We study combinatorial properties of convex sets over arbitrary valued fields. demonstrate analogs some classical results for the reals (e.g. fractional Helly theorem and B\'ar\'any's on points in many simplices), along with additional not satisfied by reals, including finite breadth VC-dimension. These are deduced from a simple description modules valuation ring spherically complete field.

Abstract We prove a relative Lefschetz–Verdier theorem for locally acyclic objects over Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal $2$ -category of cohomological correspondences. show that local acyclicity equivalent to dualisability deduce duality preserves acyclicity. As another application category correspondences, we nearby cycle functor Hens...

Let ? be a valuation of arbitrary rank on the polynomial ring K[x] with coefficients in field K. We prove comparison theorems between MacLane–Vaquié key polynomials for valuations ??? and abstract ?. Also, some results invariants associated to limit are obtained. In particular, if char(K)=0, we show that all unbounded continuous families augmentations have numerical character equal one.

Contents Introduction 2 1. Homology theory and cycle map 6 2. Kato homology 11 3. Vanishing theorem 15 4. Bertini theorem over a discrete valuation ring 19 5. Surjectivity of cycle map 22 6. Blowup formula 24 7. A moving lemma 26 8. Proof of main theorem 28 9. Applications of main theorem 31 Appendix A. Resolution of singularities for embedded curves 34 References 39