In the mathematical field of

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, 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 equals the ...

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 com ...

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 vers ...

s that ramify at single points that it happens in real codimension two). A classical result, Zariski–Nagata purity of

Masayoshi Nagata Masayoshi Nagata (Japanese: 永田雅宜 ''Nagata Masayoshi''; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra. Work Nagata's compactification theorem shows that var ...

and

Oscar Zariski , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions = ...

, 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. Mo ...

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 In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier (mathematician), Pierre Cartier ...

). 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. Prominent ...

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 sc ...

, 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 In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy t ...

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 conjecture ...

Michael Artin Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.SGA 4 SGA may refer to: * Old Irish language (ISO 639-3 code) * ''Schwarz-Gelbe Allianz'' (Black-Yellow Alliance), an Austrian political party * Second-generation antipsychotics * Séminaire de Géométrie Algébrique du Bois Marie, an influential mat ...

), 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 viewed ...

. A general statement of Alexander Grothendieck known as the

absolute cohomological purity conjecture In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states:A version of the theorem is stated at given *a regular scheme ''X'' over some base scheme, *i: Z \to X a clo ...

was proved by Ofer Gabber. It concerns a

closed immersion In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formaliz ...

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 fun ...

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