We prove a blow-up formula for cyclic homology which we use to show that inﬁnitesimal K-theory satisﬁes cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of ﬁnite type over a ﬁeld of characteristic zero. Introduction The negative algebraic K-theory of a singular variety is related to its geometry. ...
One of the major advances of science in the 20th century was the discovery of a mathematical
formulation of quantum mechanics by Heisenberg in 1925 .1 From a
mathematical point of view, this transition from classical mechanics to quantum mechanics
amounts to, among other things, passing from the commutative algebra of
classical observables to the noncommutative algebra of quantum mechanical observables.
To understand this better we recall that in classical mechanics an observable of
a system (e.g. energy, position, momentum, etc.
In this article we study several homology theories of the algebra E ∞ (X) of Whitney functions over a subanalytic set X ⊂ Rn with a view towards noncommutative geometry. Using a localization method going back to Teleman we prove a Hochschild-Kostant-Rosenberg type theorem for E ∞ (X), when X is a regular subset of Rn having regularly situated diagonals. This includes the case of subanalytic X. We also compute the Hochschild cohomology of E ∞ (X) for a regular set with regularly situated diagonals and derive the cyclic and periodic cyclic theories. ...