Fractional ideals & dual groups & dual spaces

references:

1. Deitmar A. A first course in harmonic analysis[M]. 2005.
2. Ideal quotient | encyclopedia article by TheFreeDictionary
3. Fractional ideal | encyclopedia article by TheFreeDictionary
4. Pontryagin duality | encyclopedia article by TheFreeDictionary
5. Algebraic number field | encyclopedia article by TheFreeDictionary
6. Duality and Subgroups on JSTOR
7. The Duality of Groups - Zhihu (zhihu.com)
8. Lattice password 1.6 Dual space and dual lattice - Zhihu (zhihu.com)
9. Dual space (1): space, basis and mapping - Zhihu (zhihu.com)
10. Explanation of dual lattice - Zhihu (zhihu.com)
11. Commutative Ring Theory (6) Noether Ring - Zhihu (zhihu.com)
12. Commutative ring theory (8): Dedejin whole ring - Zhihu (zhihu.com)
13. [Abstract Algebra] 18. Direct products and direct sums of modules, free modules and projective modules - Zhihu (zhihu.com)
14. Continuous mapping and homeomorphism - Zhihu (zhihu.com)
15. Metric space (metric space)-CSDN Blog
16. Algebraic Structure: Module-CSDN Blog
17. Algebraic Geometry: Irreducible Varieties-Prime Ideals-CSDN Blog

fractional ideal

Let $R$ is an integral ring, and itsfieldof fractions is denoted byqf ( R ) : = { ab ∣ a , b ∈ R } qf(R):=\{\dfrac{a}{b}| a,b \in R\}qf(R):={ ba​∣a,b∈R}

Fractional ideal : it is qf ( R ) qf(R)AnRR$of\; q$$f$$($$R$$)$$R$ -Submodule$I$ , there exists a ring element$r\in R$ , such that$rI\subseteq R$ is an (integral) ideal.

Principal fraction ideal (principal): as $R$ - submodule, consisting of a single element of the fractional domain$x\in qf\left(R\right)$ is generated, in the form$I=Rx$

Attention set $I\subseteq qf\left(R\right)$ is not an ideal of the fractional domain (only trivial ideals). Take$r=e$ is a unit, and all (integral) ideals can be regarded as special fractional ideals. If$x\in R$ use$I=Rx$ , then it is an (integer) principal ideal. RR_The fractional ideal contained in$R\; is\; the\; (integral)\; ideal.$

Similar to the ideal sum, intersection and product, define the integral ring $R$ fraction ideal$I,J$ 的运算，
I + J : = { a + b ∣ a ∈ I , b ∈ J } I J : = { ∏ i = 1 n a i b i ∣ a i ∈ I , b i ∈ J , n ∈ Z + } I ∩ J : = { a ∣ a ∈ I , a ∈ J } \begin{aligned} I+J &:= \{a+b| a \in I,b \in J\}\\ IJ &:= \{\prod_{i=1}^na_ib_i| a_i \in I,b_i \in J,n \in \mathbb Z^+\}\\ I \cap J &:= \{a|a \in I,a \in J\} \end{aligned} I+JIJI∩J​:={ a+b∣a∈I,b∈J}:={ i=1∏n​ai​bi​∣ai​∈I,bi​∈J,n∈Z+}:={ a∣a∈I,a∈J}​

In addition, we define the formal quotient of the fractional ideal , (
J : I ) : = { a ∈ qf ( R ) ∣ a I ⊆ J } (J:I) := \{a \in qf(R )| aI \subseteq J\}(J:I):={ a∈qf(R)∣aI⊆J}

易知 $I(J:I)\subseteq J.$ _ Formal quotient$(J:I)$ is$R$ - submodule, not necessarily a fractional ideal; if$I,J$ are all (integer) ideals, then$(J:I)$ is also ideal. If there exists a fractional ideal$J$ makes$IJ=R$ , we call the fractional ideal$I$ isinvertibleideal. Regardless of the fractional ideal$Whether\; I$ is invertible, we define itsformal inverse,
$$I^{-1}:=(R:I)$$

Zero ideal $The\; formal\; inverse\; of\; O$ is the fraction field$qf\left(R\right)$ (not a fractional ideal), non-zero fractional ideal$I\ne The\; formal\; inverse\; of\; O$ is a fractional ideal (and$I^{-1}I\subseteq R$ is ideal).

