aa r X i v : . [ m a t h . K T ] F e b A NOTE ON RELATIVE VASERSTEIN SYMBOL
KUNTAL CHAKRABORTY Introduction
In an unpublished work of Fasel-Rao-Swan in [6] the notion of the relative Witt group W E ( R, I ) is defined.In this article we will give the details of this construction:Let I be an ideal of R . We denote the set { α ∈ Alt ′ n ( R ) : α ≡ χ ± r ⊥ χ ± n − r (mod I ) for some r ≥ } by Alt ′ n ( R, I ) , where Alt ′ n ( R ) is the set of all skew-symmetric matrices in GL n ( R ) and χ = − ! and χ r +1 = χ r ⊥ χ for r ≥ . For m ≤ n , Alt ′ m ( R, I ) can be embedded into Alt ′ n ( R, I ) by the inclusion map i m,n : Alt ′ m ( R, I ) ֒ → Alt ′ n ( R, I ) given by α α ⊥ χ n − m . We put Alt ′ ( R, I ) = lim −→ Alt ′ n ( R, I ) . Wecan define an equivalence relation ∼ I on Alt ′ ( R, I ) as follows:Let α ∈ Alt ′ m ( R, I ) and β ∈ Alt ′ n ( R, I ) . Then α ∼ I β if and only if there exists t ∈ N and ǫ ∈ E m + n + t ) ( R, I ) such that : α ⊥ χ n + t = ǫ T ( β ⊥ χ m + t ) ǫ We denote the set
Alt ′ ( R, I ) / ∼ I by W ′ E ( R, I ) .We can also define the set Alt n ( R, I ) as the subset of Alt ′ n ( R, I ) consisting of skew-symmetric matricesof Pfaffian . We put Alt ( R, I ) := lim −→ Alt n ( R, I ) . As the relation ∼ I defines an equivalence relationon Alt ′ ( R, I ) , then ∼ I defines an equivalence relation on Alt ( R, I ) . We denote the set Alt ( R, I ) / ∼ I by W E ( R, I ) .This definition is compatible with the definition of the symplectic Witt group introduced by Vaserstein in[26]. This definition can also be realised as a kernel of a certain map between two Witt groups, namely thefollowing sequence is exact: W E ( R, I ) W E ( D ) W E ( R ) 0 i ( p ) ∗ where D is the double ring of R with respect to the ideal I and p is the first projection map.Likewise in [26], we can consider the relative Vaserstein symbol , namely, the map V R,I : U m ( R, I ) /E ( R, I ) → W E ( R, I ) . In [26], Vaserstein proved that for a commutative ring R of dimension , the Vaserstein symbol V R : U m ( R ) /E ( R ) → W E ( R ) is bijective.In [20], Rao-van der Kallen proved that the Vaserstein symbol V R : U m ( R ) /E ( R ) → W E ( R ) is bijectivefor R smooth affine algebra of dimension over a C - field k which is perfect if its characteristic is or . Also in [7], Fasel-Rao-Swan proved that the Vaserstein symbol V R : U m ( R ) /E ( R ) → W E ( R ) isinjective for R smooth affine algebra of dimension over an algebraically closed field.It is believed that these results can be generalised in relative cases. In fact we have: Theorem 1.1. ( See Corollary 5.10 ) Let R be a commutative ring of dimension and I ⊂ R be an ideal of R .Then the Vaserstein symbol V R,I : U m ( R, I ) /E ( R, I ) → W E ( R, I ) is bijective. Theorem 1.2. ( See Theorem 5.24 ) Let R be an affine non-singular algebra of dimension over a perfect C -field k and I ⊂ R be a local complete intersection ideal of R . Then the Vaserstein symbol V R,I : U m ( R, I ) /E ( R, I ) → W E ( R, I ) is bijective. Theorem 1.3. ( See Theorem 5.26 ) Let R be an affine non-singular algebra of dimension over an alge-braically closed field k and I ⊂ R be a local complete intersection ideal of R . Then the Vaserstein symbol V R,I : U m ( R, I ) /E ( R, I ) → W E ( R, I ) is injective. In fact we studied more about injectivity of Vaserstein symbol. We have considered two cases: Oneinjectivity of the Vaserstein symbol V R [ X ] ,I [ X ] where R is an affine algebra of dimension ≤ over a field k and I is an ideal of R . The next is injectivity of the Vaserstein symbol V R [ X ] , ( X − X ) where R is an affinealgebra of dimension ≤ over a perfect C -field k . In fact we have: Theorem 1.4. ( See Theorem 5.27 ) Let R be a non-singular affine algebra of dimension over a perfect C -field k , and let I ⊂ R be a local complete intersection ideal of R . Then the Vaserstein symbol V R [ X ] ,I [ X ] : M SE ( R [ X ] , I [ X ]) → W E ( R [ X ] , I [ X ])) is injective. Theorem 1.5. ( see Theorem 5.18 ) Let R be an affine algebra of dimension over a perfect C -field k with R = R . Then the Vaserstein symbol V R [ X ] , ( X − X ) : U m ( R [ X ] , ( X − X )) /E ( R [ X ] , ( X − X )) → W E ( R [ X ] , ( X − X )) is bijective. We study the above case because of Fasel-Rao-Swan. According to Fasel-Rao-Swan the relative Witt group W E ( R [ X ] , ( X − X )) of a two dimensional non-singular algebra R over a perfect C -field was a divisiblegroup prime to characteristic. And divisibility of W E ( R [ X ] , ( X − X )) gives completability of a relativeunimodular row of size over R [ X ] via a result of Suslin.Later we computed the kernel of the Vaserstein map V R,I : Theorem 1.6. ( see Theorem 6.9 ) Let R be a ring of dimension and I ⊂ R be an ideal of R . If the orbit space M SE ( R, I ) has a nice group structure, then, the Vaserstein map V R,I induces a bijection of the map φ : U m ( R, I ) / { σ ∈ SL ( R, I ) ∩ E ( R, I ) } ≡ W E ( R, I ) . It is believed that for a -dimensional affine algebra non-singularity is not necessary for establishing injec-tivity of the Vaserstein symbol. We will give an example to show this at the end of this article. In [21], it isshowed that a Rees algebra R [ It ] , with R non-singular is non-singular if and only if I = 0 , R and I is generatedby a single element. By this result we will be able to construct a singular -dimensional algebra over a perfect NOTE ON RELATIVE VASERSTEIN SYMBOL 3 C -field for which the Vaserstein symbol is injective. Moreover we can extend this result for -dimensionalsingular algebra over a field k . In fact we have: Theorem 1.7. ( See Theorem 7.13 ) Let R be a non-singular affine algebra of dimension over a field k and I ⊂ R be an ideal. Then the Vaserstein symbol V R [ It ] : M SE ( R [ It ]) → W E ( R [ It ]) is injective. We established the injectivity of Vaserstein symbol over extended Rees algebra also:
Theorem 1.8. ( See Theorem 7.15 ) Let R be a ring of dimension and I ⊂ R be an ideal of R . Then theVaserstein symbol V R [ It,t − ] : M SE ( R [ It, t − ]) → W E ( R [ It, t − ]) is bijective. Witt Group and Relative Witt Group R will denote a commutative ring with = 0 , unless stated otherwise. I will denote an ideal of R .We denote by GL n ( R, I ) by the kernel of the canonical map GL n ( R ) −→ GL n (cid:0) RI (cid:1) . Let SL n ( R, I ) denotes the subgroup of GL n ( R, I ) of elements of determinant . Definition 2.1.
The Relative Elementary group E n ( R , I ) : Let R be a ring and I ⊂ R be an ideal. Therelative elementary group is the subgroup of SL n ( R, I ) generated by the matrices of the form αe i,j ( a ) α − ,where α ∈ E n ( R ) , i = j and a ∈ I . We identify GL n ( R ) with a subgroup of GL n +1 ( R ) by associating the matrix α ! ∈ GL n +1 ( R ) , withthe element α ∈ GL n ( R ) . We set GL ( R ) = lim −→ GL n ( R ) , GL ( R, I ) = lim −→ GL n ( R, I ) SL ( R ) = lim −→ SL n ( R ) , SL ( R, I ) = lim −→ SL n ( R, I ) E ( R ) = lim −→ E n ( R ) , E ( R, I ) = lim −→ E n ( R, I ) Let
Alt ′ n ( R ) be the set of all skew-symmetric matrices in GL n ( R ) . For any r ∈ N , define χ r inductively by χ = − ! and χ r +1 = χ r ⊥ χ . Clearly χ r ∈ Alt ′ r ( R ) . For any m ≤ n , Alt ′ m ( R ) can be embedded into Alt ′ n ( R ) bythe inclusion map i m,n : Alt ′ m ( R ) ֒ → Alt ′ n ( R ) given by α α ⊥ χ n − m . Define Alt ′ ( R ) = lim −→ Alt ′ n ( R ) .For α ∈ Alt ′ m ( R ) and β ∈ Alt ′ n ( R ) define α ∼ E β if and only if there exists some t ∈ N and ε ∈ E m + n + t ) ( R ) such that α ⊥ χ n + t = ε T ( β ⊥ χ m + t ) ε . It can be shown that this relation is reflexive,symmetric and transitive i.e. ∼ E is an equivalence relation on Alt ′ ( R ) . It can be shown that Alt ′ ( R ) / ∼ E is an abelian group with respect to the operation ⊥ [26, Section 3]. Denote this group by W ′ E ( R ) . Similarlyone can consider the subset Alt n ( R ) ⊂ Alt ′ n ( R ) , the set of all skew-symmetric matrices in GL n ( R ) withPfaffian . Using the same embedding, as defined earlier, one can construct the set Alt ( R ) := lim −→ Alt n ( R ) .Also ∼ E is an equivalence relation on Alt ( R ) and Alt ( R ) / ∼ E is an abelian group with respect to the operation ⊥ . Denote this group by W E ( R ) . This group is called the elementary symplectic Witt group .Let I be an ideal of R . We denote the set { α ∈ Alt ′ n ( R ) : α ≡ χ ± r ⊥ χ ± n − r (mod I ) for some r ≥ } by Alt ′ n ( R, I ) . For m ≤ n , Alt ′ m ( R, I ) can be embedded into Alt ′ n ( R, I ) by the inclusion map i m,n : KUNTAL CHAKRABORTY
Alt ′ m ( R, I ) ֒ → Alt ′ n ( R, I ) given by α α ⊥ χ n − m . We put Alt ′ ( R, I ) = lim −→ Alt ′ n ( R, I ) . We candefine an equivalence relation ∼ I on Alt ′ ( R, I ) as follows:Let α ∈ Alt ′ m ( R, I ) and β ∈ Alt ′ n ( R, I ) . Then α ∼ I β if and only if there exists t ∈ N and ǫ ∈ E m + n + t ) ( R, I ) such that : α ⊥ χ n + t = ǫ T ( β ⊥ χ m + t ) ǫ We denote the set
Alt ′ ( R, I ) / ∼ I by W ′ E ( R, I ) .We can also define the set Alt n ( R, I ) as the subset of Alt ′ n ( R, I ) consisting of skew-symmetric matricesof Pfaffian . We put Alt ( R, I ) := lim −→ Alt n ( R, I ) . As the relation ∼ I defines an equivalence relationon Alt ′ ( R, I ) , then ∼ I defines an equivalence relation on Alt ( R, I ) . We denote the set Alt ( R, I ) / ∼ I by W E ( R, I ) . Definition 2.2.
