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 notion of relative cohomology of a morphism $f\colon Y \to X$ has been defined for some classic cohomology? (classic= singular, étale, de Rham...)

If so,

2.- Could you provide a reference?

Thank you.

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.