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Let $k$ be a field and $K$ is a extension of this field.

And let $A$ be a finite type $k$-algebra, and $M$ be a finitely generated module over $A$.

Then, is the form of submodules of $M⊗_kK$ always $N⊗_kK$? (where $N$ is an $A$-submodule of $M$)

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  • $\begingroup$ Do you want $A$-submodules or $A\otimes K$-submodules ? $\endgroup$ – Captain Lama Mar 31 '16 at 11:37
  • $\begingroup$ sorry,as A⊗K submodule . $\endgroup$ – 8Frog Mar 31 '16 at 11:54
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No. Small examples suffice: take $k=\mathbb{F}_p$, the finite field of $p$ elements, and take $K=\mathbb{F}_{p^2}$. Now, take $A=k$ and $M=k^2$. Then $M$ only has $p+1$ non-trivial submodules, while $M\otimes_k K$ (which is isomorphic to $K^2$) has $p^2+1$ non-trivial submodules. Thus $M\otimes_k K$ must have submodules which are not of the form $N\otimes_k K$, with $N$ a submodule of $M$.

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  • $\begingroup$ oh thank you for the nice counterexample! $\endgroup$ – 8Frog Mar 31 '16 at 11:56

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