For any whole ring $R$ , has the following properties

1. Fractional field $qf\left(R\right)$ Any finitely generated$R$ - submodules, all fractional ideals

2. Good luck $I$ is finitely generated, then any$(J:I)$ is a fractional ideal (so$I^{-1}$ is the inverse ideal)

3. All reversible fractional ideals are finitely generated

4. $I+J$和$Iand\; J$ are still fractional ideals, but$I\cap JNot$ necessarily

5. 令 $x\in qf\left(R\right)$ , principal fraction ideal$I=Rx$ is reversible, and the inverse ideal is$I^{-1}=R{x}^{-1}$

6. Nonzero Fractional Ideals $I$ is reversible if and only if$I\subseteq qf\left(R\right)$ is the projective module

   • Free mold (free): $R$ -$M$ has a set ofbasis(basis,$R$ - linearly independent finite generator)

   • Projective mold (projective): Projective mold $P$ , for any two modules$M,N$ and the full modular homomorphism$p:M\to N$ , for any modular homomorphism$\varphi :P\to N$ , there exists a modular homomorphism$\psi$ σ ∘$p\circ p=\varphi$

   • $R$ -die$P$ is projective if and only if there is a free module$F$ and modular homomorphism$a:F\to P,b:P\to F$ solver$a\circ b=1_P$

   • The projective mode can be written as the direct sum of the free modes

For special rings, they have the following properties

1. If $R$ is a Noether ring, then any fractional ideal is finitely generated.
   • Noetherian : commutative ring $All\; (integral)\; ideals\; of\; R$ are finitely generated
2. If $R$ is a local ring, then any fractional ideal is principal
   • Local ring (Local): exchange ring $R$ has only one great ideal

令 $F\left(R\right)$ is the collection of allnon-zero fractional ideals, let$P\left(R\right)$ is the ideal collection of allreversible fractions, let$Prin\left(R\right)$ is the ideal collection of allprincipal fractions. Regarding the product of fractional ideals:

• $F\left(R\right)$ isa commutative monoid group(closed, combined), unitary$R$
• $P\left(R\right)$ isan Abelian group(closed, associative, unitary, reversible),$Prin\left(R\right)$ is itssubgroup

Dedekind domains: Integral $All\; nonzero\; fractional\; ideals\; of\; R$ are reversible

• Number field KKThe integer ring of$K$${\mathcal{O}}_K$It's a Dede gold ring
• Ring of integer: Finite algebra extension $K=\mathbb{Q}(e_1,\cdots ,e_n)$ all integer elements$z_1{e}_1+\cdots z_n{e}_n,\forall z_i\in$The ring composed of$Z$${\mathcal{O}}_K$
• Gaussian rationals $Q\left(i\right)$ ,Cyclotomic field $Q(g_n)$
• square-free integer $d$ (any$s^2|d$ 都有 $s\in R$ belongs to the unit),Quadratic field $\mathbb{Q}(\sqrt{d})$，

Duality group

Duality of finite groups

Let $G$ isa finite Abelian group,T = { e 2 π ix ∣ x ∈ R } ⊆ C \mathbb T=\{e^{2\pi ix}|x \in \mathbb R\} \subseteq \mathbb CT={ e2πix∣x∈R}⊆C isthe unit torus(continuous multiplicative group), including the unit cyclic group$\forall n,(g_n)\subseteq T$

Character is a group homomorphism $h:G\to T$ ,
$$\chi (ab)=x\left(a\right)x\left(b\right)$$

Let the set $\hat{G}$ is the collection of all features, we define the multiplication (pointwise product) of the group homomorphism$(x,h)\mapsto \chi$，
$$chi\left(a\right)=x\left(a\right)h\left(a\right),\forall a\in G$$

Dual group : algebraic structure $(\hat{G},\cdot )$ becomes an Abelian group, also called Pontryagin dual

For the cyclic group $G=\left(g\right)$ , if$ord(g)=N$ , then the group homomorphism
$$h_l(g)=g_N^l,l=0,1,\cdots ,N-1$$

are all the characteristics, the dual group $\hat{G}=\left(x_1\right)$ is also$Cyclic\; group\; of\; order\; N$ , so there is$G\cong \hat{G}$

