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 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
¡
Let be an algebra over a commutative ring . An ¥ –filtration
¡ ¨ on a –algebra is a family © of –submodules of such that
¡ ¨ 1. for all .
¡ ¨ 2. for all ¥ .
©
3. .
#" ! 4. .
Refiltering for some noetherian rings – p.3/12 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
¡ ¡ ¨ © Assume is ¥ –filtered by
Refiltering for some noetherian rings – p.3/12 Filtered and graded rings
¡ ¢ ¢ £ ¡ 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 .
Refiltering for some noetherian rings – p.3/12 A Refiltering Theorem *)
¡ ¡ ¨ © Theorem 1. Let be an ¥ –filtration
+ © " on such that is a left noetherian ring, the ’s are % ,.- 1 ,.-.2 1 + + $
¡ /0 /0 © £ £ 2 f. g. over , and is an –graded it- -.2 - £ erated Ore extension for some homogeneous elements¥ ¥ such
+
6 -54 6 -54 0 © that 3 4 3 , where 4 3 . Then there is an –filtration + +
7 ) © " ¡ ¡ on such that , the ’s are –f.g., and ;
$ : 89 © .
Refiltering for some noetherian rings – p.4/12 Given ,
Since is a –basis of , we get from that is a –basis of .
Proof: they’re polynomials ¨ £ <=<=<=< ?=?=?=? ¨5>
BDC
E ¨ A -F4 © © 4 @ Set . 2 ¡ with . ¨ 2
¤' I G - G © 4 4 4 H and choose such that H .
Refiltering for some noetherian rings – p.5/12 Given ,
Since is a –basis of , we get from that is a –basis of .
Proof: they’re polynomials J
B ¤' I E C ¨ A -54 G ¨ ¨ -54 © © © § § § 4 4 £ ¥ 2 ¥ ¥ ¥ H %
$ Given 9 , we have an expression in
+ T ' I X 9 W W © PRQ S LNM T T ¥ O K P S LNM U¦V O T K Q
Refiltering for some noetherian rings – p.5/12 Given ,
Since is a –basis of , we get from that is a –basis of .
Proof: they’re polynomials J
B ¤' I E C ¨ A -54 G ¨ ¨ -54 © © © § § § 4 4 £ ¥ 2 ¥ ¥ ¥ H %
$ Given 9 , we have an expression in
+ T ' I X 9 W W © PRQ S LNM T T ¥ O K P S LNM U¦V O T K Q T .\ Y T[Z - \ \ - © § § § £ £ 2 ¥ ¥ 2
Refiltering for some noetherian rings – p.5/12 Since is a –basis of , we get from that is a –basis of .
Proof: they’re polynomials J
B ¤' I E C ¨ A -54 G ¨ ¨ -54 © © © § § § 4 4 £ ¥ 2 ¥ ¥ ¥ H
+ T ' I X W 9 W 9 © P S P S LNM U LNM T T ¥ O K T Q O K Q Given , V This leads to
T ' ' I I ] 9 W © PRQ S P S LNM LNM T O O K K Q P S LNM U¦V O T K Q
Refiltering for some noetherian rings – p.5/12 Since is a –basis of , we get from that is a –basis of .
Proof: they’re polynomials J
B ¤' I E C ¨ A -54 G ¨ ¨ -54 © © © § § § 4 4 £ ¥ 2 ¥ ¥ ¥ H
+ T ' I X W 9 W 9 © P S P S LNM U LNM T T ¥ O K T Q O K Q Given , V
BDC E A An induction on 9 shows that + T ] _ _ 9 © T T ¥ PRQ S L U ^ O M K T
Refiltering for some noetherian rings – p.5/12 Proof: they’re polynomials J
B ¤' I E C ¨ A -54 G ¨ ¨ -54 © © © § § § 4 4 £ ¥ 2 ¥ ¥ ¥ H
+ T ' I X W 9 W 9 © P S P S LNM U LNM T T ¥ O K T Q O K Q Given , V
+ T ! ] 9 _ _ © T T ¥ ¥ PRQ S LNM U ^ O K T % + $
! 2 T X \ Since ) is a –basis of , we get from that +
T 2 ] \ ) is a –basis of .
