Timeline for Regularity before and after completion - reference request
Current License: CC BY-SA 4.0
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| Jul 28, 2022 at 19:12 | comment | added | Johan | When $n = 1$ you could argue like this: $M$ is $x = x_1$-torsion free, so completion is equal to derived completion. Then the short exact sequence $0 \to M \to M \to M/xM \to 0$ upon derived completion becomes the distinguished triangle $M^\wedge \to M^\wedge \to (M/xM)^\wedge$. But $(M/xM)^\wedge = M/xM$ so sits only in degree $0$ hence $x$ is regular on $M^\wedge$. For larger $n$ try induction on $n$ (using the same argument essentially). Cheers! | |
| Jul 28, 2022 at 14:41 | history | asked | Neil Strickland | CC BY-SA 4.0 |