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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How much can we say about the number of nilpotents in a finite local commutative ring?
A commutative ring is local if it has a single maximal ideal. If the ring is finite, this implies that all elements are either units or nilpotents. Further, all finite local rings have prime power ...
- 1,793
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52
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Cohen-Macaulayness of inseparable isogeny k-algebras
Let $R$ and $S$ be 2 associated, commutative, and unita $k$-algebras where $k$ is an algebraically closed field of characteristic $p$. We call these algebras inseparable isogeny or $F$-isomorphism if ...
- 127
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Characterizing non-singularity of varieties through properties of their derivations
I am interested in knowing about the possible implications between the following properties of a commutative, complex algebra:
Its spectrum is non-singular.
Its derivation module is projective and ...
- 393
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87
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Algorithm for computing basis of zero dimensional ring?
If given a zero dimensional ring over a field, for example, a polynomial ring $A=k[x_1,\ldots,x_n]/(f_1,\ldots,f_n)$ such that $A$ is 0-dimensional, is there an algorithm to compute a monomial basis ...
- 1,157
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261
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On a characterization of the symbolic square of prime ideals in polynomial rings
If $R=k[x_1,...,x_n]$ is a polynomial ring in $n$ indeterminates over a field $k$ of characteristic $0$, there is a characterization of the symbolic square of a prime ideal $P$ (the $n$-th symbolic ...
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236
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Terminology question - "Chern number"
I have seen the term Chern number used to refer to the first Hilbert-Samuel coefficient, $e_{1}(I)$, of an ideal $I$ in a local ring $(R, m)$. (Where the Hilbert-Samuel polynomial agrees with $\...
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190
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"Unknot" algebraic set defined by two mutually dependent set of variables
Let $n$ be an integer $\geq 4$, and let $V \subseteq {\mathbb C}^{2n-1}$ be the set of all
$(a_1,a_2, \ldots ,a_n,b_1,b_2, \ldots ,b_{n-1}) \in {\mathbb C}^{2n-1}$ such that the derivative of the ...
- 3,595
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Sufficient Conditions for Finiteness of a Module
In order to prove that a module is finite we can find a finite set of generators, or an exact sequence in which flanking terms are finite; or use Nakayama's lemma to drive a contradiction . I'm ...
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276
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Generalizations of divided-power algebras over finite fields
In Andrews, Askey, and Roy's Special Functions, the authors state that Gauß sums are finite field analogs of the $\Gamma$-function as Jacobi sums are to B-function. The $\Gamma$-function is well-...
- 1,049
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388
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is there a notion of weakly noetherian?
A left module M over a ring is finitely presented iff the functor Hom(M,_) preserves filtered inductive limit. Meanwhile, if M is finitely generated, then for every filtered system { N_i } of left ...
- 403
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450
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Rosenlicht differentials for possibly non-reduced curves
Let $X$ be proper Cohen-Macaulay scheme of pure dimension 1 over an algebraically closed field $k$. When $X$ is moreover reduced, Rosenlicht's theory of regular differential forms gives a beautiful ...
- 1,609
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333
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Localization of module
M an A-module, $S\subset A$ a multiplicative subset. Is it possible for $S^{-1}M$ to have an $S^{-1}A$-module structure satisfying $\frac{a}{1}\cdot\frac{m}{1}=\frac{am}{1}$ other than the "usuall" ...
- 2,857
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Does there exist a commutative ring R such that SL_3(R) and SL_2(R) have the same finite subgroups?
This question is inspired, of course, by this question, and I don't know enough commutative algebra to know whether it's answered by silence dogood's answer to this follow-up question. If the answer ...
- 118k
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Flatness of a homogeneous quasi-coherent sheaf on the formal plane from flatness on lines through the origin?
(This is a follow-up to a previous question; as often happens, I didn't include enough hypotheses, so I'm asking a new, more-likely-to-be-true question).
Consider the formal plane $\operatorname{Spec}...
- 44.7k
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267
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Is there a straightforward way to solve unmixed, homogeneous systems of polynomials?
I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry.
Suppose we have a system of $k\leq n$ polynomials in $\...
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Kahler differentials and the m-adic filtration
Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies $...
- 1,038
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98
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Explicit operations with correspondences
Let $X: y^2=f(x)$ be an hyperelliptic curve over a finite field $k$. Consider two non-fibral correspondences $C,D\subset X\times X$. It is well known that they induce endomorphisms $\phi_C,\phi_D:...
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ring-valued points of locally ringed spaces
of course, one should expect that the concept of ring-valued points is not well-behaved for locally ringed spaces (LRS). I want to see examples for this.
so consider $LRS \to Set^{Ring}, X \mapsto X(-...
- 63.1k
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325
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Obstructions for reduced embedded deformation of Artinian rings
Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline f)$...
- 30.5k
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107
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Regularity in terms of jets?
Is there any criterion of regularity for rings in terms of jets?
More precisely: It is known that a local ring $B$ (with some hypothesis) is regular if and only if the module of differentials $\...
- 55
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238
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relative flatness and torsion freeness
Hi.
Question 1: let $f:X\rightarrow S$ be a proper and surjective morphism of complex reduced spaces with $X$ pure dimensional. Let $F$ be a $S$-flat coherent sheaf on $X$. Is it true that the two ...
