Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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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 $m,n,$ $...
- 127
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135
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Relation of primary decomposition of two ideals
I have a simple question: Let $R=\mathbb{C} \lbrace t,u \rbrace$ be the ring of formal series in two variables. Let $I,J \subset R$ ideals of heigth one, and $I \subset J$. What is the relation of the ...
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When does $R [x]/I $ have infinitely many idempotents in special case?
At < When does $R [x]/I $ has infinitely many idempotents? >, Er_Ro asked the following question.
Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for ...
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271
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Open covering of the Hilbert functor of points
Let $R \to A$ be a homomorphism of commutative rings. Define the functor $$\mathrm{Hilb}^n_{A/R} : \mathsf{CAlg}(R) \to \mathsf{Set}$$ as follows: If $B$ is a commutative $R$-algebra, then $\mathrm{...
- 63.1k
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740
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Vanishing of Tor
Let $(R, \mathfrak{m})$ be a commutative Noetherian local ring and $M$ a finitely generated $R$-module. Let $x_1,...,x_t$ be an $M$-regular sequence and $I = (x_1,...,x_t)$. Is it true that
$$\mathrm{...
- 2,773
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1
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167
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Given a locally nilpotent derivation over a field of characteristic 0 and a local slice, how is the ring homomorphism below defined?
Let $K$ be a field of characteristic 0 and $A$ a $K$-domain. Let $D:A\longrightarrow A$ be a locally nilpotent K-derivation, that is, $D(k)=0$ for all $k\in K$, $D(ab)=(Da)b+a(Db)$ for all $a,b\in A$, ...
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458
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R[[X]] flat as a R[X]-module?
I assume $R[X]\rightarrow R[[X]]$ is not flat in general, but I was wondering if any conditions on a commutative ring $R$ are known such that $R[[X]]$ is flat as a $R[X]$-module.
Would $R$ noetherian ...
- 31
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2
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509
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An extremal combinatorics problem over Finite Rings
Cross Posting from: https://math.stackexchange.com/questions/462016/a-combinatorics-problem-over-finite-rings
Consider the set $S$ of all non-zero vectors over $\Bbb Z_{q}$ of length $r$ whose ...
5
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answer
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Kernel of the differential in de Rham complex in positive characteristic
Roughly, I'd like to ask how does the first terms in de Rham complex behaves for singular varieties.
Let $Y$ be a potentially singular integral scheme over a perfect field $k$ of characteristic $p$ ...
- 351
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893
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Completion and localisation on noetherian rings
Let $(A,m)$ be a commutative noetherian local ring such that $m$ is principal, say $m=(t)$. Let $(\hat A,\hat m)$ be its $m$-adic completion. Let $A\subset B\subset\hat A$ be any intermediate subring ...
- 55
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4
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596
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Generalization of Jordan Decomposition for Several Commuting Operators
Recently I became curious about the following question:
Let $V$ be a finite dimensional vector space over $k$ and let $A_1, \cdots, A_n: V \rightarrow V$ be a set of commuting maps. Question: ...
- 607
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Is the positive part of an algebraic bilateral p-adic convergent power series algebraic?
Let $\mathbb{Z}_p \{ X\}$ and $\mathbb{Z}_p \{ X , X^{-1}\}$ be the henselizations of $\mathbb{Z}_p [X]$ and $\mathbb{Z}_p [ X , X^{-1}]$ with respect to the ideals $p\mathbb{Z}_p [X]$ and $p\mathbb{Z}...
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Quadratic suborders of an imprimitive quartic order
Let $Q$ be an irreducible quartic order; that is, $Q$ is a subring of the ring of integers $\mathcal{O}_K$ in a quartic extension $K$ over $\mathbb{Q}$ such that the fraction field of $Q$ is equal to $...
- 26.9k
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Deformations of associative algebras and Hochschild cohomology
I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer:
Let $(A,\mu)$ be a commutative associative algebra ...
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448
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Are dualizable modules finitely generated?
Let $A$ be a commutative noetherian ring, and assume that $A$ has a dualizing complex $R$. Let $D(-) := \operatorname{RHom}_A(-,R)$ be the associated dualizing functor, and let $M$ be an $A$-module.
