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If M is a finitely generated projective R-module then $M \bigotimes -$ is exact.

I need some help to prove this. So please give some hints.

Germain
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1 Answers1

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Proceed as follows:

  1. $M=R$ satisfies this property (which is, by the way, known as flatness).
  2. If $M,M'$ satisfy this property, then so does $M \oplus M'$.
  3. If $M \oplus M'$ satisfies this property, then so does $M$.
  4. Conclude that every finitely generated projective module satisfies this property (actually every projective module, since 2. also holds for infinitely many summands).

Alternatively, finitely generated projective modules are dualizable, and dualizable objects are flat since $M \otimes -$ is right adjoint to $M^* \otimes -$.

Martin Brandenburg
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