Quasi-Bigraduations of Modules, Slow Analytic Independence

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1 International Mathematical Forum, Vol 13, 2018, no 12, HIKRI Ltd, wwwm-hikaricom Quasi-Bigraduations of Modules, Slow nalytic Independence Youssouf M Diagana Université Nangui brogoua, UFR-SF Laboratoire Mathématiques-Informatique 02 BP 801 bidjan 02, Côte d Ivoire Copyright c 2018 Youssouf M Diagana This article is distributed under the Creative Commons ttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited bstract When f = ( I (m,n (m,n (Z 2 { } is a + quasi-bigraduation on a ring R, a f + - quasi-bigraduation of an R module M is a family g = (G (m,n (m,n (Z 2 { } of subgroups of M such that G = (0 and I (m,n G (p,q G (m+p,n+q, for all (m, n and (p, q (N N { } We will show that r elements of R are slowly J independent of order k with respect to a + quasi-bigraduation g on an R module M if and only if the two property which follow hold: they are J independent of order k with respect to the + quasi-bigraduation f 2 (, I and there exists a relation of compatibility between the + quasi-graduation on R deduced from g and g I where I is the sub- module of R constructed by these elements We give criteria of slow J independence of + quasi-bigraduations on an R module in term of isomorphisms of graded algebras Mathematics Subject Classification: 1302, 1315, 1330, 13B25, 13F20 Keywords: Quasi-bigraduations, modules, Generalized analytic independence Introduction ll rings are supposed to be commutative and unitary In 1954, D G Northcott and D Rees [7] developed a theory of integral closure and reductions of ideals in a noetherian local ring (, M In particular, they

2 548 Youssouf M Diagana introduced two notions of analytic independence with respect to an ideal in a local ring and they proved that the reduction of an ideal in such a ring is minimal if and only if it has an analytically independent generating set In 1970 one notion of independence is generalized by Valla [9] in a noetherian commutative ring He showed that the maximum number of independent elements in an ideal is bounded from above by its height Let f = (I n n Z {+ } be a filtration of an arbitrary commutative ring and R (, f = n N I n X n and R (, f = n Z I n X n be its Rees rings Let k be a positive integer which may be equal to + and let J be an ideal of such that J + I k Take u = X 1 Then the following numbers are known in the literature to be extensions to filtrations of the analytic spread - the maximum number l J (f, k of elements of the ideal J which are J independent of order k with respect to f and - the maximum number l a J (f, k of elements of the ideal J which are regularly J independent of order k with respect to f That work generalized results { of Okon [8] concerning the analytic } spread R (, f of noetherian filtrations, sup dim (u, M R (, f, M Max and established comparisons of several extensions In [4] we studied theses notions for a + quasi-graduation of a ring R We say that the family (G n of subgroups of R is a quasi-graduation (resp + quasi-graduation of R if G 0 is a subring of R, G + = (0 and G p G q G p+q p, q Z (resp N Here we need the following concept of compatibility of a family of subgroups of R with a given quasi-graduation (resp + quasi-graduation f of R and we extend this concept to quasi-bigraduations: Definition 1 Let R be a ring 1 Let f = (I n n Z {+ } be a family of subgroups of R We say that f is a quasi-graduation (resp + quasi-graduation of R if I 0 is a subring of R, I = (0 and I p I q I p+q for all p and q Z (resp N 2 Let f = (I n n Z {+ } be a quasi-graduation (resp + quasi-graduation of R, M be an R module and g = (G i i Z {+ } be a family of subgroups of M We say that g is an f + -quasi-graduation of M or that g is a + quasigraduation of M compatible with f if G = (0 and I p G q G p+q for each p and q N For a ring R, the construction of rings of polynomials R [X 1,, X n ] of n indeterminates with coefficients in R have a critical importance, since geometrical objects (curves, surfaces, etc are described by equations in several variables

