Queer Beagle Wiki - User contributions [en]

Category:Point-Set Topology

2023-05-21T22:50:35Z

Kfagerstrom: Created page with "871 material"

871 material

Compact

2023-05-21T22:50:19Z

Kfagerstrom:

=Compact (Topology)=

A topological space $$X$$ is '''compact'''' if every open covering of $$X$$ contains a finite subcollection that also covers $$X$$.

Compactness is a homeomorphism invariant.

If $$X$$ is a compact space and $$X/∼$$ is a quotient space, then $$X/∼$$ is compact.

If $$A$$ is a subspace of a compact space $$X$$ and $$A$$ is a closed subset in $$X$$, then $$A$$ is compact.

(EVT = Extreme Value Theorem) Let $$X$$ be a compact space, and let $$f: X → (ℝ,𝒯_{\rm Eucl})$$ be a continuous function. Then there exist $$c,d ∈ X$$ such that for all $$p ∈ X, f(c) ≤ f(p) ≤ f(d)$$.

A continuous image of a compact space is compact. That is, if $$X,Y$$ are topological spaces, and if $$X$$ is compact and $$f: X → Y$$ is a continuous surjective function, then $$Y$$ is compact.

Let $$A$$ be a subspace of a topological space $$(X,𝒯_X)$$. The space $$A$$ is compact if and only if for every collection $$𝒞$$ of open sets in $$X$$ satisfying $$A ⊆ ∪_{C ∈ 𝒞} C$$, there is a finite subcollection $$𝒟 ⊆ 𝒞$$ such that $$A ⊆ ∪_{D ∈ 𝒟} D$$.

[[Category:Point-Set Topology]]

Compact

2023-05-21T22:49:55Z

Kfagerstrom:

Compact

2023-05-21T22:22:23Z

Kfagerstrom:

=Compact (Topology)=

A topological space $$X$$ is '''compact'''' if every open covering of $$X$$ contains a finite subcollection that also covers $$X$$.

Compactness is a homeomorphism invariant.

If $$X$$ is a compact space and $$X/∼$$ is a quotient space, then $$X/∼$$ is compact.

If $$A$$ is a subspace of a compact space $$X and $$A$$ is a closed subset in $$X$$, then $$A$$ is compact.

(EVT = Extreme Value Theorem) Let $$X$$ be a compact space, and let $$f: X → (ℝ,𝒯_{\rm Eucl})$$ be a continuous function. Then there exist $$c,d ∈ X$$ such that for all $$p ∈ X, f(c) ≤ f(p) ≤ f(d)$$.

A continuous image of a compact space is compact. That is, if $$X,Y$$ are topological spaces, and if $$X$$ is compact and $$f: X → Y$$ is a continuous surjective function, then $$Y$$ is compact.

Let $$A$$ be a subspace of a topological space $$(X,𝒯_X)$$. The space $$A$$ is compact if and only if for every collection $$𝒞$$ of open sets in $$X$$ satisfying $$A ⊆ ∪_{C ∈ 𝒞} C$$, there is a finite subcollection $$𝒟 ⊆ 𝒞$$ such that $$A ⊆ ∪_{D ∈ 𝒟} D$$.

Topology Qualifying Syllabus

2023-05-21T22:02:58Z

Kfagerstrom: /* Homeomorphism invariants: */

==Point-set Topology:==
===Topological spaces and continuous functions:===
Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.

[[Quotients (Topology)]]

===Homeomorphism invariants:===
Separation properties (T0, T1, Hausdorff, regular, normal),
countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications.

[[Path Connected]]
[[ T2 (Hausdorff)]]
[[Compact]]

===Continuous deformations:===
Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.

==Algebraic topology:==
===Fundamental groups:===
Fundamental group, induced homomorphism; free group, group presentation, Tietze’s theorem, amalgamated product of groups, Seifert - van Kampen Theorem; cell complex, presentation complex, Classification of surfaces.
===Covering spaces:===
Covering map, Lifting theorems; covering space group action; universal
covering, Cayley complex; Galois Correspondence Theorem, deck transformation, normal
covering; applications to group theory.
===Homology:===
Simplicial homology, singular homology, induced homomorphism, homotopy invariance; exact sequence, long exact homology sequence, Mayer-Vietoris Theorem. Applications.

871 - Topology

2023-05-21T22:02:29Z

Kfagerstrom: /* Separation Properties */

==Homeomorphism Invariants==

===Metrizability===
:''see also'' [[Metrizability]]

A topological space <math> (X,\mathcal{T}_X) </math> is metrizable if there is a metric <math>d </math> on <math> X</math> such that <math>\mathcal{T}_X = \mathcal{T}_d </math>, where <math> \mathcal{T}_d </math> is the metric topology on <math> X</math> induced by <math> d</math>.

Metrizability is a homeomorphism invariant. Metrizability is not preserved by quotients, continuous images, or continuous preimages. Metrizable spaces are <math> \mathcal{T}_4 </math>.

===Connectedness===

===Path Connected===

===Compactness===
:''see also:'' [[Compact]]

===Separation Properties ===

====<math>T_1 </math> ====
A topological space <math>X </math> is <math>T_1 </math> if for any two distinct points <math> a,b\in X</math> there are open sets <math> U,V</math> in <math>X </math> with <math>a\in U, b\not \in U, a\not\in V, b\in V </math>.

====<math>T_2 </math> (Hausdorff) ====
:''See also'' [[ T2 (Hausdorff)]]

====<math>T_3 </math> (Regular) ====
A topological space <math>X </math> is <math>T_3 </math> if it is <math>T_1 </math> and for any point <math> a \in X</math> and closed set <math>B \in X </math> with <math>a \not \in B </math>, there are disjoint open sets <math>U,V \in X </math> with <math>a\in U </math> and <math>B\subseteq V </math>

====<math>T_4 </math> (Normal)====
A topological space <math>X </math> is <math>T_4 </math> if it is <math>T_1 </math>and for any two disjoint closed sets <math>A,B \in X </math> there are disjoint open sets <math>U,V \in X </math> with <math> A \subseteq U</math> and <math>B \subseteq V </math>.

==Homotopy==

==Fundamental Groups==

Quotients (Topology)

2023-05-21T21:34:58Z

Kfagerstrom:

The '''quotient''' of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes

The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$ is called the '''equivalence map'''.

The '''equivalence class''' of an element $$a$$ of $$X$$ is $$[a] := \{b | b ∼ a\}$$.

An equivalence relation $$∼$$ on a set $$X$$ is a subset $$S$$ of $$X × X$$ (where $$(a,b) ∈ S$$ is written $$a ∼ b$$) that satisfies the following for all $$a,b,c \in X$$:
# Reflexive: $$a ∼ a$$
# Symmetric: $$a ∼ b$$ implies $$b ∼ a$$
# Transitive: $$a ∼ b$$ and $$b ∼ c$$ implies $$a ∼ c$$

'''FBT = Function Building Theorem for quotient sets:''' Let $$∼$$ be an equivalence relation on a set $$X$$, and let $$f: X → Y$$ be a function satisfying the property that whenever $$x,x' ∈ X$$ and $$x ∼ x'$$ then $$f(x) = f(x')$$. Then:
# There is a well-defined function $$g:X/∼ → Y$$ defined by $$g([x]) = f(x)$$ for all in $$X/∼$$; that is, $$g ∘ q = f$$, where $$q$$ is the equivalence map.
# If $$f$$ is onto, then $$g$$ is onto.
# If $$f$$ also satisfies the property that whenever $$x,x' ∈ X $$ and $$f(x) = f(x')$$ then $$x ∼ x'$$, then $$g$$ is one-to-one.



