aarjaneiro/planetmath huggingface dataset:
Weil divisors on schemes

Weil divisors on schemes

Let

X

be a

noetherian

integral

separated scheme

such that every local ring

𝒪

x

of

X

of dimension one is

regular

(such a scheme

X

is said to be

regular in codimension one

, or

non-singular

in codimension one).

Definition

.

A

prime divisor

on

X

is a closed integral subscheme

Y

of codimension one. We define an

abelian group

Div

⁡

(

X

)

generated by the prime divisors on

X

. A

Weil divisor

is an

element of

Div

⁡

(

X

)

. Thus, a Weil divisor

W

can be written as:

𝒲

=

∑

n

Y

⁢

Y

where the sum is over all the prime divisors

Y

of

X

, the

n

Y

are

integers

and only finitely many of them are non-zero. A degree of a

divisor

is defined to be

deg

⁡

(

W

)

=

∑

n

Y

. Finally, a divisor is said to be

effective

if

n

Y

≥

0

for all the prime divisors

Y

.

For more information, see Hartshorne’s book listed in the

bibliography for algebraic geometry

.

Title

Weil divisors on schemes

Canonical name

WeilDivisorsOnSchemes

Date of creation

2013-03-22 15:34:08

Last modified on

2013-03-22 15:34:08

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 14C20

Related topic

BibliographyForAlgebraicGeometry

Defines

prime divisor

Defines

effective divisor

Defines

regular in codimension one

Generated on Fri Feb 9 17:57:43 2018 by

LaTeXML
general type

general type

A

variety

is said to be of

general type

if its

Kodaira dimension

equals its

dimension

.

Title

general type

Canonical

name

GeneralType

Date of creation

2013-03-22 12:03:19

Last modified on

2013-03-22 12:03:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 14E05

Generated on Fri Feb 9 17:57:44 2018 by

LaTeXML
Kodaira-Itaka dimension

Kodaira-Itaka dimension

Given a

projective algebraic variety

X

and a

line bundle

L

→

X

,
the

Kodaira-Itaka dimension

of

L

is defined to be the supremum of the

dimensions

of the image of

X

by the map

φ

|

m

⁢

L

|

associated to the linear system

|

m

⁢

L

|

,
when

m

is a

positive integer

, namely

κ

⁢

(

L

)

=

sup

m

∈

ℕ

⁡

{

dim

⁡

φ

|

m

⁢

L

|

⁢

(

X

)

}

.

It is a standard fact that if we consider the

graded ring

R

⁢

(

X

,

L

)

=

⊕

m

∈

ℕ

H

0

⁢

(

X

,

m

⁢

L

)

,

then

tr.deg

⁢

R

⁢

(

X

,

L

)

=

κ

⁢

(

L

)

+

1

.

When the line bundle we have is the canonical bundle

K

X

of

X

,
then its Kodaira-Itaka dimension is called

Kodaira dimension

of

X

.

In paticular, if for some

m

we have

dim

⁡

φ

|

m

⁢

L

|

⁢

(

X

)

=

dim

⁡

X

then

κ

⁢

(

L

)

=

dim

⁡

X

and

L

is called

big

.

If

κ

⁢

(

X

)

=

κ

⁢

(

K

X

)

=

dim

⁡

X

,
then

X

is said to be of

general type

.

Title

Kodaira-Itaka dimension

Canonical

name

KodairaItakaDimension

Date of creation

2013-03-22 16:12:43

Last modified on

2013-03-22 16:12:43

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

17

Author

yark (2760)

Entry type

Definition

Classification

msc 14E05

Defines

Kodaira dimension

Defines

bigness

Defines

general type

Generated on Fri Feb 9 17:57:46 2018 by

LaTeXML
direct image (functor)

direct image (functor)

If

f

:

X

→

Y

is a

continuous map

of

topological spaces

, and if

𝐒𝐡𝐞𝐚𝐯𝐞𝐬

⁢

(

X

)

is the category of sheaves of

abelian groups

on

X

(and similarly for

𝐒𝐡𝐞𝐚𝐯𝐞𝐬

⁢

(

Y

)

), then the direct image functor

f

*

:

𝐒𝐡𝐞𝐚𝐯𝐞𝐬

⁢

(

X

)

→

𝐒𝐡𝐞𝐚𝐯𝐞𝐬

⁢

(

Y

)

sends a sheaf

ℱ

on

X

to its

direct image

f

*

⁢

ℱ

on

Y

. A

morphism of sheaves

g

:

ℱ

→

𝒢

obviously gives rise to a morphism of sheaves

f

*

⁢

g

:

f

*

⁢

ℱ

→

f

*

⁢

𝒢

, and this determines a

functor

.

