# Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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### $P$ is projective if and only if $P\otimes N\cong Hom(Hom(P,R),N)$

Let $R$ be a commutative Noetherian ring and $P$ be finitely generated $R$-module.
How to prove the following.
$P$ is projective if and only if $P\otimes N\cong Hom(Hom(P,R),N)$ for all finitely ...

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78 views

### Cohomology of a chain complex over a polynomial ring

I asked this on SE but I did not get any answer; I got some progress but I hope here I can find some help to finish the problem out.
Let $R = F[x_1, \ldots, x_n]$ be a polynomial ring over a field $...

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### Valuation Rings and Ultrafilters II

See my post here: Valuation Rings and Ultrafilters
Let $K$ be a field, and let $\mathcal{S}$ be the set of pairs $(R, \mathfrak{p})$ of subrings $R$ of $K$ with designated prime ideals $\mathfrak{p}$ ...

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85 views

### When a semigroup ideal is a determinantal ideal?

Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \...

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149 views

### tangent cone and local picture

Let $X\subset \mathbb P^n_{\mathbb C}$ be a closed algebraic subset and $x\in X$.
Suppose tha tangent cone of $X$ at $x$ is the union of (say) a plane and a line (meeting at the origin). Can we ...

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121 views

### $\omega$-categorical algebra

Let us consider a 1-category $C$. For any commutative and unital ring $k$ the the free $k$-module generated by the morphisms of $C$ can be equipped with an algebra structure by setting $fg$ to be ...

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### Strongly isovariant (aka fixed point reflecting / stabiliser preserving) morphisms

I have some questions about what feels like basic topics in quotients of schemes by group actions. Consequently I suspect there are well-known references; I couldn't find them, though.
First ...

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79 views

### Tensor square of duals over a domain

The title is motivated by my needs ($M=N$ in the sequel).
Linked to the question here and there (in the case of products) is the following.
Let $M,N$ $k$-modules ($k$ a commutative ring), then we ...

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92 views

### Intersection of a reduced projective variety with a general hyperplane is reduced

Let $X\subset \mathbb{P}^n$ be a reduced closed subscheme. For a general hyperplane $H$, $X\cap H$ is again reduced (and of dimension one less). Is there an easy proof of this result?
Algebraically, ...

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207 views

### Valuation Rings and Ultrafilters

I notice there is a certain similarity between the definition of a valuation ring and the definition of an ultrafilter.
To begin, take a field $K$ and let $\mathcal{A}$ be the set of subrings of $K$. ...

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335 views

### Coordinate free expression for the determinant of a $2 \times 2$ matrix

Let $E\rightarrow X$ be a rank two vector bundle on a variety $X$. How can one write an abstract map $Sym^2(E)\rightarrow (\wedge^2 E)^{\otimes 2}$ that in local coordinates, when we fix a frame of $E$...

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128 views

### Extension of alg closed fields = limit of smooth algebras

In the proof of https://stacks.math.columbia.edu/tag/0F0B, it is a claim that if $K/k$ is an extension of algebraically closed fields, then $K$ is a limit of smooth $k$-algebras. This is justified ...

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### Absolute value on tensor product of fields

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2:=\mathbb F_p((Y))$.
Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{...

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268 views

### Inverse of the Structure Theorem for Finitely Generated Modules over PID

We know that for a PID $R$, any finitely generated module is of the form $\frac{R}{(a_1)} \oplus \dots \oplus \frac{R}{(a_s)} $.
I was wondering if the converse of this statement is true, that is, is ...

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42 views

### Dimension of a tangent space of the normalization of a domain

Let $A$ be a finitely generated (commutative) domain over an algebraically
closed field $k.$ Let $B$ be the normalization of $A$. Let $m$ be a maximal
ideal in $A$, and $m'$ be a maximal ideal in $B$ ...

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### Does linear independence imply algebraic independence for partitioned homogeneous polynomials?

