Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,316
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History behind Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$ for a commutative Noetherian ring
This is a repost. So far, I've received no answers on HSM Stack Exchange; maybe I do in MO.
In 033Q we find defined what some sources call “Serre's conditions $\mathrm{S}_k$ and $\mathrm{R}_k$” (for a ...
2
votes
1
answer
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Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
2
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0
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Help with Macaulay2 computation of invariant ring
Consider the algebraic group $G:=\operatorname{SL}_{2}\times\operatorname{SL}_{2}$ acting on $V:=\operatorname{Mat}_{2\times 2}\oplus\operatorname{Mat}_{2\times 2}$ via the action $(A,B)\,\cdot\,(X,Y)=...
8
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1
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Homological conjectures for finite dimensional commutative algebras
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $...
4
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2
answers
350
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$p$-divisibility of Picard groups
Let $p$ be a prime number and let $k$ be a field with $char(k)\neq p$ such that all finite extensions have degree coprime to $p$. (For example, we can take $k=\mathbb{R}$ and $p\neq 2$ or let $k$ the ...
3
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1
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200
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On Flat and Projective Modules over integral domain
Is this true that finitely generated flat module over an integral domain is projective.
If Yes, please provide a proof.
5
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153
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differential normal cone
$\newcommand{\Spec}{\operatorname{Spec}}$Let $X$ be a scheme, and $Y$ a closed subscheme; to simplify notation assume $X=\Spec(A)$
is affine, so $Y=\Spec(B)$, $B=A/I$. According to the standard ...
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One-step vectorial recurrence into multi-step scalar recurrence on commutative rings with boundary conditions
Introduction over unbounded domain
Consider the forward time shift $\mathsf{z}$ acting on a discrete function of time (sequence) $f=(f^n)_{n\in\mathbb{N}}$ as $(\mathsf{z} f)^{n} = f^{n+1}$. Also ...
0
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1
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141
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Existence of linear disjointness between an algebraic number field and $p$-cyclotomic field over $ \mathbb{Q}$
Suppose $ K $ be an algebraic number field and $ n $ be an even integer. Is it possible to find atleast one $ p $ such that $ p\equiv 1( \text{mod}~ n)$ and $ \mathbb{Q}(\eta_p) $ is linearly disjoint ...
2
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201
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Is a variety always contained in a hypersurface of smaller or equal degree?
(a) Let $V\subset \mathbb{A}^n$ be an affine variety (not necessarily irreducible). Write $\deg(V)$ for the sum of the degrees of its irreducible components. Must there be a hypersurface $W\subset \...
1
vote
1
answer
121
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Pure-dimensional intersection of smooth varieties
Let $X\subset \mathbb{P}^n$ be a complex smooth projective variety of degree $d$ and dimension $m$, such that $X$ is not contained in any projective subspace of $\mathbb{P}^n$. Let $P\subset \mathbb{P}...
1
vote
0
answers
65
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Relative 1 form of Frobenius morphism
Let $X$ be a connected smooth algebraic variety over a field $k$ of characteristic $p > 0$. We can consider the Frobenius morphism associated to $X \mapsto Spec(k)$. I'd like to show $\Omega^{1}_{X/...
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0
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Projective dimension and certain subideals
My question is related to this one. I thought mine could be very elementary but I'm not sure how to look into it.
Let $J$ be an ideal of $R=k[\mathbf{x}]$ where $\mathbf{x}=\{x_1,\dots,x_n\}$. Let $\{...
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51
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Completeness of formal power series rings over various linearly topologized commutative rings
Let $R$ be a commutative ring and fix an ideal $I\subseteq R$, such that $R$ is complete with respect to the $I$-adic topology. When is a formal power series ring $R[[X_{1},\dots,X_{d}]]$ over $R$ ...
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Taking polynomial inverses over a field?
Let $f \in F_p[x] / q(x)$. I know for a fact that the inverse of $f$, $g$ exists in the field. Taking the regular inverse is easy, but I'm looking for the compositional inverse.
I'm looking for ...
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Confusion regarding change of variable and irreducibility
Let $\mathbb{K}$ be an algebraically closed field of characteristics zero. Let $X$ be an irreducible affine variety, with a rational action of a linearly reductive algebraic group $G$. Also, assume ...
4
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Koszul algebras among finite dimensional commutative algebras
Given a local commutative artinian algebra $A$ of the form $K[x_1,...,x_n]/I$ with quadratic ideal $I$ and $K$ a field.
Question 1: Is there a computer algebra system that can check whether such an ...