The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ | q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.

[[Category:Point-Set Topology]]

Quotients (Topology)

2023-05-21T21:31:02Z

Kfagerstrom:

Quotients (Topology)

2023-05-21T21:30:15Z

Kfagerstrom:

The '''quotient''' of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes

The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$ is called the '''equivalence map'''.

The '''equivalence class''' of an element $$a$$ of $$X$$ is $$[a] := \{b | b ∼ a\}$$.

An equivalence relation $$∼$$ on a set $$X$$ is a subset $$S$$ of $$X × X$$ (where $$(a,b) ∈ S$$ is written $$a ∼ b$$) that satisfies the following for all $$a,b,c \in X$$:
# Reflexive: $$a ∼ a$$
# Symmetric: $$a ∼ b$$ implies $$b ∼ a$$
# Transitive: $$a ∼ b$$ and $$b ∼ c$$ implies $$a ∼ c$$

'''FBT = Function Building Theorem for quotient sets:''' Let $$∼$$ be an equivalence relation on a set $$X$$, and let $$f: X → Y$$ be a function satisfying the property that whenever $$x,x' ∈ X$$ and $$x ∼ x'$$ then $$f(x) = f(x')$$. Then:
# There is a well-defined function $$g:X/∼ → Y$$ defined by $$g([x]) = f(x)$$ for all in $$X/∼$$; that is, $$g ∘ q = f$$, where $$q$$ is the equivalence map.
# If $$f$$ is onto, then $$g$$ is onto.
# If $$f$$ also satisfies the property that whenever $$x,x' ∈ X $$ and $$f(x) = f(x')$$ then $$x ∼ x'$$, then $$g$$ is one-to-one.

() The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ | q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.

Quotients (Topology)

2023-05-21T21:27:37Z

Kfagerstrom:

The '''quotient''' of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes

The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$ is called the '''equivalence map'''.

The '''equivalence class''' of an element $$a$$ of $$X$$ is $$[a] := \{b | b ∼ a\}$$.

An equivalence relation $$∼$$ on a set $$X$$ is a subset $$S$$ of $$X × X$$ (where $$(a,b) ∈ S$$ is written $$a ∼ b$$) that satisfies the following for all $$a,b,c \in X$$:
# Reflexive: $$a ∼ a$$
# Symmetric: $$a ∼ b$$ implies $$b ∼ a$$
# Transitive: $$a ∼ b$$ and $$b ∼ c$$ implies $$a ∼ c$$

'''FBT = Function Building Theorem for quotient sets:''' Let $$∼$$ be an equivalence relation on a set $$X$$, and let $$f: X → Y$$ be a function satisfying the property that whenever $$x,x' ∈ X$$ and $$x ∼ x'$$ then $$f(x) = f(x')$$. Then:
# There is a well-defined function $$g:X/∼ → Y$$ defined by $$g([x]) = f(x)$$ for all in $$X/∼$$; that is, $$g ∘ q = f$$, where $$q$$ is the equivalence map.
# If $$f$$ is onto, then $$g$$ is onto.
# If $$f$$ also satisfies the property that whenever $$x,x' ∈ X $$ and $$f(x) = f(x')$$ then $$x ∼ x'$$, then $$g$$ is one-to-one.

 The '''quotient topology''', or identification topology on $$X/∼$$ induced by $$∼$$, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ | q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map''' induced by $$∼$$.

Quotients (Topology)

2023-05-21T21:22:39Z

Kfagerstrom:

The quotient of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes

The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$ is called the equivalence map.

The equivalence class of an element $$a$$ of $$X$$ is $$[a] := \{b | b ∼ a\}$$.

An equivalence relation $$∼$$ on a set $$X$$ is a subset $$S$$ of $$X × X$$ (where $$(a,b) ∈ S$$ is written $$a ∼ b$$) that satisfies the following for all $$a,b,c \in X$$:
# Reflexive: $$a ∼ a$$
# Symmetric: $$a ∼ b$$ implies $$b ∼ a$$
# Transitive: $$a ∼ b$$ and $$b ∼ c$$ implies $$a ∼ c$$

FBT = Function Building Theorem for quotient sets: Let $$∼$$ be an equivalence relation on a set $$X$$, and let $$f: X → Y$$ be a function satisfying the property that whenever $$x,x' ∈ X$$ and $$x ∼ x'$$ then $$f(x) = f(x')$$. Then:
# There is a well-defined function $$g:X/∼ → Y$$ defined by $$g([x]) = f(x)$$ for all in $$X/∼$$; that is, $$g ∘ q = f$$, where $$q$$ is the equivalence map.
# If $$f$$ is onto, then $$g$$ is onto.
# If $$f$$ also satisfies the property that whenever $$x,x' ∈ X $$ and $$f(x) = f(x')$$ then $$x ∼ x'$$, then $$g$$ is one-to-one.

Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). The '''quotient topology''', or identification topology on X/∼ induced by ∼, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ | q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a '''quotient space of $$X$$''', and the equivalence map $$q$$ is called the '''quotient map'' induced by $$∼$$.

Quotients (Topology)

2023-05-21T21:18:41Z

Kfagerstrom: Created page with "The quotient of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$..."

The quotient of $$X$$ with respect to $$∼$$, $$X/∼$$ denotes the set of equivalence classes

The function $$q:X → X/∼$$ defined by $$q(p) := [p]$$ for all $$p ∈ X$$ is called the equivalence map.

The equivalence class of an element $$a$$ of $$X$$ is $$[a] := \{b | b ∼ a\}$$.

An equivalence relation $$∼$$ on a set $$X$$ is a subset $$S$$ of $$X × X$$ (where $$(a,b) ∈ S$$ is written $$a ∼ b$$) that satisfies the following for all $$a,b,c \in X$$:
# Reflexive: $$a ∼ a$$
# Symmetric: $$a ∼ b$$ implies $$b ∼ a$$
# Transitive: $$a ∼ b$$ and $$b ∼ c$$ implies $$a ∼ c$$

FBT = Function Building Theorem for quotient sets: Let $$∼$$ be an equivalence relation on a set $$X$$, and let $$f: X → Y$$ be a function satisfying the property that whenever $$x,x' ∈ X$$ and $$x ∼ x'$$ then $$f(x) = f(x')$$. Then:
# There is a well-defined function $$g:X/∼ → Y$$ defined by $$g([x]) = f(x)$$ for all in $$X/∼$$; that is, $$g ∘ q = f$$, where $$q$$ is the equivalence map.
# If $$f$$ is onto, then $$g$$ is onto.
# If $$f$$ also satisfies the property that whenever $$x,x' ∈ X $$ and $$f(x) = f(x')$$ then $$x ∼ x'$$, then $$g$$ is one-to-one.

Let $$X$$ be a topological space and let $$∼$$ be an equivalence relation on $$X$$. Let $$X/∼$$ be the set of equivalence classes and let $$q: X → X/∼$$ be the equivalence map (defined by $$q(p) := [p]$$ for all $$p$$ in $$X$$). The quotient topology, or identification topology on X/∼ induced by ∼, is the topology $$𝒯∼ = 𝒯_{\rm quo}:= \{U ⊆ X/∼ | q^{-1}(U) \text{ is open in }X\}$$. The set $$X/∼$$ together with the quotient topology is called a quotient space of $$X$$, and the equivalence map $$q$$ is called the quotient map induced by $$∼$$.

