Artículos con la etiqueta ‘K-Theory and Homology (math.KT)’

Triangulated categories of motives in positive characteristic

Por • 29 may, 2013 • Category: matemática

This thesis presents a way to apply this theorem of Gabber to a large portion of Voevodsky’s work in order to lift the assumption that resolution of singularities holds. This gives unconditional versions of many of his and others’ theorems provided we work Z[1/p] linearly, where p is the exponential characteristic of the base field. One example of the many applications we give is a partial answer to a 1980 conjecture of Weibel. Another is the removal of the hypothesis of resolution of singularities from a result of Suslin that compares Bloch’s higher Chow groups and etale cohomology. Voevodsky’s main tool in applying resolution of singularities is the cdh topology. We enlarge it slightly in order to apply this theorem of Gabber, presenting in this thesis a topology that we name the ldh topology, where l is a prime. We compare the cdh and ldh topologies using the concept of a «presheaf with traces», providing conditions under which the cdh and ldh sheafifications of a presheaf agree, as well as its cdh and ldh cohomologies.

Derived Representation Schemes and Noncommutative Geometry

Por • 24 abr, 2013 • Category: Opinion

Some 15 years ago M. Kontsevich and A. Rosenberg [KR] proposed a heuristic principle according to which the family of schemes ${Rep_n(A)}$ parametrizing the finite-dimensional represen- tations of a noncommutative algebra A should be thought of as a substitute or «approximation» for Spec(A). The idea is that every property or noncommutative geometric structure on A should induce a corresponding geometric property or structure on $Rep_n(A)$ for all n. In recent years, many interesting structures in noncommutative geometry have originated from this idea. In practice, however, if an associative algebra A possesses a property of geometric nature (e.g., A is a NC complete intersection, Cohen-Macaulay, Calabi-Yau, etc.), it often happens that, for some n, the scheme $Rep_n(A)$ fails to have the corresponding property in the usual algebro-geometric sense.