Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
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Transformation of a bivariate polynomial into a homogeneous one
For a given a bivariate polynomial $P(x,y)$ with rational coefficients:
Q1. How compute such (invertible) substitutions of its variables that would transform the polynomial into a homogeneous one? In ...
- 34.3k
9
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793
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Reconstructing a polynomial from resultants
I am trying to compute a monic polynomial $f(x)$ with integer coefficients and known degree $d$. I am given $n$ pairwise coprime polynomials $g_1(x),\ldots,g_n(x)$, also with integer coefficients, ...
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296
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What is your expectation of the depth?
Let $S=k[x_1,...,x_9]$ be a polynomial ring over field $k$. Set $q_1=(x_1,x_2,x_5,x_6)$, $q_2=(x_1,x_2,x_6,x_7)$, $q_3=(x_2,x_3,x_7,x_8)$, $q_4=(x_1,x_5,x_6,x_7)$, $q_5=(x_1,x_6,x_7,x_8)$, $q_6=(x_2,...
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358
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Classes of fields and Cantor-Schröder-Bernstein
In what classes of fields does CSB hold? That is to say, in what classes of fields is it true that if there exist embeddings $F\to K$ and $K\to F$ then $F$ and $K$ must be isomorphic?
I know this ...
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457
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About subspaces of $F$-spaces
A topological space $X$ is an $F$-space, if Every finitely generated ideal in the ring of all continuous functions on $X$,denoted by $C(X)$, is principal. The text "Rings of continuous functions" ...
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718
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Bloch-Kato conjecture and Wiles' numerical criterion
I already asked this question some days ago on https://math.stackexchange.com/questions/158747/bloch-kato-conjecture-and-wiles-numerical-criterion but didn't receive any response.
In the ...
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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)\ ...
- 1,355
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203
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Flatness and intersections of Cohen-Macaulay subvarieties
There's a commutative algebra fact that I would very much like to be true but could, for all I know, be completely false. One version that would be sufficient is:
Say $A$ is a smooth projective ...
- 1,510
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477
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Comparing the homogeneous defining ideals of multiple embeddings of a projective scheme
If $X$ is a projective scheme over a field $k$ (which we may assume is algebraically closed), then under an embedding $i: X \hookrightarrow \mathbb{P}^n_k$, we may write $X = Proj(R/I)$ where $R = k[...
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Chain of ideals in a complex algebra
Suppose $\mathfrak{A}$ is an unital algebra over complex numbers and $\mathfrak{J}$ is chain of left-ideals in $\mathfrak{A}$ ordered by inclusion such that none of its elements is countably generated....
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How many DISTINCT vectors we get from pairs v_i + v_j for some set of given vectors v_i ?
Consider some set of vectors v_i i=1...N , v_i \in Z^k.
e.g. N = 10^4; k = 10
Consider all possible sums: v_i + v_j.
Is it possible to estimate how many DISTINCT vectors we get in advance without ...
- 24.9k
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Why is algebraic de Rham cohomology via completion independent of embedding?
In Hartshorne's "On the de Rham cohomology of algebraic varieties", he defines algebraic de Rham cohomology of a variety $X$ over a field $k$ of characteristic zero by choosing a closed immersion $X \...
- 1,329
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Turning a resolution of an algebra into a resolution of its field of fractions
Is there a way to turn a free resolution of a $k$-algebra $A$ into a resolution of the field of fractions $Q(A)$?
Specifically, I'm interested in the ring of polynomials in two variables: $A = k[x,y]$...
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Elementary proof that $\operatorname{Ass}(I^n)$ stabilizes for a monomial ideal $I$
For my bachelor thesis I'd like to have a short "elementary" proof that $\operatorname{Ass}(I^n)$ stabilizes for large $n$ if $I$ is a monomial ideal in a polynomial ring $K[x_1, \dots, x_r]$ over ...
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427
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Reference Request: Smith Normal Form for maps between free _graded_ modules
I feel like this should be easy, but I cannot quite find a literature reference for this:
We know (i.a. from the Kaplansky reference in Does Smith normal form imply PID?) that sufficient for Smith ...
- 2,537
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94
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Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?
Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...
- 853
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survival of a prime ideal in its Nagata transform
Let $R$ be a Noetherian normal domain with fraction field $K$. Recall that for any ideal $I \subseteq R$, its Nagata transform $T(I)$ is defined as the set of elements $f\in K$ such that $I^n f \...
