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uthor's personal copy ournal of Pure and pplied lgebra 216 (2012) 2159 2169 Contents lists available at SciVerse ScienceDirect ournal of Pure and pplied lgebra journal homepage: www.elsevier.com/locate/jpaa Excision in algebraic obstruction theory Satya Mandal a,, Yong Yang b a Department of Mathematics, University of Kansas, 1460 ayhawk Blvd., Lawrence, KS 66045, US b Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan, 410073, PR China a r t i c l e i n f o a b s t r a c t rticle history: Received 20 une 2011 Received in revised form 20 November 2011 vailable online 24 ebruary 2012 Communicated by R. Parimala MSC: 13c10 n this paper the relative algebraic obstruction groups (also known as Euler class groups) were defined and some excision exact sequences were established. n particular, for a regular domain, essentially of finite type over an infinite field k, and a rank one projective -module L 0, it was proved that E n ([T], L 0 [T]) E n (, L 0 ) whenever 2n dim + 4. 2012 Elsevier B.V. ll rights reserved. 1. ntroduction n topology, there is a well established obstruction theory for vector bundles (see [11]). germ for a parallel obstructions theory for projective modules over noetherian commutative rings was given by M.V. Nori around 1990 (see [6,8]). Originally, this program was focused on the top rank case. or projective modules P of rank d, over noetherian commutative rings with dim = d, an obstruction class e(p) was defined. Bhatwadekar and Sridharan [3] proved that e(p) = 0 if and only if P has a free direct summand. theory of obstructions for all projective modules, as complete as that of topological vector bundles, is possible. n fact, a parallel K -theoretic approach was initiated by Barge and Morel [1] in 2000. This was given a more complete shape by asel [5]. However, this approach does not seem very descriptive and two approaches must reconcile. n this paper, we are concerned with the approach of Nori. ollowing [4], obstruction groups E n (, L) were defined in [9], for all integers n 1 and rank one projective -modules L. There has been only a limited success in defining obstruction classes e(p) E n (, L) for projective -modules P with rank(p) = n < dim and det(p) L, as would be desired. When det P L, we say that P is oriented and the situation is referred to as the oriented case. Otherwise, it is referred to as the non-oriented case. Recently, in the oriented case, Yang [12] defined relative obstructions groups E n (,, ), with respect to ideals of, when n 1. When 2n dim + 3, he established some exact sequences of these groups. The purpose of this paper is to extend the results of Yang [12] to the nonoriented case. s in [12], first we define pull-back homomorphisms f : E n (, L) E n (B, B L) of the obstructions groups, corresponding to certain ring homomorphisms f : B and some integers n. Then, we define the relative obstruction groups, E n (,, L) (see 4.1), and establish some exact sequences in Theorems 4.2 and 4.3. n particular, we establish an excision exact sequence as follows. Theorem 1.1. Let be noetherian commutative ring with dim() = d and be an ideal of. Write 0 =. ssume that the quotient homomorphism q : 0 has a splitting β : 0 such that for each locally n-generated ideal 0 of 0, of height n, we have height(β( 0 )) n. Suppose L 0 is a projective 0 -module of rank one and L = L 0 β. Then, for integers n with 2n d + 3, we have a split exact sequence as follows: 0 E n (,, L) E n (, L) E n (, L 0) 0. Corresponding author. E-mail addresses: mandal@math.ku.edu (S. Mandal), yangyong@nudt.edu.cn (Y. Yang). 0022-4049/$ see front matter 2012 Elsevier B.V. ll rights reserved. doi:10.1016/j.jpaa.2012.02.008

