Intersection Theory, Homology Sequence And Excision Theorem For Algebraic Obstructions

In topology, there is a classical obstruction theory for vector bundles([1]). The germof the obstruction theory in algebra was given by Nori, around 1990([2],[3]). For a smoothvariety X =Spec(A) over a infinite field k, with dim(A) = d≥2, Nori outlined adefinition of an obstruction group E(A). Further, for a projective A?module P of rankd, with an orientationχ: A≌∧^dP, Nori outlined a definition of an obstruction classe(P,χ)∈E(A). These definitions ware later extended by S.M. Bhatwadekar and R.Sridharan([4]). Given a noetherian commutative ring A with dimA = d≥2 and a rankone projective A?module L, they defined an obstruction group E(A,L). In addition, ifQ ? A, given a projective A?module P of rank d and an orientationχ: L≌∧^dP, theydefined an obstruction class e(P,χ)∈E(A,L). They proved that P≌Q⊕A if and onlyif e(P,χ) = 0.The present dissertation includes some works we do about this obstruction theory.Our works mainly focus on two problems in the field: one is about the intersection theoryof algebraic obstructions, the another is about the homological construction of Euler classgroups.In the second chapter, we try to establish a foundation for a theory of algebraic ob-structions, in analogy to the respective characteristic classes theories in topology and inalgebraic geometry. This work is inspired by the fact that the Chern Classes of algebraicvector bundles, which is highly related with the Weak Euler Class Group, are defined bythe intersection theory in algebraic geometry[5]. Given a commutative noetherian ring AofdimensiondandarankoneprojectiveA?moduleL,wegiveadefinitionofgeneralizedEuler Class Groups Er(A,L), for r≥1. In[5] the top Chern Class of an algebraic vectorbundle of rank n is defined as a homomorphism of degree n. In the same spirit, for anyprojective A?module Q with rankQ = n and an orientationχ: L≌∧^rQ, we define aWhitney Class homomorphismin which L′is another projective A?module of rank one. Further, this homomorphismis compatible with the top Chern Class homomorphism of Q. For r≥2,s≥1, we also define bilinear maps(intersection):In the third chapter, we give a homological construction for Euler Class Group. Infact, in this section, our aim is to prove the characteristic theorems of homology theoryfor Euler Class Group. Let R be a Noetherian commutative ring with dimension d, l be anideal of R with dim Rl = d?m. For any integer n such that 2n≥d+3, we define a grouphomomorphism E(ρ) : En(R;R)→En(Rl ; Rl ), called Restriction Map of Euler ClassGroups. Further, in analogy to the relative K₀-group K₀(R,l)(denoted by K₀(l) in[6]),we define the Relative Euler Class Group En(R,l;R) and Relative Weak Euler ClassGroup E0n(R,l). Particularly, when l = R, the Relative Euler Class Group En(R,l;R)andRelativeWeakEulerClassGroupE0n(R,l)arethesameasthegeneralizedEulerClassGroup En(R;R) and Weak Euler Class Group E0n(R), respectively. Using these groups,we construct an exact sequencecalled the Homology Sequence of Euler Class Group. If the ring homomorphism l has a splittingβ, which satisfies a dimensional condition, then the above HomologySequence reduces to the following short split exact sequencecalled the Excision Sequence of Euler Class Group. Particularly, under these conditions,we have an isomorphism En(R;R)≌En(R,l;R)⊕En(Rl ; Rl ). We also investigate theEuler Class Groups of polynomial rings and Laurent polynomial rings.