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1 This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article e.g. in Word or Tex form to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving and manuscript policies are encouraged to visit:

2 Author's personal copy Journal of Pure and Applied Algebra Contents lists available at ScienceDirect Journal of Pure and Applied Algebra journal homepage: Intersection theory of algebraic obstructions Satya Mandal a, Yong Yang b, a Department of Mathematics, University of Kansas, 146 Jayhawk Blvd., Lawrence, KS 6645, USA b Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan, 4173, PR China a r t i c l e i n f o a b s t r a c t Article history: Received 9 August 29 Received in revised form 1 January 21 Available online 24 March 21 Communicated by R. Parimala MSC: 13c1 Let A be a noetherian commutative ring of dimension d and L be a rank one projective A-module. For 1 r d, we define obstruction groups E r A, L. This extends the original definition due to Nori, in the case r = d. These groups would be called Euler class groups. In analogy to intersection theory in algebraic geometry, we define a product intersection E r A, A E s A, A E r+s A, A. For a projective A-module Q of rank n d, with an orientation χ : L n Q, we define a Chern class like homomorphism wq, χ : E d n A, L E d A, LL, where L is another rank one projective A-module. 21 Elsevier B.V. All rights reserved. 1. Introduction In topology, there is a classical obstruction theory for vector bundles See [13]. The germ of the obstruction theory in algebra was given by Nori, around 199 [6,1]. For a smooth affine variety X = SpecA over an infinite field k, with dim A = d 2, Nori outlined a definition of an obstruction group EA. Further, for a projective A-module P of rank d, with an orientation isomorphism χ : A d P, Nori outlined a definition of an obstruction class ep, χ EA. The definition of Nori was later extended by Bhatwadekar and Sridharan [2]. Given a noetherian commutative ring A with dim A = d 2 and a rank one projective A-module L, they defined an obstruction group EA, L. In addition, if Q A, given a projective A-module P of rank d and an orientation χ : L d P, they defined an obstruction class ep, χ EA, L. They proved that P Q A if and only if ep, χ =. Surpassing all expectations, a solid body of work has been accomplished regarding the obstructions for projective A-modules P with rankp = d = dim A. Among them are [6,1,1,2,4,8,9]. The theory in topology is much advanced and complete. For a real smooth manifold M with dim M = d and a vector bundle V of rank r d, there is an obstruction group and an obstruction class called Whitney class wv in the obstruction group. If V has a nowhere vanishing section, then wv =. In case rankv = r = d, V has a nowhere vanishing section if and only if wv =. Obstruction theory in algebra remains incomplete in this respect. At this time, barring [3], the theory is confined to the case when rankp = d = dim A. Further, the intersection theory see [5] in algebraic geometry has also been fairly advanced and complete. Because of such an advanced status of these two theories, there has been expectations that the obstruction theory in algebra would have a similar advanced counter part. In this paper, we try to establish a foundation for a theory of algebraic obstructions, in analogy to the aforementioned theories in topology and algebraic geometry. Given a commutative noetherian ring A of dimension d and a rank one projective A-module L, in this paper we give a definition of obstruction groups E r A, L, for r 1. These groups will be called Euler class groups. The obstruction group Corresponding author. addresses: mandal@math.ku.edu S. Mandal, yangyong@nudt.edu.cn Y. Yang /$ see front matter 21 Elsevier B.V. All rights reserved. doi:1.116/j.jpaa

