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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Embedding $K((X,Y))$ into $K((X))((Y))$

This question is closely related to Explicit elements of $K((x))((y))∖K((x,y))$. Recall that $K((X,Y))$ can be embeded into $K((X))((Y))$, althouth this embedding is not surjective. A natural question ...
Yijun Yuan's user avatar
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1 answer
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Krull dimension of the smooth locus

Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R_{\mathfrak{p}}$ is a regular local ring? In ...
Shravan Patankar's user avatar
3 votes
1 answer
262 views

Finite subschemes of projective bundles

$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Proj{\mathbf{Proj}}$Let $S$ be a Noetherian scheme and $X$ finite $S$-scheme. The finite morphism $X \to S$ is projective in the sense of the ...
Jordan's user avatar
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3 votes
0 answers
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Uniformly closed ideals of smooth/real analytic functions

Consider $U\subseteq \mathbb{R}^n$ an open subset and denote by $R$ either the algebra of real-valued smooth or real analytic functions on $U$. In either case suppose that $R$ is equipped with the ...
Thomas Kurbach's user avatar
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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 ...
SKS's user avatar
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10 votes
2 answers
329 views

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 ...
P-addict's user avatar
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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 ...
atssit's user avatar
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1 answer
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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^{...
Pierre Dubois's user avatar
2 votes
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144 views

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 ...
ABB's user avatar
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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 ...
Hephaistos's user avatar
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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) ...
Tom Copeland's user avatar
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2 votes
1 answer
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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{...
atssit's user avatar
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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 ...
M.G.'s user avatar
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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} ...
CARLO's user avatar
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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 (...
mathieu_matheux's user avatar
4 votes
1 answer
291 views

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}...
Nobody's user avatar
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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 ...
Mare's user avatar
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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 ...
S.D.'s user avatar
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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\...
GreginGre's user avatar
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9 votes
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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}\...
Richard Stanley's user avatar
1 vote
0 answers
84 views

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 ...
XYC's user avatar
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4 votes
1 answer
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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 $...
Damian Rössler's user avatar
2 votes
1 answer
158 views

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 ...
H U's user avatar
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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 ...
Ethan's user avatar
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2 votes
1 answer
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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 ...
Matt's user avatar
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1 vote
0 answers
75 views

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 ...
Amir Mafi's user avatar
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4 votes
0 answers
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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 \...
wlad's user avatar
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3 votes
0 answers
256 views

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 ...
Tim Campion's user avatar
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0 votes
1 answer
114 views

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 ...
user577413's user avatar
20 votes
3 answers
1k views

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, ...
Tim Campion's user avatar
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2 votes
2 answers
199 views

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 ...
SKS's user avatar
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6 votes
2 answers
520 views

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 ...
Tim Campion's user avatar
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2 votes
0 answers
80 views

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 ...
Cubikova's user avatar
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3 votes
0 answers
104 views

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 ...
Tom Copeland's user avatar
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4 votes
0 answers
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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 ...
Tim Campion's user avatar
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3 votes
0 answers
138 views

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 (...
M-M's user avatar
  • 31
11 votes
2 answers
615 views

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 \...
walkar's user avatar
  • 223
2 votes
0 answers
132 views

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 ...
gimothytowers's user avatar
5 votes
1 answer
235 views

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 ...
Jens Hemelaer's user avatar
4 votes
1 answer
181 views

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}(...
William Sun's user avatar
4 votes
1 answer
162 views

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,...
Loïc Teyssier's user avatar
5 votes
1 answer
241 views

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 / ...
Tom Copeland's user avatar
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1 vote
1 answer
115 views

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 ...
Yuting Li's user avatar
1 vote
1 answer
96 views

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 $...
Michail Karatarakis's user avatar
3 votes
2 answers
100 views

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 ...
Schemer1's user avatar
  • 701
2 votes
2 answers
714 views

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?
B.math's user avatar
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0 votes
1 answer
207 views

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 ...
Tom Copeland's user avatar
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5 votes
2 answers
236 views

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 ...
Colin Tan's user avatar
  • 243
4 votes
1 answer
220 views

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. ...
Haldot's user avatar
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