algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the étale spectrum of a

commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...

or an E_∞-ring, denoted by Spec^ét or Spét, is an analog of the

prime spectrum In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with ...

Spec of a commutative ring that is obtained by replacing

Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...

with

étale topology In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale ...

. The precise definition depends on one's formalism. But the idea of the definition itself is simple. The usual prime spectrum Spec enjoys the relation: for a scheme (''S'', ''O''_''S'') and a commutative ring ''A'', :

\operatorname(S, \operatorname(A)) \simeq \operatorname(A, \Gamma(S, \mathcal_S))

where Hom on the left is for

morphisms of schemes In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Alth ...

and Hom on the right

ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...

s. This is to say Spec is the

right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

to the

global section In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as Set (mathematics), sets, abelian groups, Ring (mathematics), rings) attached to the open sets of a topological space and defined locally with regard to them. ...

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

(S, \mathcal_S) \mapsto \Gamma(S, \mathcal_S)

. So, roughly, one can (and typically does) simply define the étale spectrum Spét to be the right adjoint to the global section functor on the category of "spaces" with étale topology. Over a

field of characteristic zero In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widel ...

, K. Behrend constructs the étale spectrum of a

graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...

called a perfect resolving algebra. He then defines a

differential graded scheme In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutativ ...

(a type of a

derived scheme In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutativ ...

) as one that is étale-locally such an étale spectrum. The notion makes sense in the usual algebraic geometry but appears more frequently in the context of

derived algebraic geometry Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutative ...

Notes

References

* Algebraic geometry {{algebraic-geometry-stub