Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2402.12125 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Inhaltsangabe:
- The present paper deals with the investigation of the structure of general fiber product rings $R\times_TS$, where $R$, $S$ and $T$ are local rings with common residue field. We show that the Poincaré series of any $R$-module over the fiber product ring $R\times_TS$ is bounded by a rational function. In addition, we give a description of ${\rm depth}(R\times_TS)$, which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over $R\times_TS$ obtained, we provide certain cases of the Cohen-Macaulayness of $R\times_TS$ and, in particular, we show that $R\times_TS$ is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over $R\times_TS$ are also established.