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

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local rings with finite type maximal ideal

Let $A$ be a local ring with a maximal ideal $\mathfrak{m}$ finitely generated (not principal). Is there a sufficient condition for $A$ to be noetherian? For example, we know that the completion ...
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33 views

Extending and contracting an ideal by a faithfully flat homomorphism [migrated]

Let $ B $ be a faithfully flat $ A $-algebra. Let $ I \subset A $ an ideal. Shows that $ IB \cap A = I $. This is the second item of Exercise 2.6, Chapter 1, of the Qing Liu's book Algebraic Geometry ...
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99 views

Fermat's Theorem on p = a^2 + b^2 [migrated]

I have read that Fermat predicted that for an odd prime $p$, $p = a^2 + b^2$ iff $p = 1$ mod 4. I heard that such a criterion could be possible for a given integer $n$ like $p = a^2 + n b^2$ ...
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1answer
80 views

Relation between intersection multiplicities

Consider $(f_1,\dots,f_n), (g_1,\dots,g_n)\in \mathbb{C}[z_1,\dots,z_n]\ $ such that: i) $\{f_1=\dots=f_n=0\}= \{g_1=\dots=g_n=0\}=\{0\}\in \mathbb{C}^n\ $ and ii) $f_1g_1+\dots+f_ng_n\equiv0$. ...
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1answer
243 views

Valuations on tensor products

Let $A$ be a commutative ring, $B$ (resp. $C$) be a commutative $A$-algebra endowed with a valuation $v$ (resp. $w$), not necessarily of rank 1. Assume that $v$ and $w$ induce equivalent valuations on ...
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91 views

Does the ring of invariants inherit normality?

Let $A$ be a normal ring (in the sense that its localizations at prime ideals are normal domains), and suppose that a finite group $G$ acts on $A$ by ring automorphisms. Form the subring $A^G \subset ...
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1answer
262 views

When is a power of an indeterminate in an ideal with 2 generators?

If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...
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1answer
282 views

A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...
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2answers
79 views

Ascending chain condition on radical ideals

There is a basic theorem in the geometry of schemes saying that the Spec of a Noetherian ring is a Noetherian topological space. It can be formulated as the ACC condition implies the ACCR condition ...
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1answer
174 views

When two determinantal ideals together generate a power of the maximal ideal?

(A somewhat technical question, but maybe it is well known.) Consider matrices over the ring $k[[x_1,\dots,x_n]]$, whose entries vanish at the origin (i.e. belong to the maximal ideal ...
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2answers
131 views

Divisibility of the degree of an extension by the degree of its residual field

Let $A$ be an integrally closed domain whose quotient field is $K$, $L$ be a finite Galois extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M_A$ be a maximal ideal of $A$, and ...
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1answer
248 views

Is the ring of invariants Noetherian?

Let $R$ be a complete regular local ring whose residue field is perfect. Suppose that a finite group $G$ acts on $R$ by ring automorphisms in such a way that the induced action on the residue field is ...
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1answer
110 views

Properties of Integral Closure [on hold]

Definition(Integral closure): Let $R$ be a ring and $I$ an ideal of $R$. An element $x$ is said to be integral over $I$ if $x$ satisfies a monic equation $x^n + i_1x^{n−1} + ··· + i_n = 0$ such ...
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27 views

Unimodular triangulation and affine toric variety

Let $\mathcal{K}$ be a $pointed$ rational cone in $\mathbb{R}^d$ with extremal rays generated by $r_1,r_2,\dots, r_m\in \mathbb{Z}^d$. Here, pointed means that all $r_i$ lie strictly on one side of ...
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77 views

flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion. If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat. If $A$ is not noetherian, ...
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303 views

Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables

Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function $P_i = px^i + qy^i + rz^i$ where $x,y,z$ are coordinates. I have a few related ...
6
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2answers
187 views

Homological criterion for $A(B \cap C) = AB \cap AC$?

Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? I mean a statement such as "the given equation holds if and only if (some $\operatorname{Tor}$, ...
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1answer
167 views

Relation between intersection and product of ideals

Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...
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1answer
248 views

When is $X \rightarrow \text{Spec}(C(X))$ a homeomorphism?

Let $X$ be compact Hausdorff topological space. Consider the ring $C(X)$ of continuous functions $X \rightarrow \mathbb C$ (we do not consider the C* algebra structure, just consider $C(X)$ as a ring) ...
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1answer
97 views

if $R$ is Noetherian local with a finite module of finite injective dimension and if “?” , then $R$ is *Gorenstein*

I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay. Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete ...
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1answer
125 views

Noetherianess of a finite module over a noethrian ring without Axiom of Choice

All rings are assumed to be commutative with 1. We say a module over a ring is strictly noetherian if every non-empty set of submodules has a maximal member. We say a ring is strictly noetherian if it ...
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52 views

The kernel of the residue map before passing to Milnor's K-theory

Let $F$ be a field of zero characteristic. All groups are taken modulo torsion. Consider a residue map from the exterior algebra of the multiplicative group of the function field of the projective ...
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1answer
175 views

Generators vs minimal degree polynomials of ideals

Given an ideal $I$ of $\mathbb{R}[X_1,X_2,X_3,X_4,X_5]$ generated by two unknown polynomials. I know two homogenous polynomials $p_1 \in I$ and $p_2 \in I$ such that $p_1$ is of degree 2 and up to a ...
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65 views

Are all (graded) Artinian complete intersections like this?

I'm trying to prove some stuff (it's not important what) about (graded) Artinian complete intersections $R=\mathbb{C}[x_1,\ldots,x_n]/I$, where the $x_i$ have certain positive weights and where $I$ is ...
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65 views

How far finiteness dimension can be from edges? Example for $f_m(S/I)\ge depth S/I+2$

Let $ (R,m) $ be a commutative unital noetherian local ring (with $m$ as its maximal ideal), $ I $ an ideal of $ R $, and $ M $ a finite $R$-module with $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)\ ...
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0answers
67 views

“Exceptional components” of the exceptional divisor of a blow up

Assume we are blowing up an ideal $I$ on an affine variety $X$, let $E$ be the exceptional divisor, and $P$ be a (closed) point in $V$, the zero set of $I$. Is there any algorithm to check that $E$ ...
6
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1answer
257 views

Find a polynomial not in any ideal generated by polynomials of total degree $o(n)$

Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ ...
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29 views

normality of truncated arc space

Let $X=Spec(A)$, with $A$ a normal $k$-algebra of finite type, $k$ is a field. For any integer $n$, let $X(k[t]/(t^{n}))$ the $n$-th truncated arc space, is it also normal? Same question for ...
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3answers
149 views

canonical module can be identified with an ideal. how can one reach that ideal?

Let $R[[X,Y,Z]]/(X,Y)\cap (Y,Z)\cap(X,Z)$. then $R$ is Cohen-Macaulay ring and has a canonical module, $K$. By Proposition 3.3.18 of Bruns_Herzog, $K$ can be identified with an ideal in $R$. So we ...
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84 views

Weierstrass division theorem for henselian rings

Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...
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1answer
242 views

Projective Modules/Algebras: decomposition of linear functions, and the rank formula

Let $A$ be a ring, $B$ a finite projective $A$-algebra, and $C$ a finite projective $B$-algebra. We can show that $C$ is also finite and projective when regarded as an $A$-algebra (by, for instance, ...
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1answer
257 views

Invertibility of a matrix whose entries are certain binomial coefficients

Let $l$ be a positive integer. Does the matrix $$ M_l \ := \ \left( \binom{l-(2p+1)}{j} \right)_{0\leq p,j \leq[(l-1)/2]} $$ have nonzero determinant?
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1answer
103 views

Graded version of Baer's Criterion

Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a ...
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1answer
534 views

geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

Let $R$ be a local Noetherian ring. What is the geometric interpretation of: 1- Gorenstein rings 2- Complete intersections 3- Regular rings? and how can I realize differences by geometric ...
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2answers
206 views

Triangulations of special polyhedra

Let $A_1,A_2,A_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e_1,e_2,e_3\}$ is the standard basis in $\mathbb{R}^3$, we can ...
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2answers
1k views

Does completion commute with localization?

