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| In [[algebra]], the '''Gelfand–Kirillov dimension''' of a [[right module]] ''M'' over a [[K-algebra|''k''-algebra]] ''A'' is:
| | 27 yr old Valuer Deandre from Englehart, spends time with hobbies and interests which include exercise, [https://www.youtube.com/watch?v=20KQhsAFzw4&list=PLamGIcKOwjLAkEC1eNEGMRs1281D71Gu7&index=5 the holistic sanctuary] and drawing. Has been a travel maniac and in recent years arrived at Historic Centre (Chorá) with the Monastery of Saint John. |
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| :<math>\operatorname{GKdim} = \sup \limsup_{n \to \infty} \log_n \dim_k M_0 V^n</math>
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| where the sup is taken over all finite-dimensional [[Linear subspace|subspace]]s <math>V \subset A</math> and <math>M_0 \subset M</math>.
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| == In the theory of D-Modules ==
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| Given a right module ''M'' over the [[Weyl algebra]] <math>A_n</math>, the Gelfand–Kirillov dimension of ''M'' over the Weyl algebra coincides with the dimension of ''M'', which is by definition the degree of the [[Hilbert polynomial]] of ''M''. This enables to prove additivity in [[short exact sequence]]s for the Gelfand–Kirillov dimension and finally to prove [[Weyl_algebra#Properties of the Weyl algebra|Bernstein's inequality]], which states that the dimension of ''M'' must be at least ''n''. This leads to the definition of [[D-module|holonomic D-Module]]s as those with the minimal dimension ''n'', and these modules play a great role in the [[geometric Langlands program]].
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| == References ==
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| *{{cite journal|title=A remark on Gelfand–Kirillov dimension|url=http://www.math.washington.edu/~smith/Research/GK-rmk.pdf |last1=Smith |first1=S. Paul |last2=Zhang |first2=James J. |year=1998 |journal=[[Proceedings of the American Mathematical Society]] |volume=126 |number=2 |pages=349–352}}
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| * Coutinho: A primer of algebraic D-modules. Cambridge, 1995
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| {{DEFAULTSORT:Gelfand-Kirillov dimension}}
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| [[Category:Algebra]]
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| [[Category:Dimension]]
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| {{algebra-stub}}
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Latest revision as of 20:24, 30 September 2014
27 yr old Valuer Deandre from Englehart, spends time with hobbies and interests which include exercise, the holistic sanctuary and drawing. Has been a travel maniac and in recent years arrived at Historic Centre (Chorá) with the Monastery of Saint John.