‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.

Let M be a reductive monoid with unit group G. Let denote the idempotent cross-section of the G × G-orbits on M . If W is the Weyl group of G and e, f ∈ with e ≤ f , we introduce a projection map from WeW to WfW. We use these projection maps to obtain a new description of the Bruhat-Chevalley order on the Renner monoid of M . For the canonical compactification X of a semisimple group G0 with Bo...

In this paper we prove that there exists a Zariski dense open subset U defined over Q in the parameter space of one-variable rational functions with prescribed l poles with fixed orders, such that for every geometric point f in U(Q), the L-function of exponential sum of f at p has Newton polygon approaches the Hodge polygon as p approaches infinity. This result generalizes some result in [13] a...

Let G = PSL2(F) where F = R,C, and consider the space Z = (Γ1 × Γ2)\(G × G) where Γ1 < G is a co-compact lattice and Γ2 < G is a finitely generated discrete Zariski dense subgroup. The work of Benoist-Quint [2] gives a classification of all ergodic invariant Radon measures on Z for the diagonal G-action. In this paper, for a horospherical subgroup N of G, we classify all ergodic, conservative, ...