Topological Invariants for Non-Archimedean Bornological Algebras

by Devarshi Mukherjee
Doctoral thesis
Date of Examination:2020-09-24
Date of issue:2020-10-13
Advisor:Prof. Dr. Ralf Meyer
Referee:Prof. Dr. Thomas Schick
Referee:Dr. Dustin Clausen
Referee:Prof. Dr. Yakov Kremnitzer
Persistent Address: http://dx.doi.org/10.53846/goediss-8247

 

 

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Abstract

English

In this thesis, we define a cyclic homology theory for non-archimedean bornological algebras, which we call analytic cyclic homology. Let V be a complete, discrete valuation ring with uniformiser p, residue field k, and quotient field K. The material we present is divided in three parts. In the first part, we interpret Monsky and Washnitzer's weak completion using the framework of bornologies. Weakly complete algebras are used to define Monsky-Washnitzer cohomology, and have three characteristic features - bornological torsion-freeness, completeness and a certain spectral radius condition. We study the homological inheritence of each of these three conditions, and call a V-algebra with these properties a dagger algebra. In the second part, we define analytic cyclic homology for projective systems of complete, bornologically torsion-free V-algebras. The theory we develop satisfies homotopy invariance, Morita invariance and excision. We use these properties to compute our theory for (dagger completed) Leavitt path algebras, and tensor product with such algebras. We show that our theory coincides with Berthelot's rigid cohomology for smooth commutative V-algebras of relative dimension 1. In the third part, we define analytic cyclic homology for algebras over the residue field k, by lifting them to free algebras over V and then building dagger completed tube algebras. We show that under very mild assumptions on the bornology of a k-algebra A, any complete, bornologically torsion-free V-module lifting can be used to compute its analytic cyclic homology. The theory we define satisfies polynomial homotopy invariance, matricial stability, and excision for extensions of finitely generated k-algebras.
Keywords: Cyclic homology; Non-archimedean geometry; Non-commutative geometry
 
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