3 Author's personal copy 228 S. Mandal, Y. Yang / Journal of Pure and Applied Algebra EA, L mentioned above would be same as E d A, L. Now, suppose Q is a projective A-module with rankq = n and an orientation isomorphism χ : L n Q. In [5], the top Chern class was defined as a homomorphism of degree n. In the same spirit, if n d 2 and L is another projective A-module of rank one, we define a Whitney class homomorphism wq, χ : E d n A, L E d A, L L. In case rankq = n = d 1, the same homomorphism is defined for L = A. This homomorphism is compatible with the top Chern class homomorphism of Q. For r 2, s 1, we also define bilinear maps intersection: : E r A, L E s A, A E r+s A, L. For r = 1, the same is defined with L = A. 2. Euler class groups In this section we define general Euler obstruction class groups of commutative noetherian rings. Definition 2.1. Let A be a commutative noetherian ring with dim A = d and L be a rank one projective A-module. We write F r = L A r 1. For an A-module M, the group of transvections of M will be denoted by ElM see [7] for a definition. 1. A local L-orientation is a pair I, ω, where I is an ideal of A of height r and ω is an equivalence class of surjective homomorphisms ω : F r /IF r I/I 2. The equivalence is defined by ElF r /IF r -maps. Sometimes, we will simply say that ω is a local L-orientation, to mean that I, ω is a local L-orientation. By abuse of notations, we sometimes denote the equivalence class of ω, by ω. 2. Let L r A, L denote the set of all pairs I, ω, where I is an ideal of height r, such that SpecA/I is connected and ω : F r /IF r I/I 2 is a local L-orientation. Similarly, let L r A, L denote the set of all ideals I of height r, such that SpecA/I is connected and there is a surjective homomorphism F r /IF r I/I Let G r A, L denote the free abelian group generated by L r A, L and G r A, L denote the free abelian group generated by L r A, L. 4. Suppose I is an ideal of height r and ω : F r /IF r I/I 2 is a local L-orientation. By [3], there is a unique decomposition I = I 1 I 2 I k such that I i +I j = A for i j and SpecA/I i is connected. Then ω naturally induces local L-orientations ω i : F r /I i F r I i /I 2 i. Denote I, ω := I i, ω i G r A, L. Similarly, we denote I := I i G r A, L. 5. Global orientations: Let I be an ideal and ω : F r /IF r I/I 2 be a local L-orientation. We say that ω is global, if there s a surjective lift Ω of ω as follows: Ω F r I F r /IF r ω I/I Let H r A, L be the subgroup of G r A, L, generated by global L-orientations. Also, let H ra, L be the subgroup of Gr A, L, generated by I such that I is surjective image of F r. Now define the Euler class group of codimension r cycles as E r A, L = Gr A, L H r A.L. and the weak Euler class group of codimension r cycles as E r A, L = Gr A, L H ra.l. Lemma 2.2. We use the notations as in Definition 2.1. For r 1, there are natural surjective homomorphisms ζ r : E r A, L E r A, L.

4 Author's personal copy S. Mandal, Y. Yang / Journal of Pure and Applied Algebra Further, assume that A is Cohen Macaulay. For r 2, there are natural homomorphisms η r : E r A, L CHr A and there is a natural homomorphism η 1 : E 1 A, A CH1 A. Here CH r A denotes the Chow group of cycles of codimension r in SpecA. Proof. Existence of ζ r follows from the above definitions of E r A, L, E r A, L. Existence of ηr follows directly from [5, Lemma A.2.7.]. 3. Whitney class homomorphisms In this section we establish the Whitney class homomorphism of a projective module, as follows. Theorem 3.1. Let A be a commutative noetherian ring with dim A = d 2 and L, L be two rank one projective A-modules. Suppose Q is a projective A-module with rankq = n d 2 and det Q = L. For an orientation χ : L n Q, there is a canonical homomorphism wq, χ : E d n A, L E d A, L L. Also, for rankq = n = d 1, there is a canonical homomorphism wq, χ : E 1 A, A E d A, L. Proof. We will write F k = L A k 1, F k = L A k 1. Let I be an ideal of height d n and ω : F d n /IF d n I/I2 be an equivalence class of surjective homomorphisms local L-orientation. To each such pair I, ω G d n A, L, we will associate an element wq, χ I, ω E d A, L L. First, we can find an ideal Ĩ A with heightĩ d and a surjective homomorphism ψ : Q /IQ Ĩ/I. Let ψ = ψ A/Ĩ and γ : F n /ĨF n Q /ĨQ be an isomorphism such that n γ = χ A/Ĩ. Let β = ψγ and β : F n /ĨF n Ĩ/Ĩ 2 be a lift of β. The following diagram Q /IQ ψ Ĩ/I Q /ĨQ ψ Ĩ/I + Ĩ 2 γ χ β f F n /ĨF n β Ĩ/Ĩ 2 I commutes. Further, ω induces following F d n /IF d n ω I/I 2 F d n /ĨF ω d n I + Ĩ 2 /Ĩ 2 II ω Ĩ/Ĩ 2

