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I'm reading "HOMOLOGICAL ALGEBRA ON A COMPLETE INTERSECTION, WITH AN APPLICATION TO GROUP REPRESENTATIONS" by David Eisenbud

I'm wondering what would be a nice example illustrating Theorem 6.1 on page 52 of the document, where I get a periodic free resolution of $M$ after $d+1$ steps, and another example where I can see a periodic free resolution when $M$ is maximal Cohen-Macaulay.

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    $\begingroup$ Cross posted: math.stackexchange.com/q/4595497/884739 $\endgroup$
    – It'sMe
    Commented Dec 10, 2022 at 3:01
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    $\begingroup$ For instance, let $A = k[t]/t^2$, then $$\dots \to A \stackrel{t}\to A \stackrel{t}\to A \stackrel{t}\to A \stackrel{t}\to A \to k \to 0$$ is an example of 2-periodic (actually, even 1-periodic) resolution. $\endgroup$
    – Sasha
    Commented Dec 10, 2022 at 4:34
  • $\begingroup$ @Sasha thanks for your answer. Do you have any other example where the periodicity starts after "$d+1$" steps? $\endgroup$
    – It'sMe
    Commented Dec 10, 2022 at 4:44
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    $\begingroup$ For the same $A$, e.g., you can replace $k$ by $A \oplus k$ and then periodicity will start from the second step. For a longer initial interval you need a more complicated ring. $\endgroup$
    – Sasha
    Commented Dec 10, 2022 at 5:30
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    $\begingroup$ But you can use the same idea: just add to $M$ another module that has a finite free resolution --- this will spoil the first steps keeping the rest unchanged. $\endgroup$
    – Sasha
    Commented Dec 10, 2022 at 5:38

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