On uniqueness of P-twists

TL;DR

This paper proves the uniqueness of convolutions defining P-twists for P^n-functors and introduces a new concept of non-split P^n-functors, advancing the understanding of their structure.

Contribution

It establishes the isomorphism of all convolutions of the defining complex for P-twists and introduces non-split P^n-functors, providing new insights into their properties.

Findings

All convolutions of the P-twist complex are isomorphic.

Introduces the concept of non-split P^n-functors.

Advances the theoretical understanding of P-twists in derived categories.

Abstract

We prove that for any $\mathbb{P}^n$-functor all the convolutions (double cones) of the three-term complex $FHR \xrightarrow{\psi} FR \xrightarrow{tr} Id$ defining its $\mathbb{P}$-twist are isomorphic. We also introduce a new notion of a non-split $\mathbb{P}^n$-functor.