
to resolution of singularities leading the reader directly to
concrete research problems.
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."
Robin Hartshorne's classical 1966 book ""Residues and Duality""
[""RD""] developed Alexandre Grothendieck's ideas for a
pseudofunctorial variance theory of residual complexes and duality
for maps of noetherian schemes. The three articles in this volume
rework the main parts of the last two chapters in ""RD"", in
greater generality - for Cousin complexes on formal schemes, not
just residual complexes on ordinary schemes - and by more concrete
local methods which clarify the relation between local properties
of residues and global properties of dualizing pseudofunctors.A new
approach to pasting pseudofunctors is applied in using residual
complexes to construct a dualizing pseudofunctor over a fairly
general category of formal schemes, where compactifications of maps
may not be available. A theory of traces and duality with respect
to pseudo-proper maps is then developed for Cousin complexes. For
composites of compactifiable maps of formal schemes, this, together
with the above pasting technique, enables integration of the
variance theory for Cousin complexes with the very different
approach to duality initiated by Deligne in the appendix to ""RD"".
The book is suitable for advanced graduate students and researchers
in algebraic geometry.
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