The ascent-descent property for $2$-term silting complexes

Abstract

We will prove that over commutative rings the silting property of 2-term complexes induced by morphisms between projective modules is preserved and reflected by faithfully flat extensions.

Silting Complexes and Partial Tilting Modules

“…It should be noted that in Example 2.6, in the paper [15] by Breaz, a similar phenomenon is exhibited to those studied in our paper, but for a different problem.…”

Minimal silting modules and ring extensions

“…The behaviour of (co)silting modules under ring extensions has been studied in [12]. In particular, it was shown that over commutative rings every silting module extends to a silting module.…”

“…Of course, if λ : R → S is surjective with kernel I, then every silting right R-module T verifies the condition T ⊗ R S ≃ T /T I ∈ GenT and therefore extends to a silting S-module T ⊗ R S, cf. [12,Corollary 2.4]. In general, however, the condition in the theorem above can fail, even when T is minimal.…”

“…The second aspect we want to address is the behaviour of silting and cosilting modules under ring extensions. A criterion recently established in [12] ensures that every silting module T over a commutative ring R extends to a silting module T ⊗ R S along any ring epimorphism R → S. In Section 4, we give conditions under which minimality is preserved. In particular, we show that all minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism (Corollary 4.11).…”

Silting, cosilting, and extensions of commutative ring

“…It is already known that the properties of being projective, 1-tilting and 2-silting are ad-properties (cf. [29], [19], and [8]). The Zariski locality for ntilting modules was also proved in [19].…”