Dependencies

Let $A$ be a Noetherian ring, $M$ be a finitely generated torsion $A$-module. The characteristic ideal of $M$, denoted by ${\mathrm{char}}_A(M)$, or simply $\mathrm{char}(M)$ if there is no risk of confusion, is defined to be

Let $A$ be a Noetherian ring. Let $0\to M^{\prime }\to M\to M^{″}\to 0$ be a short exact sequence of finitely generated $A$-modules. Then $M$ is $A$-torsion if and only if $M^{\prime }$ and $M^{″}$ are $A$-torsion. If $M$ is $A$-torsion, then ${\mathrm{char}}_A(M)={\mathrm{char}}_A(M^{\prime }){\mathrm{char}}_A(M^{″})$.

Let $A$ be a Noetherian ring, $M$ be a finitely generated torsion $A$-module. Then for any height one prime $\mathfrak{p}$ of $A$, $M_{\mathfrak{p}}$ is an $A_{\mathfrak{p}}$-module of finite length. Moreover, there are only finitely many height one primes $\mathfrak{p}$ of $A$ such that $M_{\mathfrak{p}}\ne 0$.

Let $K$ be a number field, $p$ be a prime, $K_{\mathrm{\infty }}/K$ be a ${\mathbb{Z}}_p$-extension. Then there exist integers $\lambda ,\mu ,\nu$ such that for all sufficiently large $n$, ${\mathrm{ord}}_p(\mathrm{\#}\mathrm{Cl}(K_n))=\mu p^n+\lambda n+\nu$.

If $X$ is a finitely generated torsion $\mathrm{\Lambda }$-module such that $X/({\gamma }^{p^n}-1)X$ is finite for any $n\ge 0$, then there exists some constant $\nu =\nu (X)$ such that for all sufficiently large $n$, ${\mathrm{ord}}_p(\mathrm{\#}(X/({\gamma }^{p^n}-1)X))=\mu (X)p^n+\lambda (X)n+\nu (X)$.

If $(A,\mathfrak{m},k)$ is a local ring, then a polynomial $f(X)=\sum _{i=0}^n{a}_i{X}^i\in A[X]$ is called a distinguished polynomial if $a_n=1$ and $a_i\in \mathfrak{m}$ for all $0\le i\le n-1$.

(Mathlib: Polynomial.IsDistinguishedAt)

Let $A$ be a Noetherian ring. We say that the height one localization of $A$ is PID (?), if

Let $A$ be a domain, $S$ be a subset of $\mathrm{Spec}(A)$ such that $A_{\mathfrak{p}}$ is integrally closed for all $\mathfrak{p}\in S$ and $\bigcap _{\mathfrak{p}\in S}{A}_{\mathfrak{p}}=A$ inside $\mathrm{Frac}(A)$. Then $A$ itself is integrally closed.

(Generalization of IsIntegrallyClosed.of_localization_maximal. #24588)

Let $(A,\mathfrak{m},k)$ be a complete local ring, $g(X)=\sum _{i=0}^n{a}_i{X}^i\in A[X]$ be a polynomial such that $a_n\in A\setminus \mathfrak{m}$ and $a_i\in \mathfrak{m}$ for all $i<n$. Then the natural map $A[X]/(g)\to A[[X]]/(g)$ is an isomorphism.

The Iwasawa algebra is defined as the completed group algebra

where the transition map ${\mathbb{Z}}_p[{\mathrm{\Gamma }}_{n+1}]\to {\mathbb{Z}}_p[{\mathrm{\Gamma }}_n]$ is induced by the natural projection ${\mathrm{\Gamma }}_{n+1}\to {\mathrm{\Gamma }}_n$. Each ${\mathbb{Z}}_p[{\mathrm{\Gamma }}_n]$ is a free ${\mathbb{Z}}_p$-module of rank $p^n$, we endow it with the $p$-adic topology, and endow $\mathrm{\Lambda }$ with the subspace topology of the product topology of $\prod _{n=0}^{\mathrm{\infty }}{\mathbb{Z}}_p[{\mathrm{\Gamma }}_n]$.