uthor's personal copy 2160 S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 s an application, we prove that if is a regular domain, essentially of finite type over an infinite perfect field k, and L 0 is a projective -module of rank one, then E n ([T], L 0 [T]) E n (, L 0 ) whenever 2n dim + 4. The authors would like to thank the referee for careful reviewing and many valuable suggestions. The exposition of the paper improved due to these suggestions. The Remark 3.6 due to the referee is particularly appreciated. 2. Preliminaries or the definition of the obstructions groups the readers are referred to [9]. We recall some notations from [9]. Notations 2.1. Throughout this paper, will denote a commutative noetherian ring with dim = d and L will denote a projective -module of rank one. (1) or integers n 1, write = n = L n 1. (2) local L-orientation (of codimension n) is an equivalence class of surjective homomorphisms ω : / / 2, where is an ideal of height n. When it is clear from the context, we just call them orientations. (3) The free abelian group generated by the local orientations (, ω), where the ideal is connected, is denoted by G n (, L). (4) The obstruction group of codimension n is defined by E n (, L) := Gn (,L), where R is the subgroup generated by the R global orientations. (5) local orientation ω : / / 2, defines an element (, ω) G n (, L). We will take the liberty to use the same notation (, ω) to denote its image in E n (, L). The following lemma is proved by using standard basic element theory along with generalized dimensions (see [7]). Lemma 2.2. Suppose is a noetherian commutative ring with dim = d. Suppose, are two ideals of with height() = n and 2. Let be a projective -module of rank n and ω : / be a surjective homomorphism. lso suppose 1,..., r are finitely many ideals of. Then there is a surjective lift f : K, such that (1) + K =, (2) height(k) n and (3) for K+ 1 i r, height i i n. Proof. Similar to [9, Lemma 4.3]. Lemma 2.3. Suppose is a noetherian commutative ring with dim = d. Suppose, are two ideals of and is a projective -module of rank n. Let ω : / 2 and ϕ : + be two surjective homomorphisms such that ω + = ϕ + : ( + ) 2 +. Then, there is a surjective homomorphism Ω : Proof. Consider the fiber product diagrams: that lifts both ω and ϕ. 2 ( ) Ω 2 ω 2 + and ϕ + ( + ). 2 + By the properties of fiber product diagrams, the desired homomorphism Ω is defined in the later diagram. This completes the proof. ollowing is the non-oriented version of the theorem of Bhatwadekar and Sridharan [4, Theorem 4.2].

uthor's personal copy S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 2161 Theorem 2.4. Let be a noetherian commutative ring with dim = d and 2n d + 3. Let L be a projective -module of rank one and n = n 1 L. Let be an ideal of height n and ω : n / 2 be a local L-orientation. ssume n (, ω) = 0 E n (, L). Then there is a surjective lift θ : n of ω. Proof. Similar to that of [4, Theorem 4.2]. 2.1. Double of a ring n this subsection, we define the Double of a ring along an ideal and summarize some of the facts about it. Let be a noetherian commutative ring with dim = d and be an ideal of. The double of along the ideal, is defined as D = D(, ) = {(x, y) : x y }. This will be considered as a subring of. The projection to the first and second coordinates from D will be denoted by p 1 and p 2. Similarly, for an -module M, the double of M along the ideal, is defined as D(M, ) = {(m, n) M M : m n M}. The following are some facts: (1) The