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Results tagged with modules
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user 75
For questions on modules over rings.
6
votes
If the tensor product of two $kG$-modules is projeсtive, does either of them have to be proj...
In that case, we can understand modules in terms of the support variety $V={\rm Proj} H^{2*}(G,k)$, the projective variety associated to the (even-dimensional) cohomology ring of $G$. … Thus unless $V$ is just a single point, it is possible to have non-projective modules whose tensor product is projective. …
- 31.2k
11
votes
is the category of coherent sheaves some kind of abelian envelope of the category of vector ...
To see this, note that the category of finitely generated projective modules is self-dual via the functor $Hom(-,R)$, but the inclusion of projective modules into all modules is obviously not usually self-dual … (for instance, since projective modules are not the same as injective modules). …
- 31.2k
11
votes
Accepted
Injective flat module
More generally, a similar argument shows that flat modules over any Artinian local ring are free (replace this short exact sequence by the filtration by powers of the maximal ideal), so any such ring that …
- 31.2k