Let f : A → C be a ring homomorphism between two commutative rings A and C . Also let f : B → C be another ring homomorphism between B and C . A ring D is said to be fibre product of A and B with respect to C if the following two conditions are satisfied:1) there exist two ring homomorphisms,namely p : D → A , p : D → B such that the following diagramis commutative: D AB C p p f f
2) If there exists some ring E with two ring homomorphisms q : E → A , q : E → B such that thefollowing diagram is commutative : E AB C q q f f then there exists a unique map h : E → D such that the following diagram is commutative: E D AB C q q ∃ ! h p p f f We denote the fibre product of A and B with respect to C by A × C B . Consider the fibre product
D RR R/I p p ππ Here the ring D is called the double ring of the ring R with respect to the ideal I . D can be identified with theset { ( a, b ) ∈ R × R : a − b ∈ I } NOTE ON RELATIVE VASERSTEIN SYMBOL 5
Lemma 2.3.
Let α ∈ M n ( D ) . Then α can be associated canonically to an element ( α , α ) ∈ M n ( R ) × M n ( R ) such that α ≡ α (mod I ) . In other words M n ( D ) is isomorphic to the double ring of M n ( R ) withitself with respect to the ideal M n ( I ) . Proof : The maps p and p induces the ring homomorphisms p ∗ : M n ( D ) → M n ( R ) and p ∗ : M n ( D ) → M n ( R ) . We have a commutative diagram M n ( D ) M n ( R ) M n ( R ) M n ( R/I ) ( p ) ∗ ( p ) ∗ π ∗ π ∗ Since the fibre product is unique upto isomorphism it is enough to show that M n ( D ) ∼ = M n ( R ) × M n ( R/I ) M n ( R ) . By the universal property of the fibre product there exists a unique map h : M n ( D ) → M n ( R ) × M n ( R/I ) M n ( R ) ,which is defined by h ( A ) = ( A , A ) where A = ( a ij ) with a ij = ( a ij , a ij ) such that a ij − a ij ∈ I , A = ( a ij ) and A = ( a ij ) . Now define g : M n ( R ) × M n ( R/I ) M n ( R ) → M n ( D ) as following:Let ( B , B ) ∈ M n ( R ) × M n ( R/I ) M n ( R ) . Then ¯ B = ¯ B in M n ( R/I ) . If B = ( b ij ) and B = ( b ij ) then we have b ij − b ij ∈ I . Therefore the matrix B ∈ M n ( D ) is well-defined. We define g ( B , B ) = B .It can be shown that g and h both are ring homomorphisms. Now we have g ◦ h = 1 M n ( D ) and h ◦ g =1 M n ( R ) × Mn ( R/I ) M n ( R ) . Hence M n ( D ) ∼ = M n ( R ) × M n ( R/I ) M n ( R ) . (cid:3) Using the above canonical isomorphism we always represent the element of M n ( D ) as an element of M n ( R ) × M n ( R/I ) M n ( R ) , i.e. as a pair of two matrices satisfying certain properties. Lemma 2.4.
Let
M, N ∈ M n ( D ) be such that M = ( M , M ) and N = ( N , N ) . Then M ⊥ N = ( M ⊥ N , M ⊥ N ) in M n ( D ) . Proof : This is true in M n ( R ) × M n ( R ) hence the relation is true in M n ( R ) × M n ( R/I ) M n ( R ) . Finallythe relation is true in M n ( D ) by Lemma 2.3. (cid:3) Definition 2.5.
Let R be a ring and I ⊂ R be an ideal. The excision ring of R with respect to the ideal I is denoted by R ⊕ I and is defined by the set { ( r, i ) : r ∈ R, i ∈ I } with addition is component-wise andmultiplication is defined by ( r, i )( s, j ) = ( rs, rj + si + ij ) . Lemma 2.6. [8, Lemma 4.3]
Let ( R, m ) be a local ring with maximal ideal m . Then the excision ring R ⊕ I with respect to a proper ideal I in R is also a local ring with maximal ideal m ⊕ I . Definition 2.7.
We shall say a ring homomorphism φ : B → D is a retract if there exists a ring homomorphism γ : D ֒ → B so that φ ◦ γ is identity on D . We shall also say that D is a retract of B . Lemma 2.8. ( [1, Lemma 3.3] ) Let
B, D be rings and let D be a retract of B and let π : B ։ D . If J = ker( π ) ,then E n ( B, J ) = E n ( B ) ∩ SL n ( B, J ) , n ≥ . KUNTAL CHAKRABORTY
Proposition 2.9. ( [13, Proposition 3.1] ) Let R be a commutative ring and I ⊂ R be an ideal. Then theexcision ring R ⊕ I is the fibre product of R and R with respect to R/I . In fact, if φ : D → R ⊕ I is definedby φ ( a, b ) = ( b, a − b ) , then φ is an isomorphism. For all n ∈ N , let us define a map i n : Alt ′ n ( R, I ) → W ′ E ( D ) by i n ( α ) = ( χ ± r ⊥ χ ± n − r , α ) where α ≡ χ ± r ⊥ χ ± n − r (mod I ) for some r ≥ . It can be shown that the maps i n will induce a map i : Alt ′ ( R, I ) → W ′ E ( D ) . It can be shown that i induce a well defined map between W ′ E ( R, I ) and W ′ E ( D ) ,which we still call i . Theorem 2.10.
Let R be a commutative ring and I ⊂ R be an ideal. Then The set W ′ E ( R, I ) has an abeliangroup structure with the operation [ α ] . [ β ] = [ α ⊥ β ] . Proof : It is enough to check that each element [ α ] of W ′ E ( R, I ) has an inverse. Let α ∈ Alt ′ n ( R, I ) .Consider the element ˜ α defined by φ ◦ i n ( α ) . We have ˜ α ≡ χ ± r ⊥ χ ± n − r (mod 0 ⊕ I ) for some r ≥ .In the group W ′ ( R ⊕ I ) , we have [˜ α ⊥ ˜ α − ] = [ χ ] . Hence there exists E ∈ E n +2 t ( R ⊕ I ) such that E T (˜ α ⊥ ˜ α − ⊥ χ t ) E = χ n + t . Going modulo ⊕ I we have ¯ E T ( χ ′ ⊥ ( χ ′ ) − ⊥ χ t ) ¯ E = χ n + t where χ ′ = χ ± r ⊥ χ ± n − r . Since [ χ ± r ⊥ χ ± n − r ] = [ χ n ] in W ′ ( R, I ) , then there exist F ∈ E n +2 t ( R, I ) such that F T ( χ ′ ⊥ ( χ ′ ) − ⊥ χ t ) F = χ n + t . Also we have ˜ F T ( χ ′ ⊥ ( χ ′ ) − ⊥ χ t ) ˜ F = χ n + t where ˜ F ∈ E n +2 t ( R ⊕ I, ⊕ I ) . Now replacing E by E ¯ E − ˜ F , we may assume that E ∈ E n +2 t ( R ⊕ I, ⊕ I ) and E T (˜ α ⊥ ˜ α − ⊥ χ t ) E = χ n + t . Now projecting R ⊕ I onto R , we have E T ( α ⊥ α − ⊥ χ t ) E = χ n + t where E ∈ E n +2 t ( R, I ) . Hence we have [ α ⊥ α − ] = [ χ ] , i.e., [ α ] . [ α − ] = [ χ ] . Similarly we can showthat [ α − ] . [ α ] = [ χ ] . Hence W ′ ( R, I ) is a group with the operation [ α ] . [ β ] = [ α ⊥ β ] . To show this groupstructure is commutative: Let α ∈ Alt n ( R, I ) and β ∈ Alt m ( R, I ) . Hence we have α ≡ χ ± r ⊥ χ ± n − r (mod I ) and β ≡ χ ± s ⊥ χ ± m − s (mod I ) for some r ≥ , s ≥ . Now consider the element ˜ α, ˜ β in Alt n ( R ⊕ I ) and Alt m ( R ⊕ I ) respectively. In W ′ ( R ⊕ I ) , we have [˜ α ⊥ ˜ β ⊥ ˜ α − ⊥ ˜ β − ] = [ χ ] .Hence there exist t ∈ N and E ∈ E m + n )+2 t ( R ⊕ I ) such that E T (˜ α ⊥ ˜ β ⊥ ˜ α − ⊥ ˜ β − ⊥ χ t ) E = χ m + n )+ t . Now going modulo ⊕ I , we have ¯ E ( χ ′ ⊥ χ ′′ ⊥ ( χ ′ ) − ⊥ ( χ ′′ ) − χ t ) ¯ E = χ m + n )+ t where χ ′ = χ ± r ⊥ χ ± n − r and χ ′′ = χ ± s ⊥ χ ± m − s . Now there exist F ∈ E m + n )+2 t ( R ⊕ I, ⊕ I ) such that F T ( χ ′ ⊥ χ ′′ ) ⊥ ( χ ′ ) − ⊥ ( χ ′′ ) − ⊥ χ t ) F = χ m + n )+ t . Hence replacing E by E ¯ E − F we may assumethat E ∈ E m + n )+2 t ( R ⊕ I, ⊕ I ) and E T (˜ α ⊥ ˜ β ⊥ ˜ α − ⊥ ˜ β − ⊥ χ t ) E = χ m + n )+ t . Now projecting R ⊕ I onto R , we have E T ( α ⊥ β ⊥ α − ⊥ β − ⊥ χ t ) E = χ m + n )+ t where E ∈ E m + n )+2 t ( R, I ) .Hence [ α ⊥ β ⊥ α − ⊥ β − ] = [ χ ] in W ′ ( R, I ) . Hence [ α ⊥ β ] = [ β ⊥ α ] . Hence W ′ ( R, I ) is an abeliangroup with the operation ⊥ . (cid:3) Lemma 2.11.
The following sequence is exact W ′ E ( R, I ) W ′ E ( D ) W ′ E ( R ) 0 i ( p ) ∗ Where i ([ α ]) = [( χ ± r ⊥ χ ± n − r , α )] and p : D → R is the projection onto first factor. Proof : Let i ([ α ]) = [ χ ] in W ′ E ( D ) . Consider the isomorphism between W ′ E ( D ) and W ′ E ( R ⊕ I ) inducedfrom the isomorphism φ : D → R ⊕ I given by φ ( a, b ) = ( b, a − b ) . Then we have φ ◦ i ([ α ]) = [ χ ] . Hencewe have [˜ α ] = [ χ ] where ˜ α = φ (( χ ± r ⊥ χ ± n − r , α )) . We have ˜ α ≡ χ ± r ⊥ χ ± n − r (mod 0 ⊕ I ) . Replacing ˜ α NOTE ON RELATIVE VASERSTEIN SYMBOL 7 by F T ˜ αF for some F ∈ E n ( R ⊕ I, ⊕ I ) , we may assume that ˜ α ≡ χ n (mod 0 ⊕ I ) . Since [˜ α ] = [ χ ] in W ′ E ( R ⊕ I ) , there exists a natural number t and an elementary matrix E ∈ E n + t ) ( R ⊕ I ) such that E T (˜ α ⊥ χ t ) E = χ n + t .Now going modulo ⊕ I , we have, ¯ E T χ n + t ¯ E = χ n + t . Hence replacing E by E ¯ E − , we may assume that E T (˜ α ⊥ χ t ) E = χ n + t and E ∈ E n + t ) ( R ⊕ I, ⊕ I ) by Lemma 2.8. Hence projecting R ⊕ I onto R , wehave ε T ( α ⊥ χ t ) ε = χ n + t , where ε ∈ E n + t ) ( R, I ) . Hence i is injective.Clearly ( p ) ∗ is surjective. Now let ( p ) ∗ ([ β ]) = [ χ ] . By Lemma 2.3, we have β = ( β , β ) with β ≡ β (mod I ) . Thus [ β ] = [ χ ] in W ′ E ( R ) . Hence there exists a natural number and F ∈ E n + t ) ( R ) such that F T ( β ⊥ χ t ) F = χ n + t Now ( F, F ) T (( β , β ) ⊥ ( χ t , χ t ))( F, F ) = ( χ n + t , F T ( β ⊥ χ t ) F ) .Hence [ β ] = [( β , β )] = [( β , β ) ⊥ ( χ t , χ t )] = [( χ n + t , F T ( β ⊥ χ t ) F ] = i ([ F T ( β ⊥ χ t ) F ]) .Hence the sequence W ′ E ( R, I ) W ′ E ( D ) W ′ E ( R ) 0 i ( p ) ∗ is exact. (cid:3) Lemma 2.12. [6]
Let C be the kernel of the group homomorphism R × → ( R/I ) × induced by π , with theconvension that C = R × if I = R . Then the Pfaffian gives a split exact sequence W E ( R, I ) W ′ E ( R, I ) C j P f Proof : Clearly the homomorphism i is injective and the sequence is exact on the middle. For split exactsequence we have to show that there is a map r : C → W ′ E ( R, I ) such that P f ◦ r = id C . Define r ( a ) = " a − a ! The map r is well-defined since a ≡ I ) . (cid:3) Corollary 2.13.