4
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0
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89
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Lie bracket of general unipotent matrices
Let $k$ be a field (of characteristic $0$). Let
$$
X:=\begin{pmatrix}1&x_{1,2}&x_{1,3}&\cdots&x_{1,n}\\ &1&x_{2,3}&\cdots&x_{2,n}\\ &&1&\cdots&x_{3,n}\\ ...
2
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Multivariable Weierstrass preparation theorem
The Weierstrass preparation theorem for formal power series says the following:
Let $f(T) \in \mathbf{Z}_p [[ T ]]$ be a formal power series. Then we can write $f(T) = p^{\mu} \cdot u(T) \cdot g(T)$, ...
4
votes
1
answer
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Quadratic refinements of a bilinear form on finite abelian groups
$\DeclareMathOperator\Hom{Hom}$Let $A$ be a finite abelian group and $\text{Sym}(A)$ the (abelian) group of symmetric bilinear forms over $A$ valued in $\mathbb{R}/\mathbb{Z}$.
A quadratic function on ...
4
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1
answer
328
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Can the ring of symmetric polynomials be generated by powers of symmetric polynomials of degree 1?
I'm making a research on Galois theory, and found something interesting regarding the ring of symmetric polynomials:
At least up to 5 variables, we can rewrite the elementary symmetric polynomials ...
5
votes
1
answer
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Discovery of Hilbert polynomial
Presumably it was Hilbert who discovered Hilbert polynomials - where did they first appear?
The basic theorem is that for a finitely generated graded module $M = \bigoplus_k M_k$ over the ring of ...
6
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1
answer
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If degree $N$ polynomials always have a root, does there still exist an irreducible polynomial of degree $N+1$?
Given that $F$ is a field, let $F_n$ be the completion of $F$ with respect to roots of degree $n$ polynomials. For example this would make $\mathbb{Q}_2$ the field of (ruler and compass) constructible ...
1
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1
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79
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On "minimal presentation" of local rings essentially of finite type over a field
Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
5
votes
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280
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Reference request for the group of units of a power series ring in one variable
Let $\mathbb F_p$ denote the finite prime field of $p$ elements. What reference can be recommended for an analysis of the structure of the group of units of the power series ring $\mathbb F_p[[x]]$? ...
3
votes
2
answers
375
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Cohen-Macaulay Representations
I came across the book "Cohen-Macaulay Representations" by Graham J. Leuschke and Roger Wiegand, and now I'm wondering if this is an active area of research.
If yes, then
what are some of ...
9
votes
1
answer
492
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Isbell Duality and Dualizing Scheme Objects
I'm not sure if this question is too elementary for MO; nevertheless, I have seen many helpful discussions surrounding this topic here. I'm interested in studying the adjunction $\operatorname{Spec}\...
0
votes
1
answer
70
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Ideal membership and change of fields
Let $R=k[x_1,...,x_n]$ be a polynomial ring over a field. Let $f$ be a homogeneous polynomial and $I=(f_1,...,f_m)$ a homogeneous ideal.
With Macaulay2, one can compute the Groebner basis of $I$ when $...
2
votes
0
answers
110
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Understanding normalization algorithms
Let $R$ be a commutative and reduced ring, finitely presented over $\mathbb Z$. Let $\overline R$ be the integral closure of $R$ in its total ring of fractions. In https://arxiv.org/abs/alg-geom/...
2
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An iterative formula for the Kreweras-Voiculescu polynomials (reference request)
Let
$$N(x) = 1 + \sum_{k \ge 1} N_k(h_1,h_2,...,h_k) \;x^k$$
$$ = 1 + h_1 x + (h_1^2 + h_2) x^2 + (h_1^3 + 3h_1h_2 + h_3)x^3 + (h_1^4 + 6 h_2 h_1^2 + 4 h_3 h_1 + 2 h_2^2 + h_4) x^4 + \cdots$$
be the ...
0
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1
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320
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Can every idempotent ideal be generated by an idempotent?
This problem comes from this commutative algebra problem
Let $R$ be a commutative ring with identity, $I$ is a finite generated
ideal of $R$ such that $I^2=I$, then exists $e\in R$ such that $I=Re$.
...
3
votes
1
answer
720
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irreducibility of the polynomial $ x^4 +1 $
Let $K$ be a field. We consider the polynomial $f(x) = x^4 + 1$. It is known that $f(x)$ is irreducible over $\mathbb{Q}$ but reducible over any finite field. Thus $ f(x)$ is reducible over any field $...
8
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Singularity category of a hypersurface associated to $M_{11}$
For reasons to do with classifying spaces of finite groups, I have the following algebra. Let $k$ be a field of characteristic two, and let $R = k[x,y,z]/(x^2 y + z^2)$, as a graded $k$-algebra with $|...