Bidual : mapping $A\to \hat{\hat{A}}$义为$a\mapsto d_a$, in which $d_a$δ
$$\begin{array}{rl}{d}_a：\hat{A}& \to \mathbb{T}\\ & \mapsto x(a\end{array}$$

A → A ^ ^ A \ to $A\to \hat{\hat{A}}$is a canonical isomorphism

Compactness & LCA Group

Metric space : set$X$ and its measureddStructure composed of$d$$(X,d)$

For an (infinite) sequence (xn) in a set ⊆ X (x_n) \subseteq X$(x_n)\subseteq X$ , we say itconvergesto$x\in X$，如果
$$\forall ϵ>0,\exists N\in \mathbb{Z},\forall n>N,d(x_n,x)<ϵ$$

Two metric spaces $X,$Mapping f between$Y$$f:X\to Y$ , we say it iscontinuousif it: maps the convergent sequence to the convergent sequence and maintains the limit
$$\underset{n\to \infty }{\lim }f(x_n)=f(\underset{n\to \infty }{\lim }{x}_n)$$

We say that the two measures $d_1,d_2$is equivalent if for all $(x_n)$ , both have:$\left(x_n\right)$ in$d_1$Lower convergence $⟺$ $\left(x_n\right)$ in$d_2$Lower convergence, denoted as $d_1\sim d_2$

Homeomorphism: also called bicontinuous function . Topological space $(X,{\mathcal{O}}_X),(Y,{\mathcal{O}}_Y$Function f on$)$$f:X\to Y$ , satisfies three properties:$f$ is a bijection,$f$ is a continuous function,$f^{-1}$ is a continuous function.

For the metric space $(X,$Automorphism fon$d$$)\; :\; X\; \to \; X\; f:X\; \backslash to\; X$$f:X\to X$ , then define a new metric
$$d^{\prime }(x,y):=d(f(x),f(y))$$

It satisfies $d^{\prime }\sim d$ is equivalent.

Since $f(x)=x/(x+1)$ is$[0,\infty )\to [0,1)$ Monotone homeomorphismon, then: any metric$d$ , there exists an equivalent metric$d^{\prime }(x,y):=\frac{d(x,y)}{d(x,y)+1}$, such that $Im(d^{\prime })=[0,1)$ . According to the equivalence relationship, we combine all measures$d$ is divided into equivalence classes$\left[d\right]$ , we call$(X,[d])$ isa measurable space(metrizable space)

A measurable space $(X,[d])$ iscompactif any sequence$(x_n)$ , both contain a convergedsubsequence(from$x_n$Pick some points infinitely). Example: ${\mathbb{R}}^{A\; bounded\; closed\; subset\; of\; n}$ ; a discrete space is compact if and only if it is finite.

Two weaker compacts:

1. We say $X$ is$\sigma$ - compact, if there exists a compact subset sequence$K_n\subset K_{n+1}$Such that $X={\bigcup }_n{K}_n$, this sequence $(K_n)$ is calledcompact exhaustion. Easy to know$R$ can choose$K_n=[-n,n]$ , this sequence also exists in countable discrete spaces.
2. We say $X$ iscompactif for any point$x\in X$ , there exists$r>0$ makes the closed ball${\bar{B}}_r\left(x\right)$ is compact. Example: space${\mathbb{R}}^n$ , discrete space; counterexample: infinite-dimensional Hilbert space.
3. If the metric space simultaneously $\sigma$ -compact and locally compact, it is called$\sigma$ - locally compact. Example:$\mathbb{R}$，$R/Z$ , a countable discrete space.

Measurable abelian group : Abelian group $A$ , metric equivalence class$\left[d\right]$ , and satisfy$(x,y)\mapsto xy$ and$x\mapsto x^{-1}$ are all continuous functions (map the convergence sequence to the convergence sequence and maintain the limit)

LCA group : measurable $\sigma$ - locally compact Abelian group. Example: countable Abelian group + discrete metric, real number field$R$ , real torus$\mathbb{R}/\mathbb{Z}$

1. A compact exhaustion sequence $(K_n)$ is calledabsorbing, if for any compact subset$K\subset A$ , there exists a certain$n\in N$ makes$K\subset K_n$

   LCA groups always contain an absorbing exhaustion

2. Subset D ⊂ XD \subset X of measurable space$D\subset X$ is calleda densesubset. For any$x\in X$ , there always exists$x_n\in D$ makes the sequence$(x_n)$ converges to$x$

   The LCA group always contains a countably dense subset

Note that continuous functions and dense subsets are relative to the "convergence sequence", and the measure between different points is $d(x,y)$ can be discrete as long as$n\to$When$\infty$$0$ is enough.