Topology Qualifying Syllabus

2023-05-21T21:06:33Z

Kfagerstrom: /* Topological spaces and continuous functions: */

==Point-set Topology:==
===Topological spaces and continuous functions:===
Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.

[[Quotients (Topology)]]

===Homeomorphism invariants:===
Separation properties (T0, T1, Hausdorff, regular, normal),
countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications.

[[Path Connected]]

===Continuous deformations:===
Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.

==Algebraic topology:==
===Fundamental groups:===
Fundamental group, induced homomorphism; free group, group presentation, Tietze’s theorem, amalgamated product of groups, Seifert - van Kampen Theorem; cell complex, presentation complex, Classification of surfaces.
===Covering spaces:===
Covering map, Lifting theorems; covering space group action; universal
covering, Cayley complex; Galois Correspondence Theorem, deck transformation, normal
covering; applications to group theory.
===Homology:===
Simplicial homology, singular homology, induced homomorphism, homotopy invariance; exact sequence, long exact homology sequence, Mayer-Vietoris Theorem. Applications.

Topology Qualifying Syllabus

2023-05-21T21:05:53Z

Kfagerstrom: /* Point-set Topology: */

==Point-set Topology:==
===Topological spaces and continuous functions:===
Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.

[[Quotient Topology]]

===Homeomorphism invariants:===
Separation properties (T0, T1, Hausdorff, regular, normal),
countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications.

[[Path Connected]]

===Continuous deformations:===
Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.

==Algebraic topology:==
===Fundamental groups:===
Fundamental group, induced homomorphism; free group, group presentation, Tietze’s theorem, amalgamated product of groups, Seifert - van Kampen Theorem; cell complex, presentation complex, Classification of surfaces.
===Covering spaces:===
Covering map, Lifting theorems; covering space group action; universal
covering, Cayley complex; Galois Correspondence Theorem, deck transformation, normal
covering; applications to group theory.
===Homology:===
Simplicial homology, singular homology, induced homomorphism, homotopy invariance; exact sequence, long exact homology sequence, Mayer-Vietoris Theorem. Applications.

Path Connected

2023-05-21T21:03:38Z

Kfagerstrom:

Definition: A space $$X$$ is path-connected, or PC, if for all $$p,q ∈ X$$, there is a continuous function $$f: I → X $$ such that $$f(0) = p$$ and $$f(1) = q$$ (that is, there is a path from $$p$$ to $$q$$).

A continuous image of a path-connected space is path-connected.

Path-connectedness is a homeomorphism invariant.

If $$X_α$$ is a path-connected space for all $$α$$, then the product space $$∏_α X_α$$ is path-connected.
If $$X$$ is a path-connected space and $$∼$$ is an equivalence relation on $$X$$, then the quotient space $$X/∼$$ is path-connected.

Path-connectedness is not preserved by subspaces or continuous preimages.

If $$X$$ is a path-connected space, then $$X$$ is connected. Connectedness does not imply path-connectedness. In particular, the flea-and-comb space is connected but not path-connected.

A subspace $$Y$$ of $$(ℝ,𝒯_{\rm Eucl})$$ is path-connected if and only iff $$Y$$ is either an interval, ray, or $$ℝ$$.

Path-connectedness is a homotopy invariant.

If $$X$$ is a path-connected space, then $$π_1(X)$$ is independent of basepoint, up to isomorphism.

A space $$X$$ is 0-connected if $$X$$ is path-connected.
A space $$X$$ is 1-connected, or simply connected, if $$X$$ is path-connected and $$π_1(X) = 1.$$

If $$X$$ and $$Y$$ are path-connected spaces and $$X ≃ Y$$, then $$π_1(X) ≅ π1(Y)$$.

If $$X$$ and $$Y$$ are homotopy equivalent path-connected spaces, then $$π_1(X)$$ is abelian [respectively, finite] if and only if $$π_1(Y)$$ is abelian [respectively, finite].

872:

If $$X$$ is a path-connected space, then $$H_0^{\rm sing}(X) ≅ ℤ$$.

A CW complex $$X$$ is path-connected if and only if the 1-skeleton $$X^{(1)}$$ is path-connected.

For any PC CW complex $$X$$, $$π_1(X) ≅ π_1(X^{(2)})$$

Path Connected

2023-05-21T20:59:13Z

Kfagerstrom: Created page with "Definition: A space $$X$$ is path-connected, or PC, if for all $$p,q ∈ X$$, there is a continuous function $$f: I → X $$ such that $$f(0) = p$$ and $$f(1) = q$$ (that is,..."

Topology Qualifying Syllabus

2023-05-21T20:50:01Z

Kfagerstrom: /* Homeomorphism invariants: */

==Point-set Topology:==
===Topological spaces and continuous functions:===
Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.
===Homeomorphism invariants:===
Separation properties (T0, T1, Hausdorff, regular, normal),
countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications.
[[Path Connected]]

===Continuous deformations:===
Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.
==Algebraic topology:==
===Fundamental groups:===
Fundamental group, induced homomorphism; free group, group presentation, Tietze’s theorem, amalgamated product of groups, Seifert - van Kampen Theorem; cell complex, presentation complex, Classification of surfaces.
===Covering spaces:===
Covering map, Lifting theorems; covering space group action; universal
covering, Cayley complex; Galois Correspondence Theorem, deck transformation, normal
covering; applications to group theory.
===Homology:===
Simplicial homology, singular homology, induced homomorphism, homotopy invariance; exact sequence, long exact homology sequence, Mayer-Vietoris Theorem. Applications.

Topology Qualifying Syllabus

2023-05-21T20:46:58Z

Kfagerstrom: Created page with "==Point-set Topology:== ===Topological spaces and continuous functions:=== Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit..."

==Point-set Topology:==
===Topological spaces and continuous functions:===
Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.
===Homeomorphism invariants:===
Separation properties (T0, T1, Hausdorff, regular, normal),
countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications.
===Continuous deformations:===
Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.
==Algebraic topology:==
===Fundamental groups:===
Fundamental group, induced homomorphism; free group, group presentation, Tietze’s theorem, amalgamated product of groups, Seifert - van Kampen Theorem; cell complex, presentation complex, Classification of surfaces.
===Covering spaces:===
Covering map, Lifting theorems; covering space group action; universal
covering, Cayley complex; Galois Correspondence Theorem, deck transformation, normal
covering; applications to group theory.
===Homology:===
Simplicial homology, singular homology, induced homomorphism, homotopy invariance; exact sequence, long exact homology sequence, Mayer-Vietoris Theorem. Applications.

Algebra Qualifying Syllabus

2023-04-26T16:26:05Z

Kfagerstrom: /* Modules: */

==Group Theory:==
Groups, subgroups, homomorphisms, cosets, quotients, isomorphism theorems, direct and semidirect products, solvable groups, structure of cyclic, symmetric and
alternating groups; free groups, structure theorem for finite abelian groups.

==Group Actions:==
Groups acting on sets, cosets, and themselves; orbits and stabilizers, permutation representations, Cayley’s Theorem, the class equation, inner automorphisms and
automorphism groups, p-subgroups and the Sylow Theorems.

==Ring Theory:==
Definition and examples, homomorphisms, ideals, quotients, integral domains
and their fields of fractions, maximal and prime ideals.
Factorization in Commutative Rings: Euclidean domains, Unique Factorization Domains,
Principal Ideal Domains, Gauss’s Lemma, polynomial factorization, Eisenstein’s criterion,
Gaussian integers.

