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the adjective étale refers to several closely related concepts: Étale morphism Formally étale morphism Étale cohomology Étale topology Étale fundamental...

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients...

geometry, an étale morphism (French: [etal]) is a morphism of schemes that is formally étale and locally of finite presentation; the étale morphism is...

an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions. An étale algebra...

characteristic. The étale topology was originally introduced by Alexander Grothendieck to define étale cohomology, and this is still the étale topology's most...

The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. In algebraic...

commutative algebra and algebraic geometry, a morphism is called formally étale if it has a lifting property that is analogous to being a local diffeomorphism...

natural analog of a covering map is called an étale morphism. Despite its similarity to "étalé", the word étale [etal] has a different meaning in French....

In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties...

In algebraic geometry, a branch of mathematics, the étale spectrum of a commutative ring or an E∞-ring, denoted by Specét or Spét, is an analog of the...

étale group scheme is a certain kind of group scheme. A finite group scheme G {\displaystyle G} over a field K {\displaystyle K} is called an étale group...

algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic...

German mathematician. He specializes in arithmetic algebraic geometry (étale fundamental group, anabelian geometry and other topics). Stix studied mathematics...

of a complex algebraic variety is equivalent to the category of finite étale coverings of the variety. Let X be a compact Riemann surface, p 1 , ⋯ ,...

étale cohomology states that the higher direct images of a constructible sheaf are constructible. Here we use the definition of constructible étale sheaves...

{\displaystyle \mathbb {Z} /\ell ^{n}} -modules F n {\displaystyle F_{n}} in the étale topology and F n + 1 → F n {\displaystyle F_{n+1}\to F_{n}} inducing F n...

open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topologies. The notion also generalizes a Galois...

that is weakly equivalent to X {\displaystyle X} in a natural way. For the étale topology and other sites, these conditions fail. The idea of a hypercover...

on topos theory and étale cohomology, jointly with Alexander Grothendieck. He also collaborated with Barry Mazur to define étale homotopy theory which...

Étale is a mountain of Savoie and Haute-Savoie, France. It lies in the Aravis Range of the French Prealps and has an elevation of 2,484 metres above sea...

integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides a basic...

under the supervision of Eric Friedlander (Tubular Neighborhoods in the Etale Topology). From 1974 to 1975, he was assistant professor at Haverford College...

mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by Quillen (1975, p. 175), who...

Deligne–Mumford if its diagonal is sufficiently well behaved and if it admits an étale surjective cover by a scheme (an atlas). Pierre Deligne and David Mumford...

cohomologie étale des schémas, 1963–1964 (Topos theory and étale cohomology), Lecture Notes in Mathematics 269, 270 and 305, 1972/3 SGA4½ Cohomologie étale (Étale...

Étalle (French pronunciation: [etal]; Lorrain: Ètaule) is a municipality of Wallonia located in the province of Luxembourg, Belgium. On 1 January 2007...

inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic...

on an étale site is the union of its constructible subsheaves. Twisted sheaf Milne 2012, Remark 17.6 Milne, James S. (2012). "Lectures on Étale Cohomology"...

formalizing the heuristic. An important example of this programmatic idea is the étale topos of a scheme. Another illustration of the capability of Grothendieck...

as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes f : X → Y. The basic insight...

→ X {\displaystyle f:Y\to X} is called a Nisnevich morphism if it is an étale morphism such that for every (possibly non-closed) point x ∈ X, there exists...

theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's...

Deligne. Collaborators on the SGA projects also included Michael Artin (étale cohomology), Nick Katz (monodromy theory, and Lefschetz pencils). Jean Giraud...

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...

mod-ℓ to étale cohomology is an isomorphism. The case ℓ = 2 is the Milnor conjecture, and the case n = 2 is the Merkurjev–Suslin theorem. The étale cohomology...

use the étale topology. Michael Artin defined an algebraic space as a functor that is a sheaf in the étale topology and that, locally in the étale topology...

Knizhnik–Zamolodchikov equations. In algebraic geometry, the so-called étale fundamental group is used as a replacement for the fundamental group. Since...

finite étale morphism of adic spaces over K and Y is perfectoid, then X also is perfectoid; A morphism X → Y of perfectoid spaces over K is finite étale if...

means the étale sheaf (μm)⊗j, with μm being the mth roots of unity. This generalizes the cycle map from the Chow ring of a smooth variety to étale cohomology...

sheaves and f proper or g flat, similarly in analytic geometry, but also for étale sheaves for f proper or g smooth. A simple base change phenomenon arises...

SGA 4 his theory of hypercovers and anticipating the later development of étale homotopy by Michael Artin and Barry Mazur, following a suggestion he attributed...

groups H i(K, K*) vanish for i ≥ 1. This result is used to calculate the étale cohomology groups of an algebraic curve. The theorem was published by Chiungtze...

variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic...

isomorphisms. There is yet another projection formula in the setting of étale cohomology. Integration along fibers § Projection formula Hartshorne, Robin...

A. Grothendieck, J. L. Verdier (eds.), Théorie des Topos et Cohomologie Etale des Schémas - SGA 4. II , LNM 270 Springer Heidelberg 1972. Benjamin Frot...

which is finer than étale topology. Its main use is to define the cohomology of an algebraic stack with coefficients in, say, the étale sheaf Q l {\displaystyle...

else. A morphism of schemes f : X → Y {\displaystyle f:X\to Y} is called étale if it is flat and unramfied. These are the algebro-geometric analogue of...

The idea is that theories such as Betti cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology should arise as different realizations...

an analog of K-theory for the étale topology called étale K-theory. For varieties defined over the complex numbers, étale K-theory is isomorphic to topological...

pullback by a finite étale map—are important. This acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding...

f:X\to Y} of finite type between Noetherian schemes, f {\displaystyle f} is étale (flat and unramified) if and only if for each x in X, f is an analytically...

the r-th étale cohomology group of the scheme X with values in F, and Extr(F,G) is the group of r-extensions of the étale sheaf G by the étale sheaf F...

group is étale if and only if it is discrete; pair groupoids are never étale; unit groupoids are always étale; an action groupoid is étale if and only...