If

ℱ

is a sheaf of abelian groups (or anything else), so is

f

*

⁢

ℱ

, so likewise we get direct image functors

f

*

:

𝐀𝐛

⁢

(

X

)

→

𝐀𝐛

⁢

(

Y

)

, where

𝐀𝐛

⁢

(

X

)

is the category of sheaves of abelian groups on

X

.

Title

direct image (functor)

Canonical

name

DirectImagefunctor

Date of creation

2013-03-22 12:03:13

Last modified on

2013-03-22 12:03:13

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Definition

Classification

msc 14F05

Related topic

DirectImageSheaf

Generated on Fri Feb 9 17:57:48 2018 by

LaTeXML
presheaf of a topological basis

presheaf of a topological basis

Let

X

be a

topological space

and let

ℬ

be a basis of its

topology

. We can regard

ℬ

as a

category

with objects being the

open sets

in

ℬ

and arrows/

morphisms

between

U

,

V

∈

ℬ

to exists only if

U

⊂

V

, and
where the

only

element of

ℬ

⁢

(

U

,

V

)

is the injection map

U

↪

V

. Let now

𝒞

be a

complete
category

, we now define the

presheaf

of

C

-objects over the basis

B

of the topology of

X

to be a

contravariant functor

𝒫

:

ℬ

→

𝒞

Title

presheaf of a topological basis

Canonical

name

PresheafOfATopologicalBasis

Date of creation

2013-03-22 16:22:36

Last modified on

2013-03-22 16:22:36

Owner

jocaps (12118)

Last modified by

jocaps (12118)

Numerical id

14

Author

jocaps (12118)

Entry type

Definition

Classification

msc 14F05

Classification

msc 54B40

Classification

msc 18F20

Related topic

site

Generated on Fri Feb 9 17:57:50 2018 by

LaTeXML
subbundle

subbundle

Given a

vector bundle

π

:

ℰ

→

M

, a

subbundle

ℰ

′

is a subset of the total space,

ℰ

′

⊂

ℰ

, so that

π

|

ℰ

′

:

ℰ

′

→

M

is a vector bundle, and for each point

p

∈

M

, the fibre at

p

π

|

ℰ

′

-

1

⁢

(

p

)

=

ℰ

p

′

is a

vector subspace

of

ℰ

p

=

π

-

1

⁢

(

p

)

Title

subbundle

Canonical name

Subbundle

Date of creation

2013-03-22 16:50:54

Last modified on

2013-03-22 16:50:54

Owner

guffin (12505)

Last modified by

guffin (12505)

Numerical id

4

Author

guffin (12505)

Entry type

Definition

Classification

msc 14F05

Classification

msc 55R25

Synonym

sub-bundle

Synonym

vector sub-bundle

Synonym

vector subbundle

Synonym

sub-vector bundle

Synonym

sub vector bundle

Defines

subbundle

Generated on Fri Feb 9 17:57:51 2018 by

LaTeXML
subsheaf of abelian groups

subsheaf of abelian groups

Let

ℱ

be a sheaf of abelian groups over a

topological space

X

. Let

𝒢

be a sheaf
over

X

, such that for every

open set

U

⊂

X

,

𝒢

⁢

(

U

)

is a

subgroup

of

ℱ

⁢

(

U

)

. And further let the on

𝒢

be by those on

ℱ

.
Then

𝒢

is a

subsheaf

of

ℱ

.

Suppose a sheaf of abelian groups

ℱ

is defined as a disjoint union of stalks

ℱ

x

over points

x

∈

X

, and

ℱ

is topologized in the appropriate manner.
In particular, each stalk is an

abelian group

and the

group operations

are

continuous

.
Then a subsheaf

𝒢

is an open subset of

ℱ

such that

𝒢

x

=

𝒢

∩

ℱ

x

is a subgroup of

ℱ

x

.

