Silting theory under change of rings

TL;DR

This paper explores how silting theory for algebras over a Noetherian ring behaves under change of rings, establishing isomorphisms and descent results for silting and tilting complexes.

Contribution

It provides new insights into the relationship between silting theory of an algebra and its tensor products with other algebras, including isomorphisms of silting posets and descent properties.

Findings

Isomorphism between silting posets of an algebra and its quotient by an ideal.

Silting embedding and descent results for tensor product algebras.

Characterization of restrictions of silting complexes under change of rings.

Abstract

The main goal of this paper is to compare the silting theory of an $R$-algebra $\Lambda$ over a Noetherian ring $R$ with that of its tensor product $\Lambda \otimes \Gamma$ with another $R$-algebra $\Gamma$. In the case that the $R$-algebra $\Lambda$ is Noetherian, $R$ a complete local ring and $\mathfrak{a}$ a certain ideal of the ring $R$, we obtain an isomorphism between the silting poset of $\Lambda$ and that of its quotient $\Lambda \otimes R/ \mathfrak{a}$. Furthermore, we study the restrictions of such a bijection to tilting complexes and deduce silting embedding and descent results for the algebra $\Lambda$ and a certain family of algebras $(\Lambda \otimes \Gamma_i)_{i \in I}$.