Refiltering for some noetherian rings Noncommutative Geometry and Rings Almer´ıa, September 3, 2002 J. Gomez-T´ orrecillas Universidad de Granada Refiltering for some noetherian rings – p.1/12 Use filtrations indexed by more general (more flexible) monoids than Filtration Associated Graded Ring Re-filtering: outline Generators and Relations Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Associated Graded Ring Re-filtering: outline Generators Filtration and Relations Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline Generators Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Re- Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Re-filtering: outline Use filtrations indexed by more general (more flexible) monoids than New Generators Re- Filtration and Relations Associated Graded Ring Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Re- Filtration and Relations Associated Graded Ring Keep (or even improve) properties of Refiltering for some noetherian rings – p.2/12 Use filtrations indexed by more general (more flexible) monoids than Re-filtering: outline New Generators Re- Filtration and Relations Associated Graded Ring We will show how these ideas can be used to obtain the Cohen-Macaulay property w.r.t. the Gelfand-Kirillov dimension for the quantized enveloping algebras. The results will be published in a joint paper with F. J. Lobillo Refiltering for some noetherian rings – p.2/12 is a free abelian monoid with generators a total monoid ordering on with as minimal element Assume is –filtered by The associated –graded algebra is given by where and . Filtered and graded rings ¡ ¢¤£¦¥ ¢ § § § ¡ Let denote a free abelian monoid with generators ¥ , whose ¨ ¨ ¨ © § § § £ ¥ ¡ elements will be considered as vectors ¥ with non-negative integer components. Refiltering for some noetherian rings – p.3/12 a total monoid ordering on with as minimal element Assume is –filtered by The associated –graded algebra is given by where and . Filtered and graded rings ¡ ¢ ¢ £ ¡ is a free abelian monoid with generators ¥ ¥ ¡ Let be a monoid total ordering on such that ¨ for every ¡ ¨ . Basic examples are the lexicographical orderings Refiltering for some noetherian rings – p.3/12 Assume is –filtered by The associated –graded algebra is given by where and . Filtered and graded rings ¡ ¢ ¢ £ ¡ is a free abelian monoid with generators ¥ ¥ ¡ a total monoid ordering on with as minimal element ¡ ¥ becomes then a well-ordered monoid Refiltering for some noetherian rings – p.3/12 p.3/12 – rings –filtration noetherian some ¥ for ¡ of ¡ ¢ element ¡ Refiltering ¥ An . ¡ ¨ ¢ ¥ £ minimal . ring ¡ as ¨ e ¥ v all rings © with generators ¡ for ¨ with commutati on that all a . amily f er for © v such a o graded monoid is ordering of . algebra and abelian #" an monoid free –algebra ed be ! a total is a a –submodules 3. 2. 4. 1. ¡ on Let Filter . by en v gi is and by algebra –filtered –graded is associated The Assume where The associated –graded algebra is given by where and . 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