Gelfand-Kirillov dimension under base field extension
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LetF ⊂K be a field extension,A be aK-algebra. It is proved that, in general, GK dim F A≥GK dim K A+tr F (K). For commutative algebras or Noetherian P.I. algebras, the equality holds. Two examples are also constructed to show that: (i) there exists an algebraA such that GK dim F A=GK dim K A+tr F (K)+1; (ii) there exists an algebraic extensionF ⊂K and aK-algebraA such that GK dim F A=∞, but GK dim K A<∞.
KeywordsCommutative Algebra Free Algebra Algebraic Extension Transcendental Extension Kirillov Dimension
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- 6.M. Lorenz and L. W. Small,On the Gelfand-Kirillov dimension of Noetherian P.I. algebras, Contemp. Math. 13, Am. Math. Soc., Providence, 1982, pp. 199–205.Google Scholar