SK1-like functors for division algebras

We investigate the group valued functor G(D) = D*/F*D' where D is a division algebra with center F and D' the commutator subgroup of D*. We show that G has the most important functorial properties of the reduced Whitehead group SK1. We then establish a fundamental connection between this group, its residue version, and relative value group when D is a Henselian division algebra. The structure of G(D) turns out to carry significant information about the arithmetic of D. Along these lines, we employ G(D) to compute the group SK1(D). As an application, we obtain theorems of reduced K-theory which require heavy machinery, as simple examples of our method.

Reduced K-theory for Azumaya algebras

Abstract In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors). Since a central simple algebra splits and the functors above are “trivial” in the split case, one can prove certain calculus on these functors. The common examples are kernel or co-kernel of the maps Ki(F)?Ki(D), where Ki are Quillen K-groups, D is a division algebra and F its center, or the homotopy fiber arising from the long exact sequence of above map, or the reduced Whitehead group SK1. In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it. This, for example, immediately shows that K-theory of an Azumaya algebra over a local ring is “almost” the same as K-theory of the base ring. The main result is to prove that reduced K-theory of an Azumaya algebra over a Henselian ring coincides with reduced K-theory of its residue central simple algebra. The note ends with some calculation trying to determine the homotopy fibers mentioned above.

An 8000-year history of landscape, climate and copper exploitation in the Middle East: the Wadi Faynan and the Wadi Dana National Reserve in southern Jordan

Investigations of geomorphology, geoarchaeology, pollen, palynofacies, and charcoal indicate the comparative scales and significance of palaeoenvironmental changes throughout the Holocene at the junction between the hyper-arid hot Wadi â??Arabah desert and the front of the Mediterranean-belt Mountains of Edom in southern Jordan through a series of climatic changes and episodes of intense mining and smelting of copper ores. Early Holocene alluviation followed the impact of Neolithic grazers but climate drove fluvial geomorphic change in the Late Holocene, with a major arid episode corresponding chronologically with the â??Little Ice Ageâ?? causing widespread alluviation. The harvesting of wood for charcoal may have been sufficiently intense and widespread to affect the capacity of intensively harvested tree species to respond to a period of greater precipitation deduced for the Roman-Byzantine period - a property that affects both taphonomic and biogeographical bases for the interpretation of palynological evidence from arid-lands with substantial industrial histories. Studies of palynofacies have provided a record of human and climatic causes of soil erosion, and the changing intensity of the use of fire over time. The patterns of vegetational, climatic change and geomorphic changes are set out for this area for the last 8000 years.

An analogue of the Magnus problem for associative algebras

We prove an analogue of Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by $n+k$ generators and $k$ relations and has an $n$-element system of generators, then this algebra is a free algebra of rank $n$.