SHEAF COHOMOLOGY IN o-MINIMAL STRUCTURES

Abstract

Here we prove the existence of sheaf cohomology theory in arbitrary o-minimal structures.

Poincaré - Verdier duality in o-minimal structures

“…The isomorphism of Theorem 2.8 allowed the development of o-minimal sheaf cohomology without supports in [12] by defining concepts and also sometimes proving results via this tilde isomorphism. In this paper we will continue to use this technique but allowing now the presence of supports.…”

Invariance of o-minimal cohomology with definably compact supports

“…We will refer the reader to [10] for basic results and notions about o-minimal spectral spaces or about the tilde functor Def → Def. Note that these results were stated in [10] in the category of definable sets but are true in the category of definable spaces with exactly the same proofs. In fact most of them hold also in real algebraic spaces [3,7] and more generally in spectral topological space [6].…”

A note on generic subsets of definable groups

“…With a good cohomology theory in arbitrary o-minimal structures which generalizes the o-minimal singular cohomology in o-minimal expansion of real closed fields ( [10] and [22]) one could obtain a uniform proof of the computation of m-torsion subgroups of abelian definably compact definable groups in arbitrary o-minimal structures which would include the three cases above. The authors already have made significant advances in this direction building on previous joint work with other authors ( [6], [8] and [9]). …”