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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Global sections of exceptional divisor in normalized blow-up
Let $(R, \mathfrak{m})$ be a Noetherian normal local domain and $I$ an $R$-ideal. Write $X$ for the normalization of $\mathrm{Proj}(R[It])$ and $E$ for the effective Cartier divisor defined by the ...
- 661
7
votes
1
answer
757
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Characterizing intersection of zero sets of elementary symmetric polynomials on R^n
Stated simply, the question is:
Consider two elementary symmetric polynomials $\sigma_{k}$ and $\sigma_{k+1}$ on $\mathbb{R}^{n}$ with zero sets $U_{k}$ and $U_{k+1}$. Let $V_{i_{1}i_{2}\dotsb i_{j}...
- 83
3
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1
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928
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How exotic can DVRs be in the ring of rational functions over a local field?
Suppose that $R$ is a complete DVR with field of fractions $K$, uniformiser $\pi$ and residue field $k$.
Let $B$ be a subring of the ring $K(t)$ of rational functions over $K$. Moreover assume that $...
- 3,545
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85
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if $\Delta$ is pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$
Assume that $\Delta$ is a simplicial complex and $\Delta ^v$ is its Alexander dual.
Let in addition $\Delta$ be pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$?
Is there a ...
- 1,355
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549
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Fitting ideal sheaves and determinant bundles
I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known.
Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the ...
- 5,857
1
vote
1
answer
218
views
Does projective duality preserve arithmetic-Cohen-Macaulay-ness?
Let $V$ be a vector space over $\mathbb{C}$.
Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring $\...
- 569
1
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2
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703
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Is a reduced, torsion-free module of finite rank over an Henselian ring free?
Let $R$ be an Henselian discrete valuation ring with field of fractions $K$. Let $M$ be a torsion-free $R$-module of finite rank (i.e. $dim_K(M\otimes_RK)<+\infty$). Let $D$ be the maximal ...
- 73
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166
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For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?
For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo $...
- 403
3
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1
answer
305
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Tangent cone and embedded components
Is it possible for a reduced, equidimensional germ of complex analytic singularity to have a tangent cone with embedded components but without multiple irreducible components?
If it is, how can you ...
- 31
8
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1
answer
3k
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When a tensor product of two local rings is a local ring?
This is a follow-up to Is tensor product of local algebras local?.
Let $A, B$ and $C$ be local rings (commutative and noetherian). Suppose that we have local ring maps $C \to A$ and $C \to B$.
What ...
- 91
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0
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197
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Cohen-Macaulay fibers
Let $Y$ be a set of points in $\mathbb{P}^n$. Then we can write a resolution
$$0\rightarrow P_n \rightarrow \cdots \rightarrow P_0\rightarrow \mathcal{O}_Y$$
where each $P_i=\bigoplus_j\mathcal{O}_{\...
- 39
2
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0
answers
236
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The Euler characteristic of Hilbert series
The Hilbert series of a graded vector space $V=\bigoplus_{n\mathbb Z}V_n$ is the (ordinary) generating function of the dimensions of its homogeneous components, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim V_n$...
- 47.3k
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345
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Domains with prime ideal theorems
Let $D$ be a domain, and for prime ideals $\frak P$ of $D$ the norm is $N({\frak P}):=|D/{\frak P}|$. The prime ideal counting function of $D$ is given by $\pi_D(x)=\#\{{\frak P}\in{\rm Spec}(D):N({\...
- 441
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completion of non-finitely generated ideal
Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates.
Then we can consider the completion $\hat{A}=\varprojlim_{r,l}k[x_{1},x_{2},..]/(x_{1}^{r},..x_{l}^{r},x_{l+...
- 3,472
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some sort of 'saturation' of module quotients
Let $R$ be a local Noetherian ring over a field, with the maximal ideal $\mathfrak{m}$. (e.g. $R=k[[x_1,\dots,x_{p>1}]]$) Given two $R$-modules, $N\subset M$, of the same (finite, non-zero) rank. ...
- 2,232
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2
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1k
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Maximal ideal of codimension >1
To assuage my conscience over an unsourced statement in a paper I'm writing:
I am looking for an example of a commutative algebra over the complex numbers having a maximal ideal of codimension >1, or ...
- 131
4
votes
1
answer
932
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Relative integral closure
Definition: Let $R$ be an $A$-algebra, where $R$ and $A$ are both commutative rings with unit. Let $S \subset R$ be the (multiplicatively closed) subset consisting of those nonzerodivisors $s \in R$ ...
- 7,338
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How to find ideals of finite length in a power series ring with special properties?
