Deformations of Nilpotent Matrices over Rings and Reduction by Donald G. Babbitt
By Donald G. Babbitt
Read Online or Download Deformations of Nilpotent Matrices over Rings and Reduction of Analytic Families of Meromorphic Differential Equations PDF
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Extra resources for Deformations of Nilpotent Matrices over Rings and Reduction of Analytic Families of Meromorphic Differential Equations
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.
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.