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Let $K$ be a ring, $L$ an extension of $K$, and $I$ an ideal of $K[X_1,...,X_n]$. Is it true that

$$ L[X_1,...,X_n]/(IL[X_1,...,X_n]) \cong (K[X_1,...,X_n]/I) \otimes_K L \ \ ?$$

(additional question: does something similar hold more generally, for instance when $K$ is a ring and $L$ a $K$-algebra (and $K$ does not necessarily inject into $L$)?)

I have troubles applying general commutative algebra machinery because the situation is very specific: we have an identification $K[X_1,...,X_n] \otimes_K L \cong L[X_1,...,X_n]$, into which $K[X_1,...,X_n]$ injects.

57Jimmy
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1 Answers1

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Hint: As a scalar extension, the first step is that $L[X] \simeq K[X]\otimes_K L$.

Moreover, for the ideal, $I L[X] \simeq I\otimes_K L$.

Thus $L[X]/(IL[X]) \simeq (K[X]\otimes_K L) / (I\otimes_KL)\simeq (K[X]/I)\otimes_K L$.

Wuestenfux
  • 20,964
  • Ok thanks. So I guess you can do this (in general for algebras) whenever $K$ injects into $L$ and $L$ is $K$-flat? Or even in complete generality? – 57Jimmy Jan 08 '21 at 14:08
  • I guess it should hold in complete generality, but then $I \otimes_K L$ should be replaced by its image in the second term of the last line – 57Jimmy Jan 11 '21 at 11:00