Residually Thin Schemes

Throughout this section, S is assumed to have finite valency. The letter T will stand for a closed subset of S. 

Theorem 5.6.1 The set T is residually thin if and only if each composition factor ofT is thin. 

Let us first assume T to be residually thin, and let us pick an element U in 

Since T is assumed to be residually thin, 1 = 0 (T). Thus, as U IA, there exists a non-negative integer n such that (0 )n+1(T) C U and Uu C (i0 d)n(T).1 By Lemma 2.5.6(iii), this implies that U is strongly normal in Uu. Thus, by Lemma 4.2.5(ii), Uu//IJ is thin. Thus, for each element x in X, (Uu//U)x is thin cf. Theorem 2.1.8(iii). Thus, as U has been chosen arbitrarily in IA T, we have proved that each composition factor of T is thin cf. Theorem 5.4.2. 

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In the second section, we shall relate specific closed subsets of S containing a finite closed subset T to the corresponding closed subsets of the quotient scheme of S over T. Among other issues we focus on the relationship between commutators and quotient schemes. This leads naturally to the connection between the thin residue of S and the thin residue of quotient schemes of S. This relationship will be described in Theorem 4.2.8, a result which depends on Lemma 3.2.7. Theorem 4.2.8 turns out to be useful in Section 5.5 where we discuss residually thin schemes. 

Lemma 5.6.2 and Lemma 5.6.3(ii) say that the property of being residually thin is inherited by closed subsets and by quotient schemes over normal closed subsets. Here is a partial converse. 

Polymer/additive analysis then usually proceeds by separation of polymer and additives (cf. Scheme 2.12) using one out of many solvent extraction techniques (cf. Chapter 3). After extraction the residue is pressed into a thin film to verify that all extractables have been removed. UV spectroscopy is used for verification of the presence of components with a chromophoric moiety (phenolic antioxidants and/or UV absorbers) and IR spectroscopy to verify the absence of IR bands extraneous to the polymer. The XRF results before and after extraction are compared, especially when the elemental analysis does not comply with the preliminary indications of the nature of the additive package. This may occur for example in PA6/PA6.6 blends where... 

The location of the double bonds in unsaturated glycerophosphocholine (GPC) lipids is made difficult mainly due to the limited amounts of these compounds isolated from cells. When a thin layer of a GPC lipid is exposed to the action of ozone, partial or total ozonization of the unsaturated fatty acid residues may take place, depending on the particular compound being analyzed. The ozonized hpid is dissolved and analyzed by tandem MS. A typical injection into the instrument may be 5 xL containing 10-20 ng of the lipid. An illustrative example is the ozonized l-stearoyl-2-arachidonoyl-GPC, showing the systematic fragmentation pattern in Scheme 21, typical of the ozonized GPC lipids. Thus, the molecular peak appears increased by one nominal unit (M + H, m/e 1002),... 

Let us now look how the thin residue works together with quotient schemes. 

From Theorem 3.2.1 (i) we know that O U J ) is strongly normal in T. Thus, by Lemma 4.2.5(h), T//0 T) is thin. Thus, we obtain from Lemma 4.2.7(i) that the thin residue 0 S(T) of T is the uniquely defined smallest closed subset of S having a thin quotient scheme. 

In the third chapter of this monograph, we look at closed subsets generated by specific subsets. Our investigation leads us to the definitions of commutator subsets and thin residue of a closed subset, as well as to other characteristic closed subsets of schemes. 

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