When

𝒢

is a subsheaf of

ℱ

, then

ℱ

x

/

𝒢

x

is an abelian group. Considering this to be the stalk over

x

we have a sheaf which is denoted by

ℱ

/

𝒢

, with the

topology

being the

quotient topology

.

Example

.

Suppose

M

is a

complex manifold

.
Let

M

*

be the sheaf of germs of

meromorphic functions

which are not identically zero. That is, for

z

∈

M

,

the stalk

M

z

*

is the abelian group of germs of meromorphic functions at

z

with the group operation being

multiplication

.
Then

O

*

, the sheaf
of germs of

holomorphic functions

which are not identically 0 is a subsheaf
of

M

*

.

The sheaf

M

*

/

O

*

is then the sheaf of

divisors

. If

M

is of (

complex

)

dimension

1, then

M

*

/

O

*

is just the sheaf of

functions

into the

integers

with finite support.

References

1

Glen E. Bredon.
,
Springer, 1997.

2

Robin Hartshorne.
,
Springer, 1977.

3

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

Title

subsheaf of abelian groups

Canonical name

SubsheafOfAbelianGroups

Date of creation

2013-03-22 17:39:21

Last modified on

2013-03-22 17:39:21

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 14F05

Classification

msc 54B40

Classification

msc 18F20

Synonym

subsheaf

Synonym

subsheaves

Generated on Fri Feb 9 17:57:53 2018 by

LaTeXML
sheafification

sheafification

Let

T

be a site. Let

P

T

denote the category of presheaves on

T

(with values in the category of abelian groups), and

S

T

the category of sheaves on

T

. There is a natural

inclusion functor

ι

:

S

T

→

P

T

.

Theorem 1

The

functor

ι

has a

left adjoint

♯

:

P

T

→

S

T

, that is, for any sheaf

F

and

presheaf

G

, we have

Hom

S

T

⁡

(

G

♯

,

F

)

≅

Hom

P

T

⁡

(

G

,

ι

⁢

F

)

.

This functor

♯

is called

sheafification

, and

G

♯

is called the

sheafification of

F

.

One can readily check that this description in terms of adjoints characterizes

♯

completely, and that this definition reduces to the usual definition of sheafification (

http://planetmath.org/Sheafification

) when

T

is the

Zariski site

. It also allows derivation of various exactness properties of

♯

and

ι

.

References

1

Grothendieck et al.,

Séminaires en Gèometrie Algèbrique 4

, tomes 1, 2, and 3, available on the web at

http://www.math.mcgill.ca/ archibal/SGA/SGA.html

http://www.math.mcgill.ca/ archibal/SGA/SGA.html

Title

sheafification

Canonical name

Sheafification1

Date of creation

2013-03-22 14:13:08

Last modified on

2013-03-22 14:13:08

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Theorem

Classification

msc 14F20

Classification

msc 18F10

Classification

msc 18F20

Related topic

Sheafification

Related topic

Site

Related topic

Sheaf2

Related topic

Sheaf

Defines

sheafification

Generated on Fri Feb 9 17:57:57 2018 by

LaTeXML
ℓ-adic étale cohomology

ℓ

-adic étale cohomology

Let

X

be a scheme over a field

k

having

algebraic closure

k

¯

.
Let

(

X

⊗

k

k

¯

)

ét

be the small étale site on

X

⊗

k

k

¯

,
and let

ℤ

/

l

n

⁢

ℤ

denote the sheaf on

(

X

⊗

k

k

¯

)

ét

associated to the group scheme

ℤ

/

l

n

⁢

ℤ

for some fixed prime

l

.
Finally, let

Γ

be the global sections functor on the category of étale sheaves on

(

X

⊗

k

k

¯

)

ét

.

The

l

-adic

étale cohomology

of

X

is

H

ét

i

⁢

(

X

,

ℚ

l

)

=

ℚ

l

⊗

ℤ

l

lim

←

n

⁡

(

R

i

⁢

Γ

)

⁢

(

ℤ

/

l

n

⁢

ℤ

)

,

.

where

R

i

denotes taking the

i

-th right-derived functor.

