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In algebraic topology, one defines relative cohomology groups $H(X,A)$ of a pair of spaces $A\subset X$. Is there an analogue in algebraic geometry of cohomology of a pair of schemes?

For example, let $H$ be either etale or de Rham cohomology theory, let $X$ be a scheme and let $D$ be a divisor on $X$ or a closed subvariety of $X$. Can one define relative cohomology $H^*(X,D)$ of a pair in a useful way ? Under what assumptions?

The simplest example would be $X$ a curve and $D$ is a pair of points.

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    $\begingroup$ Relative cohomology is the cohomology of the mapping cone, so what one needs is an algebro-geometric mapping cone. At the level of schemes there doesn't seem to be a reasonable way to do this due to the lack of a reasonable substitute for the interval. Etale cohomology only depends on the etale (pro)homotopy type, so you ought to be able to take the mapping cone in (pro)homotopy types in that case. In general I guess you want something like a model structure on simplicial schemes to be able to talk about mapping cones in the latter. $\endgroup$
    – Qiaochu Yuan
    Commented May 30, 2014 at 5:19
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    $\begingroup$ If $D$ is a (well-behaved) divisor, you can take the log de Rham complex as a proxy for relative cochains. $\endgroup$
    – S. Carnahan ♦
    Commented May 30, 2014 at 15:10

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