- 435
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152
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Characterization of a "Jacobian pair" member
Consider the ring ${R}$ of Puiseux series in $Y$ with coefficients in the ring $\mathbb{C}((X^*))$ of Puiseux series in $X$ with coefficients in $\mathbb{C}$; $F \in R$ is said to be a member of a ...
- 96
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134
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Infinitesimal lifting for hensel schemes?
I have local hensel ring $A$ and a finite flat $A$-algebra $B$ (which is therefore a direct product of local henselien $A$-algebras) and I would like a section of the canonical map $B \to B / N$ where ...
- 1,347
4
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310
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Solvable subgroups of groups of polynomial automorphisms
Does every finitely generated free solvable group embed into the group of polynomial automorphisms of some C^n?
- 41
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71
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Integral Leray Number?
The Leray number of a finite simplicial complex $K$ relative to a field $\Bbbk$ is defined to be the least $d\geq 0$ such that $\widetilde H^n(C,\Bbbk)=0$ for all $n\geq d$ and all induced ...
- 38.6k
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Moduli space of modules with fixed length
Let $R$ be a (commutative) local Artinian ring, with an algebraically closed residue field $k$. I am interested in the set $L_n(R)$ of isomorphism classes of $R$-modules of length $n$.
If $R$ is a $k$...
- 30.5k
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219
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Name for a module with only one associated prime
In EGA IV2, Def. 3.2.4, Grothendieck defines a quasicoherent sheaf over a locally Noetherian scheme to be "irredondant" if it has a unique associated point. Presumeably, a module over a Noetherian ...
- 7,338
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198
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why a reduced ring can be embedded into a sum of integral rings?
Hi,
the question is exactly
"why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?"
Is this simply because in the normalization process we can have many irreducible ...
- 141
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233
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When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?
Let V be a finite dimensional symplectic vector space over $\mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is
generated by symplectic reflections, i.e. by elements $g\...
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Rational map defined over K leads to algebra question
Hello,
Concrete algebraic question : Let $K$ be a perfect field, $\bar{K}$ a fixed algebraic closure and let $f \in \bar{K}[x_1,\ldots,x_n]$. I was wondering when there exists another polynomial (non-...
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Krull rings and determinantal invariants
During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
- 22.1k
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96
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Explicit expression for multivariable meromorphic series
Let $K$ be a field. For $m>1$, elements in the ring of power series in $m$ variables over $K$, i.e., $K[[X_1, \cdots, X_m]]$ look like -
$$ \displaystyle\sum_{(i_1, \cdots, i_m)\in (\mathbb N_{0})^...
- 899
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Seek for good methods of computing the Krull dimension of a module?
Hi, everyone. Recently I am interested in computing the Krull dimensions of modules without using any software. However, it is not an easy job for me to do so only by its definition. Therefore, I ...
- 1,207
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234
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Flatness of module
$A\rightarrow B$ a ring homomorphism, $N$ a $B$-module which is flat over $A$. $\mathfrak{q}\subset B$ a prime ideal, $\mathfrak{p}\subset A$ its contraction in $A$. Then is it true that $N_{\mathfrak{...
- 2,857
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When can we discard higher order terms in a set of ideal generators?
Let $A=k[x_{1},...,x_{n}]$ be a polynomial ring over a field $k$, let $f_{1},...,f_{m} \in A$ be polynomials, and let $I=(f_{1},...,f_{m})$ be the ideal that they generate in $A$. Now suppose that we ...
- 1,329
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138
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Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
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Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
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169
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Sum of two free o-submodules in a vector space over a local field
Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...
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Standard system of parameters and an example
Let $(R,m)$ be a local Noetherian ring. A system of parameters $\bf{x}$$:=x_{1}, \dots, x_{d}$ is a standard system of parameters if $(\bf{x})H^{i}_{m}(R/(x_{1}, \dots, x_{j}))=0$ holds for all non-...
- 113
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87
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Standard Notation for Monomial Orderings?
Is there a standard way to denote a particular lexicographic (resp. reverse lexicographic) monomial ordering using subscripts or superscripts? For example, I might want to refer to the lexicographic (...
- 326
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180
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Generic Rank of R^{1/p}
Suppose $R$ is a local Noetherian domain of dimension $d$ in characteristic $p>0$. Suppose $R^{1/p}$ is a finitely generated $R$-module, and suppose $k$ is the residue field of $R$. Is the ...
- 103
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Global dimensions of orders over non-Gorenstein centers
This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\...
- 30.5k
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164
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How to design or create or generate a bijective ring map? [closed]
How to design or create or generate a bijective ring map?
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165
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Support sets along a ring homomorphism.
Let $(R,m)$ and $(S,n)$ be commutative noetherian local rings, and $f: R\rightarrow S$ be a local homomorphism (i.e., $f(m) \subseteq n$) with $S$ flat as $R$-module. If $M$ is a finite generated $R$-...
- 1,207
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Ideals weak equivalence and "finite" equivalence
Let $R$ be an order in a number field. Two $R$-ideals $I$ and $J$ are weak equivalent if there exist (necessarily invertible) ideals $X$ and $Y$ such that $I X=J$ and $J Y=I$.
This is equivalent to ...
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Formalization of the independance of products in a (commutative) semigroup
1/ It is well known that associativity implies that the result of the product of an ordered finite set of elements in a semigroup does not depend of the order of composition of the partial products.
...
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