Timeline for Is $\dim_k M/xM$ a multiple of $\dim_k R/xR$ for $M$ finitely generated, torsion-free $R$-module?
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 14, 2018 at 7:44 | vote | accept | windsheaf | ||
| Dec 11, 2018 at 17:38 | answer | added | Hailong Dao | timeline score: 8 | |
| Dec 11, 2018 at 14:55 | comment | added | Jason Starr | Even if $R$ is irreducible, you will only get an inequality: $\text{dim}_k(M/aM) \geq \text{dim}_{\text{Frac}(R)}(M\otimes_R \text{Frac}(R)) \cdot \text{dim}_k(R/aR)$. For instance, consider the case where $M$ equals the normalization of $R$ (with $R$ not normal). If $R$ is reducible, you have the same inequality if you define $\mu(M)$ to be the minimum over all minimal primes of $R$ of the rank of $M$ after localization at that prime | |
| Dec 11, 2018 at 8:57 | history | asked | windsheaf | CC BY-SA 4.0 |