An integrality theorem of Grosshans over arbitrary base ring

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring $k$. One considers a split reductive group scheme $G$ acting on a $k$-algebra $A$ and leaving invariant a subalgebra $R$. If $R^U=A^U$ then the conclusion is that $A$ is integral over $R$.Comment: 5 pages; final versio

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G be GL_N or SL_N as reductive linear algebraic group over a field k of positive characteristic p. We prove several results that were previously established only when N 2^N. Let G act rationally on a finitely generated commutative k-algebra A. Assume that A as a G-module has a good filtration or a Schur filtration. Let M be a noetherian A-module with compatible G action. Then M has finite good/Schur filtration dimension, so that there are at most finitely many nonzero H^i(G,M). Moreover these H^i(G,M) are noetherian modules over the ring of invariants A^G. Our main tool is a resolution involving Schur functors of the ideal of the diagonal in a product of Grassmannians.Comment: 22 pages; final versio

The Cohen-Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply that the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Furthermore, we show that, for a $p$-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.Comment: We removed the conjecture which appeared in previous versions: we give a counter-example. We fixed the proof of Lemma 2.2 (previously Remark 2.2). 16 page

Divalent Metal Ions Tune the Self-Splicing Reaction of the Yeast Mitochondrial Group II Intron Sc.ai5γ

Group II introns are large ribozymes, consisting of six functionally distinct domains that assemble in the presence of Mg2+ to the active structure catalyzing a variety of reactions. The first step of intron splicing is well characterized by a Michaelis–Menten-type cleavage reaction using a two-piece group II intron: the substrate RNA, the 5′-exon covalently linked to domains 1, 2, and 3, is cleaved upon addition of domain 5 acting as a catalyst. Here we investigate the effect of Ca2+, Mn2+, Ni2+, Zn2+, Cd2+, Pb2+, and [Co(NH3)6]3+ on the first step of splicing of the Saccharomyces cerevisiae mitochondrial group II intron Sc.ai5γ. We find that this group II intron is very sensitive to the presence of divalent metal ions other than Mg2+. For example, the presence of only 5% Ca2+ relative to Mg2+ results in a decrease in the maximal turnover rate k cat by 50%. Ca2+ thereby has a twofold effect: this metal ion interferes initially with folding, but then also competes directly with Mg2+ in the folded state, the latter being indicative of at least one specific Ca2+ binding pocket interfering directly with catalysis. Similar results are obtained with Mn2+, Cd2+, and [Co(NH3)6]3+. Ni2+ is a much more powerful inhibitor and the presence of either Zn2+ or Pb2+ leads to rapid degradation of the RNA. These results show a surprising sensitivity of such a large multidomain RNA on trace amounts of cations other than Mg2+ and raises the question of biological relevance at least in the case of Ca2+

Computational Symbolic Geometry

... Important relations in this representation and operators corresponding to geometric properties are also given. Where possible we will exhibit intersection formulas for constraint problems on these objects and we end by suggesting applications in applied fields. The first part is devoted to linear spaces, the second to spheres, the third to displacements and the last one to matrices, treated as non-commutative variables. The ensemble of this work aims to convince the reader that symbolic manipulations on such geometric objects can be effectively handled in practice

EPI64 interacts with Slp1/JFC1 to coordinate Rab8a and Arf6 membrane trafficking

ETOC: EPI64 is a TBC domain containing a microvillar protein that binds to Arf6 and induces actin-coated vacuole accumulation when overexpressed. EPI64 does this by stabilizing Arf6-GTP levels to induce clathrin-independent endocytosis and by preventing endosomes from maturing to the tubular endosome by lowering Rab8a GTP levels via association with JFC1