In the mathematical field of algebraic geometry, purity is a theme covering a number of results and conjectures, which collectively address the question of proving that "when something happens, it happens in a particular codimension".
A classical result, Zariski–Nagata purity of Masayoshi Nagata and Oscar Zariski,[1][2] called also purity of the branch locus, proves that on a non-singular algebraic variety a branch locus, namely the set of points at which a morphism ramifies, must be made up purely of codimension 1 subvarieties (a Weil divisor).
There have been numerous extensions of this result into theorems of commutative algebra and scheme theory, establishing purity of the branch locus in the sense of description of the restrictions on the possible "open subsets of failure" to be an étale morphism.
There is also a homological notion of purity that is related, namely a collection of results stating that cohomology groups from a particular theory are trivial with the possible exception of one index i.
[3] It concerns a closed immersion of schemes (regular, noetherian) that is purely of codimension d, and the relative local cohomology in the étale theory.