(Gorenstein) silting modules in recollements

TL;DR

This paper investigates Gorenstein silting modules using recollements and tensor products, establishing conditions for silting modules across quotient rings and demonstrating how Gorenstein silting properties can be combined in complex ring structures.

Contribution

It introduces new methods to analyze Gorenstein silting modules via recollements and tensor products, extending the understanding of silting theory in module categories.

Findings

Silting modules over A/J are equivalent to silting modules over A/J.

Tensor products of silting modules over finite dimensional k-algebras remain silting.

Gorenstein silting properties can be glued through recollements of module categories.

Abstract

In the paper, we focus on the silting properties and the combinatorial properties of silting and Gorenstein, which is called Gorenstein silting, where the main tools used are recollements of module categories and tensor products. For a ring A and its idempotent ideal J, we show that an A/J-module T is a silting A-module if and only if T is a silting A/J-module. For the finite dimensional k-algebras, with k a field, we show that the tensor products of silting modules are still silting. We also show that the (partial) Gorenstein silting properties can be glued by the recollements of module categories of Noetherian rings. As a consequence, we glue the Gorenstein silting modules of an upper triangular matrix Gorenstein ring by those of the involved rings.