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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Super-Gorenstein ideal of ${\Bbb F}_p[[X_1,\ldots,X_n]]$

Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$. Definition(Super-Gorenstein ideal): $...
Pierre's user avatar
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2 answers
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Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers

Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map $$- \circ \varphi: \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes_{\mathbb{Q}} ...
15 votes
3 answers
538 views

Direct sum of Hopf algebras

I realise that this question might be rather basic but however I was unable to find the answer in any textbook nor manage to figure out the answer. The question is the following: given two Hopf ...
truebaran's user avatar
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5 votes
1 answer
535 views

Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?

Let $K$ be a field of characteristic zero. Let $\Omega = K[x_1, \dots, x_n, dx_1, \dots, dx_n]$ be the differential ring of algebraic differential forms over $K[X_1, \dots, X_n]$. Is there an ...
Marc Nieper-Wißkirchen's user avatar
3 votes
1 answer
368 views

What is the geometric meaning of content or intersection flatness?

The polynomial extension $R \rightarrow R[X]$ ($X$ an indeterminate) has many nice properties beyond faithful flatness. The one I'm most interested in at the moment is the following. Say that a flat ...
Neil Epstein's user avatar
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3 votes
1 answer
2k