Suppose $A$ is a Noetherian (not necessarily local) ring and $\mathfrak{m}\subset A$ a maximal ideal. Then is it true that $$\hat{A}_{\hat{\mathfrak{m}}}=\widehat{A _{\mathfrak{m}}},$$ where hats ...
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2answers
330 views

Maximal ideals of the algebra of measurable functions

It is a well-known fact that for any compact topological space $\rm S$, the set its points is in bijection with the set of maximal ideals of $\mathcal C(\rm S)$, the algebra of continuous functions ...
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0answers
90 views

A Characterization of Closed Ideals in $C^{\infty}(\mathbb{R}^n)$

The space $C^{\infty}(\mathbb{R}^n)$ can be turned into a topological ring using the Whitney topology. Whitney's Spectral Theorem says that the closure of an ideal in this ring is the ideal of all ...
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168 views

intuitive interpretation of the multiplicity and a reference [migrated]

Although logically i can understand and use multiplicity, yet, the concept of multiplicity of a module is not completely clear for me. I wonder what idea is behind this definition and What is ...
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161 views

Algebraic characterization of commutative rings of Krull dimension 1,2, or 3

A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about the algebraic structure of ...
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2answers
58 views

Does $fd(M)\lt \infty$ and $id(M)\lt \infty $ imply that $R$ is Gorenstein?

$(R,m)$ is a local Noetherian ring. $M$ is a nonzero finite $R$-module of finite injective dimension($id$). It is known that if $R$ is Gorenstein, then $M$ has finite flat dimension ($fd$). I wonder ...
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5answers
3k views

Is there a “geometric” intuition underlying the notion of normal varieties?

I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot. One thing that strikes me ...
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75 views

: References on complete intersections rings

As we know, for the local ring, we have regular $\subset$ complete intersection $\subset$ Gorenstein $\subset$ Cohen-Macaulay but it seems that the refenences of complete intersection are more ...
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1answer
133 views

Extension of homomorphism to place in quotient field, or of local ring to valuation ring in quotient field

Let be given a domain $R$ (that can be supposed to be integrally closed if this can help), and $\varphi$ an homomorphism of $R$ into a field $F$. $\varphi$ extends uniquely to a homomorphism ...
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1answer
61 views

Canonical module of rees algebra

[Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if $I=(f_1,\ldots,f_g)$ is an ideal generated by a regular sequence with $g\ge 2$ then the ...
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1answer
210 views

Uniform Artin-Rees

The Artin-Rees lemma states that if $R$ is a Noetherian ring, $I \subseteq R$ is an ideal and $N \subseteq M$ are finitely generated $R$-modules, then there exists an integer $k$ such that for every ...
3
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1answer
327 views

Reference for comparison of heart cohomology with standard cohomology

I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations). Let A,B be two hearts of ...
4
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1answer
220 views

Valuation of an ideal in a two-dimensional regular local ring

Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field. What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we ...
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1answer
255 views

Simple Question on Injective Hulls

Let $R$ be a noetherian local ring with maximal ideal $\mathcal m$ and denote by $E$ the injective hull of the residue field $k$. Then, as an $R-$module, what is the support of $E$?
4
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1answer
124 views

Number of generators of colon and intersection ideals of two finitely generated ideals in a Prufer domain

Hi! I'm wanting to see why the following is true: Given 2 finitely generated ideals $B$ and $C$ in a Prufer domain $D$ with bases of $n$ and $m$ generators respectively, $B\cap C$ has a basis of ...