3 Quasi-bigraduations of modules, slow analytic independence 549 Otherwise, in Database, stored values must be accessible concurrently but consistently by multiple users This proves the importance of the Cartesian product in relational lgebra used in Relational Database Hence there is an interest to replace N by N N as set of indices We define the compatible + quasi-bigraduation of a ring as follow: Definition 2 Let R be a ring Let f = ( I (m,n be a family of (m,n (Z Z { } subgroups of R with the convention that I (p,, I (,q and I (, mean the same subgroup, denoted I Let us construct the family (S m m Z {+ } as following : S m = m 0; S = (0 and S m = I (n,m n m 0 m n 2m 1 We say that f is a quasi-bigraduation (resp + quasi-bigraduation of R if I (0,0 is a subring of R, I = (0 and I (p,q I (r,s I (p+r,q+s (p, q and (r, s Z 2 (resp N 2 2 If f is a quasi-bigraduation (resp + quasi-bigraduation of R then the family (S m m Z {+ } is a + quasi-graduation of R; it is called the + quasigraduation of R deduced from f In [6] we studied the notion of generalized analytic independence for a + quasi-bigraduation of a ring R In this paper we have two objectives : - To give a slow concept of generalized analytic independence of compatible + quasi-bigraduations of a module, - to establish some characterizations of generalized analytic independences of compatible + quasi-bigraduations of a module by the mean of isomorphisms of graded algebras 1 Compatible quasi-bigraduations of module We define the compatible + quasi-bigraduation of a module as follow Let be the abelian monoid Z 2 { } (resp N 2 { } = + Let R be a ring, be a subring of R and M be an R-module 11 Compatible quasi-bigraduations of module Definition 3 Let f = ( I (m,n (m,n be a + quasi-bigraduation of R Let H = ( G (i,j be a family of subgroups of M with the convention (i,j that G (p,, G (,q and G (, mean the same subgroup, denoted G We say that H is an f quasi-bigraduation (resp f + -quasi- bigraduation of M or that H is a quasi-graduation (resp a + quasi-graduation over of M compatible with f if G = (0 and I (p,q G (m,n G (p+m,q+n for each (m, n and (p, q

4 550 Youssouf M Diagana 12 Globally compatible quasi-bigraduations of module Let H = ( G (i,j (i,j be a family of sub--modules of M such that G = (0 Let us construct the family (N m m Z {+ } of sub--modules of M as following: N m = G (0,0 m 0; N = (0 and N m = G (n,m n m 0 m n 2m Definition 4 Let f = (I (s,t (s,t be a + quasi-bigraduation of M Let = I (0,0 and S = (S m be the quasi-graduation deduced from f (see Definition 2 Let H = ( G (i,j be a family of sub--modules of R (i,j We say that H is a + quasi-bigraduation of M globally compatible with f or that H is a global f + -quasi-bigraduation of M if G = (0 and S p N q N p+q for each p and q N Remark that if H is an f + -quasi-bigraduation of M then H is a + quasibigraduation of M globally compatible with f If H is a global f + -quasi-bigraduation of M then one denotes QG(H = (N m which is called the S + -quasi-graduation of M deduced from H 2 Compatible quasi-bigraduations of module and generalized nalytic Independence 21 Slowly generalized analytic independence Let R be a ring, be a subring of R and M be an R-module Definition 5 Let a 1,, a r be elements of R and let I be the sub--module of R that they generate Put f 2 (, I the + quasi-bigraduation (I (m,n of R such that I (j1,j 2 = if j 1 + j 2 0 I = (0 and I (j1,j 2 = I d if j 1 + j 2 = d 0 Suppose that H = ( G (i,j is a global (f2 (, I + -quasi-bigraduation of M (ie, for all m N, N m is a submodule of the module M and I (N m N m+1 Put G m = [ N m : G (0,0 ]R for all m N (G m is an S + -quasi-graduation of R One denotes QGT(H = (G m m Z {+ } which is called the S + -quasi-graduation of R transported from H Let k N and J be an ideal of 1 If JG (0,0 + N k G (0,0 G (0,0, the elements a 1,, a r of R are said to be J independent of order k with respect to H if for any homogeneous polynomial F of degree d in r indeterminates with coefficients in G (0,0, the