There is an isomorphism of topological rings ${\mathbb{Z}}_p[[T]]\stackrel{\sim }{\to }\mathrm{\Lambda }$ sending $1+T$ to $\gamma$, where ${\mathbb{Z}}_p[[T]]$ is endowed with $(p,T)$-adic topology.

For each $n\ge 0$, there is an isomorphism of ${\mathbb{Z}}_p$-algebras

They are all free ${\mathbb{Z}}_p$-modules of finite rank, we endow them with $p$-adic topology. By Proposition 4.2 we know that it is a homeomorphism of topological spaces.

(i) The $\mu$-invariant of $X$ is defined to be $\mu (X):=\sum _{j=1}^s{n}_j$, and the $\lambda$-invariant of $X$ is defined to be $\lambda (X):=\sum _{i=1}^m{b}_i\mathrm{deg}{f}_i$.

(ii) If $f\in {\mathbb{Z}}_p[[T]]$ is not zero, then define $\mu (f)$ be the integer such that $f\in p^{\mu (f)}{\mathbb{Z}}_p[[T]]$ but $f\notin p^{1+\mu (f)}{\mathbb{Z}}_p[[T]]$, define $\lambda (f)$ be the leading degree of $(p^{-\mu (f)}fmodp)\in {\mathbb{F}}_p[[T]]$.

We have $\mu (X)=\sum _{i=0}^{\mathrm{\infty }}{\mathrm{rank}}_{{\mathbb{F}}_p[[T]]}X[p^{i+1}]/X[p^i]$, and $\lambda (X)={\mathrm{rank}}_{{\mathbb{Z}}_p}X/X[p^{\mathrm{\infty }}]$.

An integral domain $A$ is called a Krull domain if it satisfies the following properties:

• $A_{\mathfrak{p}}$ is a discrete valuation ring for all height one primes $\mathfrak{p}$ of $A$,

• $A=\bigcap _{\mathfrak{p}\in \mathrm{Spec}(A),\mathrm{ht}(\mathfrak{p})=1}{A}_{\mathfrak{p}}$ inside $\mathrm{Frac}(A)$,

• any nonzero element of $A$ is contained in only a finitely many height one primes of $A$.

Let $A$ be a Noetherian ring. If $\mathfrak{a}=(a_1,\cdots ,a_n)$ is an ideal of $A$ generated by $n$ elements, and $\mathfrak{p}$ is a minimal prime over-ideal of $\mathfrak{a}$, then $\mathrm{ht}(\mathfrak{p})\le n$.

Conversely, if $\mathfrak{p}$ is a prime ideal of $A$ of height $\le n$, then there exists an ideal $\mathfrak{a}$ of $A$ generated by $n$ elements, such that $\bar{\mathfrak{p}}$ is a minimal prime in $A/\mathfrak{a}$.

(This is WIP in #23778.)

Hence by Proposition 2.3, any height $1$ prime $\mathfrak{p}$ of $\mathrm{\Lambda }$ is principal.

(i) If the generator of $\mathfrak{p}$ is in $p{\mathbb{Z}}_p[[T]]$, then we must have $\mathfrak{p}=(p)$.

(ii) If the generator of $\mathfrak{p}$ is not in $p{\mathbb{Z}}_p[[T]]$, then by Proposition 4.9 such $\mathfrak{p}$ has a unique generator which is a distinguished polynomial.

$\mathrm{\Lambda }\cong {\mathbb{Z}}_p[[T]]$ is a Noetherian regular local ring of Krull dimension $2$.

Hence by Proposition 2.4, $\mathrm{\Lambda }$ is a UFD (or maybe one can check directly that the $I$-adic completion of a UFD is a UFD).

