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

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43 views

Profinite Local Ring inside Polynomial Ring

This is a "technical" question that I came across in my research. Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...
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0answers
22 views

Depth of multigraded modules

Can any one please give me some references on depth of multigraded module over a standard multigraded ring.
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1answer
91 views

Idempotent fractional ideals of a Noetherian domain

Let $R$ be a commutative Noetherian domain, $K$ its fraction field, and $J$ a fractional ideal (i.e. a finitely generated sub-$R$-module of $K$) such that $J^2=J$. Is it true that $J=0$ or $J=R$? If ...
1
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0answers
68 views

How to show non-existence of elements in the intersection of two ideals?

Given l, k any two natural numbers, define $I_1 =\langle y^{l+2k}, (x+y)^{3l+2k} \rangle: x^{l+2k} + \langle y^{2l+3k}, (x+y)^{3l} \rangle: x^{l+2k}; $ $I_2 = \langle x^{l+k} \rangle.$ I want to ...
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1answer
405 views

Algebraic Closure of a Ring is Not a Ring?

I'm trying to motivate the notion of integrality in a ring extension. It seems that the following would be a good motivation, because it would show that the notion of algebraic elements over a ring is ...
7
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1answer
151 views

Semiring of vector bundles on $\mathbb{C}\mathbb{P}^1$

Consider the semiring $$\mathbb{N}[H,H^{-1}]/(H^p+H^q = H^{p+q}+1)_{p,q \in \mathbb{Z}}.$$ Is it finitely presentable? Is there any simplification of the relations (except for $p \geq q \geq 0$)? ...
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1answer
74 views

Variety of commutative semi group [on hold]

V is a variety of commutative semi group satisfying the identity $x^2 = x^3$. I need to prove that: $|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$. Any hints on this ? $F_V$ is V-free algebra.
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60 views

showing that a set of linear forms is closed (Bruns and Herzog, Theorem 4.2.12) [migrated]

Let $k$ be an infinite field and $R$ a homogeneous $k$-algebra, i.e. a $k$-algebra that is generated by linear forms. Let $s = \sup\left\{\dim_k h R_{n-1} : h \in R_1\right\}$, where $R_i$ denotes the ...
0
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0answers
10 views

Equating coefficients [migrated]

Heading Excuse me,i don't know how to deal with this problem,i try it for all time of last night, this equation is on "Concrete Mathematics" page 200: d(n) is the number of derangements,e^z is the ...
0
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0answers
10 views

What is the maximal ideal of $z[t,t^{-1}]\otimes Q$? [migrated]

I know the $z[t,t^{-1}]$ is a localization of $z[t]$.But I do not know the maximal ideal of $z[t,t^{-1}]\otimes Q$? Many thanks!
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2answers
128 views

Is there an intuitionistic generalized boolean algebra (of Stone)?

A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...
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0answers
50 views

Surjectivity of $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$

Let $R$ be a Noetherian ring and let $M$ is finitely generated $R$-module.Suppose $p$ is a minimal prime in $\text{Supp}_RM$. Then $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$ that $f(m)=m /1 $ is ...
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0answers
43 views

For Finite Dual when is $(A \otimes A)^o = A^o \otimes A^0$?

Let $A$ be any $k$-algebra. The finite dual or restricted dual of $A$ is $$ A^o = \{f \in A^* ~ | ~ f(I)= 0, \text{ for some ideal } I \subseteq A, \text{ such that } \text{dim}_k(A/I) < \infty\}. ...
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1answer
95 views

Can the property of essential finite type checked at a point?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra which is noetherian. Suppose that for each prime ideal $p$ of $A$, it holds that the field $k(p)$, the field of fractions of $A/p$ has ...
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6answers
936 views

Algebraic Geometry for non-mathematician [closed]

I think I sound stupid but I have heard a lot about Algebraic Geometry as a subject and wish to study it without actually studying abstract algebra. I have never studied abstract algebra since I am a ...
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0answers
148 views

Surjectivity of $f\colon\Gamma_Z(M)\rightarrow\bigoplus_{p\in Z\backslash Z'}\Gamma_{pR_p}(M_p)$

Suppose $Z'\subseteq Z\subseteq\text{Spec} R$ such that every element in $Z\backslash Z'$ is a minimal element (with respect to inclusion as ideals) in $Z$. Assume further that both $Z$ and $Z'$ are ...
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0answers
66 views

Graded-irreducible ideals are irreducible?

