Constant group scheme

Let r be a finite constant group scheme over a field k. Show that n-dimensional linear representations of T are given by ordinary homomorphisms of I (fc) into GMfc). 

If T is a finite group, show that the corresponding finite constant group scheme assigns to R the set of locally constant functions Spec R- T. 

We actually see here that Aut(fc x x k) is a constant group scheme and looking at its points in k shows it is the permutation group on n elements. 

Proof. Base-extending to k, we may assume we have a finite constant group scheme, say of order n. When we embed it as an algebraic matrix group, each g in it satisfies the separable equation X" — 1 = 0. If g is also unipotent, g — l. Thus the group is trivial. ... 

Connected affine group scheme 51 Connected component of G 51 Connected set, connected component 157 Constant group scheme 16, 45 Continuous function 157 Continuous -action 48 Coseparable coalgebra 53 Crossed homomorphism 137... 

Much is known about the lifetimes of carbocation intermediates of solvolysis, and these data have proven critical in the design of experiments to estimate absolute rate constants for reorganization of ion pairs. Consider reorganization of an ion-pair reaction intermediate that exchanges the positions of the nucleophilic atoms of the leaving group (, Scheme 9) and that occurs in competition with diffusional separation to free ions (k-d) which is much faster than addition of solvent to the ion pair. Ion-pair separation is irreversible and will result in formation of solvolysis reaction products s ). Reorganization of the ion pair will result in formation of isomerization reaction product and the yield of this reaction product will provide a measure of the relative rate constant... 

The rank of a finite locally free sheaf or a finite flat group scheme over S is a locally constant function on S. 

The proportionality of S and p made possible the adoption of a slight modification of the approach involving the reversal of the conventional Hammett procedure. The Selectivity Factor, 8, rather than p was chosen as the reference parameter. This procedure is the application of the p approach with a single substituent, the methyl group, defining the reaction constant. This scheme is useful because 8f is an experimental quantity determined by the para/meta product ratio in toluene (10). 

As already mentioned in Section II.C, the complexes [Cu2(R—XYL—H)]2+ (10) react with dioxygen, forming a kinetically describable intermediate dioxygen complex [Cu2(R—XYL—H)(02)]2+ (11), and attack by the peroxo group causes ligand hydroxylation to give the phenoxo and hydroxo-bridged complex [Cu2(R—XYL—0—)—(OH)]2+ (12). Here, we focus on the hydroxylation event, as described by the rate constant k2 (Scheme 13). 

An example of the capabilities of this software is shown in Fig. 7.1. The fluorescent molecular sensor Calix-DANS4 consisting of a cahx[4]arene bearing four dan-syl groups (Scheme 7.1) offers an excellent selectivity for lead ion. Analysis by SPECFIT of the evolution of the whole fluorescence spectrum upon addition of lead ions revealed that a 2 3 complex and a 1 1 complex were successively formed with very high stability constants (log IGs = 33.5 + 1.5 and log Kn = 10.0 + 0.5) [9, 10]. 

The cyclisation reaction of Problem 12 can be studied by investigating the effect of varying the substituent on the phenoxy leaving group (Scheme 8). Given the Bronsted coefficients Peqi = -0.07, 3eq2 = -1.86 and pLg - -1.05 calculate u,g for the first-order rate constant for cyclisation from the anion. 

Amphidinolide V (21) is a 14-membered macrolide possessing four dxo-methylenes and an epoxide, isolated from the marine dinoflagellate Amphidinium sp. (strain Y-5). The relative stereochemistry of amphidinolide V was deduced from the combination of the molecular mechanics calculation and the analyses of H- H coupling constants and NOESY data. Recently, a total synthesis of the proposed structure of amphidinolide V (21) has been achieved by Fiirstner group (Scheme 6). Comparison of the NMR data and CD spectrum of the... 

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