Dual of LCA group

ReiAA $A$ is the LCA group,and its characteristicsare defined as the continuous group homomorphism$h:A\to T$ (maps convergence sequences to convergence sequences while maintaining limits). Collection of all features$\hat{A}$，附带上乘法乘法$chi\left(a\right)=\chi \left(a\right)\eta \left(a\right)$ is calledthe dual group.

1. 群 $Z$ function:$x\mapsto e^{2\pi ixy},y\in R/Z$ , then$\hat{\mathbb{Z}}\cong \mathbb{R}/\mathbb{Z}$
2. 群 $The\; function\; of\; R/Z$ :$x\mapsto e^{2\pi ixy},y\in Z$ , then there is$\widehat{\mathbb{R}/\mathbb{Z}}\cong \mathbb{Z}$
3. Group $R$ function:$x\mapsto e^{2\pi ixy},y\in R$ , so$\hat{\mathbb{R}}\cong R$

The dual of the LAC group is still the LAC group . Given AAAn absorbing compact exhaustion of$A$$A={\bigcup }_n{K}_n$, for the elements χ , η \chi,\eta in the dual group$x,\eta$ , define$\hat{A}$ Definition:
$$\begin{gathered} {\hat{d}}_n(x,h)=\underset{x\in K_n}{\sup }|\chi (x)-h(x)| \\ \hat{d}(x,h)=\sum _n\frac{{\hat{d}}_n(x,h}{2^n} \end{gathered}$$

Group isomorphism $\mathbb{R}/\mathbb{Z}\to \hat{\mathbb{Z}}$，$\mathbb{Z}\to \widehat{\mathbb{R}/\mathbb{Z}}$，$\mathbb{R}\to \hat{R}$All are homeomorphic (bicontinuous).

ReiAA $A$ is the LCA group ($\sigma$ -compact + locally compact): if$A$ is compact, then$\hat{A}$ discrete; if$A$ is discrete, then$\hat{A}$ Compact.

Pontryagin Duality : Any LAC group, canonically isomorphic to its bidual, mapping $A\to \hat{\hat{A}}$Let
$$a\mapsto d_a,d_a\left(x\right)=x\left(a\right)$$

It is two LCA groups $A\cong \hat{\hat{A}}$group homomorphisms between.

Pontryagin Duality theorem : an LCA group $G$ , orderH$H\subset G$ is a closed subgroup, let$H^{\perp }\subset \hat{G}$ is the closed subgroup of the dual group, then the duality relation$\hat{\cdot }$ defines the bijection between the subgroup and the quotient group,
$$\hat{H}=\hat{G}/H^{\perp },\widehat{G/H}=H^{\perp }$$

dual space

Let $V$ is the domainF \mathbb FLinear space on$F$$F$ -module),the linear functionalis the linear map$V\to F$，defining operation
$$\begin{gathered} (a+b)\left(v\right)=a\left(v\right)+b\left(v\right) \\ (k\alpha )(v)=k\cdot a\left(v\right) \end{gathered}$$

Dual space : Let $V^{\ast }=Hom(V,F)$ is the collection of all linear functionals, and it is easy to verify that it constitutes a linear space.