Galois cohomology, and also by Michael Artin and Jean-Louis Verdier based on étale cohomology. Then David Mumford (and independently Yuri Manin) came up with...

easy to handle. The concept of étale morphism (and so étale cohomology) depends on the flat morphism concept: an étale morphism being flat, of finite...

lifting v, then A is said to be 0-unramified (or 0-neat). A is said to be 0-étale if it is 0-smooth and 0-unramified. The notion of 0-smoothness is also called...

geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories...

that the prestack of quasi-coherent sheaves is a stack with respect to étale (or fpqc) topology. "Vanilla" faithfully flat descent is generally false;...

on the level of derived categories only. Similar considerations apply to étale sheaves on schemes. The functors are adjoint to each other as depicted at...

Cohomologie étale des schémas Did not appear. Étale cohomology treated in detail in SGA IV, SGA V. XIII Cohomologie de Weil none Intended to cover étale cohomology...

L-function attached to the p-adic Galois representation arising from the p-adic etale cohomology of an algebraic variety defined over a global function field...

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...

absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states: given a regular scheme X over some base scheme, i...

spaces coming from the étale topology on a connected scheme S {\displaystyle S} . The underlying site consists of finite étale covers, which are finite...

Nisnevich topology Uses etale morphisms, but has an extra condition about isomorphisms between residue fields. Étale topology Uses etale morphisms. fppf topology...

characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of F with coefficients in Z/2Z. It was proved by Vladimir Voevodsky (1996...

route towards their proof, and other advances, lay in the construction of étale cohomology. With the benefit of hindsight, it can be said that algebraic...

an etale sheaf. That is, if U → X {\displaystyle U\to X} is an etale cover, and F {\displaystyle F} is a presheaf with transfers, it is an Etale sheaf...

"Site et Topos etales d'un schema, 9. Appendice. Critère d'existence de points par P. Deligne". Théorie des Topos et Cohomologie Etale des Schémas. Lecture...

fpqc ⊃ fppf ⊃ smooth ⊃ etale ⊃ Zariski {\displaystyle {\text{fpqc}}\supset {\text{fppf}}\supset {\text{smooth}}\supset {\text{etale}}\supset {\text{Zariski}}}...

topos that captures the idea of effectivity in mathematics Étale topos, the category of étale sheaves Rhetoric topos, topoi in rhetorical invention Literary...

certain algebraic cycles. The Tate conjecture makes a similar prediction for étale cohomology. Alexander Grothendieck's standard conjectures on algebraic cycles...

of the restrictions on the possible "open subsets of failure" to be an étale morphism. There is also a homological notion of purity that is related,...

Bouschet, Morrastel-bouschet à gros grains and Morrastel-bouschet à sarments étalés. J. Robinson Jancis Robinson's Guide to Wine Grapes pg 113 Oxford University...

fields, there is a cohomological classification of Azumaya algebras using Étale cohomology. In fact, this group, called the Brauer group, can be also defined...

Conference 2000 (Transcript of keynote address). Retrieved 2025-04-08. Milne, James S. (1980). Étale cohomology. Princeton University Press. pp. xiii, 66....

theory used in multiple contexts, such as with covering spaces (espace étalé). They were introduced as a mechanism for keeping track of data surrounding...

Michèle (2002). "Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique". Revêtements étales et groupe fondamental (SGA...

Erkelene Esneux Esneu Estaimpuis Timpu Steenput Estinnes L' Estene Étalle Etåle Eupen, Néau Nèyåw Eupe Eupen Faimes Faime Farciennes Fårcene Fauvillers...

Executive Committee of the Orange Democratic Movement Spokesperson Philip Etale Deputy Secretary Generals Agnes Zani Catherine Omanyo Treasurer-General...

This condition is important in Galois module theory. A finite generically étale extension B / A {\displaystyle B/A} of Dedekind domains is tame if and only...

Retrieved 2024-01-01. Wade, Ousmane. "Présidentielle 2024: Pape Djibril Fall étale son programme pour un Sénégal nouveau et prospère". Leral.net - Sénégal...

generated K-algebra. A separable flat (commutative) K-algebra A is formally étale. A theorem in the area is that of J. Cuadra that a separable Hopf–Galois...

f:X\to Y} is a local homeomorphism, X {\displaystyle X} is said to be an étale space over Y . {\displaystyle Y.} Local homeomorphisms are used in the study...

algebraic and geometric contexts, including that of quasi-coherent sheaves and étale sheaves on a scheme. Let f: X → Y be a continuous map of topological spaces...

(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...

functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Michael...

space. But the notion is more flexible; for example, the ∞-category of étale sheaves on some scheme is not the ∞-category of sheaves on any topological...

can also be expressed as motivic cohomology groups, by a variant known as étale (or Lichtenbaum) motivic cohomology. They show that the rational Hodge conjecture...

coverings are local homeomorphisms, a covering space is a special kind of étalé space. Covering spaces first arose in the context of complex analysis (specifically...

f have the same étale cohomology, locally. For example, a smooth morphism is universally locally acyclic. Milne, J. S. (1980), Étale cohomology, Princeton...

Galois groups of number fields can be constructed from representations on étale cohomology groups of varieties. Some cases of this conjecture in dimension...

X{\overset {g}{\to }}\mathbb {A} _{S}^{n}\to S} where g is étale. A morphism of finite type is étale if and only if it is smooth and quasi-finite. A smooth...

injective function ("étale" means the two possible maps s , t : R → U × U → U {\displaystyle s,t:R\to U\times U\to U} are étale.) Starting from a Deligne–Mumford...

the book Sheaf theory from the Cartan seminar he defines: Sheaf space (étale space), support of sheaves axiomatically, sheaf cohomology with support...

also been shown that Lefschetz pencils exist in characteristic p for the étale topology. Simon Donaldson has found a role for Lefschetz pencils in symplectic...

encountered in arithmetics: étale cohomology of finite, local and global fields (also known as Galois cohomology, since étale cohomology over a field is...

l'IHÉS. 32. doi:10.1007/bf02732123. MR 0238860. Universal homeomorphisms and the étale topology Do pushouts along universal homeomorphisms exist? v t e...

