Graded annihilators and tight closure test ideals
| dc.creator | Sharp, Rodney Y. | |
| dc.date | 2008-08-11 | |
| dc.date.accessioned | 2026-07-07T09:55:59Z | |
| dc.date.available | 2026-07-07T09:55:59Z | |
| dc.description | Let $R$ be a commutative Noetherian local ring of prime characteristic $p$. The main purposes of this paper are to show that if the injective envelope $E$ of the simple $R$-module has a structure as a torsion-free left module over the Frobenius skew polynomial ring over $R$, then $R$ has a tight closure test element (for modules) and is $F$-pure, and to relate the test ideal of $R$ to the smallest '$E$-special' ideal of $R$ of positive height. A byproduct is an analogue of a result of Janet Cowden Vassilev: she showed, in the case where $R$ is an $F$-pure homomorphic image of an $F$-finite regular local ring, that there exists a strictly ascending chain $0 = τ_0 \subset τ_1 \subset ... \subset τ_t = R$ of radical ideals of $R$ such that, for each $i = 0, ..., t-1$, the reduced local ring $R/τ_i$ is $F$-pure and its test ideal (has positive height and) is exactly $τ_{i+1}/τ_i$. This paper presents an analogous result in the case where $R$ is complete (but not necessarily $F$-finite) and $E$ has a structure as a torsion-free left module over the Frobenius skew polynomial ring. Whereas Cowden Vassilev's results were based on R. Fedder's criterion for $F$-purity, the arguments in this paper are based on the author's work on graded annihilators of left modules over the Frobenius skew polynomial ring. | |
| dc.description | This is to appear in the Journal of Algebra | |
| dc.identifier | https://arxiv.org/abs/0808.1483 | |
| dc.identifier | http://arxiv.org/abs/0808.1483 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/166823 | |
| dc.subject | Commutative Algebra | |
| dc.subject | 13A35, 16S36, 13D45, 13E05, 13E10, 13H10 (Primary) 13J10 (Secondary) | |
| dc.title | Graded annihilators and tight closure test ideals | |
| dc.type | text |