# Questions tagged [ac.commutative-algebra]

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

5,081
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### Vanishing theorem in local cohomology

I am using local cohomology as a blackbox and the ring I work with is commutative and Noetherian; however, is not local. Consider a commutative Noetherian ring $R$ of Krull dimension $n$. Consider the ...

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### Structure of module via a ring map

Let $Q \xrightarrow{f} Q/(a) = R$ be a local ring homomorphism, where $Q$ is complete intersection local ring and $a$ is a non-zerodivisor of $Q$ and $$ R^{n+1} \xrightarrow{d} R^n$$ is a module ...

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### What is the intersection of all ideals whose radicals are prime?

Let a fuzzy prime be an ideal of a commutative unital ring whose radical is prime (I'm not sure if this kind of ideal already has a name). Is the intersection of all fuzzy primes $\{0\}$?
Note this is ...

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### Does going down property imply a corresponding map is open without "finiteness"?

Does the following proposition hold?
Proposition
Let f:A$\rightarrow$B be a ring homomorphism
If f has going down property then the corresponding map
$f^*$:Spec B$\rightarrow$Spec A is open map.
I ...

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133
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### A question about surjective maps between quadratic algebras

Let $V$ be a finite-dimensional vector space and
$$
U \subseteq W \subseteq V \otimes V
$$
be a proper inclusion of vector subspaces. Then take the tensor algebra
$$
T(V) = \bigoplus_{i=1}^{\infty} V^{...

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### Can the equation $1+z^p+z^q+z^r=z^n$ have multiple complex roots $z$?

The math overflow post asks whether the equation $1+z^p+z^q=z^n$ can have multiple complex roots where $p<q<n$ (On the irreducibility of certain trinomials and quadrinomials).
Q. Let us ...

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255
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### flatness and exact sequences

Let $R$ be a commutative ring (with unit). Then if
$$0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow 0$$
is an exact sequence of $R$-modules, with $M''$ $R$-flat, $M$ is flat if ...

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### Sampling theorems for partition polynomials (associahedra, noncrossing partitions / parking functions)

Define the associahedra partition polynomial
$$
\begin{split}
A(x) &= 1 + A_1(u_1) z + A_2(u_1,u_2) z^2 + A_3(u_1,u_2,u_3) z^3 + \cdots\\
& \qquad\qquad = 1 + \sum_{n \geq 1} A_n(u_1,...,u_n) ...

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### Can a non-zero non-prime ideal become prime in a smaller ring?

All rings are assumed commutative and unital.
Some context (feel free to skip right to the questions below). I am trying to understand to what extent the property "being prime" for an ideal $...

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### Is the spectrum of this ring Noetherian?

Let $R = \mathbb{Z}/2\mathbb{Z}[X,X^{1/2},X^{1/4},\cdots,X^{1/{2^n}},\cdots]$
Is $\operatorname{Spec}R$ a Noetherian topological space?
Here is what I know.
$R$ is integral over $\mathbb{Z}/2\mathbb{...

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245
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### The $1$-dimensional Jacobian Conjecture over $\mathbb{Z}$-torsion free rings

I am looking for further proofs, preferably in the literature, of the following result:
Proposition: Let $R$ be a unital commutative $\mathbb{Z}$-torsion free ring. If $f(x) \in R[x]$ with $f'(x) \in ...

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111
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### Absolute integral closure of Noetherian local domain

Let $(R,\mathfrak{m})$ be a Noetherian local domain, let $K(R)$ be the total fraction field of $R$ and let $R^{+}$ be the absolute integral closure of $R$. Consider the $R^{+}$-module $K(R)\otimes_{R} ...

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### Intersection of (sub-)modules under Laurent and formal rings

I have a (hopefully quick) question regarding an intersection of two tensor modules. Let K
be a field and $A,B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...

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1
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289
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### Is a complete local ring determined by its values in local fields?

Let $A$ be a complete, Noetherian, local ring with finite residue field of characteristic $p$. If $F$ is a non-Archimedean local field, then we will denote the ring of integers of $F$ by $\mathcal{O}...

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### How many minimal relations are needed to obtain a Frobenius algebra?

Let $A_n:=K \langle x_1,x_2,...,x_n \rangle$ be the non-commutative polynomial ring in $n$-variables over the field $K$ and let $J=\langle x_1,...,x_n \rangle$ be the ideal spanned by the $x_i$.
An ...

