In
algebraic geometry, a branch of
mathematics, the étale spectrum of a
commutative ring or an
E∞-ring, denoted by Spec
ét or Spét, is an analog of the
prime spectrum Spec of a commutative ring that is obtained by replacing
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
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'',
:
where Hom on the left is for
morphisms of schemes and Hom on the right
ring homomorphism
In ring theory, a branch of abstract algebra, 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 such that ''f'' is:
:addition preser ...
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 kno ...
to the
global section
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
. 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 do. A field is thus a fundamental algebraic structure which is wi ...
, K. Behrend constructs the étale spectrum of a
graded algebra called a perfect resolving algebra.
He then defines a
differential graded scheme In algebraic geometry, a derived scheme is a pair (X, \mathcal) consisting of a topological space ''X'' and a sheaf \mathcal either of simplicial commutative rings or of commutative ring spectra on ''X'' such that (1) the pair (X, \pi_0 \mathcal) i ...
(a type of a
derived scheme In algebraic geometry, a derived scheme is a pair (X, \mathcal) consisting of a topological space ''X'' and a sheaf \mathcal either of simplicial commutative rings or of commutative ring spectra on ''X'' such that (1) the pair (X, \pi_0 \mathcal) ...
) 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 commutati ...
.
Notes
References
*
Lurie, J.,
Spectral Algebraic Geometry (under construction)'
Algebraic geometry
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