Sharma Hamsukh Gandhi 17 September 2017 (2017-09-17) 6 6 "Aa Seduce Karavu Etale Shu?" Chetan Sharma Hamsukh Gandhi 17 September 2017 (2017-09-17) 7 7 "Prachi...

cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided that any finite set of points...

notion of étale map in differential geometry. Étale morphisms form a very important class of morphisms; they are used to build the so-called étale topology...

over the complex numbers (GAGA) Artin comparison theorem - identifies the Étale cohomology of finite type schemes and the cohomology of their analytic counterparts...

the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental group. Although its existence was conjectured by Alexander Grothendieck...

for vector bundles with a non-degenerate quadratic form, taking values in etale cohomology groups or in Milnor K-theory. As a special case one can define...

maximal ideals of the local rings. A flat unramified morphism is called an étale morphism. Less strongly, if f {\displaystyle f} satisfies the conditions...

Anthony Vazzana, and Wafa Wei. Milne is also an avid mountain climber. Étale Cohomology. Princeton Mathematical Series. Vol. 33. Princeton, NJ: Princeton...

their union is not equidimensional. If a scheme is smooth (for instance, étale) over Spec k for some field k, then every connected component (which is...

notion of pro-étale site associated to an arbitrary scheme. In 2018, Dustin Clausen and Scholze arrived at the conclusion that the pro-étale site of a single...

been generalized to discrete valuation domains or finite fields, or for étale coverings of X. The theorem is often used for induction steps.[example needed]...

If moreover any such lifting is unique, then f is said to be formally étale. Formally smooth maps were defined by Alexander Grothendieck in Éléments...

a} has norm 1 {\displaystyle 1} . Milne, James S. (2013). "Lectures on Etale Cohomology (v2.21)" (PDF). p. 80. Hilbert, David (1897), "Die Theorie der...

Grothendieck, Alexander; et al. (1972), Théorie des Topos et Cohomologie Étale des Schémas, Lecture Notes in Mathematics 269, Springer Gabriel, P; Ulmer...

definition involves projective bundles that are locally trivial in the étale topology, not necessarily in the Zariski topology. In particular, a projective...

and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc; nowadays...

Pierre (1977). Séminaire de Géométrie Algébrique du Bois Marie - Cohomologie étale - (SGA 4½). Lecture Notes in Mathematics (in French). Vol. 569. Berlin;...

{\displaystyle p} -adic analogues of Q l {\displaystyle \mathbf {Q} _{l}} -adic étale sheaves, introduced by Grothendieck (1966a) and Berthelot & Ogus (1983)...

infinite Galois theory. A generalization used in algebraic geometry is the étale fundamental group. A Lie group is a group that also has the structure of...

homeomorphism for the strong topology, over the complex numbers, or as an étale morphism in general (under some slightly stronger hypotheses, on flatness...

theories including the theory of frobenioids, mono-anabelian geometry and the etale theta theory for line bundles over tempered covers of the Tate curve. On...

115–119, doi:10.1002/mana.19780820112. Grothendieck, A. (1971), Revêtements étales et groupe fondamental (SGA 1, Séminaire de Géométrie Algébriques du Bois-Marie...

cohérente et en cohomologie étale, Bull. Soc. Math. France, Memoirs Nr. 41, 1975 Théorèmes de Lefschetz en cohomologie étale des faisceaux en groupes non...

models commutes with products. The formation of Néron models commutes with étale base change. An Abelian scheme AR is the Néron model of its generic fibre...

resolution of this sheaf not by injective sheaves, but by fine sheaves. Étale cohomology is another cohomology theory for sheaves over a scheme. It is...

nonzero characteristic p Ogus (1975) showed that it is closely related to etale cohomology with mod p coefficients, a theory known to have undesirable properties...

Diffeomorphometry, metric study of shape and form in computational anatomy Étale morphism Large diffeomorphism Local diffeomorphism Superdiffeomorphism Steven...

relating de Rham cohomology to singular cohomology of complex varieties or étale cohomology of p-adic varieties. In Mochizuki (1999) and Mochizuki (2002a)...

not. A more refined definition became possible with the development of étale cohomology; this neatly explains what to do about the missing, 'bad reduction'...

University of Bonn. Retrieved 2024-09-16. Wolf, Sebastian (2020). "The Pro-Étale Topos as a Category of Pyknotic Presheaves". arXiv:2012.10502 [math.AG]...

(2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461 Étale morphism Dimension of an algebraic variety Glossary of scheme theory Smooth...

{\frac {n-\sum _{j}d_{j}}{d}}.} The Ax–Katz result has an interpretation in étale cohomology as a divisibility result for the (reciprocals of) the zeroes...

titled K-Théorie des anneaux d'entiers de corps de nombres et cohomologie étale. In 1979, he was awarded a CNRS Bronze Medal. He received the Prix J. Ponti...

ring Derivator Algebra over an operad En-ring Higher Topos Theory ∞-topos étale spectrum Khan, Adeel A. (2019). "Brave new motivic homotopy theory I". Geom...

resp. F e t {\displaystyle F_{et}} , denotes the h-sheafification, resp. etale sheafification. As mentioned above, in positive characteristic, any h-sheaf...

regular scheme that is not smooth, see Geometrically regular ring § Examples. Étale morphism Dimension of an algebraic variety Glossary of scheme theory Smooth...

modern abstract development of algebraic geometry. Over finite fields, étale cohomology provides topological invariants associated to algebraic varieties...