The set W E ( R, I ) is a group with respect to the operation ⊥ . Lemma 2.14.
The following sequence is exact. W E ( R, I ) W E ( D ) W E ( R ) 0 i ( p ) ∗ KUNTAL CHAKRABORTY
Proof : Consider the following diagram W E ( R, I ) W ′ E ( R, I ) C W E ( D ) W ′ E ( D ) D × W E ( R ) W ′ E ( R ) R ×
00 0 0 ji P fi φj ( p ) ∗ P f ( p ) ∗ ψj P f Where the maps φ : C → D × and ψ : D × → R × are defined by φ ( a ) = (1 , a ) and ψ ( d , d ) = d . Withrespect to the φ and ψ the above diagram is commutative. Each row of the above diagram is an exact sequenceby Lemma 2.12 . Observe that all these groups in the above diagram are abelian. The second column of thediagram is an exact sequence by Lemma 2.11. Also the third column of the above diagram is exact. Hence bydiagram chasing the first column is also exact. (cid:3) Divisibility of W E ( R [ X ] , I [ X ]) Lemma 3.1. ( [26, Lemma 3.1] ) ( Karoubi ) Let R be a commutative ring with . Let α ∈ W E ( R [ X ]) . Then wehave [ α ] = [ l ] , where l = ϕ + ϕ X where ϕ and ϕ are matrices over R . Lemma 3.2. ( Karoubi ) Let R be a commutative ring and I ⊂ R be an ideal of R . Let α ∈ W E ( R [ X ] , I [ X ]) .Then we have [ α ] = [ l ] , where l = ϕ + ϕ X where ϕ is a matrix over R and ϕ is a matrix over I . Proof : Let α ∈ Alt n ( R [ X ] , I [ X ]) . We may assume that α ≡ χ n (mod I [ X ]) . Consider the element ˜ α ∈ Alt n (( R ⊕ I )[ X ]) . Then we have ˜ α ≡ χ n (mod 0 ⊕ I [ X ]) . Now by Karoubi we have [˜ α ] = [ l ] , where l is a linear matrix in Alt (( R ⊕ I )[ X ]) . It is easy to check that l ∈ Alt m (( R ⊕ I )[ X ] , ⊕ I [ X ]) for some m .We have [˜ α ⊥ l − ] = [ χ ] . Conjugating ˜ α ⊥ l − with some element F ∈ E m + n ) (( R ⊕ I )[ X ] , ⊕ I [ X ]) ,we may assume that ˜ α ⊥ l − ≡ χ m + n (mod 0 ⊕ I [ X ]) . Since [˜ α ⊥ l − ] = [ χ ] , then there exist t ∈ N and E ∈ E m + n + t ) (( R ⊕ I )[ X ]) such that E T (˜ α ⊥ l − ⊥ χ t ) E = χ m + n + t . Now going modulo ⊕ I [ X ] , wehave ¯ E T χ m + n + t ¯ E = χ m + n + t . Hence replacing E by E ¯ E − , we have E ∈ E m + n + t ) (( R ⊕ I )[ X ] , ⊕ I [ X ]) and E T (˜ α ⊥ l − ⊥ χ t ) E = χ m + n + t . Now projecting ( R ⊕ I )[ X ] onto R [ X ] , we have E T ( α ⊥ l − ⊥ χ t ) E = χ m + n + t , where E ∈ E m + n + t ) ( R [ X ] , I [ X ]) and l ∈ Alt m ( R [ X ] , I [ X ]) is a linear matrix.Hence we have [ α ⊥ l − ] = [ χ ] in W E ( R [ X ] , I [ X ]) . In other words [ α ] = [ l ] in W E ( R [ X ] , I [ X ]) . (cid:3) Proposition 3.3. [18, Proposition 2.4.1]
Let R be a local ring with / k ∈ R , and let [ V ] ∈ W E ( R [ X ]) . Then [ V ] has a k -th root, i.e. there is a [ W ] ∈ W E ( R [ X ]) such that [ V ] = [ W ] k in W E ( R [ X ]) . Lemma 3.4.
Let R be a local ring and I ⊂ R be an ideal of R , / k ∈ R and let [ α ] ∈ W E ( R [ X ] , I [ X ]) .Then [ α ] has a k -th root. NOTE ON RELATIVE VASERSTEIN SYMBOL 9
Proof : Let I = R . By convension we have W E ( R [ X ] , R [ X ]) = W E ( R [ X ]) . The lemma is true for I = R by Proposition 3.3. Hence we may assume that I is a proper ideal of R . Let α ∈ Alt n ( R [ X ] , I [ X ]) .We may assume that α ≡ χ n ( mod I ) . Consider the element ˜ α ∈ W E (( R ⊕ I )[ X ]) . By Lemma 2.6, wehave R ⊕ I is a local ring. And also we have / k ∈ R ⊕ I . Hence by Proposition 3.3, we have there exists β ∈ W E (( R ⊕ I )[ X ]) such that [˜ α ] = [ β ] k . We revisit the proof of Proposition 3.3, to verify that β ≡ χ m (mod 0 ⊕ I [ X ]) for some m . We have ˜ α ∈ Alt n (( R ⊕ I )[ X ] , ⊕ I [ X ]) . By Lemma 3.2, we have there exist t ∈ N and E ∈ E n + t ) (( R ⊕ I )[ X ] , ⊕ I [ X ]) such that E T (˜ α ⊥ χ t ) E = χ n + t + rX , where r is a matrixover ⊕ I . Now let γ = I n + t ) − χ n + t rX . Clearly γ ∈ SL n + t ) (( R ⊕ I )[ X ] , ⊕ I [ X ]) . Hence χ n + t r is anilpotent matrix over R ⊕ I . Since / k ∈ R ⊕ I , we can extract a k -th root of γ . Call it δ . It is easy to checkthat δ ∈ SL n + t ) (( R ⊕ I )[ X ] , ⊕ I [ X ]) . Now M. Karoubi pointed out that E T (˜ α ⊥ χ t ) E = χ n + t γ = χ n + t δ k = ( δ k ) T χ n + t δ k . Let β = δ T χ n + t δ . Clearly β ≡ χ n + t ( mod 0 ⊕ I [ X ]) . Applying Whitehead’s lemma one can check that [˜ α ] = [ β ] k . Let m = n + t . We have [˜ α ⊥ β − ⊥ .. (k times) .. ⊥ β − ] = [ χ ] (1)in W E (( R ⊕ I )[ X ]) . Conjugating ˜ α ⊥ β − ⊥ · · · ⊥ β − by some element in E n + mk ) (( R ⊕ I )[ X ] , ⊕ I [ X ]) we may assume that ˜ α ⊥ β − ⊥ · · · ⊥ β − ≡ χ n + mk (mod 0 ⊕ I [ X ]) . By (1) we have there exist p ∈ N and F ∈ E n + mk + p ) (( R ⊕ I )[ X ]) such that F T (˜ α ⊥ β − ⊥ · · · ⊥ β − ⊥ χ p ) F = χ n + mk + p . Nowgoing modulo ⊕ I [ X ] , we have ¯ F T χ n + mk + p ¯ F = χ n + mk + p . Hence replacing F by F ¯ F − , we have F ∈ E n + mk + p ) (( R ⊕ I )[ X ] , ⊕ I [ X ]) and F T (˜ α ⊥ β − ⊥ · · · ⊥ β − ⊥ χ p ) F = χ n + mk + p . Now projecting R ⊕ I onto R we have F T ( α ⊥ β − ⊥ · · · ⊥ β − ⊥ χ p ) F = χ n + mk + p , where F ∈ E n + mk + p ) ( R [ X ] , I [ X ]) .Hence we have [ α ⊥ β − ⊥ · · · ⊥ β − ] = [ χ ] in W E ( R [ X ] , I [ X ]) . Hence [ α ] = [ β ] k in W E ( R [ X ] , I [ X ]) . (cid:3) Lemma 3.5.
Let R be a local ring of dimension d , d > , α ∈ R be a non-zero-divisor, v ∈ U m d +1 ( R [ X ] , ( α )) .Then v ∼ E d +1 ( R [ X ] , ( α )) ( a ( X ) , αa ( X ) , . . . , αc d ) such that c d is a non-zero-divisor. Proof : By Excision theorem [11, Theorem 3.21], we may assume that, R is a reduced ring. Let v ( X ) :=( v ( X ) , v ( X ) , . . . , v d ( X )) . We assume that deg v ( X ) ≥ . Let the leading coefficient of v ( X ) be a . Wemay assume that a is a non-zero-divisor of R . Let the ’overline’ denote modulo ( a ) and consider ¯ v ( X ) ∈ U m d +1 ( ¯ R [ X ] , ( α )) . By excision and usual stability estimates we have ¯ v ( X ) ∼ E d +1 ( ¯ R [ X ] , ( α )) e . Hence,we may modify v ( X ) suitably and assume that, ¯ v ( X ) = e . As, so crucially, indicated by M. Roitman, thistransformation can be performed so that every stage the row contains a polynomial which is unitary in R a . Andwe may ensure that v ( X ) is unitary in R a and deg v ( X ) ≥ . Let v ( X ) = 1 + v ′ ( X ) , v i ( X ) = αav ′ i ( X ) for i > . One has a l i v ′ i ( X ) = q i ( X ) v ( X ) + r i ( X ) ,for some l i > , r i ( X ) ∈ R [ X ] , with r i ( X ) = 0 or deg r i ( X ) < deg v ( X ) . By Excision theorem [11,Theorem 3.21], we can transform v ( X ) in the relative orbit with respect to ( aα ) and assume that deg v ′ i ( X ) < deg v ( X ) , for all i > . If deg v ( X ) = 1 , then we are done. Hence we may assume that deg v ( X ) = d ≥ . Let c , c , . . . , c d ( d − be the coefficients of , X, . . . , X d − of the polynomials v ′ ( X ) , . . . , v ′ d ( X ) . Since d ( d − ≥ . ( d +1)2 > dim R a , we can argue as in the proof of [22, Theorem 5] to conclude that the idealgenerated by the polynomials v ( X ) , v ′ ( X ) , . . . , v ′ d ( X ) contains a polynomial h ( X ) of degree ( d − whichis unitary in R a . Let leading coefficient of h ( X ) is ua k where u is a unit in R . Via Excision theorem and theargument in the proof of [22, Theorem 5], we have mse ( v ( X ) , αv ′ ( X ) , . . . , αv ′ d ( X ))= mse ( v ( X ) , a k αv ′ ( X ) , . . . , αv ′ d ( X ))= mse ( v ( X ) , α { a k v ′ ( X ) + (1 − a k .l ( v ′ ( X ))) h } , αv ′ ( X ) , . . . , αv ′ d ( X ))= mse ( v ( X ) , αv ′′ ( X ) , αv ′ ( X ) , . . . , αv ′ d ( X )) where l ( v ′ ( X )) is the leading coefficient of v ′ ( X ) . Note that v ′′ ( X ) is unitary in R a of degree d < d .Rename v ′′ ( X ) as v ′ ( X ) . Again we may ensure that deg v ′ i ( X ) < d , for i ≥ . Repeat the argument aboveto lower the degree of v ′ ( X ) to . Then lower the degree of v ′ i to zero. We then have the desired form of thevector v ( X ) in the class of M SE d +1 ( R [ X ] , ( α )) . (cid:3) Analytic isomorphismsDefinition 4.1.
