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Let $f\colon Y \to X$ be a morphism of schemes, the inverse image in $K$-theory always fit into a long exact sequence $$ \cdots \to K_i(f)\to K_i(X) \xrightarrow {f^*} K_i(Y)\to \cdots $$ where the groups $K_i(f)$ are called the relative $K$-theory of $f$ (see for example Weibel's K-book, Chapter III).

This is nothing particular of $K$-theory and a similar construction works for any reasonable cohomology. I wonder if this notation was standard in cohomology. Basic manuals only seem to use it for the immersion of closed subvariety. Therefore

1.- Do you know if the notation of relative cohomology of a morphism $f\colon Y \to X$ has been used for some classic cohomology? (classic= singular, étale, de Rham...)

If so,

2.- Could you provide a reference?

Thank you.

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    $\begingroup$ In the classical (homotopy theory) case this is the same thing as the cohomology of the mapping cone of $f$. This works also in equivariant and motivic homotopy theory. $\endgroup$
    – Denis Nardin
    Commented Mar 12, 2016 at 20:15
  • $\begingroup$ Thank you Denis, but that I already know. I took a look at Adam's "Stable homotopy..." and he doesn't seem to speak about he subject, does he? Do you know a reference where they treat this? $\endgroup$
    – Tintin
    Commented Mar 12, 2016 at 20:20
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    $\begingroup$ Uh, again in the homotopy theory setting this is just the long exact sequence associated to a cofiber sequence. See for example propositions 3.9 and 3.10 in Adams' book, but really this is treated in any homotopy theory textbook. I don't know if there's an easy way to generalize it to étale cohomology or deRham cohomology. $\endgroup$
    – Denis Nardin
    Commented Mar 12, 2016 at 20:25
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    $\begingroup$ My guess is that everyone would understand what you mean. Historically the term relative cohomology was used only when the morphism is some kind of embedding. However up to homotopy all morphisms are embeddings (just embed in the mapping cylinder!) so nowadays most people don't really make the distinction. $\endgroup$
    – Denis Nardin
    Commented Mar 12, 2016 at 22:06
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    $\begingroup$ As Denis says, relative (co)homology is widely used when talking of pairs given by a space and a(n appropriate) subspace, but it is commonly remarked in most books that if you have a map, you can do de same by taking the mapping cylinder, which turns it into an inclusion. Any book will do, e.g. Hatcher, Spanier, Switzer, Whitehead... $\endgroup$
    – Fernando Muro
    Commented Mar 13, 2016 at 11:25

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