University of Cambridge > Talks.cam > Category Theory Seminar > Abstract versions of Hilbert's Nullstellensatz, and dualities for algebraic categories

Abstract versions of Hilbert's Nullstellensatz, and dualities for algebraic categories

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Hilbert’s classical Nullstellensatz characterises the fixed (=radical) ideals in the contravariant Galois connection between affine algebraic varieties over an algebraically closed field, and ideals of the algebra of polynomials with coefficients in the given field. We first show how to abstract this 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 ground 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 the Galois connection lifts to a contravariant adjunction between “definable subsets” of powers of A, with “definable morphisms”, and “coordinate algebras”, with homomorphisms, under the sole (necessary and sufficient) assumption that A generates the algebraic category. We pause to discuss the relationship of this general adjunction with previous work, especially that of Y. Diers. Generalising further, we show that the adjunction extends under appropriate conditions to categories with no algebraic structure. If time allows, we close by discussing how duality theorems for algebraic varieties 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 compact polyhedral category of P.L. topology. (Talk based on joint work with Olivia Caramello and, independently, Luca Spada.)

This talk is part of the Category Theory Seminar series.

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