We prove that every faithfully flat Hopf-Galois object is a quantum torsor in the sense of Grunspan.

yψ ց m//G a/WV → t /WG where a ⊆ t is a subspace, WV ⊆ GL(a ) is a reflection group, and ψ is faithfully flat with irreducible generic fibers. Note that WV being a reflection group implies that a /WV is isomorphic to an affine space. The group WV is a very subtle invariant of V which plays a similar role as the Weyl group of the restricted root system for a symmetric space. The morphism ψ facto...

Let k be a commutative ring, H a faithfully flat Hopf algebra with bijective antipode, A a k-flat right H-comodule algebra. We investigate when a relative Hopf module is projective over the subring of coinvariants B = A , and we study the semisimplicity of the category of relative Hopf modules.

This paper describes an approach that uses flat-spacetime dimension estimators to estimate the manifold dimension of causal sets that can be faithfully embedded into curved spacetimes. The approach is invariant under coarse graining and can be implemented independently of any specific curved spacetime. Results are given based on causal sets generated by random sprinklings into conformally flat ...

Under the assumption that a residually finite-dimensional Hopf algebra H has an Artinian ring of fractions, it is proved flat module over any right coideal subalgebra satisfying polynomial identity and faithfully subalgebra. As consequence we find large class algebras which are all subalgebras subalgebras.

I show that a quasicompact morphism f : X → Y of schemes is an effective descent map for quasicoherent modules if and only if the map OY → f∗(OX) is injective, and remains injective after any base change. This generalizes Grothendieck’s result that faithfully flat quasicompact morphisms are effective descent maps.

We developed a hybrid nanoimprint-soft lithography technique with sub-15 nm resolution. It is capable of patterning both flat and curved substrates. The key component of the technology is the mold, which consists of rigid features on an elastic poly(dimethylsiloxane) (PDMS) support. The mold was fabricated by imprinting a reverse image onto the PDMS substrate using a UV-curable low-viscosity pr...

Let R be a discrete valuation ring with fraction field K and X a flat R-scheme. Given a faithful action of a K-group scheme GK over the generic fibre XK , we study models G of GK acting on X. In various situations, we prove that if such a model G exists, then there exists another model G′ that acts faithfully on X. This model is the schematic closure of G inside the fppf sheaf AutR(X) ; the maj...