Commutative algebra seminar Date and time: Thursday, 12 January 2023, 4 pm Venue: Ramanujan Hall Host: Manoj Keshari Speaker: Soumi Tikedar, Diamond Harbour Women's University Title: On a question of Moshe Roitman and its applications Abstract: Let A be a ring of dimension d and P be a projective A[T]-module of rank n. We say that p ∈ P is a unimodular element if there exists a homomorphism f in P* such that f(p) = 1. When n > d, then Plumstead proved that P has a unimodular element. But this is not the case for n=d and n< d. In this talk, we will discuss the following results: Theorem: Let A be a ring of dimension d containing an infinite field k, P be a projective A[T]-module of rank n such that 2n is not less than d + 3 and singular locus of Spec(A) is a closed set V(J) with ht J is atleast d − n + 2. If P_f has a unimodular element for some monic polynomial f(T). Then P has a unimodular element. Next, we will discuss some applications of Roitman's question to define the Euler class group, which serves as an obstruction group to detect the existence of unimodular elements in the Projective module with certain conditions. In this talk, we associate a stably free module to the Euler class group and show that the vanishing of this is the precise obstruction having P unimodular element.