1007/s00222-010-0235-1. Crainic, Marius; Moerdijk, Ieke (2000). "A homology theory for étale groupoids". Journal für die reine und angewandte Mathematik. 2000 (521):...

obtained by gluing affine differential graded schemes, with respect to étale topology. It was introduced by Maxim Kontsevich "as the first approach to...

non-archimedean function theory. He wrote a textbook on the Weil conjectures and étale cohomology with Eberhard Freitag. In 1970 Kiehl was an Invited Speaker at...

2 or 3 there are some extra families given by taking quotients by a non-etale group scheme; see the article on hyperelliptic surfaces for details. An...

{\textstyle \mathrm {Spec} \,R\to \mathrm {Spec} \,k} admits a section étale-locally (equivalently, after base change along S p e c L → S p e c k {\textstyle...

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...

local field E via its absolute Galois group, which is isomorphic to the étale fundamental group of Spec(E). This result was calculated while trying to...

arising at a fundamental level for the (analogue of) Poincaré duality in étale cohomology. The Euler products of the Hasse–Weil zeta-function for an algebraic...

algebraic topoloy is the study of spaces with (continuous) group action. etale étale homotopy. Euclidean A Euclidean neighborhood retract exact A sequence...

sub-category of A ( Top ) {\displaystyle A({\text{Top}})} consisting of étalé spaces of sheaves. As with vector bundles, the sheaves of groups and rings...

embedding into the flasque sheaf of all possibly discontinuous sections of the étalé space, and by repeating this we can find a canonical flasque resolution...

localization of A, then L B / A ≃ 0 {\displaystyle L_{B/A}\simeq 0} . If f is an étale morphism, then L B / A ≃ 0 {\displaystyle L_{B/A}\simeq 0} . If f is a smooth...

between algebraic cycles on algebraic varieties and Galois representations on étale cohomology groups. Virasoro conjecture: a certain generating function encoding...

compute, except in cases where they reduce to other theories, such as the étale cohomology. The following example shows why the "faithfully flat topology"...

Vladimir V. (2004), Algebras, rings and modules, vol. 1, Springer, ISBN 1-4020-2690-0 Milne, James S. (1980). Étale cohomology. Princeton University Press....

of a minus sign may appear. Freitag, Eberhard; Kiehl, Reinhardt (1988), Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer...

A. Rosenberg, editors, American Mathematical Society 1998: "Cohomologie étale et groupe de Brauer", pages 210 to 28 in Group de Brauer, Lecture Notes...

representations. In simple cases, it relates l-adic representations of the étale fundamental group of an algebraic curve to objects of the derived category...

(PDF). www.jmilne.org. Retrieved 16 July 2021. Milne, James S. Lectures on Étale cohomology Mumford, David (1999). The Red Book of Varieties and Schemes:...

the constant sheaf shifted by dim ⁡ X + 1 {\displaystyle \dim X+1} is an étale perverse sheaf. Let X be a disk around the origin in C {\displaystyle \mathbb...

generally, a similar statement holds for any quasi-finite morphism such as an étale morphism. If f is an open immersion, the exceptional inverse image equals...

{\displaystyle D} -modules for which F {\displaystyle F} is nilpotent, and étale group schemes correspond to modules for which F {\displaystyle F} is an...

Grothendieck's theory of étale cohomology (which eventually led to the proof of the Weil conjectures) can be phrased as cohomology in the étale topos of a scheme...

around. Firstly, Galois cohomology appeared as the foundational layer of étale cohomology theory (roughly speaking, the theory as it applies to zero-dimensional...

sheaf of sections of some (étalé) bundle over that space: the categories of sheaves on a topological space and that of étalé spaces over it are equivalent...

coherent sheaf. Precisely, it is specified by: an open covering in the étale topology Ui, coherent sheaves Fi over Ui, a Čech 2-cocycle θ for G m {\displaystyle...

Mathématique de France. 97: 81–128. doi:10.24033/bsmf.1675. Milne, James (1980). Étale cohomology. Princeton Univ. Press. ISBN 0-691-08238-3. Donald S. Passman...

sheaf cohomology over the étale topology to define the cohomology theory for varieties over a finite field. Using the étale topology for a variety over...

residue isomorphism theorem), relates Milnor K-theory to Galois cohomology or étale cohomology: K n M ( F ) / r ≅ H e t n ( F , Z / r ( n ) ) , {\displaystyle...

Anabelian geometry Grothendieck, Alexander; et al. (1971). SGA1 Revêtements étales et groupe fondamental, 1960–1961. Lecture Notes in Mathematics. Vol. 224...

prove Milnor's conjecture relating the Milnor K-theory of a field to its étale cohomology. For the above, he received the Fields Medal at the 24th International...

unramified and if and only if FX/S is a monomorphism. X is étale over S if and only if FX/S is étale and if and only if FX/S is an isomorphism. The arithmetic...

Frobenius elements in Galois groups involved in the Galois representations on étale cohomology. In particular there is a conjectural theory for curves of genus n > 1...

them; a key role is played by three rigidities established in Mochizuki's etale theta theory. Roughly speaking, arithmetic deformations change the multiplication...

is étale and k is algebraically closed, then each geometric fiber f−1(y) consists exactly of deg(f) points. Algebraic function Smooth morphism Étale morphisms...

Rigid cohomology Prismatic cohomology Habiro cohomology p-adic Hodge theory Étale cohomology, taking values over a ring of l-adic integers for l≠p This set...

action of the absolute Galois group on the ℓ {\displaystyle \ell } -adic étale cohomology H 3 ( X Q ¯ , Q ℓ ) {\displaystyle H^{3}(X_{\bar {\mathbb {Q}...

primes one has to refer to the Tate module of A, which is (dual to) the étale cohomology group H1(A), and the Galois group action on it. In this way one...

Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3) (in French)...

ISSN 1618-1913, MR 0340258, S2CID 123139343 Deligne, Pierre, ed. (1977), Cohomologie Etale, Lecture Notes in Mathematics (in French), vol. 569, Berlin: Springer-Verlag...

de Caumartin) porte sous son manteau toute la faculté que M. de Villeroy étale sur son baudrier" (Caumartin holds under his cloak all the power that Villeroy...

professor of mathematics in 2024. Sawin found important applications of étale cohomology to the theory of exponential sums over finite fields. He also...

