Local ring

The communication between the units is performed through a local ring network, based on twisted pair wire as shown in the figure below. 

B. R. McDonald, Geometric Algebra Over Local Rings (1976)... 

In this section, we present a unified picture of the different electronic effects that combine to determine methyl rotor potentials in the S0, Sp and D0 electronic states of different substituted toluenes. Our approach is based on analysis of ab initio wavefunctions using the natural bond orbitals (NBOs)33 of Weinhold and cowork-ers. We will attempt to decompose the methyl torsional potential into two dominant contributions. The first is repulsive steric interactions, which are important only when an ortho substituent is present. The second is attractive donor-acceptor interactions between CH bond pairs and empty antibonding orbitals vicinal to the CH bonds. In the NBO basis, these attractive interactions dominate the barrier in ethane (1025 cm-1) and in 2-methylpropene (1010 cm-1) see Figure 3. By analogy, donor-acceptor attractions are important in toluenes whenever there is a substantial difference in bond order between the two ring CC bonds adjacent to the C-CH3 bond. Viewed the other way around, we can use the measured methyl rotor potential as a sensitive probe of local ring geometry. 

In view of the clear correlation of local ring geometry with methyl rotor barrier height in the S0 and D0 states, the strong effects of S, <— S0 excitation on rotor potentials seem to indicate substantial distortion of the ring away from hexagonal symmetry in the S, state as well. There is little clear evidence of this from molecular spectroscopy. We have speculated that such a n-molecular orbital orientation effect in the S j state (similar to that in the cation) might explain the observed characteristic... 

ESR spectra in at least one matrix provide evidence for a localized ring-opened... 

Suppose k is an algebraically closed field of characteristic p > 0 and suppose x Spec(fc) — Ag>5 is a geometric point given by the polarized abelian variety (X0, A0) over Spec (A ). In [Cr] the complete local ring of Agtd at x is computed. We will describe the result. 

Let R be the complete local ring of Ag,< at the point x. In R we have the elements rjij constructed using the universal deformation of (X0>A0) over Spec( E). By [Cr] Corollary 23 we conclude that... 

Proof. We may assume R is a local ring with residue characteristic p. The exact sequences... 

Finally let us give a list of the isomorphism types of the complete local rings at geometric points x Spec(fc) — X in characteristic p ... 

Using the above what can we say about complete local rings of < (2,p) at geometric points are of the simple types listed above. In particular <5(2, p) -+ Spec(Z) is a locally complete intersection morphism of relative dimension th ree. 

Proof. The first statement is a direct consequence of Lemma 7.3. Indeed the lemma implies that the module Ms(G) over the local ring R/pR satisfies TorJypjR(J2/m, Ms(G)) = (0), i > 1 and hence is a free R/pR module. The statement on the rank follows from Proposition 7.2 by an argument as in the proof of Lemma 7.5. 

Theorem 8.5. Suppose S is the spectrum of a Noetherian complete local ring with perfect residue field. If the functor... 

Now we apply the Serre-Tate Theorem ([Me] V 2.3) to get a deformation (Y,p) over S of the pair (Yo o) such that the p-divisible group Y[p°°] is isomorphic to the one described above. Finally, after a finite flat base extension such that S is still the spectrum of a complete local ring with residue field we may assume that... 

In this section we will compare the singularities of S(g, p) with the singularities of a scheme X which can be described as a certain flag variety. We will show that any complete local ring of S(g,p) af a geometric point is isomorphic to a complete local ring of X at some geometric point of X. 

Ar] M. Artin, Algebraic approximation of structures over complete local rings. Publ. Math. I.H.E.S. 36 (1969), 23-34. 

Therefore we get our functor Ms C(l)s0 — M(l)s0 a pair ( >/s) as above exists. Suppose that So is the spectrum of a Noetherian complete local ring whose residue field has a finite p-basis. In this case the functor Ms induces an equivalence of the full subcategory of C(l)s0 consisting of group schemes which have multiplicative part equal to zero and a corresponding subcategory of At (1) s0 (Theorem 9.3). If the residue field is perfect then we can show under some additional hypotheses on (5,/s) that Ms C(l)s0 At(l)s0 is an equivalence (Theorem 10.2). 

By a theorem of Raynaud ([BBM] Theorem 3.1.1) any finite flat group scheme over S can locally in the Zariski topology on S be embedded into an abelian scheme. It follows that if S is the spectrum of a local ring then the category C(n)s is generated by the category BT(n)s-... 

Proof. We only have to prove this when S is the spectrum of a local ring. Let us assume this is so and put h = height(<2). 

Since the morphism Xo Spec(Fp) is of finite type, the completions of the local rings of Xo at its closed points are Noetherian complete local rings with perfect residue fields. Over these rings we can find a Barsotti-Tate group H with H pk] Go restricted to these rings (see [IL] Theorem 4.4). Hence it follows from Proposition 3.4 that Mx(Go) is locally free over Ox/pkOx of rank h — height(G) at all the closed points of X. ... 

Proof. If there exists a faithfully flat morphism of schemes T S such that Mt satisfies the conclusions of Proposition 7.6 then this is true for Ms also. Thus we may assume S is the spectrum of a local ring and the next lemma finishes the proof. ... 

We wish to study the local properties of Hilbr t) at z, that is the properties of the local ring oz = 0 z We will say that X is unobstructed ( resp. obstructed ) if Hiibrp(l) is nonsingular ( resp. singular ) at z. 

Let B in a be the k-algebra which prorepresents it, and let ojxj be the k-algebra which prorepresents Hilbx ( ojxj is the completion of 0 Xj, the local ring of HUb at the point (X]). The morphism b is induced by a local homomorphism ojxj — B. 

The aromatic character of fullerene anions can be better understood by monitoring the local ring currents of the 5-membered rings (5MRs) and 6-membered rings (6MRs) in... 

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