If $X$ is a finitely generated torsion $\mathrm{\Lambda }$-module, then there exists a pseudo-isomorphism

where $f_1,\cdots ,f_m$ are distinguished polynomials. The characteristic ideal ${\mathrm{char}}_{\mathrm{\Lambda }}(X)$ is generated by $p^{\sum _{j=1}^s{n}_j}\prod _{i=1}^m{f}_i^{b_i}$ which is contained in $p^{\sum _{j=1}^s{n}_j}{\mathbb{Z}}_p[[T]]$ but not in $p^{1+\sum _{j=1}^s{n}_j}{\mathbb{Z}}_p[[T]]$.

Let $A$ be a ring, $\mathfrak{a}$ be an ideal of $A$. Let $M,N$ be two $A$-modules, and $\phi :M\to N$ be an $A$-module homomorphism. Then $\phi$ is continuous if we endow $M,N$ with $\mathfrak{a}$-adic topology.

In particular, if $\phi$ is an $A$-module isomorphism, then it is a homeomorphism of topological spaces if we endow $M,N$ with $\mathfrak{a}$-adic topology.

For a Noetherian local domain $A$ of dimension one, the following are equivalent.

• $A$ is integrally closed.

• The maximal ideal of $A$ is principal.

• $A$ is a discrete valuation ring.

• $A$ is a regular local ring.

(Mathlib: IsDiscreteValuationRing.TFAE and tfae_of_isNoetherianRing_of_isLocalRing_of_isDomain.)

Let $A$ be a Noetherian ring and $M$ be a finitely generated $A$-module. Then there exists a chain of submodules of $M$

such that for each $1\le i\le n$, $M_i/M_{i-1}\cong A/{\mathfrak{p}}_i$ for some prime ideal ${\mathfrak{p}}_i\in \mathrm{Spec}(A)$.

A Noetherian ring is a Krull domain if and only if it is an integrally closed domain.

Suppose $K$ is any number field, and $K_{\mathrm{\infty }}/K$ is any ${\mathbb{Z}}_p$-extension. Let $L_{\mathrm{\infty }}$ be the maximal unramified abelian pro-$p$ extension of $K_{\mathrm{\infty }}$, let $X_{\mathrm{\infty }}:=\mathrm{Gal}(L_{\mathrm{\infty }}/K_{\mathrm{\infty }})$ which is a $\mathrm{\Lambda }$-module, where $\mathrm{\Lambda }:={\mathbb{Z}}_p[[\mathrm{\Gamma }]]$, isomorphic to ${\mathbb{Z}}_p[[T]]$ by choosing a topological generator $\gamma$ of $\mathrm{\Gamma }$, and maps $T$ to $\gamma -1$. Then $X_{\mathrm{\infty }}$ is a finitely generated torsion $\mathrm{\Lambda }$-module.

Suppose $\mathrm{\Lambda }\cong {\mathbb{Z}}_p[[T]]$, $X$ is a pro-$p$ $\mathrm{\Lambda }$-module, $x_1,\cdots ,x_t\in X$ such that their images in $X/TX$ generate $X/TX$ as a $\mathrm{\Lambda }/(T)$-module. Then $x_1,\cdots ,x_t$ generate $X$ as a $\mathrm{\Lambda }$-module.

${\mathcal{G}}^{\prime }=(g-1)\mathcal{X}$.

Suppose $X$ has rank $r$ as a $\mathrm{\Lambda }$-module, then we have

as $n\to \mathrm{\infty }$. This is left as an exercise. For example, if $X=\mathrm{\Lambda }$, then $r=1$, and ${\mathrm{rank}}_{{\mathbb{Z}}_p}(X/((1+T^{p^n})-1)X)=p^n$.

If $K$ is any number field, $K_{\mathrm{\infty }}/K$ is any ${\mathbb{Z}}_p$-extension, and $\mathfrak{l}$ is a prime of $K$ not lying over $p$. Then $\mathfrak{l}$ is unramified in $K_{\mathrm{\infty }}/K$.