following diagrams D(, ) p 1 D(M, ) p 1 M p 2 q q and p 2 M M M are fiber product diagrams, where q denotes the quotient homomorphism. (2) The kernel(p 1 ) = 0 and kernel(p 2 ) = 0. (3) n fact, the diagonal homomorphism : D splits both p 1, p 2. (4) We have D = ()+0 = ()+ 0. So, D = ()+ ()(0, x i ), where = x i. So, D is finitely generated -module. (5) So, D is noetherian and integral over. Therefore, dim = dim D. (6) or any ideal of, we have and p 1 1 () = {(x, x + z) : x, z } = () + 0 p 1 2 () = {(x + z, x) : x, z } = () + 0. (7) f Spec(), then P = ( ) + 0 = p 1 1 Lemma 2.5. Let,, L, M, D, be as above. Then, ( ) Spec(D). Similarly, P = ( ) + 0 = p 1 2 ( ) Spec(D). (1) There is a natural surjective homomorphism τ : D(, ) M D(M, ), where D is considered as an -algebra via the diagonal. (2) f Q is projective, then τ : D(, ) Q D(Q, ). Proof. t is easy to see that τ(m (x, x + z)) = (xm, (x + z)m) is a well defined homomorphism from D M D(M, ). This establishes (1). f Q = n is free, then it is obvious that τ : D(, ) n D( n, ). n the general case, note that there is a split exact sequence: 0 Q g f P 0

uthor's personal copy 2162 S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 where is free. Correspondingly, we have the following commutative diagram: 0 D(, ) Q D(, ) D(, ) P 0 0 D(Q, ) γ τ D(, ) ϕ τ D(P, ) 0. Here the rows are exact, while one needs a proof that the bottom row is exact. t is easy to see that γ is injective and ϕ is surjective and ϕγ = 0. Suppose ϕ(m, m + z) = 0 for some m, z. Then f (m) = f (z) = 0. Therefore, g(u) = m and g(v) = z for some u, v Q. Let ϵ : Q be a splitting of g. Then v = ϵg(v) = ϵ(z) Q. Hence (u, u + v) D(Q, ) and γ (u, u + v) = (m, m + z). This establishes that the bottom row is exact. Since the middle vertical map is an isomorphism, the first vertical map is injective. This completes the proof. (lternately, one could use the Snake lemma to prove the same.) The following lemma will be of our interest subsequently. Lemma 2.6. With the notations as above, let be an ideal of the double D = D(, ). f height() = n, then height(p 1 ()) n. Proof. Suppose p 1 () Spec(R) are minimal. We will prove that height( ) n. We have p 1 1 (p 1()) p 1 ( ). 1 There is a prime ideal P Spec(D) such that P is minimal over and P p 1 1 ( ). Write m = height(p). Then m n = height(). Let P 0 P 1 P 2 P m = P be a strictly increasing chain of primes in Spec(D). Write, i = P i R = 1 (P i ). Since, R D is integral, there is no inclusion relationship between two primes in D over the same prime Spec(R) (see [10, Theorem 9.3, pp. 66]). So, 0 1 2 m is a strictly increasing chain in Spec(R). urther, m p 1 (P) p 1 (p 1 1 ( )) =. Therefore, height( ) m n. The proof is complete. 3. Pull-back and functoriality n this section, we define some pull-back homomorphisms of the obstruction groups, corresponding to some suitable ring homomorphisms. irst one will correspond to the quotient homomorphisms q : /. Definition 3.1. Let be a noetherian commutative ring with dim = d and be an ideal. Let L be a rank one projective -module. or integers n, with 2n d + 3, there is a group homomorphism ρ = ρ : E n (, L) E n, L L defined as follows: Write = L n 1. Let ω : / / 2 be local L-orientation. We can find an ideal 1 and a local orientation ω 1 : / 1 1 / 2 1 such that (, ω) = ( 1, ω 1 ) and height 1 + n. Then, ω 1 induces an orientation β as in the following commutative diagram: / 1 ω 1 1 / 2 1 ( 1 + ) β 1 + 2 1 +. Define 1 + ρ(, ω) =, β. We use the notations q = E n (q) = ρ = ρ, corresponding to notation for the quotient map q : /. This homomorphism will be called a pull-back homomorphism.

