The Group of Isometries of $K_0(\mathbb P_n)$

TL;DR

This paper investigates the structure of the isometry group of the Grothendieck group of projective space, revealing it as a free abelian group of a specific rank and explicitly determining generators for low dimensions.

Contribution

It characterizes the isometry group of $K_0(P_n)$ with the Euler form, showing it is a free abelian group of rank $[rac{n+1}{2}]$ and computes generators for $n \\leq 6$.

Findings

The isometry group is essentially a free abelian group of rank $[rac{n+1}{2}]$.

Explicit generators are computed for $n \\leq 6$.

Provides structural insights into $K_0(P_n)$ isometries.

Abstract

We study the group of isometries of the Grothendieck group $K_0(\mathbb P_n)$ equipped with the standard Euler form $\chi$ defined by $\chi(E, F) = \sum_{\nu}(-1)^\nu\dim Ext^\nu(E, F)$. We prove several properties of this group, in particular, we show that it is essentially a free abelian group of rank $[\frac{n+1}{2}]$. Also, we compute explicitly its generators for $n\leqslant 6$.