Refiltering for some noetherian rings – p.5/12 for every
Proof: getting relations + + ! ` b b H H a - _ _ -54 _ _ © § § § 4 Given , ¥ ¥ we get from that and
b b H H ' I -F4 G - _Nc _ G _Nc _ © © 4 4 4 H
Refiltering for some noetherian rings – p.6/12 for every
Proof: getting relations + + ! ` b b H H a - _ _ -54 _ _ © § § § 4 Given , ¥ ¥ we get from that and
b b H H ' I -F4 G - _Nc _ G _Nc _ © © 4 4 4 H
+ ' H Since is left noetherian and is f.g., we have that H is noetherian.
Refiltering for some noetherian rings – p.6/12 for every
Proof: getting relations + + ! ` b b H H a - _ _ -54 _ _ © § § § 4 Given , ¥ ¥ we get from that and
b b H H ' I -F4 G - _Nc _ G _Nc _ © © 4 4 4 H
d d 2 e \ ¨ \ Thus there is a finite subset 4 of such that 4 for 4 and + T b H I ] _ _ G _ _ G © 4 4 T for every f T H
Refiltering for some noetherian rings – p.6/12 Proof: getting relations + T b H I ] _ G _ _ G _ © 4 4 T for every i h g H j k g l H ! `on p m m - -54 6 -54 - a © Analogously, from the relation 3 4 3 3 for every , we get T I ] W G G G G 6 © 3 4 3 4 4 3 T ¥ f T Hq
d d 2 I e ¨ ¨ \ \ 4 3 3 4 for finite subsets3 4 of such that for
Refiltering for some noetherian rings – p.6/12 Proof: getting relations + T b H I ] _ G _ _ G _ © 4 4 T for every i h g H j k g l H T I ] G G G G 6 W © 3 4 3 4 4 3 T ¥ i h g q H j r k g l l H q
Define the monoid ordering on 2 by e \ t sut
vw or \ m \ t \ t © and lex
Refiltering for some noetherian rings – p.6/12 Proof: getting relations + T b H I ] G _ _ G _ _ © 4 4 T for every i h g H xzy k g H T I ] W G G G G 6 © 3 4 3 4 4 3 T ¥ i h g q H x r k g y y H q
for the monoid ordering e \ t sut
vw or \ m \ t \ t © and lex
Refiltering for some noetherian rings – p.6/12 Then is the maximum of w. r. to . By using Carathéodory’s description for convex hulls, one can prove that does not belong to the convex hull of . Therefore, there is a real vector such that for every . Things can be easily arranged to obtain . Therefore, for and for . for every
From these relations, we can prove that the (f. g.) –submodules of defined as
form an –filtration of such that
Proof: Re-filtering { | Now, consider the finite subset of 2 defined as 5}
{ d d } } ¢ ¢ ¢ ¢ ¢ © § § § 4 3 4 4 4 3 £ ¥ c c c c c 2 ¥ ~ 4 £~ 4 ~ £ ~ 3 2 2 s | 2 2 and extend from to .
Refiltering for some noetherian rings – p.7/12 By using Carathéodory’s description for convex hulls, one can prove that does not belong to the convex hull of . Therefore, there is a real vector such that for every . Things can be easily arranged to obtain . Therefore, for and for . for every
From these relations, we can prove that the (f. g.) –submodules of defined as
form an –filtration of such that
Proof: Re-filtering { | Now, consider the finite subset of 2 defined as 5}
{ d d } } ¢ ¢ ¢ ¢ ¢ © § § § 4 3 4 4 4 3 £ ¥ c c c c c 2 ¥ ~ 4 £~ 4 ~ £ ~ 3 2 2 s s { | 2 2 and extend from to . Then is the maximum of w. r. to .
Refiltering for some noetherian rings – p.7/12 Therefore, there is a real vector such that for every . Things can be easily arranged to obtain . Therefore, for and for . for every
From these relations, we can prove that the (f. g.) –submodules of defined as
form an –filtration of such that
Proof: Re-filtering { | Now, consider the finite subset of 2 defined as 5}
{ d d } } ¢ ¢ ¢ ¢ ¢ © § § § 4 3 4 4 4 3 £ ¥ c c c c c 2 ¥ ~ 4 £~ 4 ~ £ ~ 3 2 2 s s { | 2 2 and extend from to . Then is the maximum of w. r. to . By using Carathéodory’s description for convex hulls, one can prove { that does not belong to the convex hull of .