...
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Arithmetically Cohen-Macaulay varieties
What do we mean by a variety being arithmetically Cohen-Macaulay? Is every such variety also Gorenstein?
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215
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Nilradical and Newton's identities
Let $R$ be a commutative ring with unity such that $n!$ is not a zero-divisor. Let $s_1=\sigma_1,s_2,s_3\cdots$ and $\sigma_1,\sigma_2\cdots$, (convention: if $k>n$, then $\sigma_k=0$) be elements ...
- 3,741
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scheme of commuting matrices
Let $k$ be any field. Let $r$ and $n$ be two positive integers.
Consider the functor $F$ from the category of $k$-schemes to the category of sets which sends a $k$-scheme $T$ to the set of matrices $...
- 183
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683
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Dimension of irreducible components of tangent bundle
Let $X$ be an irreducible algebraic variety over some field $k$. It is well known, that if $X$ is smooth and of dimension $d$, then the tangent bundle of $X$ is smooth, irreducible and of dimension $...
- 352
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359
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Thickness of the category of perfect complexes with finite length homology
Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...
- 3,101
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185
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Effective Nullstellensatz and bounds on the nilpotency index of reduced ideal together with linear forms
Let $K$ be an algebraically closed field of characteristic $p>0$
and let $I\subset K[x_{1},\dots,x_{n}]$ be an ideal generated by
(homogeneous) polynomials of degree $d$. Assume that $I$ is reduced,...
- 3,823
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3
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3k
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Generalized Chinese Remainder Theorem
Let $U,V$ be submodules of a $R$-module $M$. Then the diagonal induces an isomorphism
$M/(U \cap V) \to M/U \times_{M/(U+V)} M/V.$
This is a (useful!) generalization of the Chinese Remainder Theorem ...
- 63.1k
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176
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Flatness of a simple ring extension
Assume $A \subseteq B=A[b]$ are integral domains, $b \in B$ is algebraic over $A$ (but not necessarily integral over $A$), and $A$ and $B$ have the same field of fractions.
(Notice that $b=u/v$ for ...
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363
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When is $I \otimes_A \hat{A} \cong I\hat{A}$?
Let $A$ be a commutative ring and $I$ a (finitely generated) ideal in $A$. We denote by $\hat{A}$ the $I$-adic completion of $A$, i.e. $\hat{A} = \varprojlim(A/I \leftarrow A/I^2 \leftarrow \ldots)$.
...
- 189
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2
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683
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Projective Modules and their Determinants, Extended or not?
Let $A$ be a commutative noetherian ring, and let $P$ be a projective $A[T]$-module with constant rank $n$. Let $L$ be the determinant of $P$, $\wedge^n(P)$. We say that $P$ (resp. $L$) is extended ...
- 345
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Primary decomposition and finitely generated abelian groups
In a question asked by Ben Webster, Harry Gindi commented that it is possible to prove the classification theorem from finitely generated abelian groups by appealing primary decomposition.
I have ...
- 3,830
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3
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446
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A question concerning the isomorphic type of continuous functions
let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside a bounded open interval containing zero (depended on $f$). Is it possible to consider $S$ as (...
user39207
3
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104
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Are there necessary and sufficient conditions for a chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ to be Poincare?
I am looking for necessary and/or sufficient conditions for the chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ over a principal ideal domain to be Poincare in the sense that $H_0 \cong H^2$, $H_1 \...
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is there a good computer package for working with bicomplexes?
I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is ...
- 44.7k
3
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1
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280
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Composite families of formal power series over $\mathbb C$ as algebraic variety
I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...
- 5,428
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95
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Sign of bivariate polynomial evaluated over two algebraic numbers
I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
- 151
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1
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964
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When are two projective modules of equal rank isomorphic?
Let $R$ be a commutative ring and let $M,N$ be two finitely generated projective $R$-modules which have equal rank (not necessarily constant). What kind of general results are there concerning the ...