==[[Modules]]:==
Definition and examples: matrices, free modules, and bases over arbitrary commutative rings, Structure Theorem for finitely generated modules over PIDs, applications
to linear operators: Jordon and rational canonical forms.
===Matrices===
===Free modules===
Definition: An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis. A subset <math>A</math> of an <math>R</math>-module <math>M</math> is a basis of <math>M</math> if <math>A</math> is linearly independent and generates <math>M</math>.

====Examples:====
# <math>R=R\{1_R\}</math> is a free <math>R</math>-module
# <math>R^2=R\oplus R</math> has basis <math>\{(1_R,0_R),(0_R,1_R)\}</math>
# <math>R[x]</math> is a free <math>R</math>-module with (infinite) basis <math>\{1,x,x^2,...,x^i,...\}</math>
# <math>R[x,y]</math> has basis <math>\{x^n,y^m\ |\ n,m\geq 0 \}</math> as free <math>R</math>-module
# <math>R[x,y]</math> has basis <math>\{1,y,y^2, ...,y^i,...\}</math> as free <math>R[x]</math>-module

All free <math>R</math>-modules <math>M </math> have some basis <math>B=\{b_1,...,b_n\} </math> so have rank <math>n </math>, and can be written as <math>R^n\cong M </math>. Additionally every element <math>m\in M </math> can be uniquely written <math> m=\sum_{i=1}^n r_ib_i</math>.

===Bases===
over arbitrary commutative rings,
===Structure Theorem===
for finitely generated modules over PIDs, applications
to linear operators:
===Jordon and rational canonical forms===

==Basic Linear Algebra:==
vector spaces, bases, dimension, bases for infinite dimensional spaces
(Zorn’s Lemma), linear transformations, eigenvectors, characteristic polynomial, diagonalization, Cayley-Hamilton Theorem.

==Field Theory:==
Definition and examples, algebraic and transcendental extensions, degree of
a finite extension, multiplicativity of degrees, adjunction of roots, finite fields.

==Basic Galois Theory for Finite Separable Extensions:==
Definitions of Galois group and
Galois field extensions, the main theorem of Galois theory, primitive elements, Kummer
extensions, cyclotomic extensions, quintic polynomials.

===random pages to be categorized===

[[Cayley-Hamilton Theorem]]

Algebra Qualifying Syllabus

2023-04-26T16:17:22Z

Kfagerstrom:

Fall 2022 Classes

2023-04-26T16:13:41Z

Kfagerstrom:

==Course list ==
[[Algebra Qualifying Syllabus]]

[[817 - Algebra]]

[[818 - Algebra]]

[[871 - Topology]]

[[825 - Analysis]]

Category:Group Theory

2023-04-26T16:06:57Z

Kfagerstrom: Created page with "A Group is set with two binary operations"

A Group is set with two binary operations

Jordan Canonical Form

2023-04-26T16:03:53Z

Kfagerstrom:

$$ F $$ field, $$V$$ finite dim vector space, $$V\xrightarrow{t} V$$ linear transformation. Assume that the characteristic polynomial of $$t$$ completely factor into linear forms (so things that look like $$(x-r)^l$$). Then there exists a basis $$B$$ such that $$[t]_B^B=\begin{bmatrix} j_{e_1}(x_1)&\\ \; \ddots & \hspace{-12pt} j_{e_n}(x_n) \end{bmatrix}=j(t)$$.

Where each Jordan block $$J_n(r)$$ is an $$n\times n$$ matrix with entries $$a_{ij}=\begin{cases} r, & i=j \\ 1, & j=i+1 \\ 0 & \text{else} \end{cases}$$

Each $$r_i\in F$$ is a root of $$c_t$$, the characteristic polynomial, and $$e_i\geq 1$$. The polynomials $$(x-r_i)^{e_i}$$ are the elementary divisors of $$t$$, and this jordan canonical form for $$t$$ is unique up to order of the blocks

Jordan Canonical Form

2023-04-14T16:08:12Z

Kfagerstrom:

$$ F $$ field, $$V$$ finite dim vector space, $$V\xrightarrow{t} V$$ linear transforamtion. Assume that the charateristic polynomial of $t$ compleately factor into linear forms (so things that look like $(x-r)^l$). Then there exists a basis $$B$$ such that $$[t]_B^B=\begin{bmatrix} j_{e_1}(x_1)&\\ \; \ddots & \hspace{-12pt} j_{e_n}(x_n) \end{bmatrix}=j(t)$$.

Where each Jordan block $$J_n(r)$$ is an $$n\times n$$ matrix with entries $$a_{ij}=\begin{cases} r, & i=j \\ 1, & j=i+1 \\ 0 & \text{else} \end{cases}$$

Each $$r_i\in F$$ is a root of $$c_t$$, the characteristic polynomial, and $$e_i\geq 1$$. The polynomials $$(x-r_i)^{e_i}$$ are the elementary divisors of $$t$$, and this jordan canonical form for $t$ is unique up to order of the blocks

Jordan Canonical Form

2023-03-26T18:43:22Z

Kfagerstrom:

$F$ field, $V$ finite dim vector space, $V\xrightarrow{t} V$ linear transforamtion. Assume that the charateristic polynomial of $t$ compleately factor into linear forms (so things that look like $(x-r)^l$). Then there exists a basis $B$ such that $[t]_B^B=\begin{bmatrix} j_{e_1}(x_1)&\\ \; \ddots & \hspace{-12pt} j_{e_n}(x_n) \end{bmatrix}=j(t)$.

Where each Jordan block $J_n(r) is an $n\times n$ matrix with entries $a_{ij}=\begin{cases} r, & i=j \\ 1, & j=i+1 \\ 0 & \text{else} \end{cases}

Each $r_i\in F$ is a root of $c_t$, the characteristic polynomial, and $e_i\geq 1$. The polynomials $(x-r_i)^{e_i}$ are the elementary divisors of $t$, and this jordan canonical form for $t$ is unique up to order of the blocks

Jordan Canonical Form

2023-03-25T22:26:55Z

Kfagerstrom: Created page with "$F$ field, $V$ finite dim vector space, $V\xrightarrow{t} V$ linear transforamtion. Assume that the charateristic polynomial of $t$ compleately factor into linear forms (so th..."

$F$ field, $V$ finite dim vector space, $V\xrightarrow{t} V$ linear transforamtion. Assume that the charateristic polynomial of $t$ compleately factor into linear forms (so things that look like $(x-r)^l$). Then there exists a basis $B$ such that $[t]_B^B=\begin{bmatrix} j_{e_1}(x_1)&\\ \; \ddots & \hspace{-12pt} j_{e_n}(x_n) \end{bmatrix}=j(t)$.

Each $r_i\in F$ is a root of $c_t$, the characteristic polynomial, and $e_i\geq 1$. The polynomials $(x-r_i)^{e_i}$ are the elementary divisors of $t$, and this jordan canonical form for $t$ is unique up to order of the blocks

Modules

2023-03-25T22:10:38Z

Kfagerstrom:

Definition and examples: matrices, free modules, and bases over arbitrary commutative rings, Structure Theorem for finitely generated modules over PIDs, applications
to linear operators: Jordon and rational canonical forms.