Refiltering for some noetherian rings – p.7/12 Things can be easily arranged to obtain . Therefore, for and for . for every
From these relations, we can prove that the (f. g.) –submodules of defined as
form an –filtration of such that
Proof: Re-filtering { | Now, consider the finite subset of 2 defined as 5}
{ d d } } ¢ ¢ ¢ ¢ ¢ © § § § 4 3 4 4 4 3 £ ¥ c c c c c 2 ¥ ~ 4 £~ 4 ~ £ ~ 3 2 2 s s { | 2 2 and extend from to . Then is the maximum of w. r. to . By using Carathéodory’s description for convex hulls, one can prove { that does not belong to the convex hull of . Therefore, there { 2 n is a real vector such that ¥ for every .
Refiltering for some noetherian rings – p.7/12 Therefore, for and for . for every
From these relations, we can prove that the (f. g.) –submodules of defined as
form an –filtration of such that
Proof: Re-filtering { | Now, consider the finite subset of 2 defined as 5}
{ d d } } ¢ ¢ ¢ ¢ ¢ © § § § 4 3 4 4 4 3 £ ¥ c c c c c 2 ¥ ~ 4 £~ 4 ~ £ ~ 3 2 2 s s { | 2 2 and extend from to . Then is the maximum of w. r. to . By using Carathéodory’s description for convex hulls, one can prove { that does not belong to the convex hull of . Therefore, there { 2 n is a real vector such that ¥ for every . .
2
© § § § £ ¥ 2 Things can be easily arranged to obtain ¥ .
Refiltering for some noetherian rings – p.7/12 for every
From these relations, we can prove that the (f. g.) –submodules of defined as
form an –filtration of such that
Proof: Re-filtering { | Now, consider the finite subset of 2 defined as 5}
{ d d } } ¢ ¢ ¢ ¢ ¢ © § § § 4 3 4 4 4 3 £ ¥ c c c c c 2 ¥ ~ 4 £~ 4 ~ £ ~ 3 2 2 s s { | 2 2 and extend from to . Then is the maximum of w. r. to . By using Carathéodory’s description for convex hulls, one can prove { that does not belong to the convex hull of . Therefore, there { 2 n is a real vector such that ¥ for every . .
2
© § § § £ ¥ 2 Things can be easily arranged to obtain ¥ . d d I n n \ \ \ \ 3 4 4 3 4 4 ¥ Therefore, ¥ for and for .
Refiltering for some noetherian rings – p.7/12 for every
From these relations, we can prove that the (f. g.) –submodules of defined as
form an –filtration of such that
Proof: Re-filtering
2 . d
n
\ \ © 2 4 4 ¥ £ ¥ We have such that ¥ for and d I n
\ \ 4 3 3 4 ¥ for . This allows to rewrite our relations as + T b H I ] G G _ _ _ _ © 4 4 T for every T H T I ] G G 6 G G W © 3 4 4 3 3 4 T T H q
Refiltering for some noetherian rings – p.7/12 Proof: Re-filtering + T b H I ] _ G _ _ G _ © 4 4 T for every T H T I ] G G G G 6 W © 3 4 3 4 4 3 T T H q + From these relations, we can prove that the (f. g.) –submodules of defined as
+ ] 7 © ¡ ~ ¡
form an –filtration of such that 1 ,.- 1 ,
+ : - 89 /0 /0 © £ £ 2 2
Refiltering for some noetherian rings – p.7/12 1. Assume , where is a unit of for and is an automorphism for . If is Auslander-regular, then is Auslander-regular. 2. Assume, in addition, that is an algebra over a field , and that is generated by as a –algebra in such a way that the standard filtration satisfies that is a finitely presented and noetherian algebra, and . If either or is an Auslander-regular and Cohen-Macaulay algebra, then is Auslander-regular and Cohen Macuaulay.
