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Results tagged with modules
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user 4790
For questions on modules over rings.
20
votes
Accepted
If a tensor product of modules is semi-simple, are the tensor factors semi-simple ?
No; if $R$ is a DVR with uniformizing paramenter $t$, the $R$-module $(R/tR) \otimes (R/t^2R) = R/tR$ is semisimple, but $R/t^2R$ is not.
- 27k
15
votes
Accepted
flatness of power series rings
It is known that the product of flat $A$-modules is flat if and only if the ring $A$ is coherent, that is, every finitely generated ideal is finitely presented ( http://www.ams.org/journals/tran/1960-097 …
- 27k
8
votes
Accepted
A criterion for freeness over a local ring
No, this is false as soon as $n ≥ 3$. A second syzygy $M$ of the residue field $K$ gives a counterexample: each $M[1/X_i]$ is projective, hence free, and it is reflexive, so the second condition is sa …
- 27k