2016/12/02 by Mandal, Satya, Mishra, Bibekananda
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1612.00749
Let A be a commutative noetherian ring, containing a field k, with 1/2∈ k, dim A=d, and let P be a projective A-module or rank(P)=n. In continuation of \citeMM, we study Homotopy obstructions for P to split off a free direct summand. Let \mathcal LO(P) be the set of all pairs (I, ω), where I is an ideal of A and ω: P→ I/I2 is a surjective map. The homotopy relations on \mathcal LO(P), induced by \mathcal LO(P[T]), leads to a set π0(\mathcal LO(P)) of equivalence classes in \mathcal LO(P). There are two distinguished elements \bf e0, \bf e1∈ π0(\mathcal LO(P)), respectively, the images of (0, 0) and (A, 0). Define the obstruction class e(P)=\bf e0∈ π0(\mathcal LO(P)). The following results are under suitable smoothness or regularity hypotheses. When 2n≥ d+3, we prove e(P)=\bf e1 ⇔ P≅ Q⊕ A. We prove, if 2n≥ d+2, then π0(\mathcal LO(P)) has a natural structure of a monoid, which is a group if P≅ Q⊕ A. Further, we give a definition of a Euler class group E(P). Under suitable smoothness hypotheses, we prove, if P≅ Q⊕ A and 2n≥ d+3, then there is natural isomorphism E(P) → π0(\mathcal LO(P)) of groups.