uthor's personal copy S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 2163 Proof that ρ Well Defined. irst, we define a homomorphism ϕ : G n (, L) E n, L 1 + by ϕ(, ω) =, β E n L, L, L where β is as above. Two different representatives of ω 1 leads to the same, because transvections of / 1 will induce transvections of 1 /( + 1 ). irst, we prove that if ( 1, ω 1 ) is global, then so is the image. f it is global, then ω 1 lifts to a surjective homomorphism f : 1. Then, f induces a surjective lift g of β, as demonstrated by the following diagram: f 1 / g 1 + / 1 ω 1 1 / 2 1 ( 1 + ) β 1 +. 2 1 + This establishes that β is global. Now, suppose (, ω) = ( 1, ω 1 ) = ( 2, ω 2 ) E n (, L), with height( 1 /) = height( 2 /) = n. There is an ideal K and a surjective lift f : 1 K of ω 1, such that 1 + K =, height(k) n. Since 2n > d, we can also assume K + 2 =. Let ω K : /K K/K 2 be induced by f. We have, ( 1, ω 1 ) + (K, ω K ) = ( 2, ω 2 ) + (K, ω K ) = 0. By Theorem 2.4, there is a surjective homomorphism f 2 : 2 K that lifts ω 2 and ω K. So, both ( 1 K, ω 1 ω K ), ( 2 K, ω 2 ω K ) are global. Since the image of a global orientation is global, the images of both ( 1, ω 1 ), ( 2, ω 2 ) are negative of that of (K, ω K ). Therefore, the homomorphism ϕ is well defined. gain, since image of a global orientation is global, ϕ factors through a homomorphism ρ : E n (, L) E n, L. L This completes the proof that ρ is well defined. Proposition 3.2. Let,, d, n, L be as in Definition 3.1. Let 1 be another ideal. Then the diagram E n ρ (, L) E n, L L ρ +1 E n ρ (1 +)/ L + 1, (+ 1 )L commutes. Proof. Write ρ 0 = ρ +1, ρ = ρ, ρ 1 = ρ /(+1 ). Write = + 1 and = L n 1. Let ω : / / 2 be a local L- orientation. We can find 1 and a local L-orientation ω 1 : / 1 1 / 2 1 such that (1) (, ω) = ( 1, ω 1 ), (2) height 1 + n and (3) height 1 + + 1 + 1 = height 1 + n. We have, 1 + 1 + + 1 ρ 1 ρ(, ω) = ρ 1, β =, γ = ρ 0 (, ω) + 1

uthor's personal copy 2164 S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 where β, γ are induced by ω 1, according to the following commutative diagram: / 1 ( 1 + ) ( 1 + + 1 ) ω 1 1 / 2 1 β 1 + 1 + + 1 2 1 + 2 1 + + 1 γ. This completes the proof. Next, we define pull-back homomorphisms of the obstruction groups, corresponding to some suitable ring homomorphisms R. Definition 3.3. Suppose f : R is a homomorphism of two noetherian commutative rings, with dim R = d 1, dim = d 2. Let n 1 be an integer. or an ideal of R, we will denote := f (). ssume that for any ideal of R, which is locally generated by n elements and height() = n, we have height() n. Let L be a projective R-module of rank one and L = L. Then, there is a homomorphism f = E(f ) = E n (f ) : E n (R, L) E n (, L ) defined below. This homomorphism will be called a pull-back homomorphism. To define E n (f ), we proceed as follows. Write = n = L R n 1, =. Let be an ideal of R of height n and ω : / / 2 be a local L-orientation. Write =. Then ω induces a local L -orientation ω by the following commutative diagram: / ω / 2 ( ) ω 2. The association (, ω) (, ω ) E n (, L ) defines a group homomorphism ϕ 0 : G n (R, L) E n (, L ). f (, ω) is global, ω lifts to a surjection Ω :. t is easy to see that Ω d induces a surjective lift Ω : of ω. So, ϕ 0 ((, ω)) = 0. Therefore, ϕ 0 factors through a homomorphism E n (f ) : E n (R, L) E n (, L ). This completes the definition of E n (f ) and establishes that it is well defined. Proposition 3.4. Let f : R, g : B be homomorphisms of noetherian commutative rings of finite dimension, satisfying the properties of Definition 3.3. Let L be a rank one projective R-module. Then the diagram commutes. E(f ) E n (R, L) E E(gf ) n (, L ) E(g) E n (B, L B) Proof. Obviously from the Definition 3.3. Proposition 