5 Quasi-bigraduations of modules, slow analytic independence 551 relation F (a 1,, a r JN d + N d+k implies that F has all of its coefficients in JG (0,0 + N k 2 If J + G k, the elements a 1,, a r of R are said to be slowly (J, independent of order k with respect to H if for any homogeneous polynomial F of degree d in r indeterminates with coefficients in, the relation F (a 1,, a r JG d + G d+k implies that F has all of its coefficients in J + G k 3 If k = + the term of order + can be omit Remarks Elements a 1,, a r of R are J independent of order k with respect to H iff they are J independent of order k with respect to the deduced S + -quasi-graduation (N m m Z {+ } defined in section 12 2 Elements a 1,, a r of R are slowly (J, independent of order k with respect to H iff they are J independent of order k with respect to the transported S + -quasi-graduation (G m m Z {+ } Proposition 212 Let a 1,, a r be elements of R and I be the sub-module of R that they generate Let H = ( G (m,n (m,n be a global (f 2 (, I + - quasi-bigraduation of M Let k N and J be an ideal of such that J + G k Suppose that a 1,, a r are slowly (J, independent of order k with respect to H If J G k then the elements a 1,, a r are J independent (resp slowly ( (J, independent (of order + with respect to H and to f 2 (, I = I(m,n Proof Let x = F (a 1,, a r where F is an homogeneous polynomial of degree s in r indeterminates and with coefficients in Suppose that J contains G k Put I m = I m We have [x 0 (JG s or x 0 (JI s ] x JG s +G s+k F (J + G k [X 1,, X r ] Furthermore, J + G k = J thus the elements a 1,, a r are slowly (J, independent (of order + with respect to H and with respect to f 2 (, I Proposition 213 Let k N and J be an ideal of, a 1,, a r be elements of R and I be the sub--module of R that they generate Let H = ( G (i,j (i,j be a global (f 2 (, I + -quasi-bigraduation of M Suppose that J +G k, (G i+k i 0 (or (N i+k i 0 is decreasing and that a 1,, a r are slowly (J, independent of order k with respect to H Let p 1 be an integer If G k J + G pk then the elements a p 1,, a p r are slowly (J, independent of order k with respect to the global f 2 (, I p + -quasi-bigraduation H (p = (G (pm,pn (m,n Proof This is the consequence of the fact that under the hypotheses we have { n 0 JGpn + G p(n+k JG np + G np+k and J + G k J + G pk

6 552 Youssouf M Diagana Indeed, let F be an homogeneous polynomial of degree n in r indeterminates and with coefficients in F (a p 1,, a p r ( JG pn + G p(n+k G (a1,, a r ( JG pn + G p(n+k where G (X 1,, X r = F (X p 1,, X p r is homogeneous of degree np Thus G (a 1,, a r JG pn + G (pn+pk JG np + G np+k therefore G and F have their coefficients in J + G k F has all of its coefficients in (J + G k = J + G k J + G pk 22 Criteria of J independences : 221 Preliminaries Let (I n and (J n be two families of subgroups of R such that 1 I 0 J ( n I n m, n Z (resp N I n I m I n+m I n J m J n+m Let (P n and (K n be two families of subgroups of M such that K n P n ( I n P m P n+m m, n Z (resp N J n P m + I n K m K n+m Let P = m P m X m and K = m K m X m We have P K m P m K m Let (K n be a family of subgroups of R such that { K ( n I n m, n Z (resp N I m J n + I n K m K m+n Let P = m I m X m and K = m K m X m We have P K m I m K m 222 Construction of surjective morphisms relating to elements of the ring Let f = (I n n Z {+ } be a quasi-graduation of R and = I 0 Let J be an ideal of and J n = I n (JI n + I n+k for all n of R such that Denote f f the quasi-bigraduation ( U (i,j (i,j (Z Z { } U (m,n = I m I n for each (m, n Z Z and U = (0 Let a 1,, a r be elements of R and I be the sub--module of R that they generate Suppose that I n = I n for all n > 0 and I n = for all n 0 Put s i = a i + J 1 i = 1,, r There exists an isomorphism

7 ψ 1,k : n 0 Quasi-bigraduations of modules, slow analytic independence 553 I n J n R(, I R(, I ((u k, J R (, I such that ψ 1,k (s i = a i X+R(, I (( u k, J R (, I and ψ 1,k (α = α for α Furthermore, we have the products : For all α and b m I m if m = i i r then s i 1 1 s ir r = ( a i 1 r + J m and (bm + J m (α + J 0 = b m α + J m Hence s i 1 1 s ir r (α + J 0 = a i 1 r α + J m Let v i = ψ 1,k (s i i = 1,, r v i 1 1 vr ir = a i 1 r X m + ( R(, I (( u k, J R (, I and v i 1 1 vr ir (α + J 0 = a i 1 r αx m + ( R(, I (( u k, J R (, I and we have the following commutative diagram: J 0 [X 1,, X r ] ϕ 1,k S J (I, k = J 0 [s 1,, s r ] = J + I k I m J m θ 1,k ψ 1,k V J (I, k = J 0 [v 1,, v r ] 223 Surjective morphisms relating a global (f f + -quasi-bigraduation I Let H = ( G (i,j (i,j (Z Z { } be a global (f f+ -quasi-bigraduation of M, J m = I m G (0,0 ( JS m G (0,0 + S m+k G (0,0 m 0 and K m = ( I m G (0,0 (JNm + N m+k m 0 where S = (S m m Z {+ } = QG (f f is the + quasi-graduation of R deduced from f f and (N m m Z {+ } = QG (H is the S + -quasi-graduation of M deduced from H Put J = J m X m and T = S d G (0,0 X d m d a Consider N = N d X d and Q J (H, k the graded Q J (f, k module d I m G (0,0, where Q J (f, k = K m n 0 I n J n I = ( I m G (0,0 m Z {+ } is an f + -quasi-graduation of R and we have K m I m G (0,0 S m G (0,0 N m b For each m 0, put K m = I m (JG m + G m+k where (G m m Z {+ } = QGT (H is the S + -quasi-graduation of R transported from H