Let
A, B be commutative rings and φ : B → A be a ring homomorphism. Let h ∈ B . φ is saidto be analytic isomorphism along h , if the following conditions are satisfied.(i) h is a non-zero-divisor of B .(ii) φ ( h ) is a non-zero-divisor of A .(iii) φ induces an isomorphism between B/hB and
A/hA . In this situation the commutative diagram
B AB h A hφφ h (2)will be called an analytic diagram. In [23], it is checked that (1) is a cartesian square.Let P ( R ) denote the category of all finitely generated projective R -modules. In [23], it is shown that thecorresponding square for projective modules is also cartesian, i.e., P ( B ) P ( A ) P ( B h ) P ( A h ) φφ h is cartesian. NOTE ON RELATIVE VASERSTEIN SYMBOL 11
Lemma 4.2.
Let A be a commutative ring and B ⊂ A be a subring. Let h ∈ B be such that B ⊂ A isanalytically isomorphic along h . Let I be an ideal of A . Then B ⊕ I ⊂ A ⊕ I is analytically isomorphic along ( h, . Proof : Claim: ( h, is a non-zero-divisor of B ⊕ I .Let ( h, h , i ) = (0 , in B ⊕ I for some ( h , i ) ∈ B ⊕ I . Then ( hh , hi ) = (0 , . Thus we have hh = 0 , hi = 0 . Since h is a non-zero-divisor of B and of A also, then h = 0 and i = 0 . Hence ( h , i ) = (0 , and therefore ( h, is a non-zero-divisor of B ⊕ I .Similarly ( h, is a non-zero-divisor of A ⊕ I .Consider the natural map φ : ( B ⊕ I ) / ( h, B ⊕ I ) → ( A ⊕ I ) / ( h, A ⊕ I ) . Let φ (( b, i )) = 0 . Then ( b, i ) ∈ ( h, A ⊕ I ) . Hence ( b, i ) = ( h, a, j ) for some ( a, j ) ∈ A ⊕ I . This gives b = ha and i = hj .Since B ⊂ A is analytically isomorphic along h , b = ha implies that there exists b ′ ∈ B such that b = hb ′ .Hence we have ( b, i ) = ( h, b ′ , j ) . This shows that φ is injective. Let ( a, i ) ∈ ( A ⊕ I ) / ( h, A ⊕ I ) .Since B ⊂ A is analytic along h , then there exists b ∈ B and c ∈ A such that a − b = hc . This shows that φ (( b, i )) = ( a − hc, i ) = ( a, i ) . Hence φ is surjective. Thus B ⊕ I ⊂ A ⊕ I is analytically isomorphic along ( h, . (cid:3) Definition 4.3.
Let ( R, m ) be a local ring. A monic polynomial f ∈ R [ X ] is said to be Weierstrass polynomialif f = X n + a X n − + ... + a n , a i ∈ m f or i = 1 , , ..., n . Lemma 4.4. ( [16, Proposition 1.7] ) Let ( R, m ) be a local ring and f ∈ R [ X ] be a Weierstrass polynomial.Then R [ X ] ֒ → R [ X ] ( m ,X ) is an analytic isomorphism along f . Lemma 4.5. ( [27, Lemma 2.4] ) Let B ⊂ A be a subring, h ∈ B be such that B ֒ → A is an analyticisomorphism along h . Let α ∈ E r ( A h ) , r ≥ . Then there exists α ∈ E r ( A ) and α ∈ E r ( B h ) such that α = ( α ) h ( α ⊗ .Consequently if α ∈ GL r ( A ) with α h ∈ E r ( A h ) then there exist α ∈ E r ( A ) and α ∈ GL r ( B ) such that ( α ) h ∈ E r ( B h ) and α = ( α ) h ( α ⊗ . Definition 4.6.
Let R be a local ring. R is said to be a local algebra with a ground field if R is a localisationof an affine k -algebra for some field k . Definition 4.7.
Let R be a regular local algebra with a ground field. R is said to be regular local algebra witha separating ground field if R possesses a ground field K such that the residue field of R is a finite separableand hence a simple extension of K . Definition 4.8.
Regular k -spots Let k be a field. By a regular spot over a field k we mean a localisation C p ofa finitely generated k -algebra C at a regular prime p ∈ Spec C . Lindel [14, Proposition 2] analysed regular k-spots over a perfect field as ´etale extensions of rings of thetype k [ X , ..., X n ] ( ϕ ( X ) ,X ,...,X n ) , where ϕ ( X ) ∈ k [ X ] is a monic polynomial. Lemma 4.9. [2, Proposition 6.10]
Let A be a noetherian ring, B an A -algebra, q a prime ideal of B , p the traceof q in A . Suppose there exists a polynomial P ( T ) and an element t ∈ B such that the map A [ T ] /P A [ T ] → B defined by t is an isomorphism. Then B q is unramified over A p if and only if ( P ( T ) , P ′ ( T )) A [ T ] = A [ T ] .Suppose, in addition, that the leading coefficient of P is invertible. Then B q is ´etale over A p if and only if P ′ ( t ) / ∈ q . Theorem 4.10. ( [16, Theorem 2.8] ) Let ( R, m ) be a regular k -spot of dimension d with a separating groundfield K . Let g be any element of m . Then there exists a regular local subring S of R such that:(i) S = K [ X , ..., X d ] ( ϕ ( X ) ,X ,...,X d )) , where ϕ ( X ) ∈ K [ X ] is an irreducible monic polynomial.(ii) S ⊂ R is an analytic isomorphism along h , for some h ∈ gR ∩ S .(Here h depends on choice of g .) We revisit the proof of [16, Theorem 2.8] to justify the following crucial (in our context) claim:
Lemma 4.11.
The element h and g mentioned in Theorem 4.10 differ by a unit in R . Proof : The regular local subring S of R is constructed in the Theorem 4.10 in the following way:We can choose elements X , X , . . . , X d in m such that g, X , X , . . . , X d is a system of generators in R and X , X , . . . , X d are a part of a minimal generating set for m (mod m ) . Since R is a regular local ring wehave that g, X , . . . X d is a regular sequence in R . The field K is contained in R . Therefore a, X , . . . , X d arealgebraically independent over K . Thus C = K [ g, X , . . . , X d ] is a polynomial ring contained in R . Let B bethe integral closure of C in R and let m = m ∩ B . Since m contracts to the maximal ideal ( g, X , . . . , X d ) of C , it follows that m is a maximal ideal in B . It is proved in [16, Theorem 2.8] that R = B m . Viewing R as B m we may rename m as m .By the hypothesis of Theorem 4.10, we have, L := Q ( R ) = B/ m = K (¯ α ) for some α in B . Let ϕ denotethe minimal polynomial of ¯ α over K . Then ϕ ( α ) ∈ m . We can choose X = α in such a way that ( g, ϕ ( X ) , X , . . . , X d ) = m .Further, as X is integral over K [ g, X , . . . , X d ] , replacing X by X + g j for some integers j large enough,we may even assume that g is integral over K [ X , X , . . . , X d ] . We now define C = K [ X , X , . . . , X d ] and M = ( ϕ ( X ) , X , . . . , X d ) . Set S = C M .The element h is chosen in [16, Theorem 2.8] in the following way:Note that m ∩ C = M . Consider the ring C [ g ] and the maximal ideal ( M, g ) of C [ g ] . As ( M, g ) generates m in B , B is a finite C [ g ] -module and B/ m = L = C [ g ] / ( M, g ) , we conclude, using Nakayama’s lemma,that R = B m = C [ g ] ( M,g ) . It is proved in [16, Theorem 2.8], that R is faithfully flat S -algebra and also R isunramified over S . Therefore R is ´etale over S . Consider the C -algebra homomorphism σ : C [ T ] → C [ g ] by σ ( T ) = g . As g is integral over C , there exists an irreducible monic polynomial in C [ T ] : F ( T ) = T n + λ n − T n − + · · · + λ T + λ such that σ ( F ( T )) = F ( g ) = 0 . Thus σ induces an isomorphism between C [ T ] /F C [ T ] and C [ g ] . Since R is ´etale over S , then by Lemma 4.9, we have F ′ ( g ) / ∈ ( M, g ) . Hence we have λ / ∈ M . Thus λ = − g ( λ + λ g + · · · + λ n − g n − + g n − ) . In other words λ R = gR and λ ∈ S ∩ gR . Now take h = λ .Since R/gR = C [ g ] ( M,g ) /gC [ g ] ( M,g ) = S/S ∩ gR we have R = S + gR = S + hR . Since R is faithfullyflat S -algebra we have, S ∩ hR = hS . Hence we have S ⊂ R is an analytic isomorphism along h . (cid:3) Remark 4.12.
The following lemma gives a sufficient condition for a local domain to be a local algebra witha separating ground field.
NOTE ON RELATIVE VASERSTEIN SYMBOL 13
Lemma 4.13. ( [16, Lemma 2.10] ) Let R be a local domain. Suppose that R is a local algebra with a groundfield k and that k is a finitely generated field extension of a perfect field k . Then R is a local algebra with aseparating ground field. Lemma 4.14. ( Nagata ) Let k be a field and f be a polynomial in k [ X , X , ..., X d ] and φ ( X ) ∈ k [ X ] be amonic polynomial. Then there exists a change of variables, X X , X i X i + φ ( X ) r i , ≤ i ≤ d , suchthat f = c.h ( X , X , ..., X d ) , where c ∈ K ∗ and h is a monic polynomial in X over k [ X , X , ..., X d ] . Proof : Let f ( X , X , · · · , X d ) = P i a i X i X i · · · X i d d and φ ( X ) = X n + c X n − + · · · + c n . So, f ( X , X + φ ( X ) r , · · · , X d + φ ( X ) r d ) = X i a i X i ( X + φ ( X ) r ) i · · · ( X d + φ ( X ) r d ) i d = X i a i X i + nr i + ··· + nr d i d + terms of lower degree in X Let m > max { n.i k : k = 1 , · · · , d } . If we choose r j = m j − , then the integers i + nr i + · · · + nr d i d willhave different m − adic expansion for all tuple ( i , i , · · · , i d ) in the expansion of f . Hence if we choose such r i ,then the monomials a i X i + nr i + ··· + nr d i d in f ( X , X + φ ( X ) r , ..., X d + φ ( X ) r d ) will not cancel out eachother. Hence choosing the highest degree with non-zero coefficient among a i X i + nr i + ··· + nr d i d we can makeit the leading term of f ( X , X + φ ( X ) r , · · · , X d + φ ( X ) r d ) . Hence we have f = c.h ( X , X , · · · , X d ) ,where c ∈ k ∗ and h is a monic polynomial in X over k [ X , X , · · · , X d ] . (cid:3) The following is well-known. For completeness we sketch a proof.