(2))\rightarrow \mathbf {Q} /\mathbf {Z} } where the middle group is the étale cohomology group and Q/Z is the geometric part of the cohomology. Choose...

Riemann-Hilbert correspondence that asserts in particular that étale cohomology of étale sheaves with Z/p-coefficients can be computed in terms of the...

November 2023). "Après la mort du jeune Thomas à Crépol, l'extrême droite étale sa xénophobie". Libération. Retrieved 25 November 2023.. Mathieu Boudet...

1998), no. Extra Vol. I, 1998, pp. 579–604. Joseph Ayoub, A guide to (étale) motivic sheaves, Proceedings of the International Congress of Mathematicians—Seoul...

- idlebrain.com - YouTube "Taskara – An Indian economic superhero – Etale". Etale.in. 3 September 2016. Retrieved 20 November 2016. "Telugu Movie Graphic...

(here, both are power series because the x {\displaystyle x} coordinate is étale at the corresponding points in the normalization). At the smooth point (...

Brylinksi–Radon transform is the functor between appropriate derived categories of étale sheaves Rad := R p 2 , ∗ p 1 ∗ : D ( P d ) → D ( P ∨ , d ) . {\displaystyle...

constructible sheaves was to give a proof that could be adapted to the setting of étale and ℓ {\displaystyle \ell } -adic cohomology. Up to some restrictions on...

Infantry Part of South West African Territorial Force Garrison/HQ Ondangwa, Etale, Rundu (17°57′55″S 19°43′56″E / 17.96528°S 19.73222°E / -17.96528; 19...

Tate (the Honda–Tate theorem). The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture. They relate to the Galois action on...

proven with the help of some general theory; see local zeta function and étale cohomology for example. The set of points E(Fq) is a finite abelian group...

at its behavior locally in the étale topos of the stack X {\displaystyle X} . More concretely, suppose there is an étale morphism U → X {\displaystyle...

arXiv:1905.08229 Rydh, David (2010), "Submersions and effective descent of étale morphisms", Bull. Soc. Math. France, 138 (2): 181–230, arXiv:0710.2488,...

R} factors as an étale cover over some A R n {\displaystyle \mathbb {A} _{R}^{n}} equipped with the standard projection. Because "étale" means "(flat and)...

and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc. Nowadays...

2206/kyushujm.62.401, ISSN 1340-6116, MR 2464529 Mochizuki, Shinichi (2009), "The étale theta function and its Frobenioid-theoretic manifestations", Kyoto University...

homotopy theory, and in the study of homotopy properties of schemes (e.g. étale homotopy theory). Edwards, David A.; Hastings, Harold M. (1980), "Čech theory:...

isomorphism class of the variety X is contained in the knowledge of the étale fundamental group A concrete example is the case of curves, which may be...

the Grothendieck–Ogg–Shafarevich formula, which describes the behavior of étale cohomology under ramification; Ogg's formula for the conductor of an elliptic...

projective model structure on pro-simplicial sheaves, and the relative étale homotopy type", Advances in Mathematics, 291: 784–858, arXiv:1109.5477,...

algebraic geometry. In joint work with George Lusztig, Deligne applied étale cohomology to construct representations of finite groups of Lie type; with...

over a finite field correspond to eigenvalues of a Frobenius element on an étale cohomology group, the zeros of a Selberg zeta function are eigenvalues of...

("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...

§VI.1 Lang (2002), Example VI.2.6 Borceux & Janelidze (2001). See also Étale fundamental group. Gouvêa (2012), Theorem 6.4.8 Marker, Messmer & Pillay...

class formations. In turn, this led to the notion of Galois cohomology and étale cohomology (which builds on it) (Weibel 1999, p. 822). Some refinements...

symplectic involutions. If the center of A {\displaystyle A} is a quadratic étale k {\displaystyle k} -algebra K {\displaystyle K} and τ {\displaystyle \tau...

as a deacon in 1946, and as a priest in 1947. He served in parishes in Etale, Windhoek, and Omboloka. Hamutumbangela spoke out against the racism and...

of a site and any sheaf E of abelian groups. For the étale topology, this gives the notion of étale cohomology, which led to the proof of the Weil conjectures...

inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic...

S2CID 15924437, archived from the original (PDF) on 2015-11-12 Milne, James, Etale cohomology, Proposition I.3.5{{citation}}: CS1 maint: location (link); the...

06-Jan-22". L'Express. Retrieved 13 July 2024. "Affaire de vol de bois : Bhadain étale l'ombre de Lee Shim derrière la Political Conspiracy 06-Jan-22". Le Mauricien...

Études Scientifiques, pp. 11–19 Alexander Grothendieck. 1971, Revêtements Étales et Groupe Fondamental (SGA1), chapter VI: Catégories fibrées et descente...

1996–1997 Christophe Breuil Cohomologie log-cristalline et cohomologie étale de torsion Christine Lescop Autour de l’invariant de Casson 1997–1998 Andrei...

{\displaystyle \pi :P\to M} , such that locally on M {\displaystyle M} (in étale topology or fppf topology) π {\displaystyle \pi } is a direct product π...

mixed Hodge structure on singular varieties, and to deep analogies with étale cohomology. The Hodge conjecture on the 'middle' spaces Hp,p is still unsolved...

higher forms of ℓ {\displaystyle \ell } -adic representations by using the etale homotopy type π ^ ( X ) {\displaystyle {\hat {\pi }}(X)} of a scheme X {\displaystyle...

from B to A given by x → x2. Think of B as an open covering of U for the étale topology. If V is a complete scheme with a fixed point free action of a...

