Methods in complete intersections in corank one
Satya Mandal
TL;DR
The article addresses extending complete intersection methods from corank-zero to corank-one stably free modules over affine algebras by proposing Hypothesis Intro, which posits a stability condition on quotients by nonzero divisors. Under this hypothesis, it derives key implications that connect surjections onto products of ideals with surjections onto individual ideals and translate these into module-cancellation and splitting phenomena in the corank-one setting. It then details applications to complete intersection ideals, showing how surjections to IJ or to J yield corresponding surjections to I or splitting results, and frames these results as corank-one analogues of classical theorems by Murthy–M94 and Mohan Kumar–Murthy, while highlighting the limitations imposed by normality assumptions. Overall, the work lays out a structured pathway to coalesce four themes—cancellation, complete intersections, splitting, and Chern class vanishing—within a unified corank-one theory, contingent on the proposed hypothesis and its potential relaxation.
Abstract
Let $A$ denote an affine algebra over an algebraically closed field $k$, with $\dim A=d\geq 3$. In the light of availability of cancellation theorems for stably free modules $P$ with $rank(P)=d-1$ (corank one), we try to implement the methods of complete intersections theory in corank zero, to the corank one case. Our conclusion is that cancellation theorems need to clean up some of the lack of minor generalities, for such an approach to work. However, we hypothesize and derive some of the consequences to complete intersections, of such hypotheses.