This apparently appalling definition is necessary to ensure that (for

l

not equal to the

characteristic

of

k

) étale cohomology is the appropriate generalization of

de Rham cohomology

on a complex manifold.
For example, on a scheme of dimension

n

, the

cohomology groups

H

i

vanish for

i

>

2

⁢

n

and we have a version of Poincaré duality.
Grothendieck introduced étale cohomology as a tool to prove the

Weil conjectures

, and indeed it is what Deligne used to prove them.

These references are approximately in order of difficulty and of generality and

precision

.

References

1

J. S. Milne,

Lectures on Étale Cohomology

, 1998, available on the web at

http://www.jmilne.org/math/

http://www.jmilne.org/math/

2

James S. Milne,

Étale cohomology

, volume 33 of

Princeton Mathematical Series

. Princeton University Press, Princeton N.J., 1980

3

Deligne et al.,

Séminaires en Gèometrie Algèbrique 4

1

2

, available on the web at

http://www.math.mcgill.ca/ archibal/SGA/SGA.html

http://www.math.mcgill.ca/ archibal/SGA/SGA.html

4

Grothendieck et al.,

Séminaires en Gèometrie Algèbrique 4

, tomes 1, 2, and 3, available on the web at

http://www.math.mcgill.ca/ archibal/SGA/SGA.html

http://www.math.mcgill.ca/ archibal/SGA/SGA.html

Title

ℓ

-adic étale cohomology

Canonical name

elladicetaleCohomology

Date of creation

2013-03-22 14:13:39

Last modified on

2013-03-22 14:13:39

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14F20

Related topic

DerivedFunctor

Related topic

Site

Related topic

EtaleMorphism

Related topic

SmallSiteOnAScheme

Related topic

SheafCohomology

Generated on Fri Feb 9 17:57:59 2018 by

LaTeXML
étale morphism

étale morphism

\PMlinkescapephrase

one way

Definition 1

A

morphism of schemes

f

:

X

→

Y

is

étale

if it is flat and

unramified

.

This is the appropriate

generalization

of “

local homeomorphism

” from

topology

or “local isomorphism” from real differential geometry.
Equivalently,

f

is étale if and only if any of the following conditions hold:

•

f

is locally of

finite type

and formally étale.

•

f

is flat and the relative sheaf of

differentials

vanishes

.

•

f

is smooth of relative dimension zero.

•

f

locally looks like

A

⁢

[

x

1

,

…

,

x

n

]

/

(

p

1

,

…

,

p

n

)

where the

Jacobian

vanishes.

A

morphism

f

:

X

→

Y

of

varieties

over an algebraically
closed field is étale at a point

x

∈

X

if it induces an

isomorphism

between the completed local rings

𝒪

^

x

and

𝒪

^

f

⁢

(

x

)

. If

X

and

Y

are over an arbitrary field

k

, then the required
condition becomes that

k

⁢

(

x

)

is a

separable

algebraic extension

of

k

⁢

(

y

)

, where

y

=

f

⁢

(

x

)

, and

f

induces an isomorphism between

𝒪

^

y

⊗

k

⁢

(

y

)

k

⁢

(

x

)

and

𝒪

^

x

.

A morphism

f

of

nonsingular varieties

over an algebraically closed
field is étale if and only if

f

induces an isomorphism on the

tangent spaces

. In the

differentiable

category, the

implicit function theorem

implies that such a

function

is actually an isomorphism on some small

neighborhood

. On schemes, of course, the

Zariski topology

is too
coarse for this to be the case. One way to define a finer “topology”,
making the scheme into a site, is by using étale maps.

The word étale comes from French, where it can be used to describe a calm or slack sea.

References

1

Jean Dieudonné,

A Panorama of Pure Mathematics

, Academic Press, 1982.

2

Robin Hartshorne,

Algebraic
Geometry

, Springer–Verlag, 1977 (GTM

52

).

