Euler class groups, and the homology of elementary and special linear groups

We prove homology stability for elementary and special linear groups over rings with many units improving known stability ranges. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative local rings with infinite residue fields, we show that the obstruction to further stability is given by Milnor-Witt K-theory. As an application we construct Euler classes of projective modules with values in the cohomology of the Milnor Witt K-theory sheaf. For d-dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the Euler class is necessary and sufficient for a projective module P of rank d to split off a rank 1 free direct summand. Along the way we obtain a new presentation of Milnor-Witt K-theory.Comment: 64 pages. Revised Section 5. Comments welcome

Direct numerical simulation of open-channel flow over a fully-rough wall at moderate relative submergence

Direct numerical simulation of open-channel flow over a bed of spheres arranged in a regular pattern has been carried out at bulk Reynolds number and roughness Reynolds number (based on sphere diameter) of approximately 6900 and 120, respectively, for which the flow regime is fully-rough. The open-channel height was approximately 5.5 times the diameter of the spheres. Extending the results obtained by Chan-Braun et al. (J. Fluid Mech., vol. 684, 2011, 441) for an open-channel flow in the transitionally-rough regime, the present purpose is to show how the flow structure changes as the fully-rough regime is attained and, for the first time, to enable a direct comparison with experimental observations. The results indicate that, in the vicinity of the roughness elements, the average flow field is affected both by Reynolds number effects and by the geometrical features of the roughness, while at larger wall-distances this is not the case, and roughness concepts can be applied. The flow-roughness interaction occurs mostly in the region above the virtual origin of the velocity profile, and the effect of form-induced velocity fluctuations is maximum at the level of sphere crests. The spanwise length scale of turbulent velocity fluctuations in the vicinity of the sphere crests shows the same dependence on the distance from the wall as that observed over a smooth wall, and both vary with Reynolds number in a similar fashion. Moreover, the hydrodynamic force and torque experienced by the roughness elements are investigated. Finally, the possibility either to adopt an analogy between the hydrodynamic forces associated with the interaction of turbulent structures with a flat smooth wall or with the surface of the spheres is also discussed, distinguishing the skin-friction from the form-drag contributions both in the transitionally-rough and in the fully-rough regimes.Comment: 46 pages, 26 figure

Higher K-theory of forms I. From rings to exact categories

We prove the analog for the -theory of forms of the theorem in algebraic -theory. That is, we show that the -theory of forms defined in terms of an -construction is a group completion of the category of quadratic spaces for form categories in which all admissible exact sequences split. This applies for instance to quadratic and hermitian forms defined with respect to a form parameter

The Witt group of real algebraic varieties

Let $V$ be an algebraic variety over $\mathbb R$. The purpose of this paper is to compare its algebraic Witt group $W(V)$ with a new topological invariant $WR(V_{\mathbb C})$, based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of $V$, This invariant lies between $W(V)$ and the group $KO(V_{\mathbb R})$ of $\mathbb R$-linear topological vector bundles on $V_{\mathbb R}$, the set of real points of $V$. We show that the comparison maps $W(V)\to WR(V_{\mathbb C})$ and $WR(V_{\mathbb C})\to KO(V_{\mathbb R})$ that we define are isomorphisms modulo bounded 2-primary torsion. We give precise bounds for the exponent of the kernel and cokernel of these maps, depending upon the dimension of $V.$ These results improve theorems of Knebusch, Brumfiel and Mah\'e. Along the way, we prove a comparison theorem between algebraic and topological Hermitian $K$-theory, and homotopy fixed point theorems for the latter. We also give a new proof (and a generalization) of a theorem of Brumfiel

Grothendieck-Witt groups of some singular schemes

We establish some structural results for the Witt and Grothendieck–Witt groups of schemes over ℤ[1/2] , including homotopy invariance for Witt groups and a formula for the Witt and Grothendieck–Witt groups of punctured affine spaces over a scheme. All these results hold for singular schemes and at the level of spectra

Hermitian K-theory, derived equivalences and Karoubi's Fundamental Theorem

Within the framework of dg categories with weak equivalences and duality that have uniquely 2-divisible mapping complexes, we show that higher Grothendieck-Witt groups (aka. hermitian K-groups) are invariant under derived equivalences and that Morita exact sequences induce long exact sequences of Grothendieck-Witt groups. This implies an algebraic Bott sequence and a new proof and generalization of Karoubi's Fundamental Theorem. For the higher Grothendieck-Witt groups of vector bundles of (possibly singular) schemes with an ample family of line-bundles such that 2 is invertible in the ring of regular functions, we obtain Mayer-Vietoris long exact sequences for Nisnevich coverings and blow-ups along regularly embedded centers, projective bundle formulas, and a Bass fundamental theorem. For coherent Grothendieck-Witt groups, we obtain a localization theorem analogous to Quillen's K'-localization theorem.Comment: to appear in J. Pure Appl. Algebr