Regularity of some filtered algebras
+
¡ © " Theorem 2 Assume to be ¥ –filtered such that is
+ ¨ noetherian, is left –f.g. for every , and % 1 ,.- 1 ,.-2 + $ /0 /0 © § § § £ £ 2 .
Refiltering for some noetherian rings – p.8/12 If is Auslander-regular, then is Auslander-regular. 2. Assume, in addition, that is an algebra over a field , and that is generated by as a –algebra in such a way that the standard filtration satisfies that is a finitely presented and noetherian algebra, and . If either or is an Auslander-regular and Cohen-Macaulay algebra, then is Auslander-regular and Cohen Macuaulay.
Regularity of some filtered algebras
+
¡ © " Theorem 2 Assume to be ¥ –filtered such that is
+ ¨ noetherian, is left –f.g. for every , and % 1 ,.- 1 ,.-2 + $ /0 /0 © § § § £ £ 2 .
+ ! `on p m m 6 -54 6 a -54 0 © 1. Assume 3 4 3 , where 4 3 is a unit of for + + ! ` m m a 0 ) and 4 is an automorphism for .
Refiltering for some noetherian rings – p.8/12 2. Assume, in addition, that is an algebra over a field , and that is generated by as a –algebra in such a way that the standard filtration satisfies that is a finitely presented and noetherian algebra, and . If either or is an Auslander-regular and Cohen-Macaulay algebra, then is Auslander-regular and Cohen Macuaulay.
Regularity of some filtered algebras
+
¡ © " Theorem 2 Assume to be ¥ –filtered such that is
+ ¨ noetherian, is left –f.g. for every , and % 1 ,.- 1 ,.-2 + $ /0 /0 © § § § £ £ 2 .
+ ! `on p m m 6 -54 6 a -54 0 © 1. Assume 3 4 3 , where 4 3 is a unit of for + + + ! ` m m a 0 ) and 4 is an automorphism for . If is Auslander-regular, then is Auslander-regular.
Refiltering for some noetherian rings – p.8/12 in such a way that the standard filtration satisfies that is a finitely presented and noetherian algebra, and . If either or is an Auslander-regular and Cohen-Macaulay algebra, then is Auslander-regular and Cohen Macuaulay.
Regularity of some filtered algebras
+
¡ © " Theorem 2 Assume to be ¥ –filtered such that is
+ ¨ noetherian, is left –f.g. for every , and % 1 ,.- 1 ,.-2 + $ /0 /0 © § § § £ £ 2 .
+ ! `on p m m 6 -54 6 a -54 0 © 1. Assume 3 4 3 , where 4 3 is a unit of for + + + ! ` m m a 0 ) and 4 is an automorphism for . If is Auslander-regular, then is Auslander-regular. 2. Assume, in addition, that is an algebra over a field , and that + £ J is generated¥ by ¥ as a –algebra
Refiltering for some noetherian rings – p.8/12 If either or is an Auslander-regular and Cohen-Macaulay algebra, then is Auslander-regular and Cohen Macuaulay.
Regularity of some filtered algebras
+
¡ © " Theorem 2 Assume to be ¥ –filtered such that is
+ ¨ noetherian, is left –f.g. for every , and % 1 ,.- 1 ,.-2 + $ /0 /0 © § § § £ £ 2 .
+ ! `on p m m 6 -54 6 a -54 0 © 1. Assume 3 4 3 , where 4 3 is a unit of for + + + ! ` m m a 0 ) and 4 is an automorphism for . If is Auslander-regular, then is Auslander-regular. 2. Assume, in addition, that is an algebra over a field , and that + £ J is generated¥ by ¥ as a –algebra in such a way that +
7 ) the standard filtration ¡ satisfies that &
+ + + 89 © £ " ¡ ¡ ¡ ' is a finitely presented and noetherian
+ + 6 0 4 3 algebra, 4 £ £ and .
Refiltering for some noetherian rings – p.8/12 Regularity of some filtered algebras
+
¡ © " Theorem 2 Assume to be ¥ –filtered such that is
+ ¨ noetherian, is left –f.g. for every , and % 1 ,.- 1 ,.-2 + $ /0 /0 © § § § £ £ 2 .