Define a partitioned homogeneous polynomial of degree $d$ to be a polynomial in $$\mathbb Z[x_{11},\dots,x_{1n},\dots,x_{d1},\dots,x_{dn}]$$ with monomials from entries in (polynomials that are $d$-...

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128 views

### Image of smooth curve containing the image of a point as smooth point

Let $f: X \to Y$ be a finite, surjective morphism of smooth, quasi-projective varieties over a field $k$ of characteristic zero. Let $p\in X$.
If $\dim X>0$, then does there necessarily exist a ...

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### PID expressed as finite union of subrings

There is a classical theorem that no field can be expressed as finite union of proper subfields.
In contrast, there is an example of an integral domain that can be expressed as finite union of proper ...

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78 views

### Do Plucker relation follow from a subsystem of equations?

The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$,
\begin{align*}
P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} ...

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### Krull dimension of completions in non-noetherian setting (especially completed perfections)

What are some general results describing the Krull dimension of the completion of a non-Noetherian ring with a "nice" topology?
An example of the sort of "nice" topological ring I'm looking for is a ...

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69 views

### Completion of localization of completion

Let $(A,m)$ be a noetherian local ring,
and let $p \subseteq A$ be a prime ideal.
From this data, we can construct two rings:
1. We may localize $A$ at $p$, and then complete,
obtaining the $pA_p$-...

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### Is $[Im:(x)][Im:(y,z)]\subseteq Im$ in $k[x,y,z]$?

Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $I\subseteq m$ a proper homogeneous ideal in $S$. Is this true that we always have:
$$[Im:(x)][Im:(y,z)]\subseteq Im \ ?$$
In a paper we ...

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212 views

### Is the support of a flat module generically flat?

Let $X$ be an affine, complex variety, $A$ be a $\mathbb{C}$-algebra (not necessarily noetherian) and $F_A$ is a coherent sheaf over $X \times \mbox{Spec}(A)$, flat over $\mbox{Spec}(A)$. Denote by $Y ...

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167 views

### Are unique prime ideal factorization domains locally noetherian?

I asked this question on Mathematics Stack Exchange but got no answer.
Here is the question:
Let $A$ be a domain (that is, a commutative ring with one in which the condition $ab=0$ implies $a=0$ or $...

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90 views

### A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...

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### Is $A[b/a]$ a Krull domain?

Is $A[X]/(aX+b)$ a Krull domain when $A$ is and when $a\in A-\{0\}, b\in A-Aa$ are such that $Aa$ and $Aa+Ab$ are prime ideals of $A$?
This is stated as Proposition 8 in Pierre Samuel, "Sur les ...

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### Description of p-adics tensor the reals

What is $\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathbb{R}$ equivalent to?
where $\mathbb{Z}_p$ are the p-adic integers.
I am specially interested in the case $p=2$.
Do know that $\mathbb{Z}_p\otimes_{\...

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423 views

### Graded analogues of theorems in commutative algebra

Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word
commutative ...

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### Is the formal power series ring integrally closed?

Let $k$ be a field and $s$ and $t$ be variables.
Is the ring $k[s][[t]]$ integrally closed in $k[s,s^{-1}][[t]]$?

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### Are there enough curves (to connect 'points' of f.g. algebras)?

(Intuition: any two points in a connected space may be connected by a path. I would like to know if something like this holds in certain category of `connected algebraic spaces'. I formulate the ...

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### Primary decomposition with parameters

$\newcommand\QQ{\mathbb{Q}}$
Considering the polynomial
$$f = x^2 - a y$$
one notes, that it is irreducible in $\QQ[x,y]$ for all $a \neq 0 \in \QQ$ and factors for $a = 0$.
More generally, let $A=...

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### Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index

Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...

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156 views

### Behavior of Ext under base change

Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...

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240 views

### A question on dominant morphism of affine schemes

Let $A \subseteq B$ be a ring extension where $A,B$ are both finitely generated $\mathbb C$-domain of the same Krull dimension. Also assume $A$ is regular (i.e. $A_{ \mathfrak p}$ is regular local ...