One knows that graded ideals in polynomial rings over a field are primary iff they are graded-primary. What about the irreducible ideals? Let $I$ be a graded ideal in a polynomial ring over a ...
10
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1answer
172 views

Homological criteria for finite generation and finite presentation of modules?

(I'm new here; if I'm doing something wrong please help me out.) In short, my question is: There are some results on the behavior of finite generation and finite presentation in exact sequences (of ...
3
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1answer
238 views

Can non-isomorphic field extensions be isomorphic fields?

This is related to my earlier question on isomorphism of general quotients of $\:F\hspace{.02 in}[x]\:$. Let $F$ be a field, let $p$ and $q$ be (non-zero) monic irreducible polynomials, let $I$ and ...
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0answers
50 views

Prime ideals of Z[x] [migrated]

how to build three prime ideals of Z [x] (P_1, P_2, P_3) as P_1 is strictly included in P_2 and P_2 and strictly included in P_3?
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1answer
121 views

Universal coefficient theorem for local ring

Let $R$ be a commutative local artin $k$-algebra,where $k$ is a field with characteristic $0$.I wonder whether universal coefficient theorem holds in this case.Namely,if $C$ is a chain of flat ...
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2answers
262 views

An affine singular surface

Let $n$ be a positive integer and let $A$ be the subring of ${\mathbb C}[x,y]$ generated by $x,xy,...,xy^n$. Let $S=Spec(A)$. This is an affine surface, which is clearly singular if $n\neq 1$. Is ...
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1answer
73 views

Computing the minimal free resolution of a coherent sheaf on projective space

Most books on commutative algebra explain Grobner bases in the non graded case and minimal free resolutions in the local case. I like projective geometry and want to compute the minimal free ...
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1answer
218 views

Can you overcome the 6th degree obstruction?

I read and am still thinking about a 3-year old paper from the Danish-Norwegian "Niels Abel Journal". Two authors, named Somethingson (not Jacobson) and another Somethingelseson (still not Jacobson), ...
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2answers
1k views

Is every Noetherian Commutative Ring a quotient of a Noetherian Domain?

This was an interesting question posed to me by a friend who is very interested in commutative algebra. It also has some nice geometric motivation. The question is in two parts. The first, as stated ...
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0answers
79 views

Relation between dimension of Proj(S) and dimension of S

Let $S$ be Noetherian standard ${\mathbb{N}}^r$ graded ring where $S_{\underline{0}}$ is an Aritinian local ring. $$Proj(S)=\lbrace{P\in Spec S | S_{++}\not\subseteq P, P\hspace{0.1cm} ...
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1answer
196 views

Doubt in this proof of Horrocks theorem

I'm beginning to study some research papers and I need right now to understand the solution of Vaseršteĭn of Serre's theorem (simplest proof of this theorem), to do so, I'm beginning to understand ...
1
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1answer
74 views

composition of Puiseux series?

Formal series over a field (or ring) k can be composed in the following sense: given $y \in k[[x]]$, $y$ lying in the maximal ideal, there exists a unique map of topological rings $k[[x]] \to k[[x]]$ ...
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0answers
79 views

Two questions regarding $f$-adic completions of (non noetherian) rings

Lately I've looked at $f$-adic completions of commutative rings. I had posted two questions regarding the topic on math.SE which didn't receive any attention and I think they might be fit for ...
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1answer
63 views

reduction of an admissible filtration

Let $(R,m)$ be a local ring and $I$ an $m$-primary ideal of $R.$ $\lbrace I_n\rbrace_{n\in\mathbb{Z} }$ is called $I$-admissible filtration 1) if $m\geq n$ then $I_m\subset I_n.$ 2) for all $m,n,$ ...
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2answers
198 views

local cohomology mayer-vietoris sequence

(I originally asked this question on Math.SE here. As suggested on meta.MathOverflow (posting an unanswered Math.SE question on MathOverflow), I've waited about a week before reposting it here. Note ...
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1answer
87 views

local cohomology of Buchsbaum ring

Let $(R,m)$ be a Buchsbaum ring of dimension d. Can we say that $d$-th local cohomology $H_{m}^d(R)$ has finite length?
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1answer
82 views