5 Author's personal copy 2282 S. Mandal, Y. Yang / Journal of Pure and Applied Algebra commutative diagram, where ω = ω A/Ĩ. Combining ω, β we get a surjective homomorphism δ = β ω : F n /ĨF n F d n /ĨF d n = F n F d n Ĩ F n F Ĩ/Ĩ 2. d n To see that δ is surjective, consider the exact sequence I + Ĩ 2 /Ĩ 2 Ĩ/Ĩ 2 f Ĩ/I + Ĩ 2. Given x Ĩ/Ĩ 2, there is z F n /ĨF n such that f x = βz. Therefore, x β z I + Ĩ 2 /Ĩ 2. This implies that ω z = x β z for some z F d n /ĨF. d n This shows that δz + z = x. This establishes that δ is surjective. Now, let γ : LL A d 1 F n F ĨLL d n A d 1 Ĩ F n F d n LL d F n F d n χ. d LL A d 1 Let = δγ = β, ω γ. So, the diagram be an isomorphism that is consistent with the natural isomorphism χ : F n F d n Ĩ F n F d n χ γ LL A d 1 ĨLL A d 1 δ Ĩ/Ĩ 2 commutes. We have Ĩ, G d A, LL is a local LL -orientation. We will establish that the image of Ĩ, in E d A, LL is independent of choices of ψ, the lift β, the representative of ω and the choice of γ. 1. Step-I: First we prove, for a fixed ψ, Ĩ, in E d A, LL is independent of the lift β, the representative ω and the choice of γ. a Suppose ω, ω 1 are equivalent local orientation. Then ω 1 = ωɛ for some ɛ ElF d n /IF d n. Using the canonical homomorphisms ElF d n /IF d n ElF d n /ĨF d n El F n F d n /ĨF n F d n, we have ω = 1 ω ɛ. With δ 1 = β, ω 1, we have δ 1 = δ ɛ where ɛ El F n F d n /ĨF n F d n. Since ɛ 1 = γ 1 ɛγ El LL A d 1 /ĨLL A d 1, we have 1 = δ 1 γ = δ ɛγ = ɛ 1. b Also, two different lifts β of β differ in I + Ĩ 2 /Ĩ 2 and would lead to two different s that differ by El Proof. Let β : F n /ĨF n Ĩ/Ĩ 2 be another lift of β. Then φ = β β : F n /ĨF n I + Ĩ 2 /Ĩ 2. LL A d 1 ĨLL A d 1.

6 Author's personal copy S. Mandal, Y. Yang / Journal of Pure and Applied Algebra We have a lift g so that the diagram φ F n /ĨF n I + Ĩ 2 /Ĩ 2 g ω 1 F d n /ĨF d n I + Ĩ 2 /Ĩ 2 ω 1 ω 1 Ĩ/Ĩ 2 commutes. Let 1 ɛ 2 = : F g 1 n /ĨF n F d n /ĨF d n F n /ĨF n F d n /ĨF d n It follows β ω ɛ 1 2 = β ω. 1 Therefore, with 2 = β ω γ 1 and ɛ 3 = γ 1 ɛ 2 γ, we have 1 = β ω γ 1 = β ω ɛ 1 2γ = β ω γ 1 γ 1 ɛ 2 γ = 2 ɛ 3. Therefore, = 1 ɛ 1 1 = 2 ɛ 3 ɛ 1 1. c Again, if γ = γ ɛ 4 for some ɛ 4 El LL A d 1 ĨLL A d 1 then = 2 ɛ 3 ɛ 1 1 = β ω 1 γ ɛ 3 ɛ 1 1 = β ω 1 γ ɛ 4ɛ 3 ɛ 1 1 β ω 1 γ. So, the claim in Step-I is established. 2. Also note, Ĩ, is independent of the choice γ. This is because, two different choices of γ would differ by a SLF n /ĨF n - map. This will lead to choices of β that will differ by a SLF n /ĨF n -map. 3. Step-II: Now, we prove that Ĩ, E d A, LL is also independent of ψ. That means, it depends only on I, ω. a To see this, first we fix a surjective lift Ω of ω as follows: F d n Ω I K F d n /IF d n ω I/I 2 where K + I = A and K is an ideal of height d n or K = A. b We can find an ideal K with height K d, a surjective homomorphism ψ and replicate the diagram I: Q /KQ ψ K/K Q / K Q K/K + K 2 f η F n / K Fn 2 K/ K η γ χ. Here γ is an isomorphism so that n γ = χ A/ K and η is a lift of η = ψ A/ Kγ. c Let ω : F d n /KF d n K/K 2 be induced by Ω. d As above, let γ : LL A d 1 KLL A d 1 d F n F d n d LL A d 1. F n F d n K F n F d n be an isomorphism consistent with the canonical isomorphism χ :

