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Math Help - projective dimension

  1. #1
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    projective dimension

    I have to find the projective dimension of Z/pZ as Z-module, Z/pZ-module and Z/(p^2)Z-module, p is prime.Could anyone help me?
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  2. #2
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    Quote Originally Posted by Veve View Post

    I have to find the projective dimension of Z/pZ as Z-module, Z/pZ-module and Z/(p^2)Z-module, p is prime.Could anyone help me?
    let M=\mathbb{Z}/p\mathbb{Z}. clearly M as \mathbb{Z}/p\mathbb{Z} module is free and so projective. thus \text{proj} \cdot \dim_{\mathbb{Z}/p\mathbb{Z}} M = 0.

    M as \mathbb{Z} module is torsion and so it's not projective. now consider the exact sequence 0 \to p\mathbb{Z} \to \mathbb{Z} \to M \to 0. as \mathbb{Z} modules: p\mathbb{Z} \cong \mathbb{Z}. hence p\mathbb{Z} is projective. thus \text{proj} \cdot \dim_{\mathbb{Z}} M = 1.

    finally looking at M as \mathbb{Z}/p^2 \mathbb{Z} module we have this exact sequence: 0 \to p \mathbb{Z}/p^2 \mathbb{Z} \to \mathbb{Z}/p^2 \mathbb{Z} \to M \to 0. but as \mathbb{Z}/p^2 \mathbb{Z} module: p \mathbb{Z}/p^2 \mathbb{Z} \cong M. therefore: \text{proj} \cdot \dim_{\mathbb{Z}/p^2 \mathbb{Z}} M = \infty.
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