3.5. Let f : R be a homomorphism of commutative noetherian rings of finite dimension, and L be a rank one projective -module. Let r, s 1 be two integers be such that r 2, if L. ssume that f satisfies the properties of Definition 3.3 for n = r, s, r + s. Then, for x E r (, L), y E s (, ), we have E(f )(x) E(f )(y) = E(f )(x y) E r+s (, L ), where is defined as in [9, Theorem 4.5]. Proof. Write = L r 1, = s. Using bilinearity, we can assume that x = (, ω) and y = (, ω ), where, are ideals with height() = r, height() = s and ω : / / 2, ω : / / 2 are local orientations. We can assume that height( + ) = r + s. So, x y = ( +, ω 0 ) where ω 0 : is induced by ω, ω. ( + ) ( + ) ( + ) 2

uthor's personal copy S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 2165 Since f : R satisfies properties of Definition 3.3 for n = r, s, r + s the proposition follows by chasing the definitions. Remark 3.6. The referee points out that the Definitions 3.1 and 3.3 can be combined to define pull-back homomorphisms for more general ring homomorphisms f : B. or example, assume that both, B contain a field k and f is a k-algebra homomorphism. Then, the projection homomorphism p : k B is flat, since B is flat over k and it satisfies the conditions of Definition 3.3. So, for integers n 1 and any rank one projective -module L, the pull-back p : E n (, L) E n ( k B, L k B) is defined. Now consider the surjective homomorphism q : k B B given by q(x y) = f (x)y. Then f = qp. f 2n dim( B) + 3 then the pull-back q : E n ( B, L k B) E n (B, L B) is defined. So, the pull-back f := q p : E n (, L) E n (B, L B) is defined. (n fact, the argument works, if k is any commutative ring and k B is flat.) The authors are thankful to the referee for this comment. 4. Exact sequences and excision n this section, first we define the relative obstruction groups and then establish the exact sequences. Definition 4.1. Let be a commutative noetherian ring and be an ideal of. Let L be a projective -module with rank(l) = 1. Let n 1 be an integer. With D = D(, ) and other notations as in Section 2.1, by Lemma 2.6 the pullback homomorphism E(p 1 ) : E n (D, D L) E n (, L) is well defined. The relative obstruction groups, E n (,, L) are defined as E n (,, L) = Kernel (E(p 1 )). The following theorem establishes an exact sequence. Theorem 4.2. Let be a noetherian commutative ring with dim() = d and be an ideal. s before, p 1, p 2 : D(, ) will denote the projections to the first and second coordinates. Suppose L is a projective -module of rank one. or integers n with 2n d + 3, the following E n (,, L) E(p 2) E n (, L) ρ E n (, L L ) is an exact sequence. Proof. Let q : / denote the quotient homomorphism and = L n 1. irst, we prove ρe(p 2 ) = 0. Suppose (W, ω) E n (,, L) E n D (D, D L) where ω : W(D ) W/W 2 is a local orientation, with ideal W D of height n. W+0 By Lemma 2.2, we can assume that height 0 n and W+ 0 height 0 n. Since D p1 :, 0 p1 (W) + W + 0 height = height n. 0 Write 1 = p 1 (W), and 2 = p 2 (W). Since, qp 1 = qp 2, we have, 1 + = 2 +. or i = 1, 2 ω induces ω i, β i by the commutative diagram: D W(D ) p i i ( i +) ω ω i W/W 2 p i i / 2 i β i i + 2 i +. Then, for i = 1, 2 we have E(p i )(W, ω) = ( i, ω i ) E n (, L). t follows, β 1 = β 2. Since height is consistent, 2 + 1 + ρ E(p 2 )(W, ω) =, β 2 =, β 1 = ρ E(p 1 )(W, ω) = 0. Conversely, suppose x E n (, L), are such that ρ (x) = 0. Since, 2n > d, we can write x = (, ω) where is an ideal of height n and ω : / / 2 + is a local orientation. We can further assume that height n. Now, ω induces