7 Author's personal copy 2284 S. Mandal, Y. Yang / Journal of Pure and Applied Algebra e Combining ω, η we get a surjective homomorphism δ = η ω : F n / K Fn F d n / K F d n = f Write = δ γ. We fix K, and prove that Ĩ, + K, = E d A, LL. g Since I + K Ĩ + K, we have Ĩ + K = A. h We have Ψ = ψ ψ : i We lift Ψ = ψ ψ to Ψ as follows: F n F d n K F n F d n K/ K 2. Q I KQ Q IQ Q KQ Ĩ I K K Ĩ K I K. Q Ψ Ĩ K Q I KQ Ψ Ĩ K I K. If we reduce this diagram modulo Ĩ or K, we get the following Q /ĨQ ˆ Ψ Ĩ/Ĩ 2 Q / K Q ˆ Ψ K/ K 2 Q /ĨQ ψ f Ĩ I+Ĩ 2 γ χ β f F n /ĨF n β Ĩ/Ĩ 2 and Q / K Q f K ψ I+ K 2 γ χ η F n / K Fn 2 K/ K η f commutative diagrams. It follows that α 1 = β γ 1 ˆ Ψ : Q /ĨQ I + Ĩ2 Similarly, we have Ĩ 2 Ĩ Ĩ 2. α 2 = η γ 1 ˆ Ψ : Q / K Q K + K 2 K 2 K K 2. We lift α 1, α 2 to g 1, g 2 so that the diagrams α 1 Q /ĨQ I + Ĩ 2 /Ĩ 2 g 1 F d n /ĨF ω d n I + Ĩ ω 2 /Ĩ 2, Ĩ/Ĩ 2 α 2 Q / K Q K + K 2 2 / K g 2 F / d n K F Ω K + K 2 2 / K d n ω K/ K 2 commute.

8 Author's personal copy S. Mandal, Y. Yang / Journal of Pure and Applied Algebra j Let g be given by g 1, g 2 and γ be given by γ, γ. Write 1 γ Γ = : g 1 1 F n F d n Ĩ KF n F d n Γ Q F d n Ĩ KQ F d n Q F d n Ĩ KQ F d n. k Consider the surjective homomorphism Q F d n Ψ Ω Ĩ K. l We claim that the diagram Q F d n Ψ Ω Ĩ K Q F d n ˆ Ψ Ω Ĩ KQ F d n Γ F n F d n Ĩ KF n F d n δ,δ Ĩ K Ĩ K 2 commutes. Only the commutativity of the bottom triangle needs to be checked. This is done by checking on VĨ, V K separately. We check directly that ˆ Ψ Ω Γ, y = ˆ Ψ Ω, y = Ωy = δ, δ, y and ˆ Ψ Ω Γ x, = ˆ Ψ Ω γ x, g γ x = ˆ Ψ γ x + Ωg γ x = ˆ Ψ γ x + β γ 1 γ x ˆ Ψ γ xβ x = δ, δ x,. m Composing with γ, γ, it follows from [2, Cor. 4.4] that, is global. Therefore, Ĩ, + K, = E d A, LL. n Since, K, depends only on I, ω, it follows Ĩ, is independent of choice of ψ. This establishes the claim in Step-II. 4. If I, ω is global, then in the above proof, we can take K = A and it follows Now, the association I, ω Ĩ, E d A, LL defines a homomorphism Ĩ, is global. ϕq, χ : G d n A, L E d A, LL where I, ω G d n A, L are the free generators of G d n A, L and Ĩ, E d A, LL are as above. The above discussions establish that ϕq, χ is a well-defined homomorphism. By 4, it follows that ϕ factors through a homomorphism wq, χ : E d n A, L E d A, LL. This completes the proof of the theorem. By forgetting the orientation in Theorem 3.1, we have the following for weak Euler class groups.