Let $A$ be a Noetherian ring and let $M,N$ be finitely generated torsion $A$-modules. Let $\mathrm{\Sigma }=\{{\mathfrak{q}}_1,\cdots ,{\mathfrak{q}}_r\}=\{\mathfrak{q}\in \mathrm{Supp}(M)\cup \mathrm{Supp}(N)\mid \mathrm{ht}(\mathfrak{q})=1\}$ (by Proposition 3.1 this is a finite set). Let $S:=A\setminus \bigcup _{i=1}^r{\mathfrak{q}}_i$ which is a multiplicative subset of $A$. Let $f:M\to N$ be an $A$-module homomorphism. Then $f$ is a pseudo-isomorphism if and only if $S^{-1}f:S^{-1}M\to S^{-1}N$ is an isomorphism.

Let $A$ be a Noetherian ring satisfying 3.1. Let $M,N$ be finitely generated torsion $A$-modules. Then the followings are equivalent:

• There exists a pseudo-isomorphism $M\to N$.

• For any height one prime $\mathfrak{p}$ of $A$, we have $M_{\mathfrak{p}}\cong N_{\mathfrak{p}}$.

In particular, if there exists a pseudo-isomorphism $M\to N$, then there also exists a pseudo-isomorphism $N\to M$.

Let $A$ be a ring, and let $f(X)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{X}^n\in A[[X]]$ be a formal power series. Then $f(X)\in A[[X]{]}^{\times }$ if and only if $a_0\in A^{\times }$.

(Mathlib: PowerSeries.isUnit_iff_constantCoeff)

Let $A$ be a Noetherian ring.

(i) A finitely generated $A$-module $M$ is called a pseudo-null $A$-module, if $M_{\mathfrak{p}}=0$ for all prime ideals $\mathfrak{p}$ of $A$ of height $\le 1$.

(ii) An $A$-linear homomorphism $f:M\to N$ between finitely generated $A$-modules is called a pseudo-isomorphism (pis for short), if its kernel and cokernel are pseudo-null $A$-modules.

(iii) Two finitely generated $A$-modules $M,N$ are called pseudo-isomorphic (pis for short), denoted by $M{\sim }_{\mathrm{pis}}N$ or simply $M\sim N$, if there exists a pseudo-isomorphism from $M$ to $N$.

Let $A$ be a Noetherian ring, $M$, $N$ be finitely generated torsion $A$-modules.

(i) If $M$ is pseudo-null, then ${\mathrm{char}}_A(M)=0$.

(ii) If $M\sim N$, then ${\mathrm{char}}_A(M)={\mathrm{char}}_A(N)$.

Let $A$ be a Noetherian ring, $M$ be a finitely generated $A$-module.

(i) If $A$ is of Krull dimension $\le 1$, then $M$ is pseudo-null if and only if $M=0$.

(ii) If $A$ is of Krull dimension $2$, is a local ring with finite residue field, then $M$ is pseudo-null if and only if the cardinality of $M$ is finite.

A Noetherian regular domain (more generally, a Noetherian integrally closed domain) satisfies 3.1.

A regular local ring is a UFD.

Let $A$ be a semilocal ring, $M$ be an $A$-module such that for any maximal ideal $\mathfrak{m}$ of $A$, $M_{\mathfrak{m}}$ is a finitely generated $A_{\mathfrak{m}}$-module. Then $M$ is a finitely generated $A$-module.

Let $A$ be a semilocal (i.e. only finitely many maximal ideals) integral domain which is not a field, such that for every maximal ideal $\mathfrak{p}$ of $A$, $A_{\mathfrak{p}}$ is a PID. Then $A$ itself is a PID.

Let $A$ be a Noetherian ring satisfying 3.1 and let $M$ be a finitely generated torsion $A$-module. Then there exist height one primes ${\mathfrak{p}}_1,\cdots ,{\mathfrak{p}}_s$ of $A$ and positive integers $k_1,\cdots ,k_s$, such that there exists a pseudo-isomorphism $M\to \underset{i=1}{\overset{s}{⨁}}A/{\mathfrak{p}}_i^{k_i}$. Moreover, the sequence $({\mathfrak{p}}_i^{k_i}{)}_{i=1}^s$ is unique up to ordering.