Let $A$ be the power series ring $\mathbb{C}[[x,y]]$.
Assume we are given two ideals $I,J$ of finite length in $A$ such that:
$xJ\subseteq I\subseteq J$
Is it possible to find ideals of finite ...
- 1,025
0
votes
1
answer
508
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"thematic" algebras
I scoured what I could in the literature but I have yet to find the information that should be out there. Consider the property
(P1) Every local subalgebra can be embedded in a local ideal ...
- 103
3
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1
answer
1k
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How to prove that the subrings of the rational numbers are noetherian?
I have managed to prove that the aforementioned subrings are in bijection with the subsets of the primes, however, I haven't been able to prove that they are all noetherian. I need help.
- 59
1
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470
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Serre's conditions under blow-ups, Blowup and normalization
Suppose $X = \mathbb{Z}[x, y, z]/(f,g)$ is a 2-dimensional Cohen-Macaulay surface. In particular, $X$ satisfies Serre's condition $S_2$. Suppose it is irreducible, reduced but not normal.
$\bf{...
- 3,555
-1
votes
1
answer
1k
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Correspondence between submodules and quotient modules
What is the (natural) bijection between the isomorphic class of sub modules and isomorphic class of quotient modules of a finitely generated torsion module over a PID.
Is there any inclusion relation ...
- 1,269
4
votes
1
answer
425
views
Ring structrures on R^n
Consider a commutative ring $A= ( \mathbb{R}^n , + , \times) $, where $+$ is the usual one. Assume further that $ \times $ is continuous (with respect to the usual topology). Let $H$ be the set of non ...
- 7,249
6
votes
1
answer
2k
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Hochschild and cyclic homology of smooth varieties
Many of the standard sources which discuss the Hochschild Kostant Rosenberg theorem and cyclic homology for smooth varieties such as Loday and Weibel's paper "The Hodge Filtration and Cyclic Homology" ...
- 3,758
1
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2
answers
2k
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Real solutions to underdetermined system of polynomial equations
I have the system of multi-variable polynomial (quadratic) equations with real coefficients. The number of equations is given scales as $K$ and the number of unknowns goes as $K^2$. So for for large $...
- 13
1
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1
answer
639
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Question on bigraded module.
I have a question on an argument in this survey.
Let $A$ be a commutative Noetherian ring, $A[x_1,...,x_m, y_1,...,y_p]$ be the bigraded algebra over $A$, with deg$x_i=(1,0)$, deg$(y_j)=(d_j,1)$.
...
- 87
3
votes
1
answer
388
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Local ring $(R,\mathfrak m)$ such that $\mathfrak m^2$ is the unique minimal ideal
When $\mathfrak m^2$ is the unique minimal ideal in a local ring $(R,\mathfrak m)$?
Note that in this case $\mathfrak m^3=0$ in $R$. Furthermore assume that $\operatorname{char}(R)$ is finite.
- 33
6
votes
0
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181
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Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra
The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating.
A real finite-dimensional vector space $V$ defines the ...
- 3,513
6
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1
answer
530
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Does a variety contain a cartesian product of two curves?
We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C_1\times C_2$, where $C_1$ and $C_2$ are two curves in $\mathbb{A}^n$. ...
- 7,836
0
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109
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Non-local differentially smooth algebra
Let $A$ be a noetherian commutative algebra over a perfect field $k$.
The algebra $A$ is said to be differentially smooth over $k$ if
(1) $\Omega^1_{A/k}$ is a projective $A$-module, and
(2) the ...
- 657
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votes
2
answers
335
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What does a singular simplex with real coefficient mean [closed]
For an $n$-dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$-norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which represent the fundamental ...
- 689
0
votes
1
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225
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Depth formula in CM-ring involving canonical module
In this article by Iyama and Wemyss there is the following formula:
Let $R$ be a Cohen-Macaulay ring with canonical module $\omega$, let $X$ be a finitely generated $R$-module. Then
$$\mbox{depth}(X)=\...
- 73
2
votes
1
answer
1k
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transcendence degree of subring of polynomial ring
Hi all
Quick question that I think is true but haven't been able to prove.
Suppose there is a subring $R \subset k[x_1, \ldots, x_r]$ containing field $k$ and generated by homogenous polynomials $...
- 31
6
votes
2
answers
610
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What is the set of possible values of the degree of the sum of two algebraic numbers with fixed degrees?
This question is related to Degree of sum of algebraic numbers and algebraic numbers of degree 3 and 6, whose sum has degree 12.
In this last question I asked a very special case of the following ...