9 Author's personal copy 2286 S. Mandal, Y. Yang / Journal of Pure and Applied Algebra Corollary 3.2. Let A, L, L, Q be as in Theorem 3.1. If rankq d 2, then there is a canonical homomorphism w Q : E d n A, L E d A, L L E d A, A. Also for rankq = n = d 1, there is a canonical homomorphism w Q : E 1 A, A EdA, L Ed A, A. Proof. The isomorphisms at the right side were proved in [2]. The rest of the proof is similar to that of Theorem 3.1 and we give an outline. We will write F k = L A k 1, F = k L A k 1. Suppose I is a generator of G d n A, L. Here I is an ideal of height d r, SpecA/I is connected and there is a surjective homomorphism F IF n I/I 2. There is a surjective homomorphism ψ : Q /IQ Ĩ/I. For such a generator I we associate Ĩ E da, LL. We fix a local orientation, ω : F IF n I/I 2, and a surjective lift Ω : F I K of ω, where K is an ideal of height d n. Now, let ψ : Q /KQ K/K, be a surjective homomorphism, where K is an ideal of height d. As in Theorem 3.1, we prove that there is a surjective homomorphism LL A d 1 Ĩ K. This shows that Ĩ + K = E d A, LL and so Ĩ E da, LL is independent of choice of ψ. Ĩ This association I E da, LL, extends to homomorphism ϕ : G d n A, L E d A, LL. If I is global, i.e. I is surjective image of F, d n Ĩ taking K = A in the above argument, we prove is global. So, ϕ factors through a homomorphism w Q : E d n A, L E d A, LL. This completes the proof. Definition 3.3. This homomorphism wq, χ in Theorem 3.1, will be called the Whitney class homomorphism. The image of I, ω E d n A, L under wq, χ will be denoted by wq, χ I, ω. Similarly, the homomorphism w Q in 3.2 will be called the weak Whitney class homomorphism. The image of I E d n A, L under w Q will be denoted by w Q I. The following is about the compatibility of these homomorphisms, along with the Chern class homomorphisms. Corollary 3.4. We use the notations as in Theorem 3.1 and Corollary 3.2. Further assume that A is a Cohen Macaulay ring. Then, we have w Q ζ d n = ζ d wq, χ and C n Q η d n = η d w Q, where ζ d n, ζ d, η d n, η d are the natural homomorphisms as in Lemma 2.2 and C n Q denotes the top Chern class homomorphism [5] of Q. Proof. The first identity follows from the definitions of w Q and wq, χ. To prove the second identity, let I be an ideal of height d n with a surjective homomorphism L A d n 1 IL A d n 1 I/I2. By Eisenbud Evans theorem see [12], we can find a surjective homomorphism ψ : Q J, where J is an ideal of heightj = n and heighti + J = d. Write Ĩ = I + J. Note that I, J, Ĩ are locally complete intersection ideals. We have, Also We have η d w Q I = η d Ĩ = cyclea/ĩ. C n Q η d n I = C n Q cyclea/i. C n Q /IQ cyclea/i = cyclea/ĩ CH n A/I. With f : SpecA/I SpecA, apply f. By the projection formula [5], we have C n Q cyclea/i = cyclea/ĩ CH d A. So, the proof is complete.

10 Author's personal copy S. Mandal, Y. Yang / Journal of Pure and Applied Algebra The following is about vanishing of Whitney class homomorphisms. Theorem 3.5. Suppose A, L, L, Q, χ, F k, F k be as in Theorem 3.1 and its proof. Let I be an ideal of height d n and ω : F d n IF d n be a local L -orientation. If Q /IQ = P A/I, then wq, χ I, ω = E d A, LL. In particular, if Q = P A, then the homomorphism wq, χ : E d n A, L E d A, LL is identically zero. Similar statements holds for w Q. I/I 2 Proof. Proof of the last statement follows from the former assertions, by 3.4. We will enumerate the steps of the proof of the former assertions. 1. By Eisenbud Evans theorem [12], there is an ideal Ĩ A with heightĩ = d and a surjective homomorphism ψ : Q /IQ Ĩ/I. The following diagram Q /IQ ψ Ĩ/I Q /ĨQ ψ Ĩ/I + Ĩ 2 γ χ β f F n /ĨF n β Ĩ/Ĩ 2 I commutes. Here ψ = ψ A/Ĩ and γ is any isomorphism, with n γ = χ A/Ĩ, β = ψγ and β is a lift of β. 2. Let Ω be any lift of ω. Then, the following diagram F d n Ω I F d n ω I/I 2 IF d n F d n ĨF d n ω I/IĨ ω Ĩ/Ĩ 2 commutes. Here ω = ω A/Ĩ. 3. With Q /IQ = P A/I, let θ be the restriction of ψ to P. We can write ψ = θ, ā for some a Ĩ. Write ψp = J/I, for some ideal J containing I. In fact, height J = d 1 and Ĩ = J, a. 4. Since height J = d 1, there is an isomorphism γ : F n 1 / JFn 1 P / JP. 5. Define χ as in the commutative diagram: χ A/ J L/ JL n P A/I JP A/I χ. n 1 P / JP By adjusting the determinant, if needed, we can assume that n 1 γ = χ.