- 1,802
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551
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sub ring of algebra over subfield
Let $k$ be a field and $k[a]$ an algebric extension.
If $A$ is a reduced commutative algebra over $k[a]$ and $B$ is a subring which is an algebra over $k$, then is the following true: if there exist ...
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79
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Geometric/algebraic interpretation of quadratic points of rank r
In the paper of Eckl/Puhklikov (http://arxiv.org/abs/1210.3715) the following terminology is introduced:
"
Let $X \subset Y$ be a subvariety of codimension 1 in a smooth quasiprojective
complex ...
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383
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Pseudo-cauchy sequence and valuation
Let $k$ be a field and $x$ is transcendental over $k$. Can we construct a pseudo-cauchy sequence $(a_{i})$ convergent to $x$ with each $a_{i}$ is algebraic over $k$ and $k(a_{i})\subseteq k(a_{i + 1})$...
- 173
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245
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Degree principles for non-symmetric polynomials
A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...
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75
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Question related to $h$-invariant of a form
Let $k$ be a field.
Given a form $f \in k[x_1, ..., x_n]$ of degree at least $2$, we define the
Schmidt rank, also known as the $h$-invariant, $h_k(f)$ to be the
least positive integer $h$ such that $...
- 579
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Is an weakly finite R-module Serre subcategory of the category of R-modules?
A definition for weakly finite $R$-modules is as follow:
Definition: Let ($R$,$m$) a local ring. Let $S$ be the largest class of $R$-modules satisfying the following four properties:
(1) If $M \in S$...
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A fast way to decide satisfiability of a set of simple fewnomial inequalities?
Background
Considering a set of points $(x_i, y_i)$ in $\mathbb R^2$ and constraints between some triples of them, which state, whether the three points of the triple are oriented clockwise (R), ...
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359
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On the comparison of linear topologies on a local ring
Let $R$ be a local ring, $a_{\lambda}$ be a decreasing net of ideals, indexed by a directed set, such that each $a_{\lambda}$ is contained in the nilradical ideal and $\bigcap a_{\lambda}=(0)$. Then ...
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a question about Beauville-Laszlo
Hi,
let $V$ be a complete DVR with uniformizer $\pi$. Let $m$ be a NON zero integer, $a\in V[[u,v]]/(uv-\pi)^{\times}$ and $f=\pi^{m}a$. Consider $F$ as the kernel of the diagram
$$
V[[u,v]]/(uv-\pi)...
- 11
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Reference request on Leray numbers
The Leray number $L_{\Bbbk}(K)$ (relative to a field $\Bbbk$) of a simplicial complex $K$ is the least $d\geq 0$ such that $\widetilde H_n(C,\Bbbk)=0$ for all $n\geq d$ and all induced subcomplexes $C$...
- 38.6k
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Nonequivalent extensions with the same terms
I just construct an exact sequence $0\to M\to M\oplus N\to N\to0$ of $\mathbb{Z}$-modules that does not split, where $M=\mathbb{Z}$, $N=(\mathbb{Z}/2\mathbb{Z})^\\mathbb{N}$, and the map from $M$ to $...
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Relation between dimension of Proj(S) and dimension of S
Let $S$ be Noetherian standard ${\mathbb{N}}^r$ graded ring where $S_{\underline{0}}$ is an Aritinian local ring.
$$Proj(S)=\lbrace{P\in Spec S | S_{++}\not\subseteq P, P\hspace{0.1cm} homogeneous}\...
- 127
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Ring isomorphism $\Phi \colon C(X) \to C(Y)$ and zero dimensionality of $X$
We denote the ring of all continuous real-valued functions on $X$ by $C(X)$. The ring of all bounded continuous real valued functions on $X$ is denoted by $C_b(X)$. One of the goals of the study of $...
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Non-crystallographic cluster algebras
Background
Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed $(\mathbf{...
- 2,600
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krull dimension [closed]
im looking for a non-noetherian ring with infinite krull dimension.would you help?
- 15
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the "in" function: M->gr(M) from David Eisenbud's Commutative Algebra book
I'm reading David Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry (3rd ed.).
in chapter 5, about filtrations and the Artin-Rees lemma, the function $in: M \to gr(M)$ was defined ...
- 274
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Lifting results from smooth maps to essentially smooth maps.
Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.
(Note: $R\to S$ is essentially finitely presented provided that $...
