arXiv subjects

Bibekananda Mishra

Publications and source records attributed to Bibekananda Mishra.

3 recordsLinked to original sources

arXiv2019

The Monoid Structure on Homotopy Obstructions

Let $A$ be a commutative noetherian ring, containing a field $k$, with $1/2\in k$, $\dim A=d$, and let $P$ be a projective $A$-module or $rank(P)=n$. In continuation of \cite{MM}, 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 $ω: P\rightarrow I/I^2$ is a surjective map. The homotopy relations on ${\mathcal LO}(P)$, induced by ${\mathcal LO}(P[T])$, leads to a set $π_0\left({\mathcal LO}(P)\right)$ of equivalence classes in ${\mathcal LO}(P)$. There are two distinguished elements ${\bf e}_0, {\bf e}_1\in π_0\left({\mathcal LO}(P)\right)$, respectively, the images of $(0, 0)$ and $(A, 0)$. Define the obstruction class $e(P)={\bf e}_0\in π_0\left({\mathcal LO}(P)\right)$. The following results are under suitable smoothness or regularity hypotheses. When $2n\geq d+3$, we prove $e(P)={\bf e}_1 \Leftrightarrow P\cong Q\oplus A$. We prove, if $2n\geq d+2$, then $π_0\left({\mathcal LO}(P)\right)$ has a natural structure of a monoid, which is a group if $P\cong Q\oplus A$. Further, we give a definition of a Euler class group $E(P)$. Under suitable smoothness hypotheses, we prove, if $P\cong Q\oplus A$ and $2n\geq d+3$, then there is natural isomorphism $E(P) \rightarrow π_0\left({\mathcal LO}(P)\right)$ of groups.

math.AC↗

arXiv2018

Some perspective on Homotopy obstructions

Throughout $A$ will denote commutative noetherian ring, with $\dim A=d\geq 2$, and $P$ denote a projective $A$-module with $rank(P)=n$. In \cite{MM1} we considered the Homotopy obstruction sets $π_0\left({\mathcal LO}(P)\right)$, which has a structure of an abelian monoid, under suitable regularity and other conditions. In this article, we provide some perspective on these sets $π_0\left({\mathcal LO}(P)\right)$. Under similar regularity and other conditions, we prove if $P, Q$ are two projective $A$-modules, with $rank(P)=rank(Q)=d$ and $\det(P) \cong \det Q$, then $π_0\left({\mathcal LO}(Q)\right)\cong π_0\left({\mathcal LO}(P)\right)$. Further, for any projective $A$-module $P$ with $rank(P)=n$, we define a natural set theoretic map $π_0\left({\mathcal LO}(P)\right)\rightarrow CH^n(A)$, where $CH^n(A)$ Chow groups of codimension $n$ cycles.

math.AC↗

arXiv2018

The Homotopy Obstructions in Complete Intersections

Let $A$ be a regular ring over a field $k$, with $1/2\in k$ and dimension $d$. We discuss the Homotopy Conjecture of Madhav V. Nori, in the complete intersection case (meaning when the projective module in question if free, of rank at least 2). Recently, an obstruction set (sheaf) $π_0(Q_{2n})(A)$ was introduced [F] to detect when a surjective map $A^n\to I/I^2$ lifts to a surjective map $A^n\to I$. We establish that $π_0(Q_{2n})(A)$ coincides with the obstruction set of equivalence classes, originally suggested by Nori. We also establish that $π_0(Q_{2n})(A)$ has a natural groups structure, when $2n\geq d+2$. Further, we establish that, when $2n\geq d+2$, there is a surjective homomorphism $E^n(A) \to π_0(Q_{2n})(A) $, where $E^n(A)$ denotes the Euler class group defined by Bhatwadekar and Sridharan [BS2]. This homomorphism is an isomorphism, whenever triviality, in $π_0(Q_{2n})(A)$, of an orientation $(I, ω_I), guarantees that $omega_I$ lifts to a surjective map $A^n\to I$. We also give a Quadratic version of Lindel's Theorem, on extendibility of projective modules.

math.AC↗