The monoid structure on homotopy obstructions

[摘要] Let A be a commutative noetherian ring, containing a field k, with 1/2 is an element of k, dim A = d, and let P be a projective A-module, with rank(P) = n. Let LO(P) denote the set of all pairs (I, omega), where I is an ideal of A and omega : P -> I/I-2 is a surjective map. The homotopy relations on LO(P), induced by LO(P[T]), leads to a set pi(0) (GO(P)) of equivalence classes in LO(P). There are two distinguished elements e(0), e(1) is an element of (LO(P)), respectively, the images of (0, 0) and (A, 0). Define the obstruction class epsilon(P) = e(0) is an element of pi(0) (LO(P)), to be called the (Noni) homotopy class of P. The following results are under suitable smoothness or regularity hypotheses. We prove, if 2n >= d + 2, then pi(0) (LO(P)) has a natural structure of a monoid, which is a group if P = Q A. When 2n >= d + 3, we prove P congruent to Q circle plus A double left right arrow> epsilon(P) = e(1) (the additive zero). Further, we give a definition of a Euler class group E(P). Under suitable smoothness hypotheses, we prove, if P congruent to Q circle plus A and 2n >= d + 3, then there is natural isomorphism E(P) -> pi(0) (LO(P)) of groups. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-15

[关键词] Commutative algebra;Projective modules;Complete intersections