uthor's personal copy 2166 S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 ω as in the following commutative diagram: ω / / 2 (+) ω + 2 +. Since ρ(x) = +, ω = 0, we have ω lifts to a surjection f : /( ) f / 2 / ω / 2 +. We have the following fiber product diagram: / f /( + ) ω ( + )/ ( + )/( 2 + ) where f is defined by properties of fiber product diagrams. By Lemma 2.3, there is a surjective lift ϕ : K, of f, such that (1) K + 2 =, (2) height(k) K+ n and (3) height n. Now, ϕ defines an element (K, ω K ) = (, ω) E n (, L). Define, W = (K) + 0 = {(x, y) D : y K}. n fact, W is defined by the fiber product of and K as follows: W D p 1 = K + p 2 K K+ = /. By Lemma 2.5, D = D(, ). We consider the following fiber product diagram: p 1 D ω W 0 p 2 W/W 2 p 1 0 / 0 ω K K/K 2 0. Here ω K is defined by properties of fiber product diagrams. (lternately, one can prove elementwise that W/W 2 K/K 2 is an isomorphism, using the fact that x + y = 1 for some x K 2, y 2 and (1, x) = (x, x) + (y, 0) W 2.) We consider ω W as an orientation. t follows E(p 2 )(W, ω W ) = (K, ω K ) = (, ω) = x. So, x imagee(p 2 ). Since p 1 (W) =, we have (W, ω W ) ker(e(p 1 )) = E n (, L, ). This completes the proof of (4.2). With further conditions, we extend the above sequences as follows. Theorem 4.3. Use notations as in Theorem 4.2 and assume 2n d+3. Write 0 =. ssume that the quotient homomorphism q : 0 has a splitting β : 0 and L = L 0 β 0 for some rank one projective 0 -module L 0. (1) Then, the sequence 0 E n (,, L) E(p 2) E n (, L) ρ E n (, L L ) is exact.

uthor's personal copy S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 2167 (2) ssume that, for each locally n-generated ideal 0 of 0, of height n, we have height(β( 0 )) n. Then, the sequence E n (,, L) E(p 2) E n (, L) ρ E n (, L L ) 0 is exact and ρ splits. Proof. irst, we prove (2). Because of Theorem 4.2, we only need to prove that ρ is split-surjective. Since E(β) is defined, it is easy to check that ρ E(β) = d. So, ρ is surjective and splits. gain by Theorem 4.2, to prove (1), we need to prove that E(p 2 ) is injective on E n (,, L). Let x E n (,, L) E n (D, L D) such that E(p 2 )(x) = 0. Since 2n > d = dim(d), we can assume x = (W, ω), where = L n 1 and W is an ideal of D with height(w) n and ω : D W( D) W W 2 is a local orientation. Since L = L 0 β, we have L 0 = L 0. We will use the notation = D. Consider the following fiber product diagram: (W ( )) f W W 2 ( ) W ω W W 2 ( ) f W+( ) ( ) (W+( )) ω D W+( ) W 2 +( ). n this diagram, f is a surjective lift of ω D, which exists, by Theorem 2.4, because W+, ω D = ρe(p 1 )(W, ω) = 0. lso, f is given by the properties of fiber product diagrams and is surjective. By Lemma 2.2, f lifts to a surjection, Ω : W K where K is an ideal with (1) W + K = D, (2) height(k) n, (3) K+0 height 0 n (4) height K+ 0 0 n (5) K + = D. Ω induces a local orientation ω K : /K K/K 2. We have (, ω) + (K, ω ) = 0. Write K 1 = p 1 (K), K 2 = p 2 (K 2 ). Note q(k 1 ) = q(k 2 ) = image(k) = 0, and height(k i ) n for i = 1, 2. or i = 1, 2, let ω i : /K i K i /K 2 i be induced by ω K. The following are some observations: (1) Claim: p 1 1 (K 1) = K + 0 and p 1 2 (K 2) = K + 0. To see this, note K + 0 p 1 (K). 1 Conversely, let (x, y) p 1 (K). Then, (x, y ) K for some y. Therefore, (x, y) = (x, y ) + (0, (y y )) = (x, y ) + (0, (y x) + (x y )) K + 0. This establishes the claim. (2) Since K + = D we have K i + = for i = 1, 2. (3) lso E(p i )(K, ω K ) = (K i, ω i ) = 0 for i = 1, 2. Therefore, by Theorem 2.4, there are surjective lifts Ω i : K i of ω i. Write 0 = L 0 n 1 0. Then, 0 = L 0 n 1 0 0. Since K i + =, for i = 1, 2, we have, Ω i 0 surjects on to 0. Write Ω 0 i = Ω i 0 : 0 0 for i = 1, 2. Write 1 = {(a, a + z) : a K 1, z } and 2 = {(b + z, b) : b K 2, z }. Both 1 and 2 are ideals of D. Consider the following two fiber product diagrams: p 2 p 1 Γ 1 1 Ω 1 K 1 p 1 Γ 2 p 2 2 Ω 0 2 β 0 Ω 0 1 0 Ω 0 2 Ω 0 1 β 0 Ω 2 K 2 0.

