Let $V$ and $W$ be two $k$-vector spaces of the same dimension and $K/k$ any field extension. If $V\otimes_k K\cong W\otimes_k K$ as $K$-vector spaces then are $V$ and $W$ already isomorphic over $k$?
Many thanks in advance.
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10$\begingroup$ As soon as you said "of the same dimension" the spaces were isomorphic. $\endgroup$– Tobias KildetoftAug 5, 2014 at 18:17
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$\begingroup$ Ah yes. My eyes completely skipped over "of the same dimension", as it is both redundant and makes the problem trivial. $\endgroup$– user14972Aug 5, 2014 at 18:18
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2$\begingroup$ The question would be more interesting if you only require $k$ to be a commutative ring. $\endgroup$– Zhen LinAug 5, 2014 at 18:18
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$\begingroup$ @ZhenLin But then the answer would just end up being no (once we removed the part about same dimension which would no longer make sense in general). $\endgroup$– Tobias KildetoftAug 5, 2014 at 18:23
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1$\begingroup$ The answer is no in general. But then there's that business with faithfully flat descent... $\endgroup$– Zhen LinAug 5, 2014 at 18:27
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1 Answer
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Yes: your conjecture can be proven simply by looking at the dimensions of the vector spaces. In particular, $\otimes_k K$ commutes with direct sums, and
$$ \left(\bigoplus_i k \right) \otimes_k K \cong \bigoplus_i K $$
so $\dim_k V = \dim_K (V \otimes_k K)$.
(note that it is important that you asked for them to be isomorphic as $K$-vector spaces, as $K \cong K \oplus K$ as $k$-vector spaces for any infinite dimensional extension, which means $V = k$ and $W = k^2$ would be a counterexample)