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### Graded Betti numbers $\beta_{n,j}$ for points in $\mathbb{P}^n$

Let $S = \mathbb{C}[z_0, \dots, z_n]$, and let $X$ be a set of points in $\mathbb{P}^n$. I'm looking for references concerning results for the graded Betti numbers $\beta_{n,j}(S/I(X))$, i.e., the ...

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### Algebra for algebraic topology

My research is in analysis, but it moved to the area that requires algebraic topology. I have some working knowledge in that area, but I always feel that I am on a shaky ground and I need to go back ...

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### Notions of connected components in a finite family fibration

Let $\Pi_0:\mathsf{FinFam}(\mathsf C)\to \mathsf{FinSet}$ be the fibration exhibiting the free finite coproduct completion of $\mathsf C$. Suppose $\mathsf C$ has finite limits so that the extensive $\...

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### Counting the number of poles for rational functions in a coordinate ring of a curve

I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in ...

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### Relation between lifts of simple roots and lifts of idempotents (Henselian property)

Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....

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### Making explicit the local structure theorem of étale maps in a very simple case

Making explicit the local structure theorem of étale maps in a very simple case.
First I recall the following items from Stacks.
\smallskip
\textbf{Lemma 10.141.2.} Any étale ring map is standard ...

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### Module of Kahler differentials for manifolds [duplicate]

Let $A$ be a $k$-algebra and let $\mathcal{M}_A$ be the set of all $A$-modules. In $\mathcal{M}_A$, there exists a universal object $\Omega_{A/k}$, called the module of Kahler differentials, and a $k$...

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### Existence of isomorphism mod every power of the maximal ideal

This problem is a continuation of Hensel lemma and rational points in complete noetherian local ring.
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
Assume $...

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### Explicit description of injective hulls

Let $k$ be a field, let $R:=k[x_1,\ldots,x_n]$, and
consider the $R$-module $M:=R/{(x_1,\ldots,x_n)}\cong k$.
Then the injective hull $I_M$ of $M$ admits the following explicit description:
$$
I_M = k[...

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### Completion of a local ring of an arithmetic surface

Vaguely, an arithmetic surface is an algebraic curve with coefficients in some "ring of integers".
More formally it is an integral normal scheme $X$ of dimension 2 of finite type over a Dedekind ...

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### Summing complete intersections

Suppose we have polynomials $f_1,\dots,f_r\in k[X_1,\dots,X_N]$ defining a complete intersection in $\mathbb{A}^N$. I suspect that it is then true that $f_1(X)+f_1(Y),\dots,f_r(X)+f_r(Y)\in k[X_1,\...

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### Rank 2 vector bundle with trivial first chern class is self-dual

I saw a statement used in the paper that E of rank 2 with $c_1(E) = 0$ is self-dual. I was wondering, how does one prove this statement? If it makes a difference, let the underlying variety be ...

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### What's the probability a random module element has prime annihilator?

I'm going to pose two versions of my question---an ill-defined version and a well-defined version.
Ill-Defined Question (IDQ). Let $R$ be a (commutative, unital) Noetherian ring, $M$ a finitely ...

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### Are archimedean subextensions of ordered fields dense?

Let $E$ be an ordered field and let $F$ be a real closed subfield. We say that $E$ is $F$-archimedean if for each $e\in E$ there is $x\in F$ such that $-x\le e\le x$.
Is it true that if $E$ is $F$-...

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### Behavior of regularity under base change

Let $(R,\mathfrak{m})$ be a noetherian local domain. Let $(A,\mathfrak{n})$ be a regular local ring contains $R$ such that $\mathfrak{n}\cap R=\mathfrak{m}$ and $A$ is essential finitely generated as ...

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230 views

### Ideals invariant under ring automorphisms

I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:
$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...