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SUMMARY:Abstract versions of Hilbert's Nullstellensatz\, and dualities for
  algebraic categories - Vincenzo Marra\, Milan
DTSTART:20130305T141500Z
DTEND:20130305T151500Z
UID:TALK42452@talks.cam.ac.uk
CONTACT:Julia Goedecke
DESCRIPTION:Hilbert's classical Nullstellensatz characterises the fixed (=
 radical) ideals in the contravariant Galois connection between affine alge
 braic varieties over an algebraically closed field\, and  ideals of the al
 gebra of polynomials with coefficients in the given field. We first show h
 ow to abstract this Galois connection at the level of (finitary or infinit
 ary) algebraic categories\, for any choice of an algebra A that is to play
  the role of the ground field in the classical situation. Following a trad
 ition 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 the Ga
 lois connection lifts to a contravariant adjunction between "definable sub
 sets" of powers of A\, with "definable morphisms"\, and "coordinate algebr
 as"\, with homomorphisms\, under the sole (necessary and sufficient) assum
 ption that A generates the 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 un
 der appropriate conditions to categories with no algebraic structure. If t
 ime allows\, we close by discussing how duality theorems for algebraic var
 ieties flow naturally from the framework above. As three significant cases
  we select Stone duality for Boolean algebras\, Stone-Gelfand duality for 
 real C*-algebras\, and the lesser known but equally important Baker-Beynon
  duality between finitely presented unital vector lattices\, and the compa
 ct polyhedral category of P.L. topology. (Talk based on joint work with Ol
 ivia Caramello and\, independently\, Luca Spada.)
LOCATION:MR5\, Centre for Mathematical Sciences
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