Euler class groups and a theorem of Roitman

Abstract

a b s t r a c tWe define the Euler class group of a polynomial algebra in lower codimension and prove an analogue of a result of Roitman (on projective modules) for the Euler class groups.

Some results on Euler class groups

“…Using ([2], 4.11, 5.7), (2.2) and following the proof of ( [4], Theorem 4.2), we can prove the following result. This result is also proved in ( [6], Theorem 3.1). Note that regularity of the ring is used only when 2n = d + 3.…”

On triviality of the Euler class group of a deleted neighbourhood of a smooth local scheme

“…We now note that S = k[Z 1 , · · · , Z d ] (Z 1 ,Z 2 ,··· ,φ(Z d )) , where k is the residue field of W , which is same as the residue field of R and is infinite. Therefore, applying (4.2), we conclude that α 1 , · · · , α n can be lifted to a set of n generators of L. We can now apply [DS,Proposition 3.3] to see that there exist γ 1 , · · · , γ n such that I 1 = (γ 1 , · · · , γ n ) where γ i − α i ∈ I 1 2 . We note that such a conclusion holds trivially if L is not…”

On a question of Moshe Roitman and Euler class of stably free module

“…Proof. Following the proof of Das-Sridharan [8,Theorem 2.11] where it is proved for regular A and use (7.7).…”