Recollements, sinks elimination and Leavitt path algebras

TL;DR

This paper explores how removing sinks from graphs associated with Leavitt path algebras creates a recollement structure, extending understanding of algebraic relations through graph modifications.

Contribution

It establishes a general recollement framework for Leavitt path algebras based on hereditary subsets of graphs, highlighting differences between removing sources and sinks.

Findings

Removing sources yields Morita equivalence.

Removing sinks induces a recollement structure.

Recollement relates Leavitt path algebras of different graph modifications.

Abstract

For Leavitt path algebras, we show that whereas removing sources from a graph produces a Morita equivalence, removing sinks gives rise to a recollement situation. In general, we show that for a graph $E$ and a finite hereditary subset $H$ of $E^0$ there is a recollement $$\xymatrix{ L_K(E/\overline H) \rModd \ar[r] & \ar@<3pt>[l] \ar@<-3pt>[l] L_K(E) \rModd \ar[r] & \ar@<3pt>[l] \ar@<-3pt>[l] L_K(E_H) \rModd .}$$ We record several corollaries.