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171
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### On linear schemes

Let $X$ be a smooth projective curve and $T$ is any smooth variety. Let $E$ be a family of vector bundles over $X\times T$ which is flat over $T$. Then there exists a scheme $Y$ over $T$ such that for ...

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### Automorphisms of matrix algebras and Picard group

This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE).
Notation. In what follows, $R$ is a commutative ring with $1$, $n\...

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155
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### Local cohomology and residues of rational functions at 0 and $\infty$

Let $a_1,\dots,a_s$ and $b_1,\dots,b_t$ be positive integers, where
$s,t>0$. Choose $c\in\mathbb{Z}$. Let $M_c$ be the real vector
space spanned by all monomials $x^\alpha y^\beta=x_1^{\alpha_1}\...

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### Does maximally incompleteness cause nonvanishing of the extension of maximal ideal of a valuation ring by rank 1 free module?

In B. Bhatt's lecture notes[1], Remark 4.2.5 says
... $\operatorname{Ext}_R^2(k,R)$ is non-zero if $K$ is not spherically complete.
which amounts to the following pure algebraic question.
Statement ...

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### Example of a certain type of Cohen-Macaulay ring

Let $k$ be a perfect field. I am looking for a $k$-algebra $R$ with the following properties.
$R$ is of finite type over $k$ and is a domain;
for all ${\mathfrak p}\in{\rm Spec}(R)$, the local ring $...

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156
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### Normal forms of ADE singularities

Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?
Does a similar ...

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86
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### Product/intersection of two ideals

Let $I_{1}$ and $I_{2}$ be two ideals in polynomial ring $C[u,v,t]$, defined as $I_1 = \langle t-u^3, (v-u^5)^2\rangle$ and $I_2 = \langle u^{11} + v^{11} + t^{11} + 3\rangle$. Is there a method for ...

2
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138
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### Flat scheme-theoretic closure

Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.
Let $C_R$ be a flat ...

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### Determinants of perfect complexes and Hilbert polynomials

Let $X$ be a smooth projective variety over an algebraically closed field $k$, and let $K^{\bullet}$ be a perfect complex of $\mathcal{O}_X$-modules.
It is possible to define a canonical line bundle $\...

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73
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### Comparison of depth of two monomial ideals

Let $I$ and $J$ be two monomial ideals of $R=K[x_1,\dotsc,x_n]$ where $K$ is a field.
Could we say $\mid\operatorname{depth} (R/I)- \operatorname{depth}(R/(I:J))\mid\leq 1$, where $(I:J)$ is colon ...

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### Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism

Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...

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255
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### On the analogy between $p$-derivations and derivations

$\DeclareMathOperator\Spec{Spec}$Let $p$ be a prime number, and $A$ a commutative ring. Recall that a $p$-derivation on $A$, or a $\delta$-ring structure on $A$ is a set map $\delta : A \to A$ such ...

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### Are maps into a smooth curve equivalent to relative effective Cartier divisors?

Let $X$ be a smooth curve and $S$ an affine scheme, both over an algebraically closed field $k$.
Given a morphism $f : S \to X$, is it true that the graph morphism $\Gamma_f : S \to X \times S$ is a ...

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### Why the stable module category?

Let $R$ be a ring (usually assumed to be Frobenius). The stable module category is what you get when you take the category $\mathsf{Mod}_R$ of $R$-modules, and kill the projective modules. (Of course, ...

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### Examples of stretched artinian local ring

In Sally's Paper stretched artinian local ring is defined as :
Let $(R, \mathfrak{m})$ be an Artin local ring of length $\lambda.$ If $\nu$ is the embedding dimension of $R$, that is, $\nu$ is the ...

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### Is there a Hopf algebra-style description of chain complexes?

An old chestnut is that filtered objects are the same as sheaves over $\mathbb A^1 / \mathbb G_m$.
Question: Is there a similar description of chain complexes?
More precisely, if $\mathcal C$ is a ...

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### Kouchnirenko's theorem for non-generic polynomials

In Polyèdres de Newton et nombres de Milnor (Theorem 1.18), Kouchnirenko proved that given Laurent polynomials $f_1, \dotsc, f_k$ in $k$ variables, the number of isolated solutions is less than or ...

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### Intersection numbers of moduli spaces and noncrossing partitions

The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are ...

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### Is there a good notion of higher-rank archimedean norm?

Let $K$ be a field. I think I know what a norm (archimedean or not) $|-| : K \to \mathbb R_{\geq 0}$ is. In the case where the norm is nonarchimedean, it's equivalent to the data of a valuation of ...

