Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
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. –  Qiaochu Yuan May 30 at 5:19
If $D$ is a (well-behaved) divisor, you can take the log de Rham complex as a proxy for relative cochains. –  S. Carnahan May 30 at 15:10

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.