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I'm procedding with understanding Greenlees's paper "The Four Approaches to Cohomology Theories with Reality": https://arxiv.org/abs/1705.09365

Problem concerns the section 1.D. Consider the cofiber sequence; $$ X_h\to X^h\to X^t, $$ where $X^h=F(EQ_+,X)$ ($Q$ is a cyclic group of order two), $X_h=EQ_+\wedge X$ and $X^t=F(EQ_+,X)\wedge \tilde{E}Q$.

Then there is an information, that knowing $RO(Q)$-graded coefficients of $X^h$ and $X^t$ I can infer the coefficients for $X_h$ via this cofiber sequence ("which amounts to a simple Local Cohomology Theorem").

How do I do that? And what is Local Cohomology Theorem?

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    $\begingroup$ Any time you have a cofiber sequence $X \to Y \to Z$ in some stable category you get a long exact sequence $\cdots \to [W, \Sigma^{-1}Z] \to [W,X] \to [W, Y] \to [W,Z]\to\cdots$ which you can use to try to compute one of the terms if you know the other two. But it's not an exact science. (pun intended) $\endgroup$
    – Dylan Wilson
    Commented May 28, 2018 at 15:49
  • $\begingroup$ Ok, now I see, thank you. But still the question remains - what is the Local Cohomology Theorem? $\endgroup$
    – Igor Sikora
    Commented May 29, 2018 at 15:41

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