[[Free Modules]]

[[Finitely Generated Modules over PIDs]]

[[Jordan Canonical Form]]

[[Rational Canonical Forms]]

Cayley-Hamilton Theorem

2023-03-10T18:20:44Z

Kfagerstrom:

==Theorem==
Let <math>F</math> be a field, and let <math>V</math> be a finite dimensional <math>F</math>-vector space. If <math>t : V → V</math> is a linear transformation, then <math>m_t\mid c_t
</math>, and hence <math>c_t(t) = 0</math>.
Similarly, for any matrix <math>A ∈ M_n(F)</math> over a field <math>F</math> we have <math>m_A|c_A</math> and <math>c_A(A) = 0</math>.

==Proof==
Let <math> A = [t]_B^B</math> for some basis <math>B</math> of <math>V</math>. Note that the statements about <math>A</math> and <math>t</math> are equivalent, since by definition <math>c_A = c_t</math> , while <math>m_A = m_t</math> we have <math>f(A) = 0</math> if and only if <math>f(t) = 0</math>. So write <math>m = m_A = m_t</math> and <math>c = c_A = c_t</math>.
By Lemma 4.21, <math>m = g_k</math> and <math>c = g_1 · · · g_k</math>, so <math>m | c</math>. By definition, we <math>m(A) = 0</math>. Since <math>m|c</math>, we conclude that <math>c(A) = 0</math>.
<math> </math>

===Lemma 4.21===

Let <math>F </math> be a field, let <math> V</math> be a finite dimensional <math> F</math>-vector space, and <math> _V → V</math> be a linear transformation with invariant factors <math>g_1| · · · |g_k. </math> Then <math> c_t = g_1 · · · g_k</math> and mt = gk.

===Proof===
The product of the elements on the diagonal of the Smith Normal Form of <math> xI_n − A</math>
is the determinant of <math>xI_n − A</math>. Thus the product of the invariant factors <math>g_1 · · · g_k</math> of <math>V_t</math> is the characteristic polynomial <math>c_t</math> of <math>t</math>. Notice here that we chose our invariant factors <math>g_1, . . . , g_k</math> to be monic, so that <math>g_1 · · · g_k</math> is monic, and thus actually equal to <math>c_t</math> (not just up to multiplication by a unit). By Problem Set 5, <math>\text{ann}_{F[x]}(V_t) = (g_k)</math>, and since <math>g_k</math> is monic we deduce that <math>m_t = g_k</math>.

Cayley-Hamilton Theorem

2023-03-10T18:12:31Z

Kfagerstrom:

Cayley-Hamilton Theorem

2023-03-10T18:03:55Z

Kfagerstrom:

Let <math>F</math> be a field, and let <math>V</math> be a finite dimensional <math>F</math>-vector space. If <math>t : V → V</math> is a linear transformation, then <math>m_t\mid c_t
</math>, and hence <math>c_t(t) = 0</math>.
Similarly, for any matrix <math>A ∈ M_n(F)</math> over a field <math>F</math> we have <math>m_A|c_A</math> and <math>c_A(A) = 0</math>.

<math> </math>

Cayley-Hamilton Theorem

2023-03-10T18:03:44Z

Kfagerstrom: Created page with "Let <math>F</math> be a field, and let <math>V</math> be a finite dimensional <math>F</math>-vector space. If <math>t : V → V</math> is a linear transformation, then <math>m..."

Let <math>F</math> be a field, and let <math>V</math> be a finite dimensional <math>F</math>-vector space. If <math>t : V → V</math> is a linear transformation, then <math>m_t\mid c_t
</math>, and hence <math>c_t(t) = 0</math>.
Similarly, for any matrix <math>A ∈ M_n(F)</math> over a field <math>F we have <math>m_A|c_A</math> and <math>c_A(A) = 0</math>.

<math> </math>

Algebra Qualifying Syllabus

2023-03-10T18:01:17Z

Kfagerstrom:

'''Group Theory:''' Groups, subgroups, homomorphisms, cosets, quotients, isomorphism theorems, direct and semidirect products, solvable groups, structure of cyclic, symmetric and
alternating groups; free groups, structure theorem for finite abelian groups.

'''Group Actions:''' Groups acting on sets, cosets, and themselves; orbits and stabilizers, permutation representations, Cayley’s Theorem, the class equation, inner automorphisms and
automorphism groups, p-subgroups and the Sylow Theorems.

'''Ring Theory:''' Definition and examples, homomorphisms, ideals, quotients, integral domains
and their fields of fractions, maximal and prime ideals.
Factorization in Commutative Rings: Euclidean domains, Unique Factorization Domains,
Principal Ideal Domains, Gauss’s Lemma, polynomial factorization, Eisenstein’s criterion,
Gaussian integers.

'''[[Modules]]:''' Definition and examples: matrices, free modules, and bases over arbitrary commutative rings, Structure Theorem for finitely generated modules over PIDs, applications
to linear operators: Jordon and rational canonical forms.

'''Basic Linear Algebra:''' vector spaces, bases, dimension, bases for infinite dimensional spaces
(Zorn’s Lemma), linear transformations, eigenvectors, characteristic polynomial, diagonalization, Cayley-Hamilton Theorem.

'''Field Theory:''' Definition and examples, algebraic and transcendental extensions, degree of
a finite extension, multiplicativity of degrees, adjunction of roots, finite fields.

'''Basic Galois Theory for Finite Separable Extensions:''' Definitions of Galois group and
Galois field extensions, the main theorem of Galois theory, primitive elements, Kummer
extensions, cyclotomic extensions, quintic polynomials.

===random pages to be categorized===

[[Cayley-Hamilton Theorem]]

UMP for Free Modules

2023-03-09T21:56:24Z

Kfagerstrom:

==UMP for Free Modules ==

Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\to N</math> a function. Then there exists a unique <math>R</math>-module homomorphism <math>h:M\to N</math> such that <math>h(b)=j(b)\ \forall b\in B</math>.

==Proof==

As <math>m=\sum_{i=1}^n r_ib_i,\ b_i\in B</math> unique, imples <math>h(m):=\sum_{i=1}^n r_ij(b_i)</math> is well defined

We have two things to prove: existence and uniqueness.

''Existence:'' By Lemma 1.59, any <math>0 \ne m ∈ M</math> can be written uniquely as
<math>m = r_1b_1 + \cdots + r_nb_n</math>
with <math>b_i\in B </math> distinct and <math>0 \ne r_i \in R</math>. Define <math>h: M \to N</math> by <math> \begin{cases}
h(r_1b_1 + \cdots + r_nb_n) = r_1j(b_1) + · · · + r_nj(b_n) & \text{if} r_1b_1 + · · · + r_nb_n \ne 0 \\
h(0_M) = 0_N
\end{cases} </math>

One can check that this satisfies the conditions to be an R-module homomorphism (exercise!).

''Uniqueness:'' Let <math>h : M → N</math> be an <math>R</math>-module homomorphism such that <math>h(b_i) = j(b_i)</math>.
Then in particular <math>h: (M, +) → (N, +)</math> is a group homomorphism and therefore <math>h(0_M) = 0_N</math>
by properties of group homomorphisms. Furthermore, if <math>m = r_1b_1 + · · · + r_nb_n </math> then
<math> h(m) = h(r_1b_1 + · · · + r_nb_n) = r_1h(b_1) + · · · + r_nh(b_n) = r_1j(b_1) + · · · + r_nj(b_n)
</math> by the definition of homomorphism, and because <math>h(b_i) = j(b_i)</math>.

[[Category:Modules]] [[Category: Free Modules]] [[Category: Proofs]]

Uniqueness of Rank of Free Modules Over Commutative Rings

2023-03-09T21:50:52Z

Kfagerstrom:

==Theorem==
Let <math>R</math> be a commutative ring with <math>1 \ne 0</math> and let M be a free R-module. If A and B are both bases for M, then A and B
have the same cardinality, meaning that there exists a bijection <math>A → B</math>.