8 554 Youssouf M Diagana Consider G = G d X d and T J (H, k the graded Q J (f, k module I m d 0 K m We have K m I m S m G m Conditions (, ( and ( of 221 are satisfied for (I n, (J n, (P n and (K n, (resp ( and ( for (I n, (J n and (J n Hence we have I m G P (0,0 X m K = m [( Im G (0,0 (JNm + N m+k ] X m I m G (0,0 ( Im G (0,0 (JNm + N m+k and P K = I m X m [I m (JG m + G m+k ] X m I m I m (JG m + G m+k II Suppose that for each m 0, I m = I m Thus, for m 0 we have S m = I m, J m = I m G (0,0 ( JI m G (0,0 + I m+k G (0,0, Km = I m G (0,0 (JN m + N m+k and Q J (H, k is the graded Q J (f, k module I m G (0,0, where K m (N m m Z {+ } = QG (H is the S + -quasi-graduation of M deduced from H a Suppose that JG (0,0 +N k G (0,0 (resp JG (0,0 + ( I k G (0,0 G(0,0 is different from G (0,0 We have I m G (0,0 X m [ I m G (0,0 (JN m + N m+k ] X m I m G (0,0 I m G (0,0 (JN m + N m+k (resp I m G (0,0 X m [ I m G (0,0 (JI m + I m+k G (0,0 ] X m Put R ( G (0,0, I the graded -module n 0 R ( G (0,0, I [( ] u k, J N = d 0 I m G (0,0 I m G (0,0 ( (JI m + I m+k G (0,0 I n G (0,0 X n We have = R ( G (0,0, I [( ] [ I d G (0,0 (JN d + N d+k u k, J ] + N X d = d 0 K d X d (resp R ( G (0,0, I [( u k, J = d 0 R ( G (0,0, I ] = R ( G (0,0, I [( [ I d G (0,0 u k, J R ( G (0,0, I ] + ((JI d + I d+k ] G (0,0 X d = J d X d d 0 We have J + G k, JG (0,0 + ( I k G (0,0 G(0,0 G (0,0 and J + I k (resp J + I k

9 Quasi-bigraduations of modules, slow analytic independence 555 b Suppose that J + I k or that J + G k Then we have I m X m m I m (JG m + G m+k X m I m I m (JG m m + G m+k m Put R (, I the Rees ring of the ideal I We have [( ] R (, I u k, J G = [ ] I d (JG d + G d+k X d = K d X d d 0 d 0 III Let us define the products as follows : a For all α G (0,0 and b m I m if m = i i r then ( s i 1 1 s ir r = a i 1 r + J m and (b m + J m (α + = b m α + K m where = JG (0,0 + N k G (0,0 Furthermore, Hence s i 1 1 s ir r (α + = a i 1 r α + K m v i 1 1 v ir r = a i 1 r X m + R(, I (( u k, J R (, I (( v i 1 1 vr ir (α + J 0 = a i 1 1 ar ir αx m + R(G (0,0, I u k, J R ( G (0,0, I b For all α and b m I m if m = i i r then ( s i 1 1 s ir r = a i 1 r + J m and (b m + J m (α + = b m α + K m where = J + G k Hence s i 1 1 s ir r (α + = a i 1 r α + K m and Properties (, ( and ( of 221 show that the previous products are well defined IV a Suppose that N k G (0,0 JG (0,0 + ( I k G (0,0 G(0,0 G (0,0 Then = J 0 G (0,0, I m X m = I m J 0 X m R(G (0,0, I (( u k, J R ( G (0,0, I and the next product is well defined: For all α G (0,0 v i 1 1 v ir r (α + = a i 1 r αx m + R(G (0,0, I (( u k, J and R ( G (0,0, I Put S J (H, k = G (0,0 [s 1,, s r ] = (α i1,,i r + s i1 1 s ir r : α i1,,i r G (0,0 i 1,,i r and V J (H, k = G (0,0 [v 1,, v r ] We have R ( G (0,0, I R ( G (0,0, I (u k, J R ( G (0,0, I = G (0,0 [v 1,, v r ]