Proposition 4.15. ( [3, Theorem 2.2.12] ) Let R ′ ⊂ R be a faithfully flat local extension. Then R ′ is regular if R is regular. Proof : It is enough to show that pd R ′ ( M ) is finite for any finitely generated R ′ -module M . Let → Q → P m → · · · → P → M → be a resolution of M where P i ’s are projective R ′ modules. Since R is regular we have pd R ( M ⊗ R ′ R ) isfinite. Hence for large m we have Q ⊗ R ′ R is projective R module. Hence Q is a projective R ′ module since R is faithfully flat over R ′ . Hence pd R ′ ( M ) is finite. (cid:3) Theorem 4.16. ( [24, Corollary 5.7] ) Let R be a ring and f ∈ R [ X ] be a monic polynomial and α ∈ GL r ( R [ X ]) , r ≥ . If α f ∈ E r ( R [ X ] f ) , then α ∈ E r ( R [ X ]) . Theorem 4.17. [4, Theorem 3.12]
Let R be a regular k -spot of dimension d over a field k . Let α ∈ SL n ( R [ X ] , ( X − X )) ∩ E n +1 ( R [ X ] , ( X − X )) for n ≥ , then α ∈ E n ( R [ X ] , ( X − X )) . Relative Vaserstein’s TheoremDefinition 5.1.
A row ( a , a , ..., a n ) ∈ R n is said to be unimodular if there exist a row ( b , b , ..., b n ) ∈ R n such that P nk =1 a k b k = 1 . A row ( a , a , ..., a n ) ∈ R n is said to be relative unimodular row with respect tothe ideal I if it is unimodular and ( a , a , ..., a n ) ≡ (1 , , ...,
0) (mod I ) , i.e. a − , a , ..., a n all belong to I . The set of all unimodular rows is denoted by
U m n ( R ) and the set of all relative unimodular rows with respectto the ideal I is denoted by U m n ( R, I ) . Definition 5.2. (Stable range condition Sr n ( I ) ) Let I be an ideal in R . We shall say stable range condition Sr n ( I ) holds for I if for any ( a , a , ..., a n +1 ) ∈ U m n +1 ( R, I ) there exists c i in I such that ( a + c a n +1 , a + c a n +1 , ..., a n + c n a n +1 ) ∈ U m n ( R, I ) . Definition 5.3. (Stable range Sr ( I ) ) We shall define the stable range of I , denoted by Sr ( I ) , to be the leastinteger n such that Sr n ( I ) holds for I . When I = R , then the stable range of R will be denoted by Sr ( R ) . Definition 5.4. (Stable dimension Sd ( R ) ) We shall define the stable dimension of R, denoted by Sd ( R ) , to beone less than the stable range of R , i.e. Sd ( R ) = Sr ( R ) − . The group E n ( R ) acts on the set U m n ( R ) by the action E n ( R ) × U m n ( R ) −→ U m n ( R ) given by α · v = vα . We denote the orbit space U m n ( R ) /E n ( R ) by M SE n ( R ) . Similarly the group E n ( R, I ) acts onthe set U m n ( R, I ) by the same action. We denote the orbit space U m n ( R, I ) /E n ( R, I ) by M SE n ( R, I ) . The Relative Vaserstein symbol.
Let b = ( b , b , b ) ∈ U m ( R, I ) and a = ( a , a , a ) ∈ U m ( R, I ) besuch that a b + a b + a b = 1 .Denote θ ( a, b ) by the matrix θ ( a, b ) = − b − b − b b − a a b a − a b − a a Define a map s E : U m ( R, I ) → W E ( R, I ) by s G ( b ) = [ θ ( a, b )] . Lemma 5.5.
The map s E does not depend on the choice of a , i.e. if c = ( c , c , c ) ∈ U m ( R, I ) be such that cb T = ab T = 1 , then [ θ ( c, b )] = [ θ ( a, b )] in W E ( R, I ) . Proof : Let ε = d d d where d = c a − c a , d = c a − c a , d = c a − c a . Since d , d , d ∈ I , then ε ∈ E ( R, I ) . It is easy to check that θ ( c, b ) = ε T θ ( a, b ) ε . Hence [ θ ( c, b )] = [ θ ( a, b )] in W E ( R, I ) .The following theorem is proved in [26, Theorem 5.2] in the absolute case. But we need this theorem in therelative case. And the proof is similar to the absolute case. Theorem 5.6.
Let R be a commutative ring with and I ⊂ R be an ideal of R . The map s E : U m ( R, I ) −→ W E ( R, I ) possesses the following properties:(i) s E ( b ) = s E ( bα ) for all b ∈ U m ( R, I ) and α ∈ E ( R, I ) ∩ SL ( R, I ) .(ii) If e ( E ( R, I ) ∩ SL r +1 ( R, I )) =
U m r +1 ( R, I ) for all r ≥ , then s E ( U m ( R, I )) = W E ( R, I ) .(iii) If e E r ( R, I ) = e ( E ( R, I ) ∩ SL r ( R, I )) for all r ≥ , then s E ( b ) = s E ( d ) , where b, d ∈ U m ( R, I ) , only if b = dα for some α ∈ E ( R, I ) ∩ SL ( R, I ) . NOTE ON RELATIVE VASERSTEIN SYMBOL 15
Remark 5.7.
By (i) of Theorem 5.6, s E induces a map from U m ( R, I ) /E ( R, I ) to W E ( R, I ) . We call thismap by relative Vaserstein symbol and denote by V R,I . Lemma 5.8. ( [26, Theorem 5.2] ) Let A be a commutative ring for which U m r ( A ) = e E r ( A ) , for r ≥ .Then V A is surjective. If, moreover, SL ( A ) ∩ E ( A ) = E ( A ) , then V A is bijective. In particular, if Sd ( A ) ≤ ,and SL ( A ) ∩ E ( A ) = E ( A ) , then V A is bijective. In view of Lemma 5.8, we can derive the same for relative case from Theorem 5.6.
Lemma 5.9.
Let R be a commutative ring and I ⊂ R be an ideal. If U m r ( R, I ) = e E r ( R, I ) for r ≥ ,then V R,I is surjective. If, moreover, SL ( R, I ) ∩ E ( R, I ) = E ( R, I ) then V R,I is bijective. In particular, if Sd ( R ) ≤ , and SL ( R, I ) ∩ E ( R, I ) = E ( R, I ) , then V R,I is bijective.
Corollary 5.10.
Let R be a commutative ring of dimension and I ⊂ R be an ideal. Then U m ( R, I ) /E ( R, I ) is bijectively equivalent to W E ( R, I ) . Lemma 5.11. [8, Lemma 5.2]
Let R be a commutative ring and I ⊂ R be an ideal of R . Let u, v ∈ U m ( R, I ) be such that uα = v for some α ∈ SL ( R, I ) ∩ E ( R, I ) . Then u and v are elementary equivalent relative to I . Lemma 5.12.
Let R be a commutative ring and I ⊂ R be an ideal of R . Let ε ∈ E n ( R, I ) . Then there existsa matrix ˜ ε ∈ E n ( R ⊕ I, ⊕ I ) such that ϕ (˜ ε ) = ε , where ϕ : R ⊕ I → R is defined by ϕ (( r, i )) = r + i . Lemma 5.13.
Let R be a ring and I ⊂ R be an ideal of R . Assume that both M SE ( R ) and M SE ( R ⊕ I ) have Witt group structure via Vaserstein symbol. Let K be the kernel of the map M SE ( R ⊕ I ) → M SE ( R ) induced from the map pr : R ⊕ I → R defined by pr (( r, i )) = r . Then there is a bijection from M SE ( R, I ) to K . Proof : Let us define a map i : M SE ( R, I ) → K by i ([ v ]) = [˜ v ] , where v = (1 + i , i , i ) with i , i , i ∈ I and ˜ v = ((1 , i ) , (0 , i ) , (0 , i )) . This map is well defined because if [ v ] = [ w ] in M SE ( R, I ) ,then vε = w for some ε ∈ E ( R, I ) . By Lemma 5.12, there exists ˜ ε ∈ E ( R ⊕ I, ⊕ I ) such that ϕ (˜ ε ) = ε .It is easy to check that ˜ v ˜ ε = ˜ w . Lemma 5.14.
Let R be a commutative ring and I ⊂ R be an ideal of R . Then the relative Witt groups W E ( R, I ) and W E ( R ⊕ I, ⊕ I ) are isomorphic. Proof : Let us define a map ϕ : W E ( R, I ) → W E ( R ⊕ I, ⊕ I ) by ϕ ([ α ]) = [˜ α ] . Clearly this mapis well-defined. Let ϕ ([ α ]) = [ χ ] . Then there exists t ∈ N and ε ∈ E n + t ) ( R ⊕ I, ⊕ I ) such that ε T (˜ α ⊥ χ t ) ε = χ n + t . Projecting R ⊕ I onto R we have ε T ( α ⊥ χ t ) ε = χ n + t , where ε ∈ E n + t ) ( R, I ) .Hence ϕ is injective. Clearly ϕ is surjective. It is easy to check that ϕ is a group homomorphism. Hence ϕ isan isomorphism. (cid:3) Following L.N. Vaserstein in [26], one can show the following:
Lemma 5.15. ( Relative version of L.N. Vaserstein’s lemma ) Let ϕ ∈ SL n ( R ) be an alternating matrix. Then e E n ( R, I ) = e Sp ϕ ( R, I ) ∩ E n ( R, I ) . Here we have the isotropy group Sp ϕ ( R, I ) = { ε ∈ SL n ( R, I ) | ε t ϕε = ϕ } . Proof : Let ε ∈ E n ( A, I ) , and ε ( X ) ∈ E n ( A [ X ] , I [ X ]) be a homotopy of ε . Let v ( X ) = e ε ( X ) . Let m be a maximal ideal of A . By ([26], Lemma 5.5) v ( X ) m = e ε m ( X ) , for some ε m ( X ) , with ε m (0) = I n ,and with ε m ( X ) ∈ { Sp ϕ ) n ( A m [ X ] , I m [ X ]) ∩ E n ( A m [ X ] , I m [ X ]) } By imitating the proof of the action version of the Local Global Theorem [1, Theorem 4.7], one can show thatthere is a global ε ϕ ( X ) with v ( X ) = e ε ϕ ( X ) , with ε ϕ ( X ) ∈ Sp ϕ ( A [ X ] , I [ X ]) ∩ E n ( A [ X ] , I [ X ]) } . Nowput X = 1 .In fact, Chattopadhyay–Rao have shown that Theorem 5.16. ( [5] ) Let R be a commutative ring with . Let I be an ideal of R . Let n ≥ , and let v ∈ U m n ( R, I ) . Then vE n ( R, I ) = vESp n ( R, I ) . We state a lemma which can be found in ([15], Lemma 4.3) where it is attributed to A. Suslin:
Lemma 5.17.
