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Krull Dimension Forster’s Theorem We get a first-order (constructive) proof. It can be interpreted as an algorithm which produces the unimodular combination. The motivation for this Theorem comes from differential geometry If we have a vector bundle over a space of dimension d and all the fibers are of dimension r then we can find d + r generators for the module of global sections 21
Krull Dimension Forster’s Theorem The proof relies on the following consequence of Cramer formulae Proposition: If P is a n × n matrix of determinant δ and of adjoint matrix ˜P then we have D(δX − ˜ P Y ) D(P X − Y ) for arbitrary column vectors X, Y in R n×1 Corollary: If P is a n × n matrix of determinant δ and X, ˜ P Y are complementary then D(δ) D(P (δX) − Y ) 22
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Page 5 and 6: Krull Dimension Krull dimension of
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