[[Category: Free Modules]] [[Category: Theorems]]

Uniqueness of Rank of Free Modules Over Commutative Rings

2023-03-09T21:41:08Z

Kfagerstrom: Created page with "Let <math>R</math> be a commutative ring with <math>1 \ne 0</math> and let M be a free R-module. If A and B are both bases for M, then A and B have the same cardinality, meani..."

Let <math>R</math> be a commutative ring with <math>1 \ne 0</math> and let M be a free R-module. If A and B are both bases for M, then A and B
have the same cardinality, meaning that there exists a bijection <math>A → B</math>.

Free Modules

2023-03-09T21:38:30Z

Kfagerstrom:

===Definition:===
An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis. A subset <math>A</math> of an <math>R</math>-module <math>M</math> is a basis of <math>M</math> if <math>A</math> is linearly independent and generates <math>M</math>.

===Examples:===
# <math>R=R\{1_R\}</math> is a free <math>R</math>-module
# <math>R^2=R\oplus R</math> has basis <math>\{(1_R,0_R),(0_R,1_R)\}</math>
# <math>R[x]</math> is a free <math>R</math>-module with (infinite) basis <math>\{1,x,x^2,...,x^i,...\}</math>
# <math>R[x,y]</math> has basis <math>\{x^n,y^m\ |\ n,m\geq 0 \}</math> as free <math>R</math>-module
# <math>R[x,y]</math> has basis <math>\{1,y,y^2, ...,y^i,...\}</math> as free <math>R[x]</math>-module

All free <math>R</math>-modules <math>M </math> have some basis <math>B=\{b_1,...,b_n\} </math> so have rank <math>n </math>, and can be written as <math>R^n\cong M </math>. Additionally every element <math>m\in M </math> can be uniquely written <math> m=\sum_{i=1}^n r_ib_i</math>.

===Theorems:===
[[UMP for Free Modules| Theorem:]] (UMP for free modules) Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\to N</math> a function. Then there exists a unique <math>R</math>-module homomorphism <math>h:M\to N</math> such that <math>h(b)=j(b)\ \forall b\in B</math>.

Corollary: If A and B are sets of the same cardinality, and fix a bijection j : A → B.
If M and N are free R-modules with bases A and B respectively, then there is an isomorphism
of R-modules M ∼= N.

[[Uniqueness of Rank of Free Modules Over Commutative Rings|Theorem:]] (Uniqueness of rank over commutative rings) Let R be a commutative ring
with 1 6= 0 and let M be a free R-module. If A and B are both bases for M, then A and B
have the same cardinality, meaning that there exists a bijection A → B.

Theorem: Let R be a commutative ring with 1 6= 0. Let V and W be finitely generated
free R-modules of ranks n and m respectively. Fixing ordered bases B for V and C for W
gives an isomorphism of R-modules

<math>\text{Hom}_R(V,W)\cong \text{M}_{m\times n}(R) \quad f \mapsto [f]_B^C </math>

If V = W, so that in particular m = n, and B = C, then the above map is an R-algebra
isomorphism EndR(V ) ∼= Mn(R).

Lemma. Given any ring <math>R</math> with <math>1 \ne 0</math>, any direct sum of copies of R is always a free
R-module.

Theorem. Every R-module is a quotient of a free <math>R</math>-module

[[Category:Modules]] [[Category:Free Modules]]

Free Modules

2023-03-09T21:33:43Z

Kfagerstrom:

===Definition:===
An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis. A subset <math>A</math> of an <math>R</math>-module <math>M</math> is a basis of <math>M</math> if <math>A</math> is linearly independent and generates <math>M</math>.

===Examples:===
# <math>R=R\{1_R\}</math> is a free <math>R</math>-module
# <math>R^2=R\oplus R</math> has basis <math>\{(1_R,0_R),(0_R,1_R)\}</math>
# <math>R[x]</math> is a free <math>R</math>-module with (infinite) basis <math>\{1,x,x^2,...,x^i,...\}</math>
# <math>R[x,y]</math> has basis <math>\{x^n,y^m\ |\ n,m\geq 0 \}</math> as free <math>R</math>-module
# <math>R[x,y]</math> has basis <math>\{1,y,y^2, ...,y^i,...\}</math> as free <math>R[x]</math>-module

All free <math>R</math>-modules <math>M </math> have some basis <math>B=\{b_1,...,b_n\} </math> so have rank <math>n </math>, and can be written as <math>R^n\cong M </math>. Additionally every element <math>m\in M </math> can be uniquely written <math> m=\sum_{i=1}^n r_ib_i</math>.

===Theorems:===
[[UMP for Free Modules| Theorem:]] (UMP for free modules) Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\to N</math> a function. Then there exists a unique <math>R</math>-module homomorphism <math>h:M\to N</math> such that <math>h(b)=j(b)\ \forall b\in B</math>.

Corollary: If A and B are sets of the same cardinality, and fix a bijection j : A → B.
If M and N are free R-modules with bases A and B respectively, then there is an isomorphism
of R-modules M ∼= N.

Theorem:(Uniqueness of rank over commutative rings). Let R be a commutative ring
with 1 6= 0 and let M be a free R-module. If A and B are both bases for M, then A and B
have the same cardinality, meaning that there exists a bijection A → B.

Theorem: Let R be a commutative ring with 1 6= 0. Let V and W be finitely generated
free R-modules of ranks n and m respectively. Fixing ordered bases B for V and C for W
gives an isomorphism of R-modules

<math>\text{Hom}_R(V,W)\cong \text{M}_{m\times n}(R) \quad f \mapsto [f]_B^C </math>

If V = W, so that in particular m = n, and B = C, then the above map is an R-algebra
isomorphism EndR(V ) ∼= Mn(R).

Lemma. Given any ring <math>R</math> with <math>1 \ne 0</math>, any direct sum of copies of R is always a free
R-module.

Theorem. Every R-module is a quotient of a free $R$-module

[[Category:Modules]] [[Category:Free Modules]]

Category:Free Modules

2023-03-08T22:16:40Z

Kfagerstrom: Created page with "An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis."

An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis.

Category:Modules

2023-03-08T22:08:11Z

Kfagerstrom: Created page with "Let <math>R</math> be a ring with <math>1\neq0</math>. A ''Left'' <math>R</math>-mod is an abelian group <math>(M,+)</math> together with an action of <math>R</math> on <mat..."

Let <math>R</math> be a ring with <math>1\neq0</math>.

A ''Left'' <math>R</math>-mod is an abelian group <math>(M,+)</math> together with an action of <math>R</math> on <math>M</math>, <math>R\times M \to M</math>, such that for all <math>r,s\in R</math>, <math>m,n\in M</math>:

* <math>(r+s)m=rm+sm</math>
* <math>(rs)m=r(sm)</math>
* <math>r(m+n)=rm+rn</math>
* <math>1\cdot m=m</math>

UMP for Free Modules

2023-03-08T22:07:56Z

Kfagerstrom:

==UMP for Free Modules ==

Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\to N</math> a function. Then there exists a unique <math>R</math>-module homomorphism <math>h:M\to N</math> such that <math>h(b)=j(b)\ \forall b\in B</math>.

As <math>m=\sum_{i=1}^n r_ib_i,\ b_i\in B</math> unique, imples <math>h(m):=\sum_{i=1}^n r_ij(b_i)</math> is well defined

We have two things to prove: existence and uniqueness.

