Synonyms for etale or Related words with etale

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Examples of "etale"
Here, "H "("X,F") is the "r"-th etale cohomology group of the scheme "X" with values in "F," and "Ext "("F,G") is the group of "r"-extensions of the etale sheaf "G" by the etale sheaf "F" in the category of etale abelian sheaves on "X." Moreover, "G" denotes the etale sheaf of units in the structure sheaf of "X."
The necessity of localizing at a prime formula_6 suggested to Browder that there should be a variant of "K"-theory with finite coefficients. He introduced "K"-theory groups "K"("R"; Z/formula_6Z) which were Z/formula_6Z-vector spaces, and he found an analog of the Bott element in topological "K"-theory. Soule used this theory to construct "etale Chern classes", an analog of topological Chern classes which took elements of algebraic "K"-theory to classes in etale cohomology. Unlike algebraic "K"-theory, etale cohomology is highly computable, so etale Chern classes provided an effective tool for detecting the existence of elements in "K"-theory. Dwyer and Friedlander then invented an analog of "K"-theory for the etale topology called etale "K"-theory. For varieties defined over the complex numbers, etale "K"-theory is isomorphic to topological "K"-theory. Moreover, etale "K"-theory admitted a spectral sequence similar to the one conjectured by Quillen. Thomason proved around 1980 that after inverting the Bott element, algebraic "K"-theory with finite coefficients became isomorphic to etale "K"-theory.
Here, it is crucial that the cohomology of a stack is with respect to the smooth topology (not etale).
Another important class of ringed topoi, besides ringed spaces, are the etale topoi of Deligne-Mumford stacks.
This notation is misleading: the symbol Q on the left represents neither an étale sheaf nor an ℓ-adic sheaf. The etale cohomology with coefficients in the constant etale sheaf Q does also exist but is quite different from formula_7. Confusing these two groups is a common mistake.
The etale space of a sheaf, such as the sheaf of continuous real functions over a manifold, is a manifold that is often non-Hausdorff. (The etale space is Hausdorff if it is a sheaf of functions with some sort of analytic continuation property.)
For a scheme "X" over an arbitrary field "k", there is an analogous cycle map from Chow groups to (Borel–Moore) etale homology. When "X" is smooth over "k", this homomorphism can be identified with a ring homomorphism from the Chow ring to etale cohomology.
There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by etale equivalence relations, or as sheaves on a big etale site that are locally isomorphic to schemes. These two definitions are essentially equivalent.
A much deeper result, the Bloch-Kato conjecture (also called the norm residue isomorphism theorem), relates Milnor K-theory to Galois cohomology or etale cohomology:
It shows that, as far as etale (or flat) cohomology is concerned, the ring of integers in a number field behaves like a 3-dimensional mathematical object.
A similar definition applies to sheaves on topoi, such as etale sheaves. Instead of the above preimage "f"("U") the fiber product of "U" and "X" over "Y" is used.
An algebraic stack or Artin stack is a stack in groupoids "X" over the etale site such that the diagonal map of "X" is representable and there exists a smooth surjection from (the stack associated to) a scheme to X.
For a field "K", any "K"-algebra "A" is necessarily flat. Therefore, "A" is an etale algebra if and only if it is unramified, which is also equivalent to
For a general base scheme "S", weights and coweights are defined as fpqc sheaves of free abelian groups on "S". These provide representations of fundamental groupoids of the base with respect the fpqc topology. If the torus is locally trivializable with respect to a weaker topology such as the etale topology, then the sheaves of groups descend to the same topologies and these representations factor through the respective quotient groupoids. In particular, an etale sheaf gives rise to a quasi-isotrivial torus, and if "S" is locally noetherian and normal (more generally, geometrically unibranched), the torus is isotrivial. As a partial converse, a theorem of Grothendieck asserts that any torus of finite type is quasi-isotrivial, i.e., split by an etale surjection.
Let "X" be the spectrum of the ring of integers in a totally imaginary number field "K", and "F" a constructible etale abelian sheaf on "X". Then the Yoneda pairing
When "X" is a variety, the smooth cohomology is the same as etale one and, via the Poincaré duality, this is equivalent to Grothendieck's trace formula. (But the proof relies on Grothendieck's formula, so this does not subsume Grothendieck's.)
A quasi-coherent sheaf is roughly one that looks locally like the sheaf of a module over a ring. The first problem is to decide what one means by "locally": this involves the choice of a Grothendieck topology, and there are many possible choices for this, all of which have some problems and none of which seem completely satisfactory. The Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski topology so this is a good choice for schemes as Serre discovered, algebraic spaces and Deligne–Mumford stacks are locally affine in the etale topology so one usually uses the etale topology for these, while algebraic stacks are locally affine in the smooth topology so one can use the smooth topology in this case. For general algebraic stacks the etale topology does not have enough open sets: for example, if G is a smooth connected group then the only etale covers of the classifying stack BG are unions of copies of BG, which are not enough to give the right theory of quasicoherent sheaves.
The original construction of tmf uses the obstruction theory of Hopkins, Miller, and Paul Goerss, and is based on ideas of Dwyer, Kan, and Stover. In this approach, one defines a presheaf O ("top" stands for topological) of multiplicative cohomology theories on the etale site of the moduli stack of elliptic curves and shows that this can be lifted in an essentially unique way to a sheaf of E-infinity ring spectra. This sheaf has the following property: to any etale elliptic curve over a ring R, it assigns an E-infinity ring spectrum (a classical elliptic cohomology theory) whose associated formal group is the formal group of that elliptic curve.
formula_27which is generically etale. The stack quotient of the domain gives a stacky formula_28 with stacky points that have stabilizer group formula_29 at the fifth roots of unity in the formula_30-chart since these are the points where the cover ramifies.
where formula_80 is the separable closure of the field "K" and the right hand side is a finite direct sum, all of whose summands are formula_80. This characterization of etale "K"-algebras is a stepping stone in reinterpreting classical Galois theory (see Grothendieck's Galois theory).