- 777
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1k
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Max Noether's AF+BG theorem
I'm looking for an example of the following situation, related to Max Noether's AF+BG Theorem (see Bill Fulton's book on algebraic curves, page 61, at http://www.math.lsa.umich.edu/~wfulton/CurveBook....
- 53
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0
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136
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quasi-ordinary singularities on a versal deformation?
Let $V$ be a variety over $\mathbb{C}$ and suppose $O$ is a singular point of $V$. Are there conditions on $(V,O)$ such that a versal deformation $W$ of $(V,O)$ has only quasi-ordinary singularities.
...
- 761
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1
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582
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Lifting identities of formal power series
I am looking for a possibly general class of algebraic structures (maybe special topological rings) in which one can deduce identities of concrete power series from formal ones. This class should ...
- 13
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1
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1k
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Succesful applications of algebra in combinatorics
Hi. This may be a very general question.
Are there any examples of problems in combinatorics which were open, but which found a solution when stated in algebraic terms?
If yes, could somebody ...
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316
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When is $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}$?
Suppose $(R,\mathfrak{m})$ is a noetherian local ring. I am interested in ideals $\mathfrak{a}$ of $R$ for which $$\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\...
- 3,101
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Linear algebra over principal rings
Consider an extension $R\subseteq S$ of commutative rings, and suppose that $R$ is principal (i.e., $0$ is the only zero-divisor of $R$ and every ideal of $R$ has a generating set of cardinality $1$). ...
- 6,700
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Vector spaces with natural bases
Sergeib's question asks about vector spaces without a natural basis.
Actually, I would claim (apparently in accord with many comments and answers to Sergeib's question ) that this is the default ...
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0
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237
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"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 $E$ ...
- 5,359
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534
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Initial ideal of k-th power of an ideal
Hi,
Let $I$ be an ideal in a polynomial ring $S = k[x_1, \ldots, x_n]$, where $k$ is an algebraically closed field of characteristic zero. Fix a term order on
$S$ (e.g. a lexicographic order) and ...
- 893
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533
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A question on Castelnuovo-Mumford regularity
Consider a short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ of coherent sheaves on $\mathbb{P}^n$. Assume that $\mathcal{F}''$ (resp. $\mathcal{F}$) is $m-1$ (resp. $m$...
- 61
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76
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Coherence of subrings of K[[X,Y]]
Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$.
Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...
- 781
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1
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301
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Software computation with arithmetic schemes
For rings such as $\mathbb{Z}[x,y]$ is there software to compute any of:
1.) The integral closure of $\mathbb{Z}[x,y]/(f)$. de Jong has a very general algorithm that works in this context (http://...
- 3,555
2
votes
3
answers
1k
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General hyperplane sections and projection from a point
Let $k$ be an algebraically closed field, and consider some subscheme $X\subset \mathbb{P}_k^n$. Let $x$ be a closed point of $X$, and $H$ a general hyperplane containing $x$. There is a regular map $\...
- 45
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1k
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Algebraic Independence of Polynomials in n Variables with Real Coefficients
I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... ...
- 11
1
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0
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161
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Concerning the $SBI$-sequence for dihedral homology (Loday, Cyclic Homology, 5.2)
I was wondering about the signs of the $SBI$-sequence ("Connes' periodicity exact sequence") in equations $(5.2.7.2)$ and $(5.2.7.3)$ of 'Cyclic Homology' by Jean-Louis Loday. Why is the sequence ...
- 61
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3
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613
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Can different modules have the same symmetric algebra? (answered: no)
Algebraic geometry allows one to think of an $A$-module $M$ geometrically as a module of functions on the $A$-scheme $\mathrm{Spec}(\mathrm{Sym}(M))$.
I'm wondering if anything is lost in just ...
- 11.3k
6
votes
2
answers
318
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Existence of an $R$-basis with at least one unit in it?
Let $F$ be a domain and let $R\le F$ be a subring such that $F$ is a free $R$-module of finite rank $n$.
Question: Is there an $R$-basis $\lbrace e_1,...,e_n\rbrace$ of $F$ such that at least one ...
- 71
11
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2
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863
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Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 19.6k