5
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Which monomials are "leadable"?
Question: Let $k$ be a field, let $f \in k[t_1,\ldots,t_N]$ be a nonzero polynomial. Which monomials
$m_a = t_1^{a_1} \cdots t_N^{a_n}$ appearing in $f$ are leadable in the sense that they are the ...
2
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0
answers
158
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The dimension of the representation ring
Let $G$ be a compact Lie group. I am trying to characterize the algebraic properties of the representation ring $R(G)$ of $G$. In the case of the $n$-torus, the representation ring $R(T)$ is ...
10
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1
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809
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Is it a valuation ring?
It is known that a one-dimensional Noetherian local ring is a discrete valuation ring if it is integrally closed.
Then, even if it is not Noetherian, would a one-dimensional local ring become a ...
4
votes
1
answer
258
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Existence of module with periodic resolution
Let $R$ be a Gorenstein local ring. Does there always exist a module $M$ having eventually periodic minimal free resolution?
Any reference is also appreciated.
8
votes
1
answer
323
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Alterations and smooth complete intersections
Let $k$ be an algebraically closed field, and $X$ a projective variety over $k$. Let $i : X\subset \mathbf{P}^d_k$ be a closed immersion into a projective space of high enough dimension.
Is there a ...
2
votes
1
answer
159
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Cohen-Macaulay fiber products
Let $R$ be a regular local ring, $X$ and $Y$ smooth $R$-schemes, $T\to Y$ a regular closed immersion over $R$ with $T$ smooth over $R$, and $f: X\to Y$ an $R$-morphism.
Is the fiber product scheme $...
4
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0
answers
150
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Explicit construction for Cohen’s $p$-ring with imperfect residual field
Apologize if this is a below-research-level question. Asked in stack exchange but no response yet.
Let $p>0$ be a prime. Recall that a $p$-ring is a complete DVR $A$ with residual filed $k$ of ...
2
votes
1
answer
235
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Images of smooth schemes under lci morphisms
Let $S$ be a Noetherian scheme, $f : X\to S$ a smooth quasi-projective morphism, $g : X\to Y$ a morphism of finite type, and $h : Y\to S$ a smooth projective morphism with $h\circ g =f$.
Can we say ...
3
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1
answer
205
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Van der Waerden's classical paper on Hilbert polynomials and Bezout's theorem
Is it possible to obtain an electronic copy of Van der Waerden's "On Hilbert series of composition of ideals and generalisation of the Theorem of Bezout", Proceedings of the Koninklijke ...
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72
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Combinatorial proofs for derivational and binomial identities of partition polynomials for compositional inversion
The set $[N]$ of partition polynomials (ParPs) $N_n(u_1,...,u_n)$ of OEIS A134264 have numerous manifestations and applications in diverse areas of mathematics and physics (see the OEIS entry and ...
4
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77
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Theta functions in acyclic cluster algebras
Setup: Let $B$ be a skew-symmetrisable integer matrix and $\mathcal{A}$ be the cluster algebra with principal coefficients at a seed with mutation matrix $\tilde{B}$ whose principal part is equal to $...
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73
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Partial fraction decompositions for integral domains
I've recently been involved in a math conversation regarding partial fraction decompositions for rational numbers, but we seem to lack a formal definition and are unsure about whether there is some ...
5
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0
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97
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Size of minimal generating set of ideal over Laurent polynomial ring
Recently in attacking a problem in algebraic topology relating to the construction of stably-free non-free modules over integral group rings I’ve noticed that it is often fairly easy to reduce to ...
4
votes
1
answer
254
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Are polynomial algebras over fields (that are not algebraically closed) tame?
Let $A$ be an algebra over a field $K$. Loosely speaking, an algebra is said to be tame if for each $d \in \mathbb{Z}_{>0}$ all but finitely-many of the indecomposable $A$-modules of $K$-dimension $...
3
votes
2
answers
336
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Zeros of higher Ext functors
I'm trying to understand module classes that are defined as the kernels of higher Ext functors (e.g., arising here; as this paper suggests, I'm coming at this problem outside of module theory). One ...
4
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0
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79
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Dimension of a positively graded ring after a suitable localization
Quesion- Let $R=\bigoplus_{i\ge 0} R_i$ be a (non-trivial) positively graded commutative Noetherian ring with $1(\not=0)$ of (Krull) dimension $d\ge 0$. Let $S\subset R_0$ be a multiplicative set such ...
1
vote
0
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109
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List of automorphism groups of low-dimensional complex commutative algebras?
Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group ...