SWAPO and served as a deputy minister in the Cabinet. Shihepo was born in Etale, in Ohangwena Region. He worked as a teacher and school principal. From...

of the Weil conjectures. Étale cohomology The search for a Weil cohomology (q.v.) was at least partially fulfilled in the étale cohomology theory of Alexander...

of Weil. The scheme H G , n {\displaystyle {\mathcal {H}}_{G,n}} is an étale cover of the algebraic configuration space U n {\displaystyle {\mathcal...

life". Slow Movement. 2014. "Learning About the Slow Education Movement". Etale. 31 August 2013. Hodgkinson, Tom (16 December 2012). "We need to take education...

of height n) and the map X → M p ( n ) {\displaystyle X\to M_{p}(n)} is etale, then this presheaf can be refined to a sheaf of E ∞ {\displaystyle E_{\infty...

International Centre for Theoretical Physics. Trieste - Italy. Mathew, Akhil. "Étale π1 OF A SMOOTH CURVE" (PDF). "Formal Gaga on Artin Stacks" (PDF). 2005....

Fontaine, Jean-Marc; Messing, William (1987), "p-adic periods and p-adic étale cohomology", Current trends in arithmetical algebraic geometry (Arcata,...

ncatlab.org. Retrieved 2020-08-07. Artin, Michael; Mazur, Barry (1969). Etale Homotopy. Lecture Notes in Mathematics. Vol. 100. doi:10.1007/bfb0080957...

{Q} _{\ell })).} This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, of X ¯ {\displaystyle {\bar {X}}} with...

"Théorèmes de Lefschetz en cohomologie des faisceaux cohérents et en cohomologie étale. Application au groupe fondamental". Annales scientifiques de l'École Normale...

locally Hausdorff (it is in fact locally metrizable) but not Hausdorff. The etale space for the sheaf of differentiable functions on a differential manifold...

October 2021. Retrieved June 28, 2012. "Dakar Fashion Week: Adama Paris étale ses ambitions". Leral.net. 13 June 2012. Retrieved June 28, 2012. Gleye...

affine varieties (see Serre's theorem on affineness). But also all of the étale cohomology groups on affine space are trivial. In particular, every line...

4171/EMSS/4. https://ncatlab.org/nlab/show/higher+stack David Carchedia, On the étale homotopy type of higher stacks, Higher Structures 5(1):121–185, 2021. [1]...

Birkhauser 1983. For the étale cohomology version, see the chapter on monodromy in Freitag, E.; Kiehl, Reinhardt (1988), Etale Cohomology and the Weil...

ISSN 1245-530X, MR 0591388 Thomason, Robert W. (1985), "Algebraic K-theory and étale cohomology" (PDF), Annales Scientifiques de l'École Normale Supérieure,...

sheaves on topological spaces or more advanced cohomology theories like étale cohomology or group cohomology. In order to apply this technique, one has...

of points on C means that π restricted to the complement of S in C is an étale morphism. This is in analogy with the case of Riemann surfaces. In Abhyankar's...

resolving these. A generation later Pierre Deligne completed the proof. (See étale cohomology for the basic formulae of the general theory.) It is a consequence...

Submanifolds. Edited by Enrico Bombieri 1984-01-21 368 978-0691083193 104 Etale Homotopy of Simplicial Schemes Eric M. Friedlander 1982-12-01 191 978-0691083179...

numbers, with the formulation that they are generally birational to a finite étale covering of a moduli space of curves. Using the notion of stable vector...

Institute of Technology, under the supervision of Michael Artin, (Fibrations in Étale Homotopy Theory). He was a postdoctoral instructor at Princeton University:...

for instance define log-smoothness and log-étaleness, generalizing the notions of smooth morphisms and étale morphisms. This then allows the study of deformation...

noetherian, and Ah is manifestly algebraic as it is constructed as a limit of étale neighbourhoods. This means that Ah is usually much smaller than the completion...

X is the Weil restriction to k of the projective line over a quadratic étale algebra K. Since Q is not split over k, an auxiliary argument with special...

Séminaire de Géométrie Algébrique du Bois Marie – 1960–61 – Revêtements étales et groupe fondamental (SGA 1) (Documents Mathématiques 3), Paris: Société...

of the definition from the choice of compactification. In the context of étale sheaves, this idea was carried out by Deligne in SGA 4, Exposé XVII. In...

A_{3}=D_{3}} relates unitary groups of degree 4, defined using a quadratic étale algebra and a unitary involution, to spin groups in dimension 6, via Clifford...

generated algebra. étale 1.  A morphism of rings is called étale if it is formally etale and locally finitely presented. 2.  An étale algebra over a field...

Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269) (in French)...

( X ) e t ) {\displaystyle D({\text{Qcoh}}(X)_{et})} of quasi-coherent étale sheaves on X, and a morphism φ : E → L X {\displaystyle \varphi \colon E\to...

Likewise strict Henselian rings are the local rings of geometric points in the étale topology. For any local ring A there is a universal Henselian ring B generated...

Society, ISBN 978-0-8218-1534-2, MR 1070709 Berkovich, Vladimir G. (1993), "Étale cohomology for non-Archimedean analytic spaces", Publications Mathématiques...

ISBN 978-3-7643-6064-1, section IV.3.2 Artin, Michael; Mazur, Barry (1969), Etale homotopy, Lecture Notes in Mathematics, No. 100, Berlin, New York: Springer-Verlag...

standardized. If orbifolds are viewed as differentiable stacks presented by étale proper Lie groupoids, then there is a functor from the underlying 1-category...

fields, this conjecture was shown by Clozel for abelian varieties with etale l-adic cohomology, for an infinite set of prime numbers l (which can be...

quadric cone B in affine 3-space, then f×f:A×A→B×B cannot be produced by an étale local resolution procedure, essentially because the exceptional locus has...

semisimplicity of étale cohomology of quaternionic Shimura varieties" (2018) "Semisimplicity of certain Galois representations occurring in étale cohomology...

V\to X} of the canonical inclusion x → X {\displaystyle x\to X} via an étale morphism V → X {\displaystyle V\to X} . The collection { x ∗ } {\displaystyle...

Grothendieck, A.; Verdier, J. L. (1972), "Topos", Théorie des Topos et Cohomologie Etale des Schémas, Lecture Notes in Mathematics, vol. 269, Springer, pp. 299–518...

