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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Fixing a mistake in "An introduction to invariants and moduli"
On page 13 of the book "An introduction to invariants and moduli" of Mukai
http://catdir.loc.gov/catdir/samples/cam033/2002023422.pdf there is a mistake, in the end of the proof of Proposition 1.9. ...
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K-theory and completion
I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community.
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of ...
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200
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for which truth-operations f can f-membership in a prime ideal be represented by a polynomial?
Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital.
Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by $[\phi]\in B$ it's ...
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Direct image sheaf and tensor product (is the projection formula an isomorphism?)
Assume we have two "nice" schemes $X$ and $Y$ over $k=\mathbb{C}$, a finite flat map $f:X\rightarrow Y$ and a k-algbera $A$. Then we get an induced finite flat map $f_A:X\times_k A \rightarrow Y\...
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On the computational complexity of the Hilbert polynomial of numerical semigroup rings
Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that $\ell(R/\...
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Are valuation rings regular?
This question is short, and to the point:
Valuation rings are certainly integrally closed, but are they regular?
The motivation is that I'm trying to understand the resolution of singularities of ...
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3
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3
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670
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Algebraic, analytic, formal modules
Consider torsion free modules over the germ of a fixed isolated algebraic hypersurface singularity {$f=0$}$\subset\mathbb{C}^n$.
There are natural functors (using categories of finitely generated ...
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1
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On the functoriality of scalar extensions of local rings (edited)
Note. I have edited my question to make it more transparent, following some very good comments that I received. I am sorry if it is a bit long.
A local homomorphism of local rings $(A,\mathfrak{m})\...
- 3,101
9
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Isolated hypersurface singularities, Chow groups and D-branes
Say a ring $R$ is an isolated hypersurface singularity if $R = k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}/(W)$, where $k$ is a field and $W \in k[x_1, \ldots, x_n]$ is such that the ideal $(\partial_1 W,...
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What is known about this short exact sequence in Lie algebra cohomology?
In its most general form, I look at the following. Let $g$ be a dg Lie algebra and $Z$ be its center. The sequence $ Z \to g \to g/Z $ gives me a short exact
$ 0\to C(g,Z) \to C(g,g) \to C(g,g/Z) \to ...
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1
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Some references for f-ring
A commutative ring $R$ is said to be an $f-ring$ if every pure ideal is generated by idempotents. (Recall that the ideal $I$ is said to be pure if for each $a\in I$ there
is a $b\in I$ such that $ab = ...
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1
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648
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Pullbacks and Inclusions of Smooth function algebras of manifolds.
Let $M$ and $N$ be two smooth finite dimensional manifolds and
$C^\infty(M)$ as well as $C^\infty(N)$ their smooth function algebras.
Is the following true:
Let $\imath: M \to N$ be an embedding. ...
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1
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Sums of Squares and Totally Positive Numbers
In Van der Waerden B L. Algebra Vol.I[M]. Springer, 2003, Pro. Waerden announced in page 256 that if an element $\gamma$ of a formally real field K is not a sum of squares, there exist an ordering of ...
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Projecting solutions of Hermitian forms over local rings
Let $R$ be a local ring (commutative and with $1$) with maximal ideal $M$, with an involution $\theta$. Let $h$ be a Hermitian form on $R^n$, i.e. $h:R^n\times R^n\rightarrow R$ such that $h$ is $R$-...
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Determining Inconsistency of (first-order) Non-linear System of Equations [closed]
Is there a way I can figure out what values of the coefficients of some system of non-linear equations makes the system inconsistent?
Take the following system of equations as an example. The ...
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3
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2
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393
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How to make a function depending on some operation?
Let $S$ be an infinite set and $f:S\times S\rightarrow\mathbb R$ be bounded.
Question: Are there simple hypotheses on $f$ such that there is a commutative and associative operation $\cdot$ on $S$ ...
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6
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2
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A reference: the splitting principle for exterior powers of coherent sheaves?
It's well known that if E is a vector bundle with Chern roots $a_1,\ldots, a_r$,
then the Chern roots of the $p$th exterior power of E consist of all sums of $k$ distinct $a_i$'s. I would like to say ...