Let $A$ be a Noetherian domain. Then $A$ is a UFD if and only if every prime ideal of height $1$ is principal (recall that a prime ideal $\mathfrak{p}$ is of height $1$ if $\mathfrak{p}\ne 0$ and there is no prime ideal lies in between $0$ and $\mathfrak{p}$).

(We only need “$\Rightarrow$” in our project.)

Let $(A,\mathfrak{m},k)$ be a complete local ring, $g(X)=\sum _{i=0}^{\mathrm{\infty }}{a}_i{X}^i\in A[[X]]\setminus \mathfrak{m}[[X]]$ be a formal power series such that not all of its coefficients are in $\mathfrak{m}$. Let $n\ge 0$ be the integer such that $a_n\in A\setminus \mathfrak{m}=A^{\times }$ and $a_i\in \mathfrak{m}$ for all $0\le i\le n-1$. Then for any $f\in A[[X]]$, there exists a unique formal power series $q(X)\in A[[X]]$ and a unique polynomial $r(X)\in A[X]$ of degree $\le n-1$ such that $f=gq+r$.

Let $(A,\mathfrak{m},k)$ be a complete local ring. Let $g(X)\in A[[X]]\setminus \mathfrak{m}[[X]]$ be a formal power series such that not all of its coefficients are in $\mathfrak{m}$. Then there is a unique distinguished polynomial $f(X)\in A[X]$ and a unique invertible formal power series $h(X)\in A[[X]{]}^{\times }$ such that $g=fh$.

(This is incorrect ...) For each $n\ge 0$ there is an isomorphism $X_{\mathrm{\infty }}/({\gamma }^{p^n}-1)X_{\mathrm{\infty }}\stackrel{\sim }{\to }{X}_n$.

Let $K$ be a field, $p$ be a prime.

(i) An extension $K_{\mathrm{\infty }}/K$ is called a ${\mathbb{Z}}_p$-extension, if it is Galois with $\mathrm{\Gamma }=\mathrm{Gal}(K_{\mathrm{\infty }}/K)\cong {\mathbb{Z}}_p$ as topological groups.

(ii) If $K_{\mathrm{\infty }}/K$ is a ${\mathbb{Z}}_p$-extension, then for any $n\ge 0$, define $K_n:=K_{\mathrm{\infty }}^{{\mathrm{\Gamma }}^{p^n}}$.

$L_{\mathrm{\infty }}^{\prime }=L_{\mathrm{\infty }}$.

(i) Let $L_{\mathrm{\infty }}:=\bigcup _{n\ge 0}{L}_n=\bigcup _{n\ge 0}{L}_n{K}_{\mathrm{\infty }}$, then it is an unramified abelian pro-$p$ extension of $K_{\mathrm{\infty }}$, because each $L_n{K}_{\mathrm{\infty }}/K_{\mathrm{\infty }}$ is finite unramified abelian $p$-extension.

(ii) Let $L_{\mathrm{\infty }}^{\prime }$ be the maximal unramified abelian pro-$p$ extension of $K_{\mathrm{\infty }}$, that is, the compositum of all finite unramified abelian $p$-extensions of $K_{\mathrm{\infty }}$.

(i) For each $n\ge 0$ let $L_n$ be the $p^{\mathrm{\infty }}$-Hilbert class field of $K_n$. That is, the maximal unramified abelian extension of $K_n$ of exponent $p^{\mathrm{\infty }}$.

(ii) Let $X_n:=\mathrm{Gal}(L_n/K_n)\cong \mathrm{Cl}(K_n)(p)$, the maximal quotient of the class group $\mathrm{Cl}(K_n)$ which is $p^{\mathrm{\infty }}$-torsion.