- 19.6k
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In what generality is the natural map $\operatorname{Hom}_R(L,M)\otimes S\to\operatorname{Hom}_{R\otimes S}(L\otimes S,M\otimes S)$ an isomorphism?
Let $k$ be a commutative ring, $R$ and $S$ commutative $k$-algebras. Let $L$ and $M$ be $R$-modules. Consider the natural map
$$\operatorname{Hom}_R(L,M)\otimes_k S \to \operatorname{Hom}_{R \...
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Extending properties of commutative rings to schemes
I'm trying to pin down the various ways we can extend a property of commutative rings to a corresponding property for schemes. Let $P$ be a property of commutative rings. We could define a scheme $(X,\...
- 2,814
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Top degree local cohomology under action by a non-zerodivisor
Let $R$ be a noetherian commutative ring of dimension $n$, and let $M$ be a faithful finite $R$-module. Let $I$ be a proper ideal of $R$, and let $x\in I$ be a non-zerodivisor on $M$.
When does ...
- 19.6k
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example re torsionless quotients of abelian groups
Recall that an abelian group $G$ is $Z$-torsionless if for all $a\in G$ ($\neq 0$) there is a homomorphism of $\phi\in Hom(G,Z) = G^*$ so that $\phi(a)\neq 0$
Suppose $S$ is a subgroup of torsionless ...
- 345
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Generalizing Krull's Principal Ideal Theorem to Modules
Let $R = \mathbf{C}[x_1, \ldots, x_n]$ and let $M$ be a graded $R$-module which is finite-dimensional over $\mathbf{C}$ and suppose
$
0 \leftarrow M \leftarrow R^g \leftarrow R^d \leftarrow \cdots
$
...
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707
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Are all henselian fields algebraic over complete fields?
Motivations and Terminology
The term "henselian field" is ambiguous. What I mean when I say that $K$ is a henselian field is that there exists a henselian DVR $R$, such that $K=Frac(R)$. What I mean ...
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how to prove that localisation preserves Hom's [closed]
Can anyone tell me where I can read a proof that the natural map
$Hom_{A}(M,N)[S^{-1}]\rightarrow Hom_{A[S^{-1}]}(M[S^{-1}],N[S^{-1}])$
is an isomorphism if $M$ is finitely presented?
- 2,125
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130
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Explicit equations for conormal bundle to an affine toric variety
Let $L \subset \mathbb{Z}^n$ be a lattice and let $X_L$ be the closed toric subvariety of $\mathbb{C}^n$ cut out by the lattice ideal $I_L = \{x^{l_+} - x^{l_-} \,| \, l_+, l_- \in \mathbb{N}^n \text{ ...
- 1,543
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Could a non-algebraically closed PAC field be a finite extension of an ordered field?
Is there such an example? Or it is known that a pseudo algebraically closed field which is a finite extension of a formally real field is algebraically closed?
2
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307
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On a Strongly F-regular Pair (X, \Delta)
Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. ...
- 165
1
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268
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Flatness on the formal plane from flatness on lines through the origin?
Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of ...
- 44.7k
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425
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Are maximal Cohen-Macaulay modules supported everywhere?
Let $A$ be a local CM ring, and $M$ a maximal CM $A$-module. Is it true that $\operatorname{Supp}M=\operatorname{Spec}A$ ? This suspicion stems from such statements as:
If $\omega$ is a canonical ...
- 2,857
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578
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When does grading pass to (co)-homology?
Let $G$ be an abelian group and let $R$ be a $G$-graded commutative ring, i.e., $R=\oplus_{g\in G} R_g$ with $R_gR_h\subseteq R_{g+h}$. Let $M$ be a $G$-graded $R$-module
i.e. $M=\oplus_{g\in G}M_g$ ...
- 7,609
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1
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633
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Determining if a ring satisfies Serre's condition S_{n}
Given a specific ring $R$ (eg, $R=k[x_{1}, \cdots, x_{n}]/I)$ is there a (simple) way to determine whether or not $R$ satisfies Serre's condition $S_{n}$? In particular, is there a way to do this in ...
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76
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Minimal rank of a permutation resolution of a $G$-lattice
Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...
- 14.2k
1
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1
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927
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Torsion-free and torsionless abelian groups
This question is motivated by my most spectacular answer on MO (:
Let $A$ be a module over $\mathbb Z$. $A$ is said to be torsion-free if $na=0$ implies $n=0$ or $a=0$ for any $n\in \mathbb Z, a\...
- 30.5k