uthor's personal copy 2168 S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 Here Γ 1, Γ 2 are obtained by the properties of fiber product diagrams and they are surjective by the same. We gather some facts below: (1) We have 1 + 2 = D. Proof. We have K + = D. So, (x, y) + (u, v) = (1, 1) for some (x, y) K and u, v. So, (1, y) = (y + v, y) 2 and (x, 1) = (x, x + u) 1. Therefore, (1, 1) = (1, y) + (x, 1)(0, v) 1 + 2. This establishes 1 + 2 = D or (1). (2) We have K = 1 2. Proof. Clearly, for (a, b) K, we have (a, b) = (a, a + (b a)) 1. Similarly, (a, b) 2. Therefore, K 1 2. Now, let (a, b) 1 2. Looking at the description of i, it follows a K 1 and b K 2. So, (a, c) K for some c. So, (a, b) = (a, c) + (0, b c). Since (a, c) K 1 2, we have (0, b c) 1 2. Therefore, b c K 2 = K 2. So, b c = y i z i for some (x i, y i ) K and z i. Hence (a, b) = (a, c) + (0, b c) = (a, c) + (x i, y i )(0, z i ) K. This establishes (2). (3) Since K = 1 2, we have height( i ) n. So, Γ i induce local orientation γ i : / i i / 2 i for i = 1, 2. ndeed, they are global. So, ( 1, γ 1 ) = ( 2, γ 2 ) = 0 E n (D, L D). (4) n fact, ω K D/ i = γ i for i = 1, 2. Proof. We will check for i = 1. We will check that Ω(m) Γ 1 (m) 2, 1 for m. Since K 2 + 1 2 =, we have x + c = 1 for some (x, y) K 2 and c 2. So, (x, 1) = (x, y) + (0, (1 x) + (x y)) 2 1. Therefore, for any z we have (0, z) = (x, 1)(0, z) 2. 1 Let m and Γ 1 (m) = (a, b), Ω(m) = (x, y). Since the first coordinates of both agree in K 1 /K 2 1, we have c = x a K 2. 1 Therefore, (c, d) K 2 2 1 for some d. So, (x, y) (a, b) = (c, y b) = (c, d) + (0, y b d) 2 1. This establishes (4). t follows from above, (K, ω K ) = ( 1, γ 1 ) + ( 2, γ 2 ) = 0. Hence (W, ω) = (K, ω K ) = 0. The proof is complete. s in the paper [12] of Yang, immediate application of Theorem 4.3 would provide exact sequences for Euler class groups of polynomial rings and Laurent polynomial rings. Corollary 4.4. Let R be a commutative ring with dim R = d. Let = R[X] be the polynomial ring and B = R[X, X 1 ] be the Laurent polynomial ring. Let L 0 be a projective R-module of rank one. Write L = L 0, L = L 0 B. ssume that 2n d + 4. We have the following. (1) The sequence, 0 E n (, (X), L) E(p 2) E n (, L) ρ X E n (R, L 0 ) 0 is a split exact sequence. (2) The sequence, 0 E n B, (X 1), L E(p 2 ) E n B, L ρ X 1 E n (R, L 0 ) 0 is a split exact sequence. (3) urther, if R is a regular domain that is essentially of the finite type over an infinite field k, then ρ X : E n (, L) E n (R, L 0 ) is an isomorphism. n particular, the relative group E n (, (X), L) = 0. Proof. Obviously, (1) and (2) are direct consequences of Theorem 4.3. To prove (3), we need to show that