+ ! `on p m m 6 -54 6 a -54 0 © 1. Assume 3 4 3 , where 4 3 is a unit of for + + + ! ` m m a 0 ) and 4 is an automorphism for . If is Auslander-regular, then is Auslander-regular. 2. Assume, in addition, that is an algebra over a field , and that + £ J is generated¥ by ¥ as a –algebra in such a way that +
7 ) the standard filtration ¡ satisfies that &
+ + + 89 © £ " ¡ ¡ ¡ ' is a finitely presented and noetherian
+ + + 89 6 0 4 3 algebra, 4 £ £ and . If either or ,.- 1 ,.-.2 1 + /0 /0 § § § £ £ 2 is an Auslander-regular and Cohen-Macaulay algebra, then is Auslander-regular and Cohen Macuaulay.
Refiltering for some noetherian rings – p.8/12 Since this is also true for the grade number (Björk), the proof of 2 can be now easily finished.
Proof
Let ¡ be the filtration on given by Theorem 1 with ,.- 1 , 1 + E - /0 /0 © § § § £ £ 2 2
Well-known results by Björk and Ekström give 1.
Refiltering for some noetherian rings – p.9/12 Since this is also true for the grade number (Björk), the proof of 2 can be now easily finished.
Proof
E To prove 2, define a filtration on by
+ E X © P S 4 ¡ ¥ 4 ~ ¡
BDC B E E C - -2 © £ where ¥ ¥ . The associated graded algebra is , 1 1 ,.-2
+ : E E E - /0 /0 © § § § £ £ 2
Refiltering for some noetherian rings – p.9/12 Since this is also true for the grade number (Björk), the proof of 2 can be now easily finished.
Proof
E To prove 2, define a filtration on by
+ E X © P S 4 ¡ ¥ 4 ~ ¡
BDC B E E C - -2 © £ where ¥ ¥ . The associated graded algebra is , 1 1 ,.-2
+ : E E E - /0 /0 © § § § £ £ 2
+ E E E Since is finitely presented and noetherian, so is and, thus, the filtration is in the hypotheses of a theorem of McConnell and Stafford which reduces the computation of the Gelfand-Kirillov dimen-
E sion of –modules to its associated graded modules over .
Refiltering for some noetherian rings – p.9/12 Proof
E To prove 2, define a filtration on by
+ E X © P S 4 ¡ ¥ 4 ~ ¡
BDC B E E C - -2 © £ where ¥ ¥ . The associated graded algebra is , 1 1 ,.-2
+ : E E E - /0 /0 © § § § £ £ 2
Since this is also true for the grade number (Björk), the proof of 2 can be now easily finished.
Refiltering for some noetherian rings – p.9/12 Proof: De Concini, Kac and Procesi gave an –filtration on such that the associated graded ring satisfies the hypotheses of Theorem 2.
Regularity of quantum groups
{ Theorem 3. The quantized enveloping 6 –algebra associated to a Cartan matrix { is Auslander-regular and Cohen-Macaulay.
Refiltering for some noetherian rings – p.10/12 Regularity of quantum groups
{ Theorem 3. The quantized enveloping 6 –algebra associated to a Cartan matrix { is Auslander-regular and Cohen-Macaulay.
¡ m* Proof: De Concini, Kac and Procesi gave an ¥ –filtration on
{
such that the associated graded ring satisfies the hypotheses of Theorem 2.
Refiltering for some noetherian rings – p.10/12 AR algebras
Let a finitely generated module over a noetherian ring . The grade number of is defined as 4
¤
¢¡ ` p £ } I¦¥ © © ; ¥ ` We say that satisfies the Auslander condition if for every § and 4
¨ ¨ p ` § £ ¡ ; ; every submodule of ¥ , one has that . When the global homological dimension of is finite and every finitely generated module satisfies the Auslander condition, is said to be Auslander regular.
Refiltering for some noetherian rings – p.11/12 CM algebras
For an algebra over a field, we can define the Gelfand-Kirillov dimension of every module. The algebra is said to be Cohen-Macaulay if
© © B B ª ª p I A A © ;
for every finitely generated module . «
Refiltering for some noetherian rings – p.12/12