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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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1answer
269 views

$P$ 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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0answers
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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0answers
65 views

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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0answers
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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1answer
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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0answers
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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53 views

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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0answers
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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0answers
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, ...
6
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1answer
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 $K$. ...
3
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1answer
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 ...
5
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1answer
143 views

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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1answer
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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0answers
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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0answers
118 views

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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1answer
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 ...
12
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1answer
194 views

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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0answers
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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72 views

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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0answers
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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1answer
123 views

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 ...
2
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1answer
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 ...
2
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1answer
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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0answers
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 ...
3
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1answer
154 views

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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1answer
261 views

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_{\...
15
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3answers
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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1answer
566 views

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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191 views

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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0answers
42 views

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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0answers
80 views

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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0answers
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 ...
5
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1answer
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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0answers
128 views

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 ...
5
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2answers
657 views

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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0answers
74 views

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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1answer
158 views

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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0answers
64 views

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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0answers
101 views

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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0answers
64 views

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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1answer
188 views

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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0answers
124 views

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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0answers
324 views

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 ...
3
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1answer
92 views

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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0answers
146 views

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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0answers
54 views

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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3answers
574 views

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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0answers
119 views

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 ...
6
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2answers
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}...