11 Author's personal copy 2288 S. Mandal, Y. Yang / Journal of Pure and Applied Algebra For our purpose, γ is a choice. So, we can assume that γ is the reduction of γ A commutative diagram similar to I is induced by θ as follows: θ P J/I θ P / JP J/I + J 2 γ χ ζ F n 1 2 / JFn 1 J/ J ζ II. Here θ = θ A/ J and ζ is a lift of ζ = θγ. If we tensor this diagram with A/Ĩ, we get the following commutative diagram: θ P /ĨP J/ I + JĨ θ P /ĨP J/ I + JĨ Ĩ/ I + Ĩ 2 ζ γ χ f β Fn 1 F n 1 /ĨF n 1 J/ JĨ Ĩ/Ĩ 2 ζ III. 8. This shows that ζ, ā is a lift of β. Since wq, χ I, ω is independent of the lift β we can assume that β = ζ, ā. 9. We lift ζ to a homomorphism δ : F n 1 J. 1. The homomorphism δ, a, Ω : F n 1 A F d n Ĩ lifts β, ω = ζ, ā, ω. 11. Let J = δf n 1 + ΩF d n. Then J = J + J 2. To see this, let y J. Then, from diagram II, there is x Fn 1 such that δx y = y 1 + z where y 1 I, z J 2 Now, there is x 1 F d n such that y 1 Ωx 1 = z 1 I 2 J 2. Therefore δx Ωx 1 y = z + z There is an isomorphism γ : LL A d 2 J LL A d 2 F n 1 Fd n J F n 1 F d n such that the determinant is given by the commutative diagram: LL / JLL LL A d 2 JLL A d 2 d 1 γ F n 1 F d n J F n 1 F d n.

12 Author's personal copy S. Mandal, Y. Yang / Journal of Pure and Applied Algebra Let Γ : LL A d 2 F n 1 F d n be any lift of γ and γ = γ 1 A/Ĩ : LL A d 1 Ĩ LL A d 1 F n Fd n. Ĩ F n F d n 14. Let Γ = δ, ΩΓ. Write J = Γ LL A d 2. By 6, J = J + J There is an element ɛ J 2 such that 1 ɛ J J. So, as in [11], we have J J =, ɛ and Ĩ = J, J a =, ɛ + 1 ɛa. 16. It follows, with b = ɛ + 1 ɛa, that Γ, b : LL A d 2 A Ĩ is a surjective homomorphism. 17. Now, we have which is Γ, b A/Ĩ = δ, ΩΓ, b A/Ĩ = δ, Ω, bγ 1 A/Ĩ = δ, Ω, b A/Ĩ γ = β, ω γ. The last equality follows from 6. Therefore is global. Therefore, wq, χ I, ω = Ĩ, = EA, LL. The proof is complete. In analogy to results in Chern class theory [5], we have the following. Corollary 3.6. Let A be a noetherian commutative ring with dim A = d 2. Suppose L, L 1, L 2 PicA. Then, for x E d 1 A, L, we have w L 1 L 2 x = w L 1 x + w L 2 x E d A, L Ed A, A. Proof. We can assume x = I where I is an ideal of height d 1 and there is a surjective homomorphism L Ad 2 IL A d 2 I/I2. There are ideals I1, I2 of height d and surjective homomorphisms L i /IL i Ĩ i /I A/I, for i = 1, 2. Since, dim A/I 1, we can assume that I1 + I2 = A. Therefore, So, L 1 L 2 A/I L 1 /IL 1 L 2 /IL 2 w L 1 L 2 I = maps onto I1 /I I2 /I = I1 I2 /I. I1 I2 = I1 + I2 = w L 1 I + w L 2 I. The proof is complete. 4. Intersections in Euler class groups We define an intersection product in the Euler class groups as follows. Definition 4.1. Suppose A is a noetherian commutative ring with dim A = d 2 and L, L be two projective A-modules of rank one. We write F = L A r 1, F = L A s 1. Let I be an ideal of height r and J be an ideal of height s. Assume heighti + J r + s and suppose ω : F/IF I/I 2 and ω : F /JF J/J 2 are two local orientations. Then ω, ω induce a surjective homomorphism F F η : I + JF F I + J I + J 2

