Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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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 ...
Elías Guisado Villalgordo's user avatar
2 votes
1 answer
139 views

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 ...
uno's user avatar
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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)=...
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8 votes
1 answer
335 views

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 $...
Mare's user avatar
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4 votes
2 answers
350 views

$p$-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 ...
Boaz Moerman's user avatar
3 votes
1 answer
200 views

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.
Amit Phogat's user avatar
5 votes
0 answers
153 views

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 ...
Roman's user avatar
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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 ...
94thomas's user avatar
0 votes
1 answer
141 views

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 ...
Sky's user avatar
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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 \...
H A Helfgott's user avatar
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1 vote
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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}...
Jooh's user avatar
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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/...
Analyse300's user avatar
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80 views

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 $\{...
LeviathanTheEsper's user avatar
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51 views

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 $R$ ...
user141099's user avatar
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157 views

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 ...
mtheorylord's user avatar
1 vote
0 answers
117 views

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 ...
It'sMe's user avatar
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4 votes
0 answers
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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 ...
Mare's user avatar
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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}\\ ...
Li Guanyu's user avatar
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0 answers
169 views

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 $f(T) = p^{\mu} \cdot u(T) \cdot g(T)$, ...
Adithya Chakravarthy's user avatar
4 votes
1 answer
207 views

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 ...
Andrea Antinucci's user avatar
4 votes
1 answer
328 views

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 ...
Simón Flavio Ibañez's user avatar
5 votes
1 answer
1k views

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 ...
pinaki's user avatar
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6 votes
1 answer
229 views

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 ...
Sam Forster's user avatar
1 vote
1 answer
79 views

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 ...
strat's user avatar
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1 answer
280 views

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 $\mathbb F_p[[x]]$? ...
Peter Kropholler's user avatar
3 votes
2 answers
375 views

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 ...
It'sMe's user avatar
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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}\...
LiminalSpace's user avatar
0 votes
1 answer
70 views

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 $...
T C's user avatar
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0 answers
110 views

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/...
Thibault Poiret's user avatar
2 votes
0 answers
66 views

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 ...
Tom Copeland's user avatar
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0 votes
1 answer
320 views

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 $I=Re$. ...
GuoJi's user avatar
  • 245
3 votes
1 answer
720 views

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 $...
Sky's user avatar
  • 913
8 votes
0 answers
173 views

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 $|...
Dave Benson's user avatar
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5 votes
0 answers
125 views

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 ...
Pete L. Clark's user avatar
2 votes
0 answers
158 views

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 ...
Markuss Schmuckler's user avatar
10 votes
1 answer
809 views

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 ...
George's user avatar
  • 465
4 votes
1 answer
258 views

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.
SKS's user avatar
  • 81
8 votes
1 answer
323 views

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 ...
user avatar
2 votes
1 answer
159 views

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 $...
user avatar
4 votes
0 answers
150 views

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 ...
aaa acb's user avatar
  • 141
2 votes
1 answer
235 views

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 ...
user avatar
3 votes
1 answer
205 views

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 ...
pinaki's user avatar
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0 votes
0 answers
72 views

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 ...
Tom Copeland's user avatar
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4 votes
0 answers
77 views

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 $...
Antoine de Saint Germain's user avatar
1 vote
0 answers
73 views

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 ...
Daniel Asimov's user avatar
5 votes
0 answers
97 views

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 ...
William Thomas's user avatar
4 votes
1 answer
254 views

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 $...
Iteraf's user avatar
  • 482
3 votes
2 answers
336 views

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 ...
Will Boney's user avatar
4 votes
0 answers
79 views

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 ...
Sourjya Banerjee's user avatar
1 vote
0 answers
109 views

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 ...
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