10 556 Youssouf M Diagana Let ϕ 1,k = ϕ 1,J (H, k be the graded morphism of graded modules from G (0,0 [X 1,, X r ] onto G (0,0 [s 1,, s r ] such that ϕ 1,k (αx i = αs i for each i and ϕ 1,k (α = α for α G (0,0 and α = α + There exists an isomorphism ψ 1,k of graded modules from S J (H, k onto R ( G (0,0, I V J (H, k = R ( G (0,0, I [ (u k, J R ( ] such that G (0,0, I ψ 1,k (αs i = αv i = αa i X + R ( G (0,0, I [( u k, J R ( G (0,0, I ] and ψ 1,k (α = α for α G (0,0 and α = α + Put θ1,k = ψ 1,k ϕ 1,k Hence the following diagram commutes: G (0,0 [X 1,, X r ] ϕ 1,k G (0,0 [s 1,, s r ] θ 1,k ψ1,k R(G (0,0, I R(G (0,0, I (u k, JR(G (0,0, I b Suppose that G k J + I k and that J + I k Then = J + I k = J 0 and I m X m R(, I (( u k, J R (, I Hence we have the commutative diagram of 222 Proposition 221 a If JG (0,0 + ( I k G (0,0 G(0,0 G (0,0 then the following statements are equivalent: (i a 1,, a r are J independent of order k with respect to f 2 ( G(0,0, I (ii ϕ 1,k is an isomorphism of G (0,0 [X 1,, X r ] over G (0,0 [s 1,, s r ] J 0 J 0 (iii θ R ( G (0,0, I 1,k is an isomorphism over R ( G (0,0, I (u k, J R ( G (0,0, I (iv The elements s 1,, s r are algebraically independent over G (0,0 J 0 (v The elements v 1,, v r are algebraically independent over G (0,0 J 0 a If N k G (0,0 JG (0,0 + ( I k G (0,0 G(0,0 and JG (0,0 + ( I k G (0,0 G(0,0 is different from G (0,0 then replacing J 0 by we have a result similar to a b If G k J + I k then the following statements are equivalent : (i a 1,, a r are slowly (J, -independent of order k with respect to f 2 (, I (resp f I (ii ϕ 1,k is an isomorphism of [X 1,, X r ] over [s 1,, s r ]

11 Quasi-bigraduations of modules, slow analytic independence 557 (iii θ 1,k is an isomorphism of [X 1,, X r ] over R (, I R (, I (u k, J R (, I (iv The elements s 1,, s r are algebraically independent over (v The elements v 1,, v r are algebraically independent over Proof See Theorem 241 of [6] V a With the assumption that N k G (0,0 JG (0,0 + ( I k G (0,0 G(0,0 G (0,0 put the canonical morphisms V J (H, k = R ( G (0,0, I R ( G (0,0, I [ (u k, J R ( G (0,0, I ] θ2,k R ( G (0,0, I R ( G (0,0, I [(u k, J N] and δ k = θ 2,k ψ 1,k We have for α = α + (( v i α = a i αx + R(G (0,0, I u k, J R ( G (0,0, I = ψ 1,k (s i α where α G (0,0 Thus δ k (s i α = θ [( ] 2,k (v i α = a i αx + R(G (0,0, I u k, J N Put G 0 = G (0,0 The following diagram commutes: G 0 [X 1,, X r ] ϕ 1,k G 0 [s 1,, s r ] = I m G 0 J m ψ1,k δ k θ 1,k δ k θ 1,k G 0 [v 1,, v r ] = θ k R(G 0, I R(G 0, I (u k, JR(G 0, I θ 2,k R(G 0, I R(G 0, I (u k, JN b Suppose that G k J + I k Put the canonical morphisms V J (I, k = R (, I R(, I ((u k, J R (, I θ 2,k R (, I R (, I [(u k, J G] and δ k = θ 2,k ψ 1,k With v i = ψ 1,k (s i = a i X + R(, I (( u k, J R (, I we have v i α = a i αx + ( R(, I (( u k, J R (, I = ψ 1,k (s i α for α and α = α + Thus [( ] δ k (s i = θ 2,k (v i = a i X + R (, I u k, J G and [( ] ( [( ] δ k (s i α = θ 2,k (v i α = a i αx+r (, I u k, J G = a i X + R (, I u k, J G α