Dual basis: For a finite-dimensional space $\dim V=n$，Nyogaei$e_i$It’s $A\; set\; of\; bases\; of\; V$ , then$a^i$ is$V^{}$A set of basis for${}^{\ast }$$a^i(e_j)=d_{ij}$, where $d_{ij}$is an indicator function. It can be verified that $\dim V=\dim V^{\ast }$ , then$V\cong V^{\ast }$ ,and$e_i\mapsto a^i$

Dual map: two spaces $V,$Linear mapping f between$W$$f:V\to W$ , α : V → F , β : W → Fin their dual spaces$a:V\to \mathbb{F},b:W\to F$ can be mapped$f^{\ast }:W^{\ast }\to V^{\ast }$ Correlation, defined as$f^{\ast }\left(b\right)=b\circ f$ , which is also a linear map and satisfies
$$\begin{gathered} (f+g{)}^{\ast }=f^{\ast }+g^{\ast } \\ (kf{)}^{\ast }=k{f}^{\ast } \\ (fg{)}^{\ast }=g^{\ast }{f}^{\ast } \end{gathered}$$

Mapping of linear space to its bidual $\varphi :V\to (V^{\ast }{)}^{\ast }$ , so that it satisfies
$$\left(\varphi \right(v))\left(a\right)=a\left(v\right)$$

An injection that can be proven to be an injection and whose construction does not depend on the choice of basis is called a canonical injection . For finite dimension $\dim V=\dim (V^{\ast }{)}^{\ast }$ , mapping$\varphi$ is also a bijection (not infinitely dimensional), which is calledcanonicalisomorphism. $V\cong V^{\ast }$ is not canonical, but$V\cong (V^{\ast }{)}^{\ast }$ is exemplary.

Fixed mapping, the original image and the image have a certain "mapping" relationship (each original image only corresponds to one image); similarly, when we fix the original image, there is also a certain mapping relationship between the mapping and the image. Therefore, there is a certain symmetrical relationship between the original image and the mapping. We call this symmetrical relationship "duality" and abstract the space where it is located into the dual space.

The dual relationship is symmetric, and the linear functional can be written as $f(x)=⟨f,xbetter$ represents itsbilinearity(Bilinear). For the finite-dimensional linear space$V$ , its dual space$V^{\ast }$ Isomorphism with it, we can loosely define$x\in V$ 和 $f\in V^{\ast }$ Place them in the same space.

Duality

A lattice is a linear space ${\mathbb{R}}^{}$discrete subspace in${}^{m\; ,\; then\; it\; is\; a\; linear\; space\; (infinite\; Abelian\; group).}$

格$L\subseteq R^m$ ，格基 $B\in R^{m\times n}$ ,$\dim L=n\le m$ , its dual is defined as the set
Λ ⊥ = { y ∈ S pan ( Λ ) : ∀ x ∈ Λ , ⟨ x , y 〉 ∈ Z } \Lambda^\perp=\{y \in Span(\Lambda): \forall x \in \Lambda,\langle x,y \rangle \in \mathbb Z\}L⊥={ y∈Span ( L ) _:∀x∈L ,⟨x,y⟩∈Z}

It is a dual lattice, the basis of the lattice is $D=B(B^{T}B{)}^{-1}\in {\mathbb{R}}^{m\times n}$ ,denote$\dim L^{\perp }=\dim L$

The relationship between the two:

1. The lattice bases are pseudo-inverses of each other, $B^{T}D=D^{T}B=I$
2. constant $c>0$ ,free$(c\cdot L{)}^{\perp }=\frac{1}{c}\cdot L$
3. Then $Span(L\_{}_1)=Span(L\_{}_1)$ , then$L_1\subseteq L_2⟺L_1^{\perp }\supseteq L_2^{\perp }$
4. The sparser the grid, the denser its dual, $\mathrm{let}({\Lambda }^{\perp })=1/\mathrm{let}\left(\Lambda \right)$
5. Unlike the dual space, the bidual of the lattice is exactly itself, $(L^{\perp }{)}^{\perp }=L$

Understanding of the dual case,

• Treated as LAC group: linear space $G={\mathbb{R}}^m$ is a LAC group, lattice$\Lambda$ is a closed subgroup, consider the quotient group$\mathcal{P}=G/\Lambda$ (parallelepiped of lattice). According to Pontryagin’s duality theorem, the dual lattice$L^{\perp }$ is defined as: satisfies$\hat{P}\cong L^{}$Dual group closed subgroup Λ of${}^{\perp }$$L^{\perp }\subset \hat{G}$
• Considered as a dual space: discrete subspace $L\subset {\mathbb{R}}^m$ is a$Z$ - submodule, all linear maps$y:x\in L\mapsto ⟨x,y⟩\in Z$ forms the dual space$Hom(R^m,$Discrete Z of$R$$)$$Z$ - submodule, defined as dual lattice.