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Math Help - example, tensor product

  1. #1
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    example, tensor product

    Give an example of a ring A and A-modules B, C, D such that 0 \rightarrow B \rightarrow C is exact, yet

    0 \rightarrow B \otimes_A D \rightarrow C \otimes_A D

    is not exact.

    I cannot think of an example where this would be true. Initially, I was thinking to use A=\mathbb{Z} and then use ideals from that. However, I do not think that my example actually works. I need some help here. Thanks.
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  2. #2
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    Quote Originally Posted by pascal4542 View Post
    Give an example of a ring A and A-modules B, C, D such that 0 \rightarrow B \rightarrow C is exact, yet

    0 \rightarrow B \otimes_A D \rightarrow C \otimes_A D

    is not exact.

    I cannot think of an example where this would be true. Initially, I was thinking to use A=\mathbb{Z} and then use ideals from that. However, I do not think that my example actually works. I need some help here. Thanks.
    You can find the similar example of your problem in Example 3 of Dummit p 401.

    You see that 0 \rightarrow \mathbb{Z} \overset{2}{\rightarrow} \mathbb{Z} is an exact sequence. The kernel of the second arrow is 0.
    However, the induced map 0 \rightarrow \mathbb{Z} \otimes_{\mathbb{Z}}\frac{\mathbb{Q}}{\mathbb{Z}} \rightarrow \mathbb{Z}\otimes_{\mathbb{Z}} \frac{\mathbb{Q}}{\mathbb{Z}} is not exact since the kernel of the second map is not 0. Take 1 \otimes (1/2 + \mathbb{Z}) as a counter example.
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