- 594
3
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0
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211
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the annihilator of cokernel in a particular case
Let $A\in Mat(m,n;R)$ for $m\le n$ and $R$ a local ring. Consider the $mn\times(m^2+n^2)$ matrix $A\otimes 1_{nn}\oplus 1_{mm}\otimes A^T$, here $1_{mm}$, $1_{nn}$ are identity matrices. I'd like to ...
- 2,232
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1
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Detecting and counting free direct summands
If $M$ is a finitely generated module over a local ring $(R, \mathfrak{m})$, we can detect whether $M$ has a nonzero free direct summand as follows: Consider the natural map
$$\phi_M\colon \mathrm{Hom}...
- 5,846
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1
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159
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Chain of Ideals of same height
I have been wondering about the following (and allready posted a similar question, see Dimension of ring completion wrt to a decreasing chain of ideals):
Let $R$ be the ring of formal power series ...
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1
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345
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Is the ideal of denominators preserved under flat pullback?
Let $\phi \colon A \to B$ be a flat homomorphism of rings (commutative, with unit). Let $R$ be the total ring of fractions of $A$ (obtained by inverting all nonzerodivisors), and let $S$ be the total ...
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2
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804
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A problem for finite dimensional commutative algebra
Let $(A,m)$ be a local commutative associative algebra over the field of complex numbers, $m^n\ne 0$, $m^{n+1}=0$ for some $n>0$, and
(1) $A$ is finite dimensional as vector space
(2) for any ...
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1
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172
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$I/N$ is finitely presented module
Let $R$ be a commutative ring and $N = Nil(R)$ the set of its nilpotent elements. Suppose that $N$ is a divided prime ideal, i.e. for any ideal $I$ of $R$ either $I \subseteq N$ or $N \subseteq I$.
...
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7
votes
1
answer
801
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Extensions of torsion modules
Given a regular local ring $R$ and an $R$-algebras $S$, which is torsion free and finitely generated (even free if needed) as an $R$-module.
Assume we have a nontrivial surjective map $f: M \...
- 1,391
3
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1
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374
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Non-Noetherian Stable Homotopy
There seems to be quite a bit of theory developed to deal with "stable homotopy" in the sense of the derived category of a Noetherian ring, or just any situation where the endomorphism ring of the ...
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Syzygies in integral domains
Let $f$ be a homogeneous irreducible element in a graded commutative Noetherian ring.
What is the possible set of elements $f_1$ and $f_2$ such that $f=f_1g_1+ f_2g_2$?
Even in very particular cases ...
9
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644
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Conceptual proofs for the computation of the structure sheaf
The following lemma in commutative algebra is important for the foundations of algebraic geometry:
If $A$ is a commutative ring, $U \subseteq A$ is a finite subset generating the unit ideal, then ...
- 63.1k
4
votes
1
answer
318
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finiteness of local cohomology
Well-known Theorem:
Let $a$ be an ideal of the noetherian ring $R$ and let $M$ be a finitely generated
$R$-module. Let $i \in \Bbb N_0$ be such that $H^j_a
(M)$ is finitely generated for all $j < ...
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2
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1
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399
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Noether Normalization in $\mathbb{C}[[x_1,...,x_n]]$
Hello everyone,
though I am quite sure my question is not on a research level, I am a bit helpless as of where to ask it. I posted it on math.stackexchange here, but there has been no answer helping ...
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5
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0
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interpretation of homology of "non-commutative Koszul complex"
Let $A = Sym^*(V)$ be a polynomial ring. The Koszul complex
$\cdots \to \wedge^2 V \otimes A \to V \otimes A \to A$
gives a resolution of the residue field $k$, so for any $A$-module $M$, the ...
- 10.7k
1
vote
1
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Embedded associated prime
$\underline{\textbf{Embedded associated prime}}$
I am reading the book "Joins and Intersections". In the proof of Rees theorem I have some doubt.
Let $\mathbf M$ be a finitely generated $\mathbf A$-...
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118
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Maximal elements for ideals and subrings ordered by inclusion with fixed number of minimal generating polynomials
Let $R=\mathbb{R}[X_1,\dots,X_n]$, and
$$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\}$$
...
- 1,256
0
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2
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232
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Commutation of $GL_{n}$ with projective limits
Let $A$ denote a unital commutative ring. Given a system of ideals $(I_p)_{p\in P}$ indexed by a partially ordered set $P$ such that if $p \leq q$, then $I_p$ is contained in $I_q$, when is
$$GL_n ...