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137
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### Intersection of two modules (and sub-modules) under tensors

I have a (hopefully quick) question regarding an intersection of two tensor modules. Let $K$ be a field and $A, B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...

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2
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612
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### Hilbert polynomials of graded algebras evaluated at negative numbers

Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...

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### Interpretation of completed tensor product of algebras over lower base

Let $\mathbb F$ be a finite field of order $q = p^n$. It is known that
$$\mathbb F[[x_1]] \mathbin{\widehat{\otimes}_{\mathbb F}} \mathbb F[[x_2]] = \mathbb F[[x_1, x_2]].$$ Geometrically, this is the ...

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### Alternative description of strict henselization

Let $R$ be a local ring with field of fractions $K$, maximal ideal $\mathfrak{m}$ and residue field $\kappa = R/\mathfrak{m}$. Let $R^\mathrm{sh}$ be a strict henselization of $R$, and let $L$ be the ...

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### Finite type injective ring map between domains preserves the open point $(0)$

I am looking for a proof of the following statement without using full power of Chevalley's theorem on constructible sets. We say a domain $A$ is $0$-open if $\{(0)\}$ is open in $\operatorname{Spec}(...

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### Effective bound on "Jacobian rank" for (regular) planar algebraic curves

Let an irreducible, square-free complex polynomial $f\in \mathbb C[x,y]$ be given. It is well known that the curve $\mathcal C:=\{f=0\}$ is nonsingular if and only if $\mathbb C[x,y]=<f,\partial_xf,...

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### Cataland: Facets and partition polynomials of cluster complexes

Figure 25 on pg. 101 of "Cataland: Why the Fuss?" by Christian Stump, Hugh Thomas, and Nathan Williams depicts cluster complexes (CCs) associated with generalized $(m)$-Narayana / ...

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### Sufficient conditions to guarantee finite intersection points in Bezout's Theorem

Bezout's Theorem concludes that if $f_1,f_2,\cdots,f_n\in k[x_1,x_2,\cdots,x_n]$ have finite intersection points, then they have at most $d_1d_2\cdots d_n$ intersection points, where $d_i$ is the ...

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### The map from the ring of integers to the residue field of a valuation subring is surjective

Let $L$ be a number field and let ${\mathcal{O}}_L$ be its ring of integers. Let $B$ be a valuation subring of $L$ and let $k_B$ be the residue field of $B$. Then the map from ${\mathcal{O}}_L$ to $...

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### When do there exist $ n $ commuting, derivations which locally generate $ T_{A/k} $ as an $ A $-module?

Let $ \operatorname{Spec}(A) $ be a non-singular, $ n $-dimensional, affine variety over a field $ k $ of arbitrary characteristic. For the definition of "variety" I use Hartshorne's ...

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714
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### Excellent property of rings

Let $A$ be a commutative ring. If $A$ is an excellent ring, is the reduced ring $A/\sqrt{(0)}$ also an excellent ring?

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### Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)

Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...

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### Modulo $x^2 + y^2 - 1$, is every homogeneous polynomial that is a square of a polynomial, necessarily of sum of squares of homogeneous polynomials?

I am hoping this question is alright for Math Overflow. I didn't get a definitive solution in Math Stack Exchange.
Let $f(x, y) \in \mathbb{R}[x, y]$ be a homogeneous polynomial with real coefficients ...

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220
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### Determining when quotient of a polynomial ring is a Gorenstein ring

I would like to be able to look at the ring $R=\mathbb{Z}[x_1,x_2,\ldots,x_n]/\mathcal{I},$ where $\mathcal{I}$ is generated by a finite number of monomials and say whether $R$ is a Gorenstein ring. ...

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### Extending prime ideals that lie over the same prime ideal via monomials

Let $(R, \mathfrak{m})$ be a local (Noetherian) ring containing the rationals. Then the formal power series ring $\mathbb{Q}[[x_1, \ldots, x_n]]$ naturally forms a subring of $R[[x_1, \ldots, x_n]]$. ...

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### A sequence of polynomials that the variety defined by every $n$ of them is small

Let $\mathbb{C}[x_1,x_2,\cdots,x_n]_{= d}$ denote the set of polynomials in $\mathbb{C}[x_1,x_2,\cdots,x_n]$ of total degree $d$.
Is there exists a sequence of polynomials $f_1,f_2,f_3,\cdots$ in $\...