Farthing; Aidan Sims (14 November 2013). "Simplicity of algebras associated to étale groupoids". Semigroup Forum. 88 (2): 433–452. doi:10.1007/S00233-013-9546-Z...

( V , q ) {\displaystyle \operatorname {Cl} ^{0}(V,q)} is the quadratic étale F {\displaystyle F} -algebra Z ( q ) = F [ x ] / ( x 2 − δ ( q ) ) {\displaystyle...

internautes". 11 September 2020. "Accusé de viol et conduite déplacée, Patrick Poivre d'Arvor étale sa défense contradictoire à "Quotidien"". Official website...

{\widehat {\mathbf {G} }}_{m}} , and if G is a torus, then its Cartier dual is étale and torsion-free. For loop groups of tori, Cartier duality defines the tame...

Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1. Lecture notes in mathematics (in French)...

pages long, it depended on background results in algebraic geometry and étale cohomology that Deligne estimated to be about 2000 pages long. 1974 – 4-color...

Géométrie Algébrique du Bois Marie – 1963-64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – vol. 1 (Lecture Notes in Mathematics 269) (in French)...

theorems, via motivic methods, between de Rham and p {\displaystyle p} -adic étale cohomologies of algebraic varieties over p {\displaystyle p} -adic fields...

crossings in a scheme X if X is regular and for every point p in X, there is an étale morphism X ′ → X {\displaystyle X'\to X} with p in the image such that the...

Berlin 1973. ISBN 3-540-06434-6 Bloch, Spencer; Kato, Kazuya (1986). "p-adic etale cohomology". Publications Mathématiques de l'IHÉS. 63: 107–152. doi:10.1007/bf02831624...

resolution of S {\displaystyle {\mathcal {S}}} . Recall that a sheaf space (or étalé space) is a triple ( E , π , X ) {\displaystyle (E,\pi ,X)} where E {\displaystyle...

145–182, doi:10.2307/2946630, JSTOR 2946630, MR 1259367 Pop, Florian (1995). "Étale Galois covers of affine smooth curves. The geometric case of a conjecture...

uniqueness implies that O {\displaystyle {\mathcal {O}}} , when viewed as étalé space, is a Hausdorff topological space. Oka's coherence theorem states...

{\displaystyle X/{\overline {\mathbf {Q} }}_{p}} , they constructed an action of the etale fundamental group π1(X) on the fibers on certain vector bundles, including...

1-forms. S. Bloch conjectures a formula for the Artin conductor of the ℓ-adic étale cohomology of a regular model of a variety over a local field and proves...

Alexander; Verdier, Jean-Louis (eds.), Théorie des Topos et Cohomologie Etale des Schémas, Lecture Notes in Mathematics (in French), vol. 270, Berlin;...

the maximal connected subgroup scheme. Then G is an extension of a finite étale group scheme by G0. G has a unique maximal reduced subscheme Gred, and if...

Science un jour, Science toujours !, Delcourt, 2014. Professeur Moustache étale sa science !, Delcourt, 2015. Quand y'en a plus, y'en a encore !, Delcourt...

Zbl 0848.13001 N. Jacobson, Basic Algebra I, p. 233 Sutherland, Andrew. "18.785 Number Theory I; Lecture 4: Étale algebras, norm, and trace" (PDF)....

Steinberg 1968, Theorem 10.1 This implies that f a {\displaystyle f_{a}} is étale. Springer, T. A. (1998). Linear algebraic groups (2nd ed.). Birkhäuser....

(see slice theorem (differential geometry).) Luna, Domingo (1973), "Slices étales", Sur les groupes algébriques, Bull. Soc. Math. France, Paris, Mémoire,...

character is the dual of the first p-adic étale cohomology group of Gm. It can also be found in the étale cohomology of a projective variety, namely...

Roche Perfia, 2,499 m (8,199 ft) Aiguille de Borderan, 2,489 m (8,166 ft) Étale (mountain), 2,484 m (8,150 ft) Pointe d'Areu, 2,478 m (8,130 ft) Mont Charvin...

with respect to the smooth topology (not etale). When X is a variety, the smooth cohomology is the same as etale one and, via the Poincaré duality, this...

the characteristic of K, the p-adic Tate module of G is the dual of the étale cohomology H et 1 ( G × K K s , Z p ) {\displaystyle H_{\text{et}}^{1}(G\times...

commutative algebra and algebraic geometry, appearing in the study of descent, étale cohomology, and stacks. Israel Prize in Exact Sciences (1953), shared with...

(mathematics) Projective scheme Spectrum of a matrix Serre's theorem on affineness Étale spectrum Ziegler spectrum Primitive spectrum Stone duality Hartshorne (1977)...

semigroup is precisely a groupoid in the category of posets that is an étale groupoid with respect to its (dual) Alexandrov topology and whose poset...

of the axioms for Betti and de-Rham cohomology) Milne, James S. (1980), Étale cohomology, Princeton, NJ: Princeton University Press, ISBN 978-0-691-08238-7...

principal G-bundle over k with respect to the fppf topology on k, or the étale topology if G is smooth over k. The pointed set of isomorphism classes of...

used as the indexing category. Moreover, the sites underlying the Zariski, Etale, etc., topoi of ( X ) τ {\displaystyle (X)_{\tau }} for some scheme or algebraic...

deprecated archival service (link) Espargilière, Loup (20 October 2017). "Le FN étale ses divisions sur l'Europe". Mediapart (in French). Archived from the original...

0 {\displaystyle (s,t):G_{1}\to G_{0}\times G_{0}} is a proper map, and étale if both the source and the target maps are local diffeomorphisms. An orbifold...

the sheaf theory notes from the Cartan seminar he defines: Sheaf space (étale space), support of sheaves axiomatically, sheaf cohomology with support...

Michèle (2002). "Revêtements étales et groupe fondamental§XII. Géométrie algébrique et géométrie analytique". Revêtements étales et groupe fondamental (SGA...

