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 In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equal ...

Purity of the branch locus

For example, ramification is a phenomenon of codimension 1 (in the geometry of

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...

s, reflecting as for

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...

s that ramify at single points that it happens in real codimension two). A classical result, Zariski–Nagata purity of Masayoshi Nagata and Oscar Zariski, called also purity of the branch locus, proves that on a non-singular

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...

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 Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...

and

scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different s ...

, 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.

Cohomological purity

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''. Such results were established in

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...

by Michael Artin (included in SGA 4), and were foundational in setting up the theory to contain expected analogues of results from

singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

. A general statement of Alexander Grothendieck known as the absolute cohomological purity conjecture was proved by Ofer Gabber. It concerns a closed immersion of schemes (regular, noetherian) that is purely of codimension ''d'', and the relative

local cohomology In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES written up as SGA2 - , republished as . Given a f ...

in the étale theory. With coefficients mod ''n'' where ''n'' is invertible, the cohomology should occur only with index 2''d'' (and take on a predicted value).As formulated in http://www.math.utah.edu/~niziol/icm20062.pdf, p. 4.

Notes

{{Reflist Algebraic geometry