12 558 Youssouf M Diagana Put u i = θ 2,k (v i and U J (I, k = [u 1,, u r ] = (α i1,,i r + u i1 1 u ir r : α i1,,i r i 1,,i r We have R (, I R (, I (u k, J R (, I = [v 1,, v r ] Put ϕ 1,k = ϕ 1,J (I, k and θ 1,k = ψ 1,k ϕ 1,k The following diagram commutes: [X 1,, X r ] ϕ 1,k [s 1,, s r ] = I m J m ψ 1,k θ 1,k θ 1,k δ k δ k [v 1,, v r ] = θ k R(, I R(, I (u k, JR(, I θ 2,k R(, I R(, I (u k, JG 224 Properties of independence Under the previous hypotheses we show the following results as in [6]: Theorem 222 Under the notations and hypotheses of 223 and with the assumption that G k J + I k the following assertions are equivalent : (i a 1,, a r are slowly (J, independent of order k with respect to H { a a1,, a r are J independent of order k with respect to f 2 (, I and (ii b I p (JG p + G p+k = JI p + (I p+k I p p 0 a The family {s (iii 1,, s r } is algebraically free over and K 0 b R (, I [( u k, J G ] = R (, I ( u k, J R (, I a θ 1,k is an isomorphism from R (, I [X 1,, X r ] onto K (iv 0 R (, I (u k, J R (, I and R (, I b θ 2,k is an isomorphism from R (, I (u k, J R (, I onto R (, I R (, I [(u k, J G] a ϕ 1,k is an isomorphism from [X 1,, X r ] onto [s 1,, s r ] and (v b δ k is an isomorphism from R (, I [s 1,, s r ] onto R (, I [(u k, J G] (vi θ k = θ 2,k θ 1,k is an isomorphism from R (, I [X 1,, X r ] onto R (, I [(u k, J G]

13 Quasi-bigraduations of modules, slow analytic independence 559 Proof (ii (iii (iv (v By Proposition (221 ( b the elements a 1,, a r are J independent of order k with respect to f 2 G(0,0, I iff the family {s 1,, s r } is algebraically free over J + G k It is equivalent to the fact that θ 1,k (resp ϕ 1,k is an isomorphism Moreover, I p (JG p + G p+k = JI p + (I p+k I p p 0 if and only if R (, I [( u k, J G ] = R (, I ( u k, J R (, I if and only if θ 2,k (resp δ k is a graded ring isomorphism (iv (vi We have: θ 1,k and θ 2,k are surjective and θ k = θ 2,k θ 1,k Therefore θ k is an isomorphism if and only if both θ 1,k and θ 2,k are isomorphisms (i (vi s in Theorem 241 of [6], [the elements a 1,, a r are J independent of order k with respect to H] iff θ k = θ 2,k θ 1,k is an isomorphism from R (, I [X 1,, X r ] onto R (, I [(u k, J G] Corollary 223 If G k J +I k then the following assertions are equivalent: (i a 1,, a r are slowly (J, independent of order k with respect to H a a 1,, a r are J independent of order k with respect to f 2 (, I (ii R (, I b δ k is an isomorphism of [s 1,, s r ] over R (, I [(u k, J G] a ϕ 1,k is an isomorphism of [X 1,, X r ] over [s 1,, s r ] (iii R (, I b δ k is an isomorphism of [s 1,, s r ] over R (, I [(u k, J G] R (, I a θ 1,k is an isomorphism of [X 1,, X r ] over K (iv 0 R (, I (u k, J R (, I R (, I b θ 2,k is an isomorphism of R (, I (u k, J R (, I over R (, I R (, I [(u k, J G] a The elements s (v 1,, s r are algebraically independent over (vi b R (, I [( u k, J G ] = R (, I ( u k, J R (, I a The elements v 1,, v r are algebraically independent over b R (, I [( u k, J G ] = R (, I ( u k, J R (, I pplying this result to case k = + we have the following Corollary: Corollary 224 The following assertions are equivalent : (i a 1,, a r are slowly (J, independent (of order + with respect to H (ii The elements a 1,, a r are J independent with respect to f 2 (, I and I p (JG p = JI p for all p 0 (iii The family {s 1,, s r } (resp {v 1,, v r } is algebraically free over J R (, I JG = JR (, I and