''Existence:'' By Lemma 1.59, any <math>0 \ne m ∈ M</math> can be written uniquely as
<math>m = r_1b_1 + \cdots + r_nb_n</math>
with <math>b_i\in B </math> distinct and <math>0 \ne r_i \in R</math>. Define <math>h: M \to N</math> by <math> \begin{cases}
h(r_1b_1 + \cdots + r_nb_n) = r_1j(b_1) + · · · + r_nj(b_n) & \text{if} r_1b_1 + · · · + r_nb_n \ne 0 \\
h(0_M) = 0_N
\end{cases} </math>

One can check that this satisfies the conditions to be an R-module homomorphism (exercise!).

''Uniqueness:'' Let <math>h : M → N</math> be an <math>R</math>-module homomorphism such that <math>h(b_i) = j(b_i)</math>.
Then in particular <math>h: (M, +) → (N, +)</math> is a group homomorphism and therefore <math>h(0_M) = 0_N</math>
by properties of group homomorphisms. Furthermore, if <math>m = r_1b_1 + · · · + r_nb_n </math> then
<math> h(m) = h(r_1b_1 + · · · + r_nb_n) = r_1h(b_1) + · · · + r_nh(b_n) = r_1j(b_1) + · · · + r_nj(b_n)
</math> by the definition of homomorphism, and because <math>h(b_i) = j(b_i)</math>.

[[Category:Modules]]

UMP for Free Modules

2023-03-08T22:04:21Z

Kfagerstrom: /* UMP for Free Modules */

==UMP for Free Modules ==

Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\to N</math> a function. Then there exists a unique <math>R</math>-module homomorphism <math>h:M\to N</math> such that <math>h(b)=j(b)\ \forall b\in B</math>.

As <math>m=\sum_{i=1}^n r_ib_i,\ b_i\in B</math> unique, imples <math>h(m):=\sum_{i=1}^n r_ij(b_i)</math> is well defined

We have two things to prove: existence and uniqueness.

''Existence:'' By Lemma 1.59, any <math>0 \ne m ∈ M</math> can be written uniquely as
<math>m = r_1b_1 + \cdots + r_nb_n</math>
with <math>b_i\in B </math> distinct and <math>0 \ne r_i \in R</math>. Define <math>h: M \to N</math> by <math> \begin{cases}
h(r_1b_1 + \cdots + r_nb_n) = r_1j(b_1) + · · · + r_nj(b_n) & \text{if} r_1b_1 + · · · + r_nb_n \ne 0 \\
h(0_M) = 0_N
\end{cases} </math>

One can check that this satisfies the conditions to be an R-module homomorphism (exercise!).

''Uniqueness:'' Let <math>h : M → N</math> be an <math>R</math>-module homomorphism such that <math>h(b_i) = j(b_i)</math>.
Then in particular <math>h: (M, +) → (N, +)</math> is a group homomorphism and therefore <math>h(0_M) = 0_N</math>
by properties of group homomorphisms. Furthermore, if <math>m = r_1b_1 + · · · + r_nb_n </math> then
<math> h(m) = h(r_1b_1 + · · · + r_nb_n) = r_1h(b_1) + · · · + r_nh(b_n) = r_1j(b_1) + · · · + r_nj(b_n)
</math> by the definition of homomorphism, and because <math>h(b_i) = j(b_i)</math>.

[[Category|Modules]]

UMP for Free Modules

2023-03-08T18:47:29Z

Kfagerstrom:

==UMP for Free Modules ==

Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\to N</math> a function. Then there exists a unique <math>R</math>-module homomorphism <math>h:M\to N</math> such that <math>h(b)=j(b)\ \forall b\in B</math>.

As <math>m=\sum_{i=1}^n r_ib_i,\ b_i\in B</math> unique, imples <math>h(m):=\sum_{i=1}^n r_ij(b_i)</math> is well defined

We have two things to prove: existence and uniqueness.

''Existence:'' By Lemma 1.59, any <math>0 \ne m ∈ M</math> can be written uniquely as
<math>m = r_1b_1 + \cdots + r_nb_n</math>
with <math>b_i\in B </math> distinct and <math>0 \ne r_i \in R</math>. Define <math>h: M \to N</math> by <math> \begin{cases}
h(r_1b_1 + \cdots + r_nb_n) = r_1j(b_1) + · · · + r_nj(b_n) & \text{if} r_1b_1 + · · · + r_nb_n \ne 0 \\
h(0_M) = 0_N
\end{cases} </math>

One can check that this satisfies the conditions to be an R-module homomorphism (exercise!).

''Uniqueness:'' Let <math>h : M → N</math> be an <math>R</math>-module homomorphism such that <math>h(b_i) = j(b_i)</math>.
Then in particular <math>h: (M, +) → (N, +)</math> is a group homomorphism and therefore <math>h(0_M) = 0_N</math>
by properties of group homomorphisms. Furthermore, if <math>m = r_1b_1 + · · · + r_nb_n </math> then
<math> h(m) = h(r_1b_1 + · · · + r_nb_n) = r_1h(b_1) + · · · + r_nh(b_n) = r_1j(b_1) + · · · + r_nj(b_n)
</math> by the definition of homomorphism, and because <math>h(b_i) = j(b_i)</math>.

UMP for Free Modules

2023-03-08T17:23:30Z

Kfagerstrom:

UMP for free modules

Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\to N</math> a function. Then there exists a unique <math>R</math>-module homomorphism <math>h:M\to N</math> such that <math>h(b)=j(b)\ \forall b\in B</math>.

As <math>m=\sum_{i=1}^n r_ib_i,\ b_i\in B</math> unique, imples <math>h(m):=\sum_{i=1}^n r_ij(b_i)</math> is well defined

We have two things to prove: existence and uniqueness.
Existence: By Lemma 1.59, any <math>0 \ne m ∈ M</math> can be written uniquely as
<math>m = r_1b_1 + \cdots + r_nb_n</math>
with <math>b_i\in B </math> distinct and 0 6= ri ∈ R. Define h: M → N by
(
h(r1b1 + · · · + rnbn) = r1j(b1) + · · · + rnj(bn) if r1b1 + · · · + rnbn 6= 0
h(0M) = 0N
One can check that this satisfies the conditions to be an R-module homomorphism (exercise!).
Uniqueness: Let h : M → N be an R-module homomorphism such that h(bi) = j(bi).
Then in particular h: (M, +) → (N, +) is a group homomorphism and therefore h(0m) = 0N
by properties of group homomorphisms. Furthermore, if m = r1b1 + · · · + rnbn then
h(m) = h(r1b1 + · · · + rnbn) = r1h(b1) + · · · + rnh(bn) = r1j(b1) + · · · + rnj(bn)
by the definition of homomorphism, and because h(bi) = j(bi).

Free Modules

2023-03-08T17:18:20Z

Kfagerstrom:

===Definition:===
An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis. A subset <math>A</math> of an <math>R</math>-module <math>M</math> is a basis of <math>M</math> if <math>A</math> is linearly independent and generates <math>M</math>.

