Locally nilpotent derivations on affine surfaces with a \C^*-action

We give a classification of normal affine surfaces admitting an algebraic group action with an open orbit. In particular an explicit algebraic description of the affine coordinate rings and the defining equations of such varieties is given. By our methods we recover many known results, e.g. the classification of normal affine surfaces with a `big' open orbit of Gizatullin and Popov or some of the classification results of Danilov-Gizatullin, Bertin and others.Comment: Date of writing: 2/03/200

Dynamic heterogeneity in a glass forming fluid: susceptibility, structure factor and correlation length

We investigate the growth of dynamic heterogeneity in a glassy hard-sphere mixture for volume fractions up to and including the mode-coupling transition. We use an 80 000 particle system to test a new procedure to evaluate a dynamic correlation length xi(t): we determine the ensemble independent dynamic susceptibility chi_4(t) and use it to facilitate evaluation of xi(t) from the small wave vector behavior of the four-point structure factor. We analyze relations between the alpha relaxation time tau_alpha, chi_4(tau_alpha), and xi(tau_alpha). We find that mode-coupling like power laws provide a reasonable description of the data over a restricted range of volume fractions, but the power laws' exponents differ from those predicted by the inhomogeneous mode-coupling theory. We find xi(tau_alpha) ~ ln(tau_alpha) over the full range of volume fractions studied, which is consistent with Adams-Gibbs-type relation.Comment: 4 pages, 4 figures, To be published in Physical Review Letter

Power series rings and projectivity

We show that a formal power series ring $A[[X]]$ over a noetherian ring $A$ is not a projective module unless $A$ is artinian. However, if $(A,{\mathfrak m})$ is local, then $A[[X]]$ behaves like a projective module in the sense that $Ext^p_A(A[[X]], M)=0$ for all ${\mathfrak m}$-adically complete $A$-modules. The latter result is shown more generally for any flat $A$-module $B$ instead of $A[[X]]$. We apply the results to the (analytic) Hochschild cohomology over complete noetherian rings.Comment: Mainly thanks to remarks and pointers by L.L.Avramov and S.Iyengar, we added further context and references. To appear in Manuscripta Mathematica. 7 page