- 3,595
6
votes
1
answer
858
views
Exotic isomorphism of matrix rings
Let R and S be commutative rings with a 1 different from zero. Let m and n be positive integers. Assume the ring of m-by-m matrices over R is isomorphic to the ring of n-by-n matrices over S. Does ...
user5810
5
votes
1
answer
243
views
A question of Allan on infinite divisibility
Loosely speaking the question is: If an element of a commutative ring is infinitely divisible by $x$, is it the product of $x$ with an infinitely divisible element?
More preceisely:
For a fixed ...
- 16.5k
2
votes
1
answer
918
views
Is a formally smooth morphism a filtered inductive limit of smooth algebras?
Given a unital commutative ring $A$ (not necessarily noetherian) and a formally smooth morphism of rings $f:A \to B$, where $B$ is not necessarily noetherian, is (or when is) $B$ a filtered inductive ...
6
votes
1
answer
182
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Checking whether modules are isomorphic, via a computer algebra software
Hi
Let $R = K[X_1,\ldots, X_n]$ where $K$ is a computable field.
Suppose we are given two modules with presentations
$$ R^n \rightarrow R^m \rightarrow M \rightarrow 0 $$
and
$$ R^l \rightarrow R^p \...
6
votes
1
answer
1k
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Explicit injective resolutions of (Laurent) polynomial rings
Hi,
Despite being nothing but the dual notion of projective resolution, injective resolutions seem to be harder to grasp. For example, the general form of Poincaré-Lefschetz duality given in Iversen'...
- 558
8
votes
2
answers
537
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Prime avoidance in adjacent degrees
Let $\mathfrak{p}_1, \dotsc, \mathfrak{p}_k$ be relevant homogeneous primes ideals in the graded ring $R := \Bbbk[x_0, \dotsc, x_n]$, where $\Bbbk$ is a field. Prime avoidance (in Eisenbud's ...
- 7,338
1
vote
1
answer
360
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Representations over $\mathbb{Z}_p$
Hi,
I would like to work with indecomposable representations over a commutative ring and start with $R=\mathbb{Z}_p$
Notations: $G$ a finite group, $S$ a subgroup, $R=\mathbb{Z}_p$ the p-adic ring, ...
- 11
5
votes
3
answers
981
views
What is the coordinate ring of symmetric product of affine plane?
The symmetric product of a variety $M$ is the quotient of $M^n/S_n$ where $S_n$ is the symmetric group permuting components of n-fold product $M^n$. IF $M$ is an affine plane $C^k$ over complex ...
2
votes
3
answers
395
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On the equation defining a surface
Fix a curve $C$ and a line both in $\mathbb{P}^3$ not intersecting one another. Suppose that the ideal of the curve is generated by $P_i$ and the line is definied by linear polynomials $l_1, l_2$. Is ...
- 1,040
1
vote
1
answer
325
views
Frobenius splitting and smoothness
Let $R\rightarrow S$ be a morphism of rings in characteristic $p$ which is formally smooth.
Is it true that $R$ is Frobenius splitting if and only if $S$ is Frobenius splitting?
One direction seems ...
- 43
46
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0
answers
1k
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Enriched Categories: Ideals/Submodules and algebraic geometry
While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...
- 3,249
1
vote
1
answer
223
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Can one pick generators for the ring of invariants of binary n-ic forms which have rational coefficients?
The problem of determining a set of generators of the ring of invariants of the group $\textrm{SL}_2$ acting on the complex $n+1$-dimensional vector space of binary $n$-ic forms is known to be very ...
4
votes
2
answers
789
views
Criteria for system of parameters in polynomial rings
Let $k$ be a field and let $R=k[x_1,...,x_n]$ be a polynomial ring over $k$. A subset $\lbrace y_1,...,y_n\rbrace$ of $R$ is called a homogenous system of parameters (hsop) of $R$, if
the $y_i$ are ...
- 16.2k
1
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0
answers
224
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Computing the bourbaki ideals
By virtue the Griffith's paper and subsequently e.g. Goto's paper several examples of several desired class of Noetherian normal domains with specific finite length local cohomologies are constructed ...
- 591
1
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0
answers
242
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Criterion for normality of a schematic image
Consider a projective flat morphism
$$
f\colon X\to Y
$$
between normal varieties. Let's say over the complex numbers. The geometric fibers of $f$ are all irreducible.
I would like a criterion to ...
- 2,384
5
votes
1
answer
4k
views
Dimension of module
Does dimension of a module (say, dimension of its support) have anything to do with the supremum length of chains of prime submodules like rings?
Let's restrict to finitely generated modules over ...
- 2,857