Corollary 6.3.2 The scheme S is schurian if and only if S is a quotient scheme of a thin scheme. [Pg.114]

For each finite closed subset T of S, we call S//T the quotient scheme of S over T. [Pg.65]

The following result describes the relationship between quotient schemes of subschemes of S and subschemes of quotient schemes of S. Note that its first part generalizes Theorem 4.1.3(i). [Pg.67]

It is well known (and easy to see) that schemes are not necessarily isomorphic to quotient schemes of thin schemes. However, it seems that a general scheme theoretic characterization of such quotient schemes is out of reach. It is for this reason that one might instead ask for specific sufficient conditions for a scheme to be isomorphic to a quotient scheme of a thin scheme. [Pg.291]

Let us now look how the thin residue works together with quotient schemes. [Pg.72]

In the first section, we compute the structure constants of quotient schemes of S in terms of those of S. We relate the complex multiplication in S to the one in quotient schemes of S, and we look at the relationship between subschemes of quotient schemes and quotient schemes of subschemes. [Pg.63]

The following lemma relates the complex multiplication in S with the complex multiplication in quotient schemes of S over closed subsets of finite valency. [Pg.66]

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. [Pg.99]

The following two propositions show that the property of being faithfully embedded is inherited to closed subsets and to quotient schemes. [Pg.107]

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. [Pg.63]

The Schur group is the subject of the third section of this chapter. We shall see, in particular, that S is schurian if and only if S is isomorphic to a quotient scheme of a thin scheme, and this is the case if and only if, modulo the group [Pg.103]

In the third section, we assume S to have finite valency. We investigate the arithmetic between the structure constants of S and the structure constants of the quotient schemes of S. [Pg.63]

A subset R of S is called naturally valenced if each element of R has finite valency. The present chapter starts with the observation that naturally valenced schemes give rise to quotient schemes over finite closed subsets. After the definition of quotient schemes we shall always assume S to be naturally valenced. [Pg.63]

The second section of this chapter deals with faithfully embedded closed subsets of S. We mainly discuss the question to which extent the property of being faithfully embedded is inherited from given quotient schemes of closed subsets of S to other quotient schemes of closed subsets of S. [Pg.103]

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. [Pg.73]

Morphisms are related to faithful maps, which lead naturally to the notion of a faithfully embedded closed subset. Such subsets provide an appropriate language for an attempt to establish so-called recognition theorems. These theorems deal with the question of which schemes are quotient schemes of thin schemes. We shall come back to recognition theorems and their role in scheme theory later in this preface. [Pg.290]

See also in sourсe #XX -- [ Pg.65 ]

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