[Paper Review] Euler class group of a Noetherian ring

Manoj K. Keshari|arXiv (Cornell University)|Aug 12, 2014

Commutative Algebra and Its Applications7 references22 citations

TL;DR

This paper introduces the Euler class group $E(A)$ for a Noetherian ring $A$ of dimension $n \geq 2$ containing $\mathbb{Q}$, providing a cohomological invariant that detects whether a projective $A$-module of rank $n$ with trivial determinant admits a unimodular element. The key result establishes that such a module has a unimodular element if and only if its Euler class in $E(A)$ vanishes, generalizing earlier results over smooth affine algebras and resolving a question posed by Nori.

ABSTRACT

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.

Motivation & Objective

• To define an invariant that detects the existence of unimodular elements in projective modules of rank equal to the dimension of a Noetherian ring.
• To generalize Nori's Euler class group construction from smooth affine algebras to arbitrary Noetherian rings containing $\mathbb{Q}$.
• To prove that the vanishing of the Euler class in $E(A)$ characterizes the existence of unimodular elements for projective modules of rank $n$ with trivial determinant.
• To extend the theory to a weak Euler class group $E_0(A)$, relating it to stable freeness and $K_0$-classes.

Proposed method

• Define $E(A)$ as a quotient of the free abelian group on pairs $(J, w_J)$, where $J \subset A$ is an ideal of height $n$ with $J/J^2$ generated by $n$ elements and $w_J$ is a generating set.
• Establish addition and subtraction principles in $E(A)$ that allow manipulation of such pairs via local and global patching techniques.
• Construct the Euler class $e(P, \chi) \in E(A)$ for a projective $A$-module $P$ of rank $n$ with a trivialization $\chi: A \to \wedge^n P^*$.
• Prove that $P$ has a unimodular element if and only if $e(P, \chi) = 0$ in $E(A)$, using the existence of surjections and stably free approximations.
• Introduce the weak Euler class group $E_0(A)$ as a quotient of $E(A)$, and relate the vanishing of the weak Euler class to $K_0$-equivalence with $Q \oplus A$.
• Use the canonical homomorphism $E(A) \to E_0(A)$ and properties of ideals of height $n$ to derive $K_0$-theoretic consequences.

Experimental results

Research questions

• RQ1 Can an invariant be defined for a Noetherian ring $A$ of dimension $n$ that detects whether a projective $A$-module of rank $n$ with trivial determinant has a unimodular element?
• RQ2 Does the Euler class group $E(A)$, defined for Noetherian rings containing $\mathbb{Q}$, vanish if and only if such a module is stably free or has a unimodular element?
• RQ3 How does the weak Euler class group $E_0(A)$ relate to the $K_0$-group of $A$, particularly in the context of $[P] = [Q \oplus A]$?
• RQ4 Under what conditions is an ideal $J$ of height $n$ a surjective image of a stably free module of rank $n$?
• RQ5 Is the Euler class $e(P, \chi)$ in $E(A)$ independent of the choice of trivialization $\chi$?

Key findings

• The Euler class $e(P, \chi)$ of a projective $A$-module $P$ of rank $n$ with trivial determinant vanishes in $E(A)$ if and only if $P$ has a unimodular element.
• For a Noetherian ring $A$ of even dimension $n$, the weak Euler class $e(P)$ vanishes in $E_0(A)$ if and only if $[P] = [Q \oplus A]$ in $K_0(A)$ for some projective $A$-module $Q$ of rank $n-1$.
• If $(J, w_J) \in E(A)$ lies in the kernel of $E(A) \to E_0(A)$, then $(J, w_J)$ is the Euler class of a stably free $A$-module of rank $n$, implying $J$ is a surjective image of such a module.
• The existence of a surjection $P \to J$ for an ideal $J$ of height $n$ with $J/J^2$ generated by $n$ elements implies that $J$ is a surjective image of a stably free module if and only if $e(P) = 0$ in $E_0(A)$.
• The construction of $E(A)$ and $E_0(A)$ is valid for Noetherian rings of dimension $n \geq 2$ containing $\mathbb{Q}$, generalizing earlier results over smooth affine algebras.
• The group $E(A)$ detects obstructions to unimodular elements in projective modules of rank equal to the dimension, even when the top Chern class fails to do so over non-algebraically closed fields.