Descent (category theory) Grothendieck's Galois theory Algebraic stack Gerbe Étale cohomology Motive (algebraic geometry) Motivic cohomology A¹ homotopy theory...

Sys Y {\displaystyle \operatorname {Sys} _{Y}} in the category Top. The étalé space of a sheaf is topologized by a final topology. A first-countable Hausdorff...

matrix of an algebraic curve over a finite field determines the cyclic étale coverings of degree p of a curve in characteristic p. In the 1970s, Witt...

Paris 2014, ISBN 978-2-7560-6183-2 Marion Montaigne: Professeur Moustache étale sa science! Delcourt, Paris 2015, ISBN 978-2-7560-7317-0 "Deutsche Synchronkartei...

Kiasi', 'Lazizi' and theatre life". TNX Africa. Retrieved 14 November 2025. Etale, Martha (5 November 2025). "Meet Isaac Okoyo: The Breakout Star of Reuben...

fields, the Chern classes can take values in cohomology theories such as etale cohomology or l-adic cohomology. For varieties V over general fields the...

In algebraic geometry, the cohomology of a stack is a generalization of étale cohomology. In a sense, it is a theory that is coarser than the Chow group...

195–229. MR 1358975 1990: "Real components of algebraic varieties and étale cohomology", Colliot-Thélène, J.-L.; Parimala, R., Invent. Math. 101, no...

Alexandre; Verdier, Jean-Louis (eds.). Théorie des Topos et Cohomologie Étale des Schémas I. Séminaire de Géométrie Algébrique du Bois Marie 1963–64 (SGA...

Academia Europaea. His doctoral students include Moritz Kerz. Continuous étale cohomology, Mathematische Annalen vol. 280, no. 2 1988, pp. 207–245 doi:10...

cohomology Crystalline cohomology De Rham cohomology Deligne cohomology Étale cohomology Floer homology Galois cohomology Group cohomology Hodge structure...

EGA III, Part 1, Théorème 3.4.2. SGA1 Revêtements étales et groupe fondamental, 1960–1961 (Étale coverings and the fundamental group), Lecture Notes...

about p {\displaystyle p} -torsion in cohomology groups better than p-adic étale cohomology. This makes it a natural backdrop for much of the work on p-adic...

[1963], "Catégories dérivées: quelques résultats (état 0)", Cohomologie étale (SGA 4 1/2) (PDF), Lecture Notes in Mathematics, vol. 569, Springer, pp...

is the functor taking a field to the isomorphism classes of dimension n etale algebras over it. The cohomological invariants with coefficients in Z/2Z...

in the classical topology over C {\displaystyle \mathbb {C} } (or in the étale topology, over a general field). For example, let C be the nodal cubic curve...

functions which is maximal in the sense that it is not contained (as an etale space) within any other path connected sheaf of germs of analytic functions...

II-F Savoie FR 1066 Étale (mountain) 2483 997 45°51′00″N 06°26′52″E / 45.85000°N 6.44778°E / 45.85000; 6.44778 (1066. Étale (mountain) (2483 m))...

geometric Langlands program, the p-adic Waldspurger theorem, and the study of étale cohomology on Artin stacks. He received a Sloan Research Fellowship in 2017...

codepth of M at s, that is, to n − depth(M ⊗ k(s)). Then there exist affine étale neighborhoods X′ of x and S′ of s, together with points x′ and s′ lifting...

Springer-Verlag, 1991, ISBN 3540547045; 2006 pbk reprint with Reinhardt Kiehl: Etale Cohomology and the Weil Conjecture, Springer Verlag, 1988, ISBN 978-0387121758...

every profinite group occurs as an absolute Galois group of a field. The étale fundamental groups considered in algebraic geometry are also profinite groups...

simplicial object in the category of sheaves on the site. Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf Hom ⁡ ( −...

Lie 2-groupoids, and Hao Xu (2025), whose dissertation was entitled On Étale Algebras and Fusion 2-Categories. Under her mentorship, the Göttingen team...

Secrétariat Math., MR 0146040 Grothendieck, A.; Raynaud, M. (1971), Revêtements Etales et Groupe Fondamental, Lecture Notes in Mathematics, vol. 224, Springer...

2016. "La grève de Lohic et l'île des Souris, près de Lorient. La mer étale". Retrieved 20 August 2016. "seaweed". materialbrittany.blogspot.ca. July...

mistakenly claimed that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. The results depending on this were repaired by Olsson (2007). Frege's...

Steinberg symbol to higher algebraic K-theory. It takes a Milnor K-group to an étale cohomology group. Contou-Carrère symbol Mennicke symbol Neukirch, Jürgen...

during this period to the following bases: Mahanene Rundu Okalongo Oshigambo Etale Ondangwa Tsintsabis Okongo Ogongo Ombalantu Operation Moduler (1987), Lead...

X be a smooth complete algebraic curve.[clarification needed] Then the étale fundamental group of X is defined as: π 1 ( X ) = lim ← L / K ⁡ Gal ⁡ (...

1215/kjm/1250523277. Alper, Jarod; Hall, Jack; Rydh, David (2015). "A Luna étale slice theorem for algebraic stacks". Annals of Mathematics. 191 (3). arXiv:1504...

1007/s00209-003-0490-6. Lewis, D. W.; McGarraghy, S. (2000). "Annihilating polynomials, étale algebras, trace forms and the Galois number". Archiv der Mathematik. 75...

September 1987 to November 1989. Various bases were used for the HQ, i.e. Etale, Namibia, Eenhana, Ondangwa, Okatope and Rooikop (Walvisbay). 62 Mech Group...

J.; Lanquetin, P. (1983). "Basidiomycètes Aphyllophorales epitheloides étalés". Mycotaxon (in French). 16 (2): 461–499. Wang, H.C.; Wu, S.H.; Dai, Y.C...

pdf#nameddest=0BQ6 Definition 7.2.1. in Bhatt, Bhargav; Scholze, Peter (2015), "The pro-étale topology for schemes", Astérisque: 99–201, arXiv:1309.1198, Bibcode:2013arXiv1309...