13 Author's personal copy 229 S. Mandal, Y. Yang / Journal of Pure and Applied Algebra according to the following F F I+JF F ω ω I I+JI J I+JJ η I+J I+J 2 commutative diagram, where ω = ω A/I + J, ω = ω A/I + J. We will continue to abuse notations and denote elements in G r A, L their images in E r A, L by same notations. 1. If L = A, define intersection I, ω J, ω := I + J, η E r+s A, L. The right hand side is interpreted as zero, if I + J = A. 2. If r + s = d, define intersection where γ : I + J = A. I, ω J, ω := I + J, ηγ E r+s A, LL LL A d 1 I+JLL A d 1 F F I+JF F is any isomorphism with d γ = Id. The right hand side is interpreted as zero, if Subsequently, we prove that the above 4.1 are well defined in the general set-up. The following lemma is about intersections of global orientations. Lemma 4.2. With all notations as in Definition 4.1, suppose ω is global. If s 2 or L = A. Then there is a surjective lift Θ : F F I + J of η. In particular, 1. in case L = A, the local orientation η is global; 2. and also in case L A, and r + s = d, then ηγ is global. Proof. If I +J = A, there is nothing to prove. So, we assume heighti +J = r +s. Suppose f = g, a : F J is a surjective lift of ω. Also, let f : F I be any lift of ω. There is an e I 2 such that 1 ei f F. We claim that [11], with b = e + 1 ea, I + J = f F, gl A s 2, b =: K. First, 1 e f F e. So, 1 ea K e. So, e = b 1 ea K e. Therefore, A e = I +J e = K e. If s > 1, let F = L A s 2 and if s = 1, let F =. We have K 1 e = f F1 e, gf 1 e, b = I 1 e, gf 1 e, b = I 1 e, gf 1 e, a = I + J 1 e. This shows that Θ = f, g, b is a surjective lift of η. Now we prove 2. Write θ = f, g and I = θf F. Since dim A/I 1, there is an isomorphism γ : LL A d 2 F F ILL A d 2 IF F with d 1 γ = 1. Since γ is a choice, we can assume that γ = γ 1 A/I + J. Let Γ : LL A d 2 F F be any lift of γ. Let K = θγ LL A d 2. Then I = K + I 2 and I + J = I, b. There is ɛ I 2 such that 1 ɛi K. Write c = ɛ + 1 ɛb. Then I + J = I, b = K, c. Therefore θγ, c : LL A d 2 I + J is a surjective lift of ηγ. So, ηγ is global. This completes the proof. The following moving lemma is a tool for the rest of this paper. Lemma 4.3. Suppose A is noetherian commutative ring with dim A = d. Let I, J be two ideals of height r and s, respectively. Suppose F is a projective A-module of rank r and ω : F/IF I/I 2 is a surjective homomorphism. Also assume that J is locally generated by s elements. Then, there is a surjective lift f : F I K of ω such that 1 I + K = A 2 heightk r and 3 heightk + J r + s. Proof. The proof is done by the use of standard generalized dimension theory. First, there is a lift f : F I of ω. Then I = f F, a for some a I 2. Let P r 1 SpecA, be the set of all prime ideals, with height r 1 and a /. Also, let Q r 1 SpecA be the set of all prime ideals such that J, a / and height /J r 1. Write P = P r 1 Q r 1. Let d 1 : P r 1 N be the restriction of the usual dimension function and d 2 : Q r 1 N be the dimension function induced by that on SpecA a /J a. Then d 1, d 2 induce a generalized dimension function d : P N, [12] or see [7]. Now, f, a F A is a basic element on P. Since, rankf = r > d for all P, there is a g F, such that f = f + ag is basic on P. Clearly, f is a lift of ω and I = f F, a.

14 Author's personal copy S. Mandal, Y. Yang / Journal of Pure and Applied Algebra Since, f is a lift of ω, we can write f F = I K, such that I + K = A. It is routine to check now that heightk r and J+K J+K J+K height r. Since is locally r generated, height = r and for any minimal prime over J + K we have J J J height = height = r. J+K J J Also, since J is locally s generated ideal of height s, height = heightj, for any minimal prime over J. If J + K = A, there is nothing to prove. Suppose J + K 1 be a minimal prime over J + K such that heightj + K = height 1, and J be a minimal prime over J, such that height 1 / = height 1 /J. We have, height 1 height 1 / + height = height 1 /J + heightj = r + s. This completes the proof. The following lemma establishes the consistency of Definition 4.1. Lemma 4.4. We use all the notations in Definition 4.1. Assume r 2 or L = A. Further, let I 1 be an ideal of height r with heighti 1 + J r + s and let ω 1 : F/I 1 F I 1 /I 2 1 be a local orientation. Suppose I, ω = I 1, ω 1 E r A, L. 1. If L = A, then I, ω J, ω = I 1, ω 1 J, ω E r+s A, L. 2. If r + s = d, then I, ω J, ω = I 1, ω 1 J, ω E d A, LL. Proof. We will only prove 1. The proof of 2 is similar. We have either heighti 1 + J = r + s or I 1 + J = A. We will assume L = A. By Lemma 4.3, we can find a surjective lift θ : F I K of ω, where I + K = A and heightk + J r + s. Since I K, θ A/I K is global, by Lemma 4.2, we have I, ω J, ω + K, θ J, ω = I K + J, θ ω = E r+s A, L. This also works when K + J = A, in this case I K + J = I + J. We will prove I 1, ω 1 J, ω + K, θ J, ω = E r+s A, L. Since I, ω = I 1, ω 1 E r A, L we have I 1, ω 1 + K, θ = E r A, L. There exist, ideals {I t : 1 t i + j} of height r and global orientations θ t : F/I t F I t /I 2 t such that I 1, ω 1 + K, θ + i+j t=i+1 I t, θ t = i I t, θ t G r A, L I. t=1 Again, by Lemma 4.3, there is a surjective lift Ω : F J J of ω, such that heightj s, J + J = A and height J + I K i+j I t r + s. t=1 The identity I is a purely formal identity. Note that Definition 4.1 applies to all the terms in the following expression and by formal argument, it follows that from I that I 1, ω 1 J, Ω + K, θ J, Ω + = i I t, θ t J, Ω t=1 in G r+s A, L. By Lemma 4.2, we have i+j t=i+1 I t, θ t J, Ω I 1, ω 1 J, Ω + K, θ J, Ω = E r+s A, L II. We also have I 1, ω 1 J, ω + K, θ J, ω + I 1, ω 1 J, Ω + K, θ J, Ω = I 1, ω 1 J, ω + I 1, ω 1 J, Ω + K, θ J, ω + K, θ J, Ω = I 1, ω 1 J J, Ω + K, θ J J, Ω =.