Title

étale morphism

Canonical

name

etaleMorphism

Date of creation

2013-03-22 14:08:40

Last modified on

2013-03-22 14:08:40

Owner

mps (409)

Last modified by

mps (409)

Numerical id

14

Author

mps (409)

Entry type

Definition

Classification

msc 14F20

Classification

msc 14A15

Synonym

étale

Related topic

site

Related topic

Site

Related topic

FlatMorphism

Related topic

EtaleFundamentalGroup

Related topic

EtaleCohomology

Related topic

CoveringSpace

Generated on Fri Feb 9 17:58:00 2018 by

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Serre duality

Serre duality

The most general version of

Serre duality

states that on certain schemes

X

of

dimension

n

, including all

projective varieties

over any algebraically closed field

k

, there is a natural

perfect

pairing

(

http://planetmath.org/BilinearMap

)

Ext

i

⁢

(

ℱ

,

ω

)

×

H

n

-

i

⁢

(

X

,

ℱ

)

→

k

,

where

ℱ

is any

coherent sheaf

on

X

and

ω

is a sheaf, called the

dualizing sheaf

. Here “perfect” means that the natural map above induces an

isomorphism

Ext

i

⁢

(

ℱ

,

ω

)

≅

Hom

⁢

(

H

n

-

i

⁢

(

X

,

ℱ

)

,

k

)

.

In special cases, this reduces to more approachable forms. If

X

is

nonsingular

(or more generally, Cohen-Macaulay), then

ω

is simply

⋀

n

Ω

, where

Ω

is the sheaf of differentials on

X

.

If

ℱ

is

locally free

, then

Ext

i

⁢

(

ℱ

,

ω

)

≅

Ext

i

⁢

(

𝒪

X

,

ℱ

*

⊗

ω

)

≅

H

i

⁢

(

X

,

ℱ

*

⊗

ω

)

,

so that we obtain the somewhat more familiar looking fact that there is a perfect pairing

H

i

⁢

(

X

,

ℱ

*

⊗

ω

)

×

H

n

-

i

⁢

(

X

,

ℱ

)

→

k

.

While Serre duality is not in a strict sense a

generalization

of Poincaré duality, they are philosophically

similar

, and both fit into a larger pattern on duality results.

Title

Serre duality

Canonical name

SerreDuality

Date of creation

2013-03-22 13:51:24

Last modified on

2013-03-22 13:51:24

Owner

mps (409)

Last modified by

mps (409)

Numerical id

12

Author

mps (409)

Entry type

Definition

Classification

msc 14F25

Related topic

DualityInMathematics

Defines

dualizing sheaf

Generated on Fri Feb 9 17:58:02 2018 by

LaTeXML
sheaf cohomology

sheaf cohomology

Let

X

be a

topological space

. The category of sheaves of

abelian groups

on

X

has enough injectives. So we can define the

sheaf cohomology

H

i

⁢

(

X

,

ℱ

)

of a sheaf

ℱ

to be the right

derived functors

of the global sections functor

ℱ

→

Γ

⁢

(

X

,

ℱ

)

.

Usually we are interested in the case where

X

is a scheme, and

ℱ

is a

coherent sheaf

. In this case, it does not matter if we take the derived functors in the category of sheaves of abelian groups or coherent sheaves.

Sheaf cohomology can be explicitly calculated using Čech

cohomology

(

http://

planetmath

.org/CechCohomologyGroup2

). Choose an

open cover

{

U

i

}

of

X

. We define

C

i

⁢

(

ℱ

)

=

∏

ℱ

⁢

(

U

j

0

⁢

⋯

⁢

j

i

)

where the

product

is over

i

+

1

element

subsets of

{

1

,

…

,

n

}

and

U

j

0

⁢

⋯

⁢

j

i

=

U

j

0

∩

⋯

∩

U

j

i

. If

s

∈

ℱ

⁢

(

U

j

0

⁢

⋯

⁢

j

i

)

is thought of as an element of

C

i

⁢

(

ℱ

)

,
then the

differential

∂

⁡

(

s

)

=

∏

ℓ

(

∏

k

=

j

ℓ

+

1

j

ℓ

+

1

-

1

(

-

1

)

