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CATEGORIES:Category Theory Seminar
SUMMARY:Abstract versions of Hilbert's Nullstellensatz\, a
nd dualities for algebraic categories - Vincenzo M
arra\, Milan
DTSTART;TZID=Europe/London:20130305T141500
DTEND;TZID=Europe/London:20130305T151500
UID:TALK42452AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/42452
DESCRIPTION:Hilbert's classical Nullstellensatz characterises
the fixed (=radical) ideals in the contravariant G
alois connection between affine algebraic varietie
s over an algebraically closed field\, and ideals
of the algebra of polynomials with coefficients i
n the given field. We first show how to abstract t
his Galois connection at the level of (finitary or
infinitary) algebraic categories\, for any choice
of an algebra A that is to play the role of the g
round field in the classical situation. Following
a tradition that can be traced back to G. Birkhoff
\, we prove in this context an analogue of Hibert'
s Nullstellensatz. We then proceed to show that th
e Galois connection lifts to a contravariant adjun
ction between "definable subsets" of powers of A\,
with "definable morphisms"\, and "coordinate alge
bras"\, with homomorphisms\, under the sole (neces
sary and sufficient) assumption that A generates t
he algebraic category. We pause to discuss the rel
ationship of this general adjunction with previous
work\, especially that of Y. Diers. Generalising
further\, we show that the adjunction extends unde
r appropriate conditions to categories with no alg
ebraic structure. If time allows\, we close by dis
cussing how duality theorems for algebraic varieti
es flow naturally from the framework above. As thr
ee significant cases we select Stone duality for B
oolean algebras\, Stone-Gelfand duality for real C
*-algebras\, and the lesser known but equally impo
rtant Baker-Beynon duality between finitely presen
ted unital vector lattices\, and the compact polyh
edral category of P.L. topology. (Talk based on jo
int work with Olivia Caramello and\, independently
\, Luca Spada.)
LOCATION:MR5\, Centre for Mathematical Sciences
CONTACT:Julia Goedecke
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