Relation between local cohomology and koszul cohomology of multigraded ring

Let $R$ be a ${\mathbb {Z}}^k$-graded Noetherian ring, $J=(x,y)$ an ideal of $R$ where $x,y\in R_{(1,\ldots,1)}.$ Is this following true $$H_{J}^i(R)=\underset{n}\varinjlim{H^i((x^n,y^n),R)},$$ where ...
5
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1answer
569 views

A naive algebraic geometry question

Suppose $X$ is a scheme over a ring $A$, $B$ is an $A$-algebra, and $X\times_AB$ is affine. I am looking for conditions on $A$ and $B$ (and perhaps the structure morphism of $X$ over $A$) that will ...
2
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1answer
96 views

Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...
17
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2answers
342 views

Rings for which no polynomial induces the zero function

For any commutative ring $R$ let $R[x]$ denote the ring of polynomials with coefficients in $R$. Any polynomial $p \in R[x]$ naturally induces a function $\hat{p} :R \rightarrow R$. In some cases, a ...
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0answers
205 views

Spectral sequences and Koszul complexes in Deformation Theory

I'm reading this paper of A. Vistoli and I have some questions about the discussion in page 5. This is the context (If you don't want to download the paper): Let $A'$ be a noetherian local ring with ...
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2answers
90 views

Factorisation of a biquadratic polynomial

Let $u,v\in\mathbb{Z},$ and let $f=X^4+uX^2+v.$ Let $p$ be a prime number, and let $r\geq 1.$ In a paper I'm reading, one can find the following result. $\bf{Proposition. }$If $f$ is reducible ...
5
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1answer
198 views

Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...
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0answers
65 views

Computing toric ideals via saturation and Groebner bases of toric ideals

About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer. Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...
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136 views

Dimension of a commuting nilpotent variety

Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...
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131 views

a net of quadrics and the corresponding intersection

Let $Q_i(i=1,2,3)$ be quadric hypersurface in $\mathbb{P}^4$. Consider a net of quadrics $\Lambda=(Q_1,Q_2,Q_3)$. I can't understand some part of proof of Corollary 2.8(p.11) in Stability of genus 5 ...
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1answer
326 views

Study of convex polytopes via commutative algebra

Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...
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93 views

Submodul of finite ring extension

Let $R \hookrightarrow S$ be a finite extension of noetherian rings. Let $I \subseteq S$ be an $R$-submodule of $S$. Are there any sufficient criteria on $I$ such that it is in fact an ideal of $S$? ...
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1answer
127 views

Direct image of an ideal sheaf along a blow-up

Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
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24 views

Saturation of a subalgebra over the Tate-algebra inside the power series ring

Let $A$ be a discrete valuation ring and $\pi$ a uniformizer. Over $A$ we consider the Tate-algebra $$A\langle t \rangle =\{ f=\sum_{n=0}^\infty a_nt^n \mid a_n\in A, \lim_{n\to \infty} \lvert ...
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0answers
83 views

The standard topology of a module over a noetherian local ring

Let $A$ be a noetherian local ring with maximal ideal $m$. One says that an $A$-module $X$ is discrete if for every $x\in X$, there is a natural number $n$ such that $m^n.x=0$. My question is: Given ...
2
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1answer
151 views

Name and references for a “twisted” addition in a ring

This question may seem a little unmotivated, but it actually isn't something that just occurred to me out of nowhere. It stems from a discussion I had the other day with a friend and is directly ...
5
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2answers
278 views

Formal completion of the normal bundle

Let me for simplicity start with affine case. If $X=\operatorname{Spec}(A)$ is an affine variety $Z \subset X$ is a closed affine subvariety $Z=\operatorname{Spec}(A/I)$. What conditions are ...
5
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2answers
220 views

Hochschild homology of upper triangular matrix algebra?

Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$? Is there any ...