48,453 research outputs found
Relation between two twisted inverse image pseudofunctors in duality theory
Grothendieck duality theory assigns to essentially-finite-type maps f of
noetherian schemes a pseudofunctor f^\times right-adjoint to Rf_*, and a
pseudofunctor f^! agreeing with f^\times when f is proper, but equal to the
usual inverse image f^* when f is etale. We define and study a canonical map
from the first pseudofunctor to the second. This map behaves well with respect
to flat base change, and is taken to an isomorphism by "compactly supported"
versions of standard derived functors. Concrete realizations are described, for
instance for maps of affine schemes. Applications include proofs of reduction
theorems for Hochschild homology and cohomology, and of a remarkable formula
for the fundamental class of a flat map of affine schemes.Comment: Final version, to appear in Compositio Math. A few misprints fixed.
30 page
Modular Theory, Non-Commutative Geometry and Quantum Gravity
This paper contains the first written exposition of some ideas (announced in
a previous survey) on an approach to quantum gravity based on Tomita-Takesaki
modular theory and A. Connes non-commutative geometry aiming at the
reconstruction of spectral geometries from an operational formalism of states
and categories of observables in a covariant theory. Care has been taken to
provide a coverage of the relevant background on modular theory, its
applications in non-commutative geometry and physics and to the detailed
discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields
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