Let A be any commutative ring with . Let I be an ideal of A such that K ,n ( A ) −→ K ,n ( A/I ) ,for some n ≥ , is surjective. Then E n ( A, I ) = E n ( A ) ∩ GL n ( A, I ) . (cid:3) Theorem 5.18.
Let R be a commutative ring of dimension . Then the Vaserstein symbol Um ( R [ T ] , ( T − T ))E ( R [ T ] , ( T − T )) −→ W E(R[T] , (T − T)) ( R [ T ] , ( T − T )) is surjective. If R is an affine algebra over a perfect C -field then V is also injective if R = R . Proof : Let A = R [ T ] , t = T − T , x = X − X . The surjectivity will follow by ( ii ) of Theorem 5.6.For the injective, the argument of L.N. Vaserstein in [26] says, in view of this, that it suffices to show that if σ ∈ Sl ( A, ( t )) ∩ E( A, ( t )) then σ ∈ E ( A, ( t )) . By stability estimates, σ ∈ E ( A, ( t )) .Choose a homotopy ρ ( X ) ∈ E ( A [ X ] , ( tX )]) of (1 ⊥ σ ) . Clearly, e ρ ( X ) ∈ U m ( A [ X ] , ( tx )) . By ([19],Proposition 5.4) there is a ε ( X ) ∈ E ( A [ X ] , ( tx )) such that e ρ ( X ) ε ( X ) is a factorial vector; whence it canbe completed to a β ( X ) which is stably elementary and congruent to identity modulo ( tx ) (see ([19], Remark5.5)). If ρ ( X ) ε ( X ) β ( X ) − = ∗ ρ ∗ ( X ) ! , then ρ ∗ ( X ) is a (stably elementary) homotopy of σ . By ([24], Theorem 6.3), ρ ∗ ( X ) is an elementary matrix. NOTE ON RELATIVE VASERSTEIN SYMBOL 17
For any maximal ideal m of A , we have ρ ∗ ( X ) m ∈ SL ( A m [ X ] , ( t ) ∩ E ( A m [ X ]) . Hence by Lemma 5.17,we have ρ ∗ ( X ) m ∈ E ( A m [ X ] , ( t )) . Hence, by the Local-Global theorem for an extended ideal [1, Theorem4.5], we have ρ ∗ ( X ) ∈ E ( A [ X ] , ( t )) .Substituting X = 1 , gives σ ∈ E ( A, ( t )) . (cid:3) An alternative approach to Theorem 5.18 in case R is a non-singular affine algebra of dimension over aperfect C -field is as follows: Theorem 5.19. [9, Theorem 4.7]
Let R be an affine non-singular algebra of dimension d over a perfect C -field k , and I ⊂ R be a principal ideal of R . Let α ∈ SL d +1 ( R, I ) ∩ E ( R, I ) , then α ∈ E d +1 ( R, I ) . Corollary 5.20.
Let R be a non-singular affine algebra of dimension over a perfect C -field k . Then theVaserstein symbol V R [ X ] , ( X − X ) : M SE ( R [ X ] , ( X − X )) → W E ( R [ X ] , ( X − X )) is bijective. Proof : The surjectivity is clear from ( ii ) of Theorem 5.6. In view of ( iii ) of Theorem 5.6, V is injectiveif we can show that SL ( R [ X ] , ( X − X )) ∩ E ( R [ X ] , ( X − X )) = E ( R [ X ] , ( X − X )) . Which followsfrom Theorem 5.19. (cid:3) Theorem 5.21. ( [9, Theorem 4.8] ) Let A be a non singular affine algebra of dimension d over an algebraicallyclosed field k , d ≥ , d ! ∈ k ∗ and I = ( a ) a principal ideal of A . Then SL d ( A, I ) ∩ E ( A, I ) = E d ( A, I ) . Corollary 5.22.
Let R be a non-singular affine algebra of dimension over an algebraically closed field k , ∈ k ∗ . Then the map V R [ X ] , ( X − X ) : M SE ( R [ X ] , ( X − X )) → W E ( R [ X ] , ( X − X )) is injective. Proof : Start with the map s E : U m ( R [ X ] , ( X − X )) → W E ( R [ X ] , ( X − X )) . Let s E ( b ) = s E ( d ) ,for some b, d ∈ U m ( R [ X ] , ( X − X )) . In view of Theorem 5.6, the map s E is injective if b = dα for some α ∈ SL ( R [ X ] , ( X − X )) ∩ E ( R [ X ] , ( X − X )) . By Theorem 5.21, we have α ∈ SL ( R [ X ] , ( X − X )) ∩ E ( R [ X ] , ( X − X )) . But by Lemma 5.11, we have b, d are elementary equivalent relative to ( X − X ) . Hencethe map s E induces an injective map V R [ X ] , ( X − X ) : M SE ( R [ X ] , ( X − X )) → W E ( R [ X ] , ( X − X )) . (cid:3) Theorem 5.23. [9, Theorem 4.1]
Let R be an affine algebra of dimension d , d ≥ , over a perfect C -field k .Then U m d +1 ( R, I ) = e SL d +1 ( R, I ) , for any ideal I of R . Theorem 5.24.
Let R be an affine non-singular algebra of dimension over a perfect C -field k and I ⊂ R be a local complete intersection ideal of R . Then the map V R,I : M SE ( R, I ) → W E ( R, I ) is injective. Proof : In view of Lemma 5.9, it is enough to show that SL ( R, I ) ∩ E ( R, I ) = E ( R, I ) . Let α ∈ SL ( R, I ) ∩ E ( R, I ) . By classical stability estimates, we have ⊥ α ∈ E ( R, I ) . Hence there exists α ( X ) ∈ E ( R [ X ] , I [ X ]) such that α (0) = I and α (1) = 1 ⊥ α . Let v ( X ) = e α ( X ) . We have v ( X ) ∈ U m ( R [ X ] , ( X − X ) I [ X ]) . Since R [ X ] is affine algebra of dimension over perfect C -field k , we have v ( X ) ∈ e SL ( R [ X ] , ( X − X ) I [ X ]) by Theorem 5.23. By [19, Proposition 5.4], we have there exists ε ( X ) ∈ E ( R [ X ] , ( X − X ) I [ X ]) and β ( X ) ∈ SL ( R [ X ] , ( X − X ) I [ X ]) ∩ E ( R [ X ] , ( X − X ) I [ X ]) , suchthat v ( X ) ε ( X ) β ( X ) = e . Hence by arguing as in [20, Theorem 3.4], we have a relative stably elementary homotopy for α . Hence there exists α ( X ) ∈ SL ( R [ X ] , I [ X ]) ∩ E ( R [ X ] , I [ X ]) . By Local-Global principlefor an extended ideal, we may assume that R is a local ring with the maximal ideal m . Let S = R \ { } .Then we have α ( X ) S ∈ SL ( R S [ X ] , I S [ X ]) ∩ E ( R S [ X ] , I S [ X ]) = E ( R S [ X ] , I S [ X ]) . Hence there exists h ∈ R , such that α ( X ) h ∈ E ( R h [ X ] , I h [ X ]) . We may assume that h ∈ m . Further we may assume that h ∈ m . By Theorem 4.10, we have, there exists L , a subring of R such that(i) L ֒ → R is analytic isomorphism along h ′ for some h ′ ∈ hR ∩ L .(ii) L = K [ X , X , X ] ( ϕ ( X ) ,X ,X ) , where ϕ ( X ) ∈ K [ X ] is a monic irreducible polynomial.By Lemma 4.11, we have h and h ′ differ by a unit in R , hence therefore we may assume that h = h ′ .Therefore we have , L [ X ] ֒ → R [ X ] is analytic isomorphism along h .Further we may assume that the ideal I is an ideal of L and I is generated by a subset of the set { X , X } .This is possible since I is a local complete intersection. Also by Lemma 4.2, we have L [ X ] ⊕ I [ X ] ֒ → R [ X ] ⊕ I [ X ] is analytic isomorphism along ( h, .Let ˜ α ( X ) be a lift of α ( X ) in SL ( R [ X ] ⊕ I [ X ]) . Also we have ˜ α ( X ) ( h, ∈ E ( R h [ X ] ⊕ I h [ X ]) , i.e., ˜ α ( X ) ( h, ∈ E (( R [ X ] ⊕ I [ X ]) ( h, ) . Then by Lemma 4.11, we have there exists ˜ γ ( X ) ∈ E ( R [ X ] ⊕ I [ X ]) and ˜ β ( X ) ∈ SL ( L [ X ] ⊕ I [ X ]) such that ˜ α ( X ) = ˜ γ ( X ) ˜ β ( X ) .Now going modulo ⊕ I [ X ] , we have I = ¯˜ γ ( X ) ¯˜ β ( X ) , where ¯˜ γ ( X ) ∈ E ( R [ X ]) ⊆ E ( R [ X ] ⊕ I [ X ]) and ¯˜ β ∈ SL ( R [ X ]) ⊆ SL ( R [ X ] ⊕ I [ X ]) . Hence replacing ˜ γ ( X ) by ˜ γ ( X ) ¯˜ γ ( X ) − and ˜ β ( X ) by ¯˜ β ( X ) − ˜ β ( X ) ,we may assume that ˜ γ ( X ) ∈ SL ( R [ X ] ⊕ I [ X ] , ⊕ I [ X ]) ∩ E ( R [ X ] ⊕ I [ X ]) = E ( R [ X ] ⊕ I [ X ]) and ˜ β ( X ) ∈ SL ( L [ X ] ⊕ I [ X ] , ⊕ I [ X ]) .Hence projecting R [ X ] ⊕ I [ X ] onto R [ X ] and L [ X ] ⊕ I [ X ] onto L [ X ] we have α ( X ) = γ ( X ) β ( X ) , where γ ( X ) ∈ E ( R [ X ] , I [ X ]) and β ( X ) ∈ SL ( L [ X ] , I [ X ]) . Also we have β ( X ) h ∈ E ( L h [ X ] , I h [ X ]) .We may assume that h belongs to the maximal ideal of L . Also by multiplying unit of L we may assumethat h ∈ ( φ ( X ) , X , X ) . Now by Lemma 4.14, we have, there is a transformation of K [ X , X , X ] , namely X X , X i X i + φ ( X ) r i , i ≥ , for some r i such that h becomes a monic polynomial in X over K [ X , X ] .Let L ′ = K [ X , X ] ( X ,X ) [ X ] . Since the above transformation takes the maximal ideal ( φ ( X ) , X , X ) of K [ X , X , X ] to itself, then, the polynomial h ∈ L ′ is a Weierstrass polynomial. Hence by Lemma 4.4, wehave L ′ ֒ → L is analytic isomorphism along h .Hence we have L ′ [ X ] ֒ → L [ X ] is analytic isomorphism along h . Since β ( X ) h ∈ E ( L h [ X ] , I h [ X ]) ,then again by Lemma 4.11, there exists δ ( X ) ∈ E ( L [ X ] , I [ X ]) and θ ∈ SL ( L ′ [ X ] , I [ X ]) such that β ( X ) = δ ( X ) θ ( X ) and θ ( X ) h ∈ E ( L ′ h [ X ] , I h [ X ]) . Since I [ X ] is an extended ideal of L ′ [ X ] = ( k [ X , X ] ( X ,X ) [ X ])[ X ] ,and h is monic polynomial in X of L ′ [ X ] , then by Lemma 4.16, we have, θ ∈ E ( L ′ [ X ] , I [ X ]) .Hence β ( X ) ∈ E ( L [ X ] , I [ X ]) ⊆ E ( R [ X ] , I [ X ]) . Finally we have α ( X ) ∈ E ( R [ X ] , I [ X ]) .Hence by Local-Global principle for an extended ideal, we have α ( X ) ∈ E ( R [ X ] , I [ X ]) , where R isaffine non-singular algebra of dimension over perfect C -field k . Hence evaluating at X = 0 , we have α ∈ E ( R, I ) .Hence the map V R,I is injective. (cid:3)
NOTE ON RELATIVE VASERSTEIN SYMBOL 19
Theorem 5.25. [9, Theorem 4.4]
Let R be a non-singular affine algebra of dimension d ≥ over an alge-braically closed field k , / ( d − ∈ k and I ⊂ R be an ideal of R . Let v ∈ U m d ( R, I ) . Then v can betransformed to a factorial row ψ ( d − ( w ) = ( w , w , w . . . , w d − d − ) for some w = ( w , w , . . . , w d − ) ∈ U m d ( R, I ) by elementary operations relative to I . In particular U m d ( R, I ) = e SL d ( R, I ) . The proof of the following theorem is same as the proof of Theorem 5.24, only we have to use Theorem5.25, instead of Theorem 5.23.