- 91
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Antichains defining facets of a certain cone
Let $(P,<)$ be a finite poset. Let $V$ be the free $\mathbb{R}$-vector space on $P \times \{0,1\}$; I'll write elements as sums of pairs of the form $(p,0)$ and $(0,q)$, so a general element is $$v ...
- 720
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1
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Polynomials mapping the twisted cubic power series ring to itself
If $f(X) \in \mathbb{F}_2((T))[X]$ and $f(\mathbb{F}_2[[T^2,T^3]]) \subseteq \mathbb{F}_2[[T^2,T^3]]$, then does it follow that $f'(0) \in \frac{1}{T}\mathbb{F}_2[[T]]$?
This might seem like an ...
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320
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Invariants of the Determinant Form
Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...
- 801
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3
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When is the radical of the extension of a prime ideal prime?
(All rings assumed to be commutative and unital)
Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial ...
3
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4
answers
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Matrix factorization categories for ADE singularities
What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example this. For ...
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Bass' stable range condition for principal ideal domains [duplicate]
Do you know a characterization of commutative rings $R$ whose every prime factor ring of $R$ is a principal ideal domain?
- 321
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182
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Zariski open set of linear forms
Let $I$ be a graded homogeneous ideal over $k[x_1, ... ,x_n]$ and $h$ a linear form, let $H$ be the corresponding hyperplane and $I_H$ the restriction of $I$ to $H$.
I am looking for a Zariski open ...
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139
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Questions on prime integral ideal congruences
Suppose that we are given a fixed pair $a_1, a_2$ of non-zero irrational algebraic integers in some number field $K$ which are independent over $\mathbb{Q}$. Suppose that $\mathcal{P}$ is a prime ...
- 26.9k
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2
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the generators of $I$ when $\mathbb C[x,y]/I$ is Gorenstein with zero dimension
Let $I$ be any ideal in $R=\mathbb C[x,y]$ of height 2. If we know the localization of $I$ at any maximal ideal $m\supseteq I$ is generated by a regular sequence of length two in $R_m$. Is this true ...
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Irreducibility of $x^m-g(y)$
Let $g(y)\in \mathbb{C}[y]$, $ m\in \mathbb{Z}_{\ge 2}$. Are there some results on the irreducibility of $x^m-g(y)$ in $\mathbb{C}[x,y]$?
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980
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Spectral sequence for Ext
Why is $H^p(X, \mathcal{E}xt^q(F, I))=0$ for $p>0$ and $I$ an injective sheaf and $F$ an arbitrary sheaf? I'm trying to check the hypothesis in the grothendieck spectral sequence applied to the ...
- 23
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1
answer
470
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Status of Gao's Conjecture
In his 2001 paper titled "On The Deterministic Complexity of Factoring Polynomials", Shuhong Gao makes the following conjecture:
For any $a \in \mathbf{F}_q$ ($q$ is some prime power), we can write $...
- 11
1
vote
1
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524
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primary ideal of regular local ring
Let $(R,\mathfrak{m})$ be a regular local ring of dimension $d$. Let $P$ be a prime ideal of height $d-1$. I want to know if $P^2$ is always a $P$ primary ideal ie if $P/P^2$ is torsion free as $R/P$ ...
- 221
3
votes
1
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388
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Term for an "almost regular" sequence
Let $R$ be a ring (commutative, with unit), $M$ an $R$-module, and $x_1, \dotsc, x_n \in R$. Consider the following two conditions:
For all $i$, the homomorphism $$\frac{M}{(x_1, \dotsc, x_{i-1})M}...
- 7,338
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0
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168
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Bound for the height of equations defining the singular locus of a variety
Fix positive integers $m, n, d$.
In what follows, the height of an algebraic number will mean the absolute multiplicative height.
Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...
- 1,500
2
votes
0
answers
536
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Completion versus henselization.
For a normal local ring $(R,m)$ over complex number field C, consider the henselization $R^h$
and the completion $\widehat{R}$.
Question: Is $R^h$ algebraically closed in $\widehat{R}$?
I think ...
2
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0
answers
119
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Are all (graded) Artinian complete intersections like this?
I'm trying to prove some stuff (it's not important what) about (graded) Artinian complete intersections $R=\mathbb{C}[x_1,\ldots,x_n]/I$, where the $x_i$ have certain positive weights and where $I$ is ...
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