arXiv subjects
Publications and source records attributed to M. K. Keshari.
Let $A$ be a regular ring of dimension $d$ essentially of finite type over an infinite field $k$ of characteristic $\neq 2$. Let $P$ be a projective $A$-module of rank $n$ with $2n\geq d+3$. Let $I$ be an ideal of $A[T]$ of height $n$ and $ϕ:P[T]\twoheadrightarrow I/I^2$ be a surjection. If $ϕ\otimes A(T)$ has a surjective lift $θ:P[T]\otimes A(T)\twoheadrightarrow IA(T)$, then $ϕ$ has a surjective lift $Φ:P[T]\twoheadrightarrow I$.
Let R be a commutative Noetherian ring of dimension d and B=R[X_1,\ldots,X_m,Y_1^{\pm 1},\ldots,Y_n^{\pm 1}] a Laurent polynomial ring over R. If A=B[Y,f^{-1}] for some f\in R[Y], then we prove the following results: (i) If f is a monic polynomial, then Serre dimension of A is \leq 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^p(A) of A, defined by Bhatwadekar and Raja Sridharan, is trivial for p\geq max \{d+1, \dim A -p+3\}. In case m=n=0, this result is due to Mandal-Parker.
Let $(R,\mm,K)$ be a regular local ring containing a field $k$ such that either char $k=0$ or char $k=p$ and tr-deg $K/\BF_p\geq 1$. Let $g_1,\ldots,g_t$ be regular parameters of $R$ which are linearly independent modulo $\mm^2$. Let $A=R_{g_1\cdots g_t} [Y_1,\ldots,Y_m,f_1(l_1)^{-1},\ldots, f_n(l_n)^{-1}]$, where $f_i(T)\in k[T]$ and $l_i=a_{i1}Y_1+\ldots+a_{im}Y_m$ with $(a_{i1},\ldots,a_{im})\in k^m-(0)$. Then every projective $A$-module of rank $\geq t$ is free. Laurent polynomial case $f_i(l_i)=Y_i$ of this result is due to Popescu.
In this paper, we prove the moving lemma, addition and subtraction principles, in a more general setup than the available ones. We apply these results to explore a question of Nori on homotopy of sections of projective modules. As another application, we investigate the Segre class of ideals, extending the existing results.
This is my PhD thesis from 2004 under Prof. S.M. Bhatwadekar. Here we answer a question of Nori and prove the following result. Let $A$ be a smooth affine domain of dimension $d$ over an infinite perfect field. Let $I$ be an ideal of $A[T]$ of height $n$ such that $2n \geq d+3$. Given surjections $ϕ:(A[T]/I)^n \rightarrow I/I^2$ and $ρ:A^n \rightarrow I(0)$ such that $ϕ(0)=ρ\otimes A/I(0)$, then there exist a surjection $Φ:A[T]^n \rightarrow I$ such that $Φ(0)=ρ$ and $Φ\otimes A/I =ϕ$. This is a joint work with S.M. Bhatwadekar.
We will study the Euler class group of a commutative Noetherian ring $A$ of dimension $d$, developed by S.M. Bhatwadekar and Raja Sridharan. This is my M.Phil thesis from 2001.