Theorem 5.26.
Let R be a non-singular affine algebra of dimension over an algebraically closed field k , andlet I ⊂ R be a local complete intersection ideal of R . Then the Vaserstein symbol V R,I : M SE ( R, I ) → W E ( R, I ) is injective. The proof of the following theorem is same as the proof of Theorem 5.24.
Theorem 5.27.
Let R be a non-singular affine algebra of dimension over a perfect C -field k , and let I ⊂ R be a local complete intersection ideal of R . Then the Vaserstein symbol V R [ X ] ,I [ X ] : M SE ( R [ X ] , I [ X ]) → W E ( R [ X ] , I [ X ])) is injective. Kernel of the Vaserstein symbolTheorem 6.1. [11, Theorem 3.6]
Let R be a noetherian ring of dimension d , and I ⊂ R be an ideal of R .Then M SE d +1 ( R, I ) is an abelian group with the following operation:If [ v ] , [ w ] ∈ M SE d +1 ( R, I ) , choose representatives ( a , a , . . . , a d ) ∈ [ v ] , ( b , b , . . . , b d ) ∈ [ w ] with a i = b i for i ≥ , and choose p ∈ R such that a p ≡ a R + a R + · · · + a d R ) . Then [ w ] . [ v ] = [( a ( b + p ) − , ( b + p ) a , a , . . . , a d )] . Definition 6.2.
Let R be a ring and I ⊂ R be an ideal of R . Then excision ring Z ⊕ I is defined by the set { ( n, i ) : n ∈ Z , i ∈ I } with the following operations: (1) ( m, i ) + ( n, j ) = ( m + n, i + j ) (2) ( m, i )( n, j ) = ( mn, mj + ni + ij ) Theorem 6.3 (Excision theorem) . [11, Theorem 3.21] Let R be a ring and I ⊂ R be an ideal od R . Then for n ≥ , the natural maps M SE n ( Z ⊕ I, ⊕ I ) f → M SE n ( R, I ) defined by f ( mse ( v )) = mse ( f ( v )) and M SE n ( Z ⊕ I, ⊕ I ) i → M SE n ( Z ⊕ I ) defined by i ( mse ( v )) = mse ( v ) are bijective. Theorem 6.4 (van der Kallen) . [12, Theorem 4.1] Let R be a ring of stable dimension d , d ≤ n − , n ≥ . Then the universal weak Mennicke symbol wms : M SE n ( R ) → W M S n ( R ) is bijective and hence M SE n ( R ) has an abelian group structure. Remark 6.5.
Let R be a ring of stable dimension d , d ≥ and I be an ideal of R . Then by [11, 3.19] , wehave the maximal spectrum of Z ⊕ I is the union of finitely many subspaces of dimension ≤ d . Therefore byTheorem 6.4, we have M SE n ( Z ⊕ I ) has a group structure for n ≥ max { , d/ } . Hence we can say thegroup structure of M SE n ( R, I ) for n ≥ max { , d/ } via the excision theorem 6.3. Definition 6.6.
Let R be a ring with I ⊂ R be an ideal of R . We call the group structure of M SE n ( R ) givenby van der Kallen (Theorem 6.4) is nice if it satisfies the following ’coordinate-wise multiplication’ formula: [( b, a , . . . , a n )] . [( a, a , . . . , a n )] = [( ab, a , . . . , a n )] . Similarly we call the group structure of
M SE n ( R, I ) given by van der Kallen is nice if it satisfies thefollowing ’coordinate-wise multiplication’ formula: [( b, a , . . . , a n )] . [( a, a , . . . , a n )] = [( ab, a , . . . , a n )] . Theorem 6.7. [12, Theorem 2.2]
Let R be a commutative ring with sdim ( R ) ≤ n − , n ≥ and I be anideal of R . Let i, j be non-negative integers. Then for every g ∈ GL n + i ( R ) ∩ E n + i + j +1 ( R, I ) , there existmatrices u, v, w, M with entries in I and q with entries in R such that I i +1 + uq vwq I n − + M ! ∈ gE n + i ( R, I ) , I j +1 + qu qvw I n − + M ! ∈ E n + j ( R, I ) . Corollary 6.8.
Let R be a commutative ring of dimension d with orbit space M SE n ( R, I ) , n ≥ max { , d/ } has a nice group structure. Let σ ∈ SL n ( R, I ) ∩ E n +1 ( R, I ) , then [ e σ ] = 1 in M SE n ( R, I ) . Proof : Putting i, j = 0 in the Theorem 6.7, we have, uq vwq I d + M ! ∈ σE n ( R, I ) , qu qvw I d + M ! ∈ E n ( R, I ) .Hence in the orbit space M SE n ( R, I ) we have [ e σ ] = [(1 + uq, v )]= [(1 + uq, v )][1 + uq, q ]; since [(1 + uq, q )] is the identity = [(1 + uq, vq )]; since M SE n has nice group structure = [ e ] . (cid:3) Theorem 6.9.
Let R be a ring of dimension and I ⊂ R be an ideal of R . If the orbit space M SE ( R, I ) hasa nice group structure, then, the Vaserstein map V R,I induces a bijection of the map φ : U m ( R, I ) / { σ ∈ SL ( R, I ) ∩ E ( R, I ) } ≡ W E ( R, I ) . NOTE ON RELATIVE VASERSTEIN SYMBOL 21
Proof : Since dimension of R is , then φ is surjective by Theorem 5.6. Let v ∈ ker ( φ ) . Then there exists τ ∈ SL ( R, I ) ∩ E ( R, I ) such that τ T V ( v, w ) τ = χ . Since M SE ( R, I ) has a nice group structure, then,by Corollary 6.8, we have e τ is elementarily completable. Hence we have τ ε = 1 ⊥ ρ , with ρ ∈ SL ( R, I ) ∩ E ( R, I ) and ε ∈ E ( R, I ) . Now from [26, Chapter 5], we have vρε = e for some ε ∈ E ( R, I ) as desired. (cid:3) Theorem 6.10. [8, Theorem 4.2]
Let R be an affine algebra of dimension d over a perfect C -field k , with char k = 2 . Let I ⊂ R be an ideal of R . Then the group structure of M SE d +1 ( R, I ) is nice. Corollary 6.11.
Let R be an affine algebra of dimension over a perfect C - field k with char k = 2 , then theVaserstein map V R,I induces a bijection φ : U m ( R, I ) / { σ ∈ SL ( R, I ) ∩ E ( R, I ) } ≡ W E ( R, I ) . The Vaserstein symbol for Rees algebras and extended Rees algebrasDefinition 7.1.
Rees algebra and extended Rees algebras : Let R be a commutative ring of dimension d and I an ideal of R . Then the algebra R [ It ] := { n X i =0 a i t i : n ∈ N , a i ∈ I i } = ⊕ n ≥ I n t n is called the Rees algebra of R with respect to I . The extended Rees algebra of R with respect to I , denoted by R [ It, t − ] ,is defined by R [ It, t − ] := { n X i = − n a i t i : n ∈ N , a i ∈ I i } = ⊕ n ∈ Z I n t n where I n = R for n ≤ . Clearly the Rees algebra R [ It ] is a graded ring. The following graded version of Quillen’s Local-GlobalPrinciple is well-known: Theorem 7.2. ( [10, Theorem 4.3.11] ) Let S = S ⊕ S ⊕ S ⊕ ... be a commutative graded ring and let M be a finitely presented S-module. Assume that for every maximal ideal m of S , M m is extended from ( S ) m .Then M is extended from S . Lemma 7.3. ( [21, Lemma 3.1] ) Let R be a commutative ring and I, J ideals of R . Then the natural map φ : R [ It ] /JR [ It ] → ¯ R [ ¯ It ] , where ¯ R = R/J , ¯ I = ( I + J ) /J , defined by φ ( r + a t + a t + · · · + a n t n + JR [ It ]) = ¯ r + ¯ a t + · · · + ¯ a n t n , is an isomorphism. Theorem 7.4. ( [25, Theorem 1.3] ) Let R be a Noetherian ring of dimension d and I be an ideal of R . Thendimension of R [ It ] ≤ d + 1 . Moreover if I is not contained in any minimal primes of R , then dim ( R [ It ]) = d + 1 . Theorem 7.5. ( [21, Theorem 2.1] ) Let R be a Noetherian regular ring. Then R [ It ] is regular if and only if I = (0) or (1) or generated by single element. Theorem 7.6. ( [21, Theorem 4.2] ) Let R be a ring of dimension d and I ⊂ R an ideal of R . Then for n ≥ max { , d + 2 } , the natural map φ : GL n ( R [ It ]) /E n ( R [ It ]) → K ( R [ It ]) is an isomorphism. Theorem 7.7. [17, Theorem 5.1]
Let R be a commutative ring of dimension d and A be a ring lying be-tween R [ X ] and S − R [ X ] , where S is a multiplically closed set of non-zero-divisors in R [ X ] , then for n ≥ max { , d + 2 } , the natural map φ : GL n ( A ) /E n ( A ) → K ( A ) is an isomorphism. Corollary 7.8.
Let R be a ring of dimension d and I ⊂ R be an ideal of R . Then for n ≥ max { , d + 2 } , thenatural map φ : GL n ( R [ It, t − ]) /E n ( R [ It, t − ]) → K ( R [ It, t − ]) is an isomorphism. Proof : We have R [ t − ] ⊂ R [ It, t − ] ⊂ R ( t − ) , where R ( t − ) is the total quotient ring of R [ t − ] . Nowapply Theorem 7.7.Here we are giving an example of a -dimensional algebra for which the Vaserstein symbol is injectivethough the algebra is singular. Theorem 7.9.