the first homomorphism is zero. s in Theorem 4.3, p 1, p 2 will denote the projection maps and q : R will be the quotient homomorphism. The first homomorphism in the sequence is E(p 2 ). Suppose x E n (, (X), L). We will prove E(p 2 )(x) = 0. We can assume that x = (, ω) where is an ideal of D = D(, (X)), with height() n, height(p 1 ()) n and height(p 2 ()) n. Write p 1 () = 1, p 2 () = 2 and 0 = q( 1 ) = q( 2 ). Therefore, 0 = ( 1, X) = ( 2, X).

uthor's personal copy S. Mandal, Y. Yang / ournal of Pure and pplied lgebra 216 (2012) 2159 2169 2169 or i = 0, 1, 2, let ω i : / i i / 2 i ( 0, ω 0 ) = ρ X ( 2, ω 2 ) = ρ X E(p 2 )(, ω) = 0. be induced by ω. rom the exactness of the sequence, we have Therefore, by Theorem 2.4, ω 0 lifts to a surjective homomorphism f 0 : L 0 R n 1 0. Now, by [2, Theorem 4.13], there is a surjective lift f 2 : L n 1 2 such that f 2 /(X) = f 0. So, ( 2, ω 2 ) = 0. This completes the proof. cknowledgement The first author was partially supported by a grant from General Research und of University of Kansas. References [1] ean Barge, Morel, abien Groupe de Chow des cycles orients et classe d Euler des fibres vectoriels (The Chow group of oriented cycles and the Euler class of vector bundles), C. R. cad. Sci. Paris Sr. Math. 330 (4) (2000) 287 290 (in rench). [2] S.M. Bhatwadekar, M.K. Keshari, question of Nori: projective generation of ideals, K-Theory 28 (2003) 329 351. [3] S.M. Bhatwadekar, Raja Sridharan, The Euler class group of a Noetherian ring, Compos. Math. 122 (2) (2000) 183 222. [4] S.M. Bhatwadekar, Raja Sridharan, On Euler classes and stably free projective modules, in: lgebra, rithmetic and Geometry, Part, (Mumbai, 2000), in: Tata nst. und. Res. Stud. Math., vol. 16, Tata nst. und. Res., Bombay, 2002, pp. 139 158. [5] ean asel, Groupes de Chow-Witt, Mém. Soc. Math. r. (N.S.) 113 (2008). [6] Satya Mandal, Homotopy of sections of projective modules,. lgebraic Geom. 1 (4) (1992) 639 646. [7] Satya Mandal, Projective modules and complete intersections, in: Lecture Notes in Mathematics, vol. 1672, Springer-Verlag, Berlin, 1997, viii+114 pp. [8] Satya Mandal, Raja Sridharan, Euler Classes and Complete ntersections,. Math. Kyoto Univ. 36 (3) (1996) 453 470. [9] Satya Mandal, Yong Yang, ntersection theory of algebraic obstructions,. Pure ppl. lgebra 214 (2010) 2279 2293. [10] Hideyuki Matsumura, Commutative ring theory, in: Cambridge Studies in dvanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986, xiv+320 pp. [11] N.E. Steenrod, The Topology of ibre Bundles, Princeton University Press, 1951. [12] Yong Yang, Homology sequence and excision theorem for Euler class group, Pacific. Math. 249 (1) (2011) 237 254.