ℓ

⁢

s

|

U

j

0

⁢

⋯

⁢

j

ℓ

⁢

k

⁢

j

ℓ

+

1

⁢

⋯

⁢

j

i

)

makes

C

*

⁢

(

ℱ

)

into a

chain complex

. The cohomology of this complex is denoted

H

ˇ

i

⁢

(

X

,

ℱ

)

and called the

Čech cohomology

of

ℱ

with respect to the cover

{

U

i

}

. There is a natural map

H

i

⁢

(

X

,

ℱ

)

→

H

ˇ

i

⁢

(

X

,

ℱ

)

which is an

isomorphism

for sufficiently fine covers. (A cover is

sufficiently fine

if

H

i

⁢

(

U

j

,

ℱ

)

=

0

for all

i

>

0

, for every

j

and for every sheaf

ℱ

). In the

category

of schemes, for example, any cover by open

affine schemes

has this property. What this means is that if one can find a finite fine enough cover of

X

, sheaf cohomology becomes computable by a finite process. In fact in

[

2

]

, this is how the cohomology of

projective space

is explicitly calculated.

References

1

Grothendieck, A.

Sur quelques points d’algèbre homologique

,
Tôhoku Math. J., Second Series, 9 (1957), 119–221.

2

Hartshorne, R.

Algebraic Geometry

, Springer-Verlag Graduate Texts in Mathematics 52, 1977

Title

sheaf cohomology

Canonical name

SheafCohomology

Date of creation

2013-03-22 13:50:59

Last modified on

2013-03-22 13:50:59

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 14F25

Related topic

EtaleCohomology

Related topic

LeraysTheorem

Related topic

AcyclicSheaf

Related topic

DeRhamWeilTheorem

Defines

sufficiently fine

Generated on Fri Feb 9 17:58:04 2018 by

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étale fundamental group

étale fundamental group

Recall that in

topology

, the

fundamental group

π

1

⁢

(

T

)

of a

connected

topological space

T

is defined to be the group of loops based at a point modulo

homotopy

. When one wants to obtain something

similar

in the

algebraic category

, this definition encounters problems. One cannot simply attempt to use the same definition, since the result will be wrong if one is working in positive characteristic. More to the point, the topology on a scheme fails to capture much of the stucture of the scheme. Simply choosing the “loop” to be an

algebraic curve

is not appropriate either, since in the most familiar case (over the

complex numbers

) such a “loop” has two real dimensions rather than one.

In the

classification of covering spaces

, it is shown that the fundamental group is exactly the group of

deck transformations

of the

universal covering space

. This is more promising:

surjective

étale

morphisms

are the appropriate

generalization

of covering spaces. Unfortunately, the universal covering space is often an

infinite

covering

of the orignal space, which is unlikely to yield anything manageable in the algebraic category. Finite coverings, on the other hand are tractable, so one can define the algebraic fundamental group as an

inverse limit

of

automorphism groups

. Note also that finite étale morphisms are closed maps as well as open maps.

Definition 1

Let

X

be a scheme, and let

x

be a geometric point of

X

. Then let

C

be the

category

of pairs

(

Y

,

π

)

such that

π

:

Y

→

X

is a finite étale morphism. Morphisms

(

Y

,

π

)

→

(

Y

′

,

π

′

)

in this category are morphisms

Y

→

Y

′

as schemes over

X

. If

Y

′

factors through

Y

as

Y

′

→

Y

→

X

then we obtain a morphism from

Aut

C

⁡

(

Y

)

→

Aut

C

⁡

(

Y

′

)

. This allows us to

construct

the

étale fundamental group

π

1

⁢

(

X

,

x

)

=

lim

←

Y

∈

C

⁡

Aut

C

⁡

(

Y

)

.

This explanation follows

[

1

]

.

References

1

James Milne,

Lectures on Étale

Cohomology

,

http://www.jmilne.org/math/CourseNotes/math732.html

online course notes.

Title

étale fundamental group

Canonical name

etaleFundamentalGroup

Date of creation

2013-03-22 14:11:46

Last modified on

2013-03-22 14:11:46

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

5

Author

archibal (4430)

Entry type

Definition

Classification

msc 14F35

Related topic

EtaleMorphism

Related topic

TopologicalSpace

Related topic

FundamentalGroup

Related topic

ClassificationOfCoveringSpaces

Generated on Fri Feb 9 17:58:06 2018 by

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