Let R be an affine algebra of dimension and I ⊂ R be an ideal. Then the Vaserstein symbol V R [ It ] : M SE ( R [ It ]) → W E ( R [ It ]) is bijective. Proof : By Theorem 7.4 we have dimension of R [ It ] ≤ . Hence we have Sd ( R [ It ]) ≤ . By Theorem5.8, it is enough to show that SL ( R [ It ]) ∩ E ( R [ It ]) = E ( R [ It ]) . By Theorem 7.6, we have for n ≥ , thenatural map φ : GL n ( R [ It ]) /E n ( R [ It ]) → K ( R [ It ]) is an isomorphism. So in particular for n = 4 , we have SL ( R [ It ]) ∩ E ( R [ It ]) = E ( R [ It ]) . (cid:3) Notation 7.10.
Let R be a commutative ring and I ⊂ R be an ideal. Let S ⊂ R be a subring of R . Then wedenote the algebra S [ It ] by S [ It ] := { n X i =0 a i t i : n ∈ N , a i ∈ I i } = ⊕ n ≥ I n t n where I = S . S [ It ] is an algebra since the subring S induces an S module structure on R . Hence I is an S module. Lemma 7.11.
Let R be a commutative ring and I ⊂ R be an ideal. Let S ⊂ R be a subring of R . If thereexists some h ∈ S for which S ֒ → R is an analytic isomorphism along h , then S [ It ] ֒ → R [ It ] is an analyticisomorphism along h . Proof : Clearly h is a non-zero divisor of S [ It ] as well as of R [ It ] . Since S ֒ → R is an analyticisomorphism along h , then the natural map i : S → R induces an isomorphism between S/hS and
R/hR .Therefore the induced map ¯ i : S/hS → R/hR induces an isomorphism between ¯ S [ ¯ It ] and ¯ R [ It ] , where ¯ S = S/hS , ¯ R = R/hR and ¯ I = ( I + hR ) /hR . Consider the following diagram S [ It ] /hS [ It ] R [ It ] /hR [ It ]¯ S [ ¯ It ] ¯ R [ ¯ It ] ¯ i ∗ ψ φ ¯ i ∗ NOTE ON RELATIVE VASERSTEIN SYMBOL 23
The diagram is commutative. By Lemma 7.3 we have the map φ is an isomorphism. It can be shown that themap ψ is also an isomorphism. Therefore by commutativity of the above diagram we have ¯ i ∗ is an isomorphism. (cid:3) The same result holds for extended Rees algebra and the proof is same as Lemma 7.11 has.
Lemma 7.12.
Let R be a commutative ring and I ⊂ R be an ideal. Let S ⊂ R be a subring of R . If thereexists some h ∈ R for which S ֒ → R is an analytic isomorphism along h , then S [ It, t − ] ֒ → R [ It, t − ] is ananalytic isomorphism along h . Theorem 7.13.
Let R be a non-singular affine algebra of dimension over a field k and I ⊂ R be an ideal.Then the Vaserstein symbol V R [ It ] : M SE ( R [ It ]) → W E ( R [ It ]) is injective. Proof : By Lemma 5.8, it is enough to check that SL ( R [ It ]) ∩ E ( R [ It ]) = E ( R [ It ]) . Let α ∈ SL ( R [ It ]) ∩ E ( R [ It ]) . By Local Global principle (Theorem 7.2) we may assume that, R is a regular localring. First we assume that R is regular local algebra with a separating ground field K . Let S = R \ { } .By stability we have α S ∈ E ( S − R [ It ]) . Hence there exists g ∈ R such that α h ∈ E ( R g [ It ]) . We mayassume that g ∈ m , the maximal ideal of R . Further we may assume that g ∈ m . By Theorem 4.10, thereexists a subring L of R and an element h ∈ L ∩ gR such that L = K [ X , X , . . . , X d ] ( ϕ ( X ) ,X ,...,X d ) , where ϕ ( X ) is an irreducible monic polynomial, and L ֒ → R is an analytic isomorphism along h . By Lemma 7.11, L [ It ] ֒ → R [ It ] is an analytic isomorphism along h . Thus we have a patching diagram L [ It ] R [ It ] L h [ It ] R h [ It ] By Theorem 4.10, we have g and h are differed by a unit in R . Hence we have α h ∈ E ( R h [ It ]) . Henceby Lemma 4.11 we have there exists β ∈ SL ( L [ It ]) with β h ∈ E ( L h [ It ]) and γ ∈ E ( R [ It ]) such that α = γβ . Therefore it is enough to show that β ∈ E ( L [ It ]) .We may assume that h belongs to the maximal ideal of L . Also by multiplying unit of L we may assumethat h ∈ ( φ ( X ) , X , ..., X d ) . Now by Lemma 4.14, we have, there is a transformation of K [ X , X , ..., X d ] ,namely X X , X i X i + φ ( X ) r i , i ≥ , for some r i such that h becomes a monic polynomial in X over K [ X , X , ..., X d ] .Let L ′ = K [ X , X , ..., X d ] ( X ,X ,...,X d ) [ X ] . Since the above transformation takes the maximal ideal ( φ ( X ) , X , ..., X d ) of K [ X , X , ..., X d ] to itself, then, the polynomial h ∈ L ′ is a Weierstrass polynomial.Hence by Lemma 4.4, we have L ′ ֒ → L is analytic isomorphism along h .Hence we have L ′ [ It ] ֒ → L [ It ] is analytic isomorphism along h . Since β h ∈ E ( L h [ It ]) , then againby applying Lemma 4.11, we have, there exists δ ∈ E ( L [ It ]) and θ ∈ SL ( L ′ [ It ]) such that β = δθ and θ h ∈ E ( L ′ h [ It ]) . Since L ′ [ It ] = ( k [ X , X , ..., X d ] ( X ,X ,...,X d ) [ It ])[ X ] and h is monic polynomial in X of L ′ [ It ] , then by Proposition 4.16, we have, θ ∈ E ( L ′ [ It ]) .Hence β ∈ E ( L [ It ]) ⊆ E ( R [ It ]) . Finally we have α ∈ E ( R [ It ]) .For the arbitrary ground field, we follow the following treatment suggested by Swan, given in [14]. Let R = C p where C = k [ X , X , . . . , X m ] / ( f , f , . . . f t ) , p is a prime ideal of C . Let k be theprime subfield of k . Choose a field extension K of k such that K is finitely generated over k and K contains all coefficients of the f i ’s and all the elements of k such that α is defined over K . Set B = K [ X , X , . . . X m ] / ( f , f , . . . , f t ) , q = p ∩ B , R ′ = B q . We have B ֒ → B ⊗ K k = C . Now note that R is a flat extension of C ; C is a flat(free) extension of B . Hence R is flat over B . Hence R is flat over R ′ .Since R ′ is local ring, then R is faithfully flat over R ′ . Moreover R ′ ⊂ R is a local extension, i.e, m R ′ ⊂ m R .Hence R ′ is a regular K -spot by Proposition 4.15. Hence R ′ is a regular local algebra with a separating groundfield K by Lemma 4.13. Hence we have α ∈ SL ( R ′ [ It ]) ∩ E ( R ′ [ It ]) with R ′ is a regular local algebra witha separating ground field. This reduces to the case we have already considered. (cid:3) Theorem 7.14.
Let R be a regular k -spot of dimension d , d ≤ , and I ⊂ R be an ideal. Then The Vasersteinsymbol V R [ It,t − ] : M SE ( R [ It, t − ]) → W E ( R [ It, t − ]) is injective. Proof : By Lemma 5.8, it is enough to check that SL ( R [ It, t − ]) ∩ E ( R [ It, t − ]) = E ( R [ It, t − ]) . Let α ∈ SL ( R [ It, t − ]) ∩ E ( R [ It.t − ]) . First assume that R is a regular local algebra with a separating ground field K .Let S = R \{ } . By stability we have α S ∈ E (( S − R )[ It, t − ]) since dim ( S − R )[ It, t − ] ≤ . Hence thereexists g ∈ R such that α h ∈ E ( R g [ It, t − ]) . We may assume that g ∈ m , the maximal ideal of R . Further wemay assume that g ∈ m . By Theorem 4.10, there exists a subring L of R and an element h ∈ L ∩ gR suchthat L = K [ X , X , . . . , X d ] ( ϕ ( X ) ,X ,...,X d ) , where ϕ ( X ) is an irreducible monic polynomial, and L ֒ → R is an analytic isomorphism along h . By Lemma 7.12, L [ It ] ֒ → R [ It, t − ] is an analytic isomorphism along h .Thus we have a patching diagram L [ It, t − ] R [ It, t − ] L h [ It, t − ] R h [ It, t − ] By Theorem 4.10, we have g and h are differed by a unit in R . Hence we have α h ∈ E ( R h [ It, t − ]) . Henceby Lemma 4.11 we have there exists β ∈ SL ( L [ It, t − ]) with β h ∈ E ( L h [ It, t − ]) and γ ∈ E ( R [ It, t − ]) such that α = γβ . Therefore it is enough to show that β ∈ E ( L [ It, t − ]) .We may assume that h belongs to the maximal ideal of L . Also by multiplying unit of L we may assumethat h ∈ ( φ ( X ) , X , ..., X d ) . Now by Lemma 4.14, we have, there is a transformation of K [ X , X , ..., X d ] ,namely X X , X i X i + φ ( X ) r i , i ≥ , for some r i such that h becomes a monic polynomial in X over K [ X , X , ..., X d ] .Let L ′ = K [ X , X , ..., X d ] ( X ,X ,...,X d ) [ X ] . Since the above transformation takes the maximal ideal ( φ ( X ) , X , ..., X d ) of K [ X , X , ..., X d ] to itself, then, the polynomial h ∈ L ′ is a Weierstrass polynomial.Hence by Lemma 4.4, we have L ′ ֒ → L is analytic isomorphism along h .Hence we have L ′ [ It, t − ] ֒ → L [ It, t − ] is analytic isomorphism along h . Since β h ∈ E ( L h [ It, t − ]) ,then again by applying Lemma 4.11, we have, there exists δ ∈ E ( L [ It, t − ]) and θ ∈ SL ( L ′ [ It, t − ]) suchthat β = δθ and θ h ∈ E ( L ′ h [ It, t − ]) . Since L ′ [ It, t − ] = ( k [ X , X , ..., X d ] ( X ,X ,...,X d ) [ It, t − ])[ X ] and h is monic polynomial in X of L ′ [ It, t − ] , then by Theorem 4.16, we have, θ ∈ E ( L ′ [ It, t − ]) . NOTE ON RELATIVE VASERSTEIN SYMBOL 25
Hence β ∈ E ( L [ It, t − ]) ⊆ E ( R [ It, t − ]) . Finally we have α ∈ E ( R [ It, t − ]) .The case for the arbitrary ground field can be reduced to the case of the separating ground field by the sameargument as given at the end of the proof of Theorem 7.13. (cid:3) Theorem 7.15.
Let R be a ring of dimension and I ⊂ R be an ideal of R . Then the Vaserstein symbol V R [ It,t − ] : M SE ( R [ It, t − ]) → W E ( R [ It, t − ]) is bijective. Proof : Since dim R [ It, t − ] ≤ , we have V R [ It,t − ] is surjective. For injectivity we must show that SL ( R [ It, t − ]) ∩ E ( R [ It, t − ]) = E ( R [ It, t − ]) . By Corollary 7.8, we have the natural map φ : GL n ( R [ It, t − ]) /E n ( R [ It, t − ]) → K ( R [ It, t − ]) is an isomorphism for n ≥ . Hence for n = 4 ,we have SL ( RIt, t − ) ∩ E ( R [ It, t − ]) = E ( R [ It, t − ]) . (cid:3) R EFERENCES[1] H. Apte, P. Chattopadhyay, R. Rao;
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