# Deformations of Nilpotent Matrices over Rings and Reduction by Donald G. Babbitt

By Donald G. Babbitt

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**Extra resources for Deformations of Nilpotent Matrices over Rings and Reduction of Analytic Families of Meromorphic Differential Equations**

**Sample text**

I*. Such results would provide a foundation for developing a theory of analytic deformations, and in particular, to obtain suitable analogues of the results of §§^-5 below. We do not take up these matters here. 3. We now place ourselves in the framework of n°2 with R valuation ring and study certain families of submodules of V^ R t e T. Let Z be the set of open neighborhoods of t . LemmaJ.. G. S. VARADARAJAN are elements u. e V„ (1 < j < r) such that U(t) for all t c N-. (t))_ ___ Let i(t) = [V:U(t)].

A) For some neighborhood (t) N of t Q , A(L(t)) <^(L(t Q )) (t e N) (b) For a suitable open neighborhood N of t , ^(L(t)) takes only finitely many values as t varies in N; and if these are arranged as £- < i2 < ... ,q. (t) have the same filtration in V^ Proof. of all t for t in a neighborhood of t n . The assertion (c) is just Lemma 3 for j = 1. Hence it is enough to prove (a) and (b). (a) We shall prove (a) by a double induction on m of the filtrations of the L(t). We assume that (k) has been proved for "parameter spaces" of dimension < dim(T), spread is

X. A. r . e S. Then i-j_> • • • > i r 0 / un (t) A • • • A u (t) = D a. K i , < • • • * l . (t) x. A ... A x. ' r 1 r . (t)) . 3. For a family t -» U(t) as above, the index strictly greater than the generic value that i. i(^n) will as a rule be It is clear that the requirement i(tn) = i imposes strong restrictions on the family. In the results to be proved now we shall make this a little more precise. k* i C O = i iff i(t) = i for all t in some N e Z. *

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