Methods in complete intersections in corank one

TL;DR

This paper explores the extension of complete intersection methods to corank one cases in algebraic geometry, highlighting the need for refined cancellation theorems and hypothesizing their implications.

Contribution

It investigates the adaptation of complete intersection techniques to corank one scenarios, emphasizing the necessity of improved cancellation theorems and proposing related hypotheses.

Findings

Cancellation theorems require refinement for corank one cases

Hypotheses on complete intersections lead to new algebraic insights

Potential extensions of complete intersection theory are suggested

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.