14 560 Youssouf M Diagana 225 J independence and isomorphisms We have the equivalent of Theorem 222 for independence of a compatible quasibigraduation of module: Theorem 225 Under the notations and hypotheses of 223 and with the assumption that N k G 0 JG 0 + (I k G 0 G 0 the following assertions are equivalent: (i The elements a 1,, a r are J independent of order k with respect to H { a a1,, a r are J independent of order k with respect to f 2 (G 0, I and (ii b I p G 0 (JN p + N p+k = JI p G 0 + (I p+k I p G (0,0 p 0 a The family {s (iii 1,, s r } is algebraically free over G 0 and K 0 b R (G 0, I [( u k, J N ] = R (G 0, I ( u k, J R (G 0, I a θ R (G 0, I 1,k is an isomorphism over R (G (iv 0, I (u k and, J R (G 0, I b θ R (G 0, I 2,k is an isomorphism over R (G 0, I [(u k, J N] (v a ϕ 1,k is an isomorphism from G 0 [X 1,, X r ] onto G 0 b δ k is an isomorphism from G 0 [s 1,, s r ] onto [s 1,, s r ] and R ( G (0,0, I R (G 0, I [(u k, J N] (vi θ k = θ 2,k θ 1,k is an isomorphism from G 0 R (G 0, I [X 1,, X r ] onto R (G 0, I [(u k, J N] 226 Other properties of independence With the assumption that N k G (0,0 JG (0,0 + ( I k G (0,0 G(0,0 G (0,0 let t i = a i + K 1 = a i + I (J G 1 + G 1+k for i = 1,, r Put T J (H, k = [t 1,, t r ] = (α i1,,i r + t i1 1 t ir r : α i1,,i r i 1,,i r We have t i I I (J G 1 + G 1+k and R (, I R (, I [(u k, J G] m I m I m (JG m + G m+k = T J (H, k S J (I, k = [s 1,, s r ] ϕ 2,k T J (H, k = [t 1,, t r ] Let ϕ J (H, k be the graded morphism from [X 1,, X r ] onto T J (H, k such that ϕ J (H, k (X i = t i for each i and ϕ J (H, k (α = α for α There exists an isomorphism ψ k from T J (H, k onto R (, I R (, I [(u k, J G] that ψ k (t i = a i X + R (, I [( u k, J G ] and ψ k (α = α for α We have such

15 Quasi-bigraduations of modules, slow analytic independence 561 u i = θ 2,k (v i = ψ k (t i for all i Put ϕ k = ϕ J (H, k and θ k = ψ k ϕ k Thus R (, I R (, I [(u k, J G] = [u 1,, u r ], θ 2,k ψ 1,k = ψ k ϕ 2,k and ϕ k = ϕ 2,k ϕ 1,k Hence the following diagrams commute: [X 1,, X r ] ϕ k [t 1,, t r ] θ k [u 1,, u r ] θ k = ψ k R(, I R(, I (u k, JG ϕ k [X 1,, X r ] ϕ 1,k [s 1,, s r ] ϕ 2,k [t 1,, t r ] ψ 1,k θ 1,k θ 1,k δ k ψ k [v 1,, v r ] = θ k R(, I R(, I (u k, JR(, I Under these assumptions, we have the following Theorem as in [3]: θ 2,k R(, I R(, I (u k, JG Theorem 226 (see 124 of [3] The following statements are equivalent : (i a 1,, a r are slowly (J, independent of order k with respect to H (ii ϕ k is an isomorphism of J + G k [X 1,, X r ] over J + G k [t 1,, t r ] (iii θ k is an isomorphism of J + G k [X R (, I 1,, X r ] over R (, I [(u k, J G] (iv The elements t 1,, t r are algebraically independent over J + G k (v The elements u 1,, u r are algebraically independent over J + G k

