Second differentials in the Quillen spectral sequence

TL;DR

This paper introduces generalized first Chern classes for coherent sheaves on algebraic varieties and uses them to explicitly compute the second differentials in the Quillen spectral sequence, advancing understanding of algebraic K-theory.

Contribution

It defines generalized first Chern classes for sheaves supported in codimension p and derives an explicit formula for the d_2 differentials in the Quillen spectral sequence.

Findings

Explicit formula for d_2 differentials in Quillen spectral sequence

Generalized first Chern classes for sheaves with support in codimension p

Enhanced computational tools for algebraic K-theory

Abstract

For an algebraic variety $X$ we introduce generalized first Chern classes, which are defined for coherent sheaves on $X$ with support in codimension $p$ and take values in $CH^p(X)$. We use them to provide an explicit formula for the differentials ${d_2^p: E_2^{p,-p-1} \to E_2^{p+2, -p-2} \cong CH^{p+2}(X)}$ in the Quillen spectral sequence.