15 Author's personal copy 2292 S. Mandal, Y. Yang / Journal of Pure and Applied Algebra Combining with II, we have I 1, ω 1 J, ω + K, θ J, ω = E r+s A, L. Therefore, the proof is complete. The following is our final result on intersection product. Theorem 4.5. We use all the notations as in Definition With L = L = A, the association I, ω, J, ω I, ω J, ω E r+s A, A defines a bilinear homomorphism : E r A, A E s A, A E r+s A, A. 2. With L = A and r 2, the association I, ω, J, ω I, ω J, ω E r+s A, L defines a bilinear homomorphism : E r A, L E s A, A E r+s A, L. 3. If L A, r + s = d and r 2, the association I, ω, J, ω I, ω J, ω E r+s A, LL defines a bilinear homomorphism : E r A, L E s A, L E d A, LL. Proof. Proofs of 3, 1 are similar to that of 2. We only prove 2. Fix x = J, ω G s A, A as in Definition 4.1. Suppose I 1 is an ideal of height r and ω 1 : There is an ideal I and ω as in 4.1, such that I 1, ω 1 = I, ω E r A, L. Define ϕ x I 1, ω 1 = I, ω J, ω E r+s A, L. This is well defined by Lemma 4.4. By Lemma 4.2, ϕ x factors through a homomorphism ϕ x : E r A, L E r+s A, L. Now, the association x ϕ x defines a homomorphism ϕ : G s A, A Hom E r A, L, E r+s A, L. By Lemma 4.2, if x is a global orientation, then ϕ x =. So, ϕ, factors through a homomorphism : E s A, A Hom E r A, L, E r+s A, L. This completes the proof. Forgetting the orientations, we get the following. F I 1 F I 1 I 2 1 is a local orientation. Corollary 4.6. There is an intersection product for the weak Euler class groups, corresponding to the same in Theorem 4.5. Remark 4.7. From the definitions, it follows that the intersection products defined in Theorem 4.5 and Corollary 4.6 are commutative and associative, whenever they are defined. Acknowledgement The first author was partially supported by a grant from General Research Fund of University of Kansas. References [1] S.M. Bhatwadekar, Raja Sridharan, Projective generation of curves in polynomial extensions of an affine domain and a question of Nori, Invent. Math [2] S.M. Bhatwadekar, Raja Sridharan, The Euler class group of a Noetherian ring, Compos. Math [3] S.M. Bhatwadekar, Raja Sridharan, On Euler classes and stably free projective modules, in: Algebra, Arithmetic and Geometry, Part I, II Mumbai, 2, in: Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res, Bombay, 22, pp [4] S.M. Bhatwadekar, Mrinal Kanti Das, Satya Mandal, Projective modules over smooth real affine varieties, Invent. Math [5] W. Fulton, Intersection Theory, second ed., Springer-Verlag, 1998.

16 Author's personal copy S. Mandal, Y. Yang / Journal of Pure and Applied Algebra [6] Satya Mandal, Homotopy of sections of projective modules, J. Algebraic Geom [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, Albert J.L. Sheu, Obstruction theory in algebra and topology, J. Ramanujan Math. Soc [9] Satya Mandal, Albert J.L. Sheu, Local Coefficients and Euler Class Groups, preprint. [1] Satya Mandal, Raja Sridharan, Euler classes and complete intersections, J. of Math. Kyoto University [11] N. Mohan Kumar, Complete intersections, J. Math. Kyoto Univ [12] B. Plumstead, The conjectures of Eisenbud and Evans, Amer. J. Math [13] N.E. Steenrod, The Topology of Fibre Bundles, Princeton University Press, 1951.