16 562 Youssouf M Diagana 227 Particular case of global quasi-graduations of a ring Let R be a ring and be a subring of R Let a 1,, a r be elements of R, I be the sub--module of R that they generate, f = (I n n Z {+ } be the quasi-graduation of R with I n = for all n 0 and J n = I n ( JI n + I n+k for all n where J is an ideal of Denote f f the quasi-bigraduation ( U (i,j of R such that U(m,n = I m I n for each (m, n Z Z and U = (0 Suppose that H = ( G (i,j is a global f f + quasi-bigraduation of R and (N m = QG (H is the + quasi-graduation of R deduced from H Hence G m = N m and for = G (0,0, the elements a 1,, a r of R are J independent of order k with respect to H iff they are slowly (J, independent of order k with respect to H Moreover, considering the equalities f = f everywhere, G i = N i, = K and N = G, Theorems 222 and 225 coincide Example 227 Let R = R [X, Y ] be the ring of polynomials of two indeterminates X and Y with coefficients in R, = Z [X, Y ] and M = iq [X, Y ] Let f = ( I (m,n, where I(m,n = (X n Y m Z [X, Y ] for all m, n N Let H = ( G (p,q such that G(p,q = (X p Y q iq [X, Y ] for all (p, q N N f is a quasi-bigraduation of R and a bifiltration of R H is an f + -quasi-bigraduation of M Let a 1 = X and a 2 = Y, J = (X, Y Z [X, Y ] We have N d = ( X d, X d 1 Y, X d 2 Y 2,, Y d iq [X, Y ] N 1 = (X, Y iq [X, Y ] and N 0 = G (0,0 = iq [X, Y ] 1 Let F = α i,j X1 ixj 2 [X 1, X 2 ] an homogeneous polynomial of degree d i+j=d F (a 1, a 2 = α i,j X i Y j JN d i+j=d α i,j X i Y j (X, Y ( X d, X d 1 Y, X d 2 Y 2,, Y d iq [X, Y ] i+j=d α i,j N 1 = (X, Y iq [X, Y ] = (X, Y iz [X, Y ] Q [X, Y ] = JG (0,0 Thus α i,j (JN 0 G (0,0 = JG (0,0 Therefore, X and Y are J independent with respect to H We have: X 2 and Y 2 are J independent with respect to H 2 Put a 3 = XY and F (X 1, X 2 = iy X 1 ix 2 = α 1 X 1 + α 2 X 2 where α 1 = iy and α 2 = i Let F = iy X 1 ix 2 N 0 [X 1, X 2 ] F is an homogeneous polynomial of degree 1 F (a 1, a 3 = iy a 1 ia 3 = iy X ixy = 0 JN 1 But α 2 = i / JG (0,0 Therefore, X and XY aren t J independent with respect to H Example 228 Let R = R [X, Y ] be the ring of polynomials of two indeterminates X and Y with coefficients in R, = Z [X, Y ] and M = iq [X, Y ] Let f = ( I (m,n, where I(m,n = (X n Y m Z [X, Y ] for all m, n N f is a quasi-bigraduation of R and a bifiltration of Let H be the family ( G (p,q such that * G (p,q = (X p Y q iq [X, Y ] for all (p, q N N * G (p,q = (Y p+q iq [X, Y ] for all (p, q Z Z with q p < 0 * G (p,q = (X p+q iq [X, Y ] for all (p, q Z Z with p q < 0 * G (p,q = iq [X, Y ] for all (p, q Z Z with p 0 and q 0

17 Quasi-bigraduations of modules, slow analytic independence 563 Let a 1 = X and a 2 = Y, J = (X, Y N d = ( X d, X d 1 Y, X d 2 Y 2,, Y d iq [X, Y ], N 1 = (X, Y iq [X, Y ] and N 0 = G (0,0 = iq [X, Y ] (N m is decreasing H is an f + quasi-bigraduation of M α i,j G (0,0 = iq [X, Y ] Put k = + Put b 1 = XY, b 2 = X 2 and b 3 = Y 2 We have: b 2 1 b 2b 3 = 0 Let F (X 1, X 2, X 3 = ix 2 1 ix 2X 3 F (X 1, X 2, X 3 = α 1 X α 2X 2 X 3 G (0,0 [X 1, X 2, X 3 ] is an homogeneous polynomial of degree 2, where α 1 = i and α 2 = i F (b 1, b 2, b 3 = 0 JN 1 but α 1 = i / JG (0,0 Therefore, X 2, XY and Y 2 aren t J independent (with respect to H and to f References [1] PK Brou and YM Diagana, Quasi-graduations of modules and extensions of analytic spread, frika Matematika, 28 (2017, [2] Y Diagana, H Dichi and D Sangaré, Filtrations, Generalized analytic independence, nalytic spread, frika Matematika, Série 3, 4 (1994, [3] YM Diagana, Regular analytic independence and Extensions of nalytic spread, Communications in lgebra, 30 (2002, no 6, [4] YM Diagana, Quasi-graduations of Rings, Generalized nalytic Independence, Extensions of the nalytic Spread, frika Matematika, Série 3, 15 (2003, [5] YM Diagana, Quasi-graduations of Rings and Modules, Criteria of Generalized nalytic Independence, nnales Mathématiques fricaines, 3 (2012, [6] YM Diagana, Quasi-Bigraduations of Rings, Criteria of Generalized nalytic Independence, International Journal of lgebra, 12 (2018, no 5, [7] DG Northcott and D Rees, Reductions of ideals in local rings, it Mat Proc Cambridge Philos Soc, 50 (1954, [8] JS Okon, Prime divisors, analytic spread and filtrations, Pacific Journal of Mathematics, 113 (1984, no 2, [9] G Valla, Elementi independenti rispetto ad un ideale, Rend Sem Mat Univ Padova, 44 (1970, Received: October 29, 2018; Published: December 18, 2018