===Examples:===
# <math>R=R\{1_R\}</math> is a free <math>R</math>-module
# <math>R^2=R\oplus R</math> has basis <math>\{(1_R,0_R),(0_R,1_R)\}</math>
# <math>R[x]</math> is a free <math>R</math>-module with (infinite) basis <math>\{1,x,x^2,...,x^i,...\}</math>
# <math>R[x,y]</math> has basis <math>\{x^n,y^m\ |\ n,m\geq 0 \}</math> as free <math>R</math>-module
# <math>R[x,y]</math> has basis <math>\{1,y,y^2, ...,y^i,...\}</math> as free <math>R[x]</math>-module

All free <math>R</math>-modules <math>M </math> have some basis <math>B=\{b_1,...,b_n\} </math> so have rank <math>n </math>, and can be written as <math>R^n\cong M </math>. Additionally every element <math>m\in M </math> can be uniquely written <math> m=\sum_{i=1}^n r_ib_i</math>.

===Theorems:===
[[UMP for Free Modules| Theorem:]] (UMP for free modules) Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\to N</math> a function. Then there exists a unique <math>R</math>-module homomorphism <math>h:M\to N</math> such that <math>h(b)=j(b)\ \forall b\in B</math>.

Corollary: If A and B are sets of the same cardinality, and fix a bijection j : A → B.
If M and N are free R-modules with bases A and B respectively, then there is an isomorphism
of R-modules M ∼= N.

Theorem:(Uniqueness of rank over commutative rings). Let R be a commutative ring
with 1 6= 0 and let M be a free R-module. If A and B are both bases for M, then A and B
have the same cardinality, meaning that there exists a bijection A → B.

Theorem: Let R be a commutative ring with 1 6= 0. Let V and W be finitely generated
free R-modules of ranks n and m respectively. Fixing ordered bases B for V and C for W
gives an isomorphism of R-modules

<math>\text{Hom}_R(V,W)\cong \text{M}_{m\times n}(R) \quad f \mapsto [f]_B^C </math>

If V = W, so that in particular m = n, and B = C, then the above map is an R-algebra
isomorphism EndR(V ) ∼= Mn(R).

Lemma. Given any ring <math>R</math> with <math>1 \ne 0</math>, any direct sum of copies of R is always a free
R-module.

Theorem. Every R-module is a quotient of a free R-module

[[Category:Modules]] [[Category:Free Modules]]

UMP for Free Modules

2023-03-08T17:17:48Z

Kfagerstrom: Created page with "UMP for free modules Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\..."

Free Modules

2023-03-08T17:15:30Z

Kfagerstrom: /* Theorems: */

===Definition:===
An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis. A subset <math>A</math> of an <math>R</math>-module <math>M</math> is a basis of <math>M</math> if <math>A</math> is linearly independent and generates <math>M</math>.

===Examples:===
# <math>R=R\{1_R\}</math> is a free <math>R</math>-module
# <math>R^2=R\oplus R</math> has basis <math>\{(1_R,0_R),(0_R,1_R)\}</math>
# <math>R[x]</math> is a free <math>R</math>-module with (infinite) basis <math>\{1,x,x^2,...,x^i,...\}</math>
# <math>R[x,y]</math> has basis <math>\{x^n,y^m\ |\ n,m\geq 0 \}</math> as free <math>R</math>-module
# <math>R[x,y]</math> has basis <math>\{1,y,y^2, ...,y^i,...\}</math> as free <math>R[x]</math>-module

All free <math>R</math>-modules <math>M </math> have some basis <math>B=\{b_1,...,b_n\} </math> so have rank <math>n </math>, and can be written as <math>R^n\cong M </math>. Additionally every element <math>m\in M </math> can be uniquely written <math> m=\sum_{i=1}^n r_ib_i</math>.

===Theorems:===
[[UMP for Free Modules| Theorem:]] (UMP for free modules) Let R be a ring, M be a free R-module with basis B, N be any R-module,
and let j : B → N be any function. Then there is a unique R-module homomorphism
h : M → N such that h(b) = j(b) for all b ∈ B.

Corollary: If A and B are sets of the same cardinality, and fix a bijection j : A → B.
If M and N are free R-modules with bases A and B respectively, then there is an isomorphism
of R-modules M ∼= N.

Theorem:(Uniqueness of rank over commutative rings). Let R be a commutative ring
with 1 6= 0 and let M be a free R-module. If A and B are both bases for M, then A and B
have the same cardinality, meaning that there exists a bijection A → B.

Theorem: Let R be a commutative ring with 1 6= 0. Let V and W be finitely generated
free R-modules of ranks n and m respectively. Fixing ordered bases B for V and C for W
gives an isomorphism of R-modules

<math>\text{Hom}_R(V,W)\cong \text{M}_{m\times n}(R) \quad f \mapsto [f]_B^C </math>

If V = W, so that in particular m = n, and B = C, then the above map is an R-algebra
isomorphism EndR(V ) ∼= Mn(R).

Lemma. Given any ring <math>R</math> with <math>1 \ne 0</math>, any direct sum of copies of R is always a free
R-module.

Theorem. Every R-module is a quotient of a free R-module

[[Category:Modules]] [[Category:Free Modules]]

Free Modules

2023-03-08T17:12:46Z

Kfagerstrom: /* Theorems: */

===Definition:===
An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis. A subset <math>A</math> of an <math>R</math>-module <math>M</math> is a basis of <math>M</math> if <math>A</math> is linearly independent and generates <math>M</math>.

===Examples:===
# <math>R=R\{1_R\}</math> is a free <math>R</math>-module
# <math>R^2=R\oplus R</math> has basis <math>\{(1_R,0_R),(0_R,1_R)\}</math>
# <math>R[x]</math> is a free <math>R</math>-module with (infinite) basis <math>\{1,x,x^2,...,x^i,...\}</math>
# <math>R[x,y]</math> has basis <math>\{x^n,y^m\ |\ n,m\geq 0 \}</math> as free <math>R</math>-module
# <math>R[x,y]</math> has basis <math>\{1,y,y^2, ...,y^i,...\}</math> as free <math>R[x]</math>-module

All free <math>R</math>-modules <math>M </math> have some basis <math>B=\{b_1,...,b_n\} </math> so have rank <math>n </math>, and can be written as <math>R^n\cong M </math>. Additionally every element <math>m\in M </math> can be uniquely written <math> m=\sum_{i=1}^n r_ib_i</math>.

===Theorems:===
Theorem: (UMP for free modules) Let R be a ring, M be a free R-module with basis B, N be any R-module,
and let j : B → N be any function. Then there is a unique R-module homomorphism
h : M → N such that h(b) = j(b) for all b ∈ B.

Corollary: If A and B are sets of the same cardinality, and fix a bijection j : A → B.
If M and N are free R-modules with bases A and B respectively, then there is an isomorphism
of R-modules M ∼= N.

Theorem:(Uniqueness of rank over commutative rings). Let R be a commutative ring
with 1 6= 0 and let M be a free R-module. If A and B are both bases for M, then A and B
have the same cardinality, meaning that there exists a bijection A → B.

Theorem: Let R be a commutative ring with 1 6= 0. Let V and W be finitely generated
free R-modules of ranks n and m respectively. Fixing ordered bases B for V and C for W
gives an isomorphism of R-modules

<math>\text{Hom}_R(V,W)\cong \text{M}_{m\times n}(R) \quad f \mapsto [f]_B^C </math>

If V = W, so that in particular m = n, and B = C, then the above map is an R-algebra
isomorphism EndR(V ) ∼= Mn(R).

Lemma. Given any ring <math>R</math> with <math>1 \ne 0</math>, any direct sum of copies of R is always a free
R-module.

Theorem. Every R-module is a quotient of a free R-module

[[Category:Modules]] [[Category:Free Modules]]

Free Modules

2023-03-08T16:56:11Z

Kfagerstrom:

Free Modules

2023-03-08T16:54:57Z

Kfagerstrom: