Manoj K. Keshari

A Question of Nori: Projective Generation of Ideals

Let A be a smooth affine domain of dimension d over an infinite perfect field k and let n be an integer such that 2n ≥ d + 3. Let I ⊂A[T] be an ideal of height n. Assume that I = (f 1 ,...,f n ) + (I 2 T). Under these assumptions, it is proved in this paper that I = (g 1 ,...,g n ) with f i - g i ⊂ (I 2 T), thus settling a question of Nori affirmatively.

Serre dimension and Euler class groups of overrings of polynomial rings

Let R be a commutative Noetherian ring of dimension d and B = R[X1, . . . , Xm, Y ±1 1 , . . . , Y ±1 n ] a Laurent polynomial ring over R. If A = B[Y, f−1] for some f ∈ R[Y ], then we prove the following results: (i) If f is a monic polynomial, then Serre dimension of A is ≤ d. In case n = 0, this result is due to Bhatwadekar, without the condition that f is a monic polynomial. (ii) The p-th Euler class group E(A) of A, defined by Bhatwadekar and Raja Sridharan, is trivial for p ≥ max{d+ 1, dimA− p+ 3}. In case m = n = 0, this result is due to Mandal-Parker.

A note on rigidity and triangulability of a derivation

Let A be a

Cancellation problem for projective modules over affine algebras

\mathfrak Q

Projective modules over overrings of polynomial rings

-domain, K=frac(A), B=A^{[n]} and D\in \lnd_A(B). Assume rank D= rank D_K=r, where D_K is the extension of D to K^{[n]}. Then we show that (i) If D_K is rigid, then D is rigid. (ii) Assume n=3, r=2 and B=A[X,Y,Z] with DX=0. Then D is triangulable over A if and only if D is triangulable over A[X]. In case A is a field, this result is due to Daigle.