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Results tagged with tensor-products
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user 450
132
votes
3
answers
21k
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When is the tensor product of two fields a field?
Consider two extension fields $K/k, L/k$ of a field $k$.
A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by so …
18
votes
Duals and Tensor products
For those interested in general statements, here is a summary of assumptions under which the canonical morphisms of $A$-modules below are isomorphisms:
If $P$ is finitely generated projective:
$$P\ …
- 47.5k
5
votes
reduced ⊗ reduced = reduced; what about connected?
Ad (1): It is perfectly true that the tensor product of two reduced algebras over a perfect field is reduced. You can find a proof in Bourbaki's Algebra (Chapter V; §15; 4,5), but of course this vener …
- 47.5k
9
votes
Characterisation for separable extension of a field
Let $k\subset K$ be a completely arbitrary extension of fields. This extension is said to be separable if equivalently
a) For all extensions $k\subset L$, the ring $K \otimes _k L$ is reduced. [A r …
- 47.5k