Algebraic K-theory, A^1-homotopy and Riemann-Roch theorems

In this article, we show that the combination of the constructions done in SGA 6 and the A^1-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann-Roch theorems.Comment: 39 pages; minor changes (typos, references

Opérations sur la K-théorie algébrique et régulateurs via la théorie homotopique des schémas

This thesis contributes to the homotopy theory of schemes. In the first part, we carry onward the constructions by Fabien Morel and Vladimir Voevodsky: we define the stable homotopy categories of hanging sites with intervals. This construction is more general than the one of John F. Jardine: this enables us to provide a precise definition of the "complex points" functors in the homotopy theory of schemes.In the second part, we prove that in the homotopy category of a regular scheme S, the set of endomorphisms of the infinite Grassmannian (which gives a model for algebraic K-theory by a theorem by Morel and Voevodsky) is naturally isomorphic to the set of natural transformations from the Grothendieck group functor (considered as presheaf of sets on the category of smooth schemes over S) to itself. This enables us to build a special lambda-ring structure on higher K-groups and to check that this construction is the same as the ones that were constructed before. Additive operations on algebraic K-theory are studied carefully and stable versions of the theorems are provided either with integer or rational coefficients. The technique also allows us to define Chern classes on higher K-groups with values in motivic cohomology (and other cohomological theories) and to assert the existence of "superphantom" maps in the homotopy theory of schemes.Cette thèse est une contribution à la théorie homotopique des schémas. Dans la première partie, on poursuit les constructions de Fabien Morel et Vladimir Voevodsky en définissant la catégorie homotopique stable des sites suspendus avec intervalles. La généralité, plus grande que celle permise par la définition de John F. Jardine, permet de donner une construction rigoureuse des foncteurs " points complexes " en théorie homotopique des schémas.Dans la seconde partie, on montre qu'au-dessus d'un schéma de base régulier S, se donner un endomorphisme dans la catégorie homotopique de S de la grassmannienne infinie (donnant un modèle de la K-théorie algébrique d'après un théorème de Morel et Voevodsky) revient à se donner une application fonctorielle K_0(X) -> K_0(X) où X parcourt la catégorie des schémas lisses sur S. Ceci permet de construire une structure de lambda-anneau spécial sur les groupes de K-théorie algébrique supérieure et de vérifier que cette structure coïncide avec les constructions antérieures. Les opérations additives sur la K-théorie algébrique sont étudiées en détail et des versions stables de ces énoncés sont obtenues, à coefficients entiers ou rationnels. La technique utilisée permet également de construire des classes de Chern sur la K-théorie algébrique supérieure à valeurs dans la cohomologie motivique (et dans d'autres théories cohomologiques) et de montrer très explicitement l'existence de morphismes stablement fantômes en théorie homotopique des schémas

Intramyocardial protein therapy with vascular endothelial growth factor (VEGF-165) induces functional angiogenesis in rat senescent myocardium.

International audienceMyocardial capillary density and angiogenesis are impaired during aging but whether growth factor therapy is able to induce functional neovascularization in senescent heart have never been studied. In 3, 24, 28 and 32 mo male Wistar rats, cardiac hemodynamic measurements indicated heart failure at 28 and 32 mo, associated with left ventricular hypertrophy. VEGF/VEGF-R2, Ang-1/Ang-2/Tie-2 and PTN levels, quantitated in left ventricle by western blotting and immunohistochemistry, showed that VEGF and VEGF-R2 levels were specifically decreased during aging. In vitro angiogenesis ± rhVEGF-165 (5 and 50 ng/mL) was measured in aortic segments in 3D-collagen. Aortic sprouting was decreased during aging but restored by VEGF treatment (P<0.001), similarly in 3 and 24 mo with 50 ng/mLVEGF. Finally, 3 and 24 mo rats were submitted to in vivo intramyocardial rhVEGF-165 (10 micrograms) or saline solution injection and angiogenesis was measured by SPECT imaging of the alpha(v)beta(3) integrin-targeted tracer (99m)Tc-RAFT-RGD, capillary fluorescence staining in isolated perfused heart and vWF and alpha smooth muscle actin immunohistochemistry, 7 and 21 days later. VEGF administration increased capillary density in 3 but also in 24 mo rats at days 7 (+26%, P<0.01) and 21 (+41%, P<0.01) and arteriolar density at day 21 (+36%, P<0.01). Activity of (99m)Tc-RAFT-RGD and capillary fluorescence labeling indicated that new formed capillaries were functional. Cardiac aging was associated with strong VEGF/VEGF-R2 pathway downregulation. VEGF-165 protein therapy was able to induce in vitro and in vivo angiogenesis during aging. In 24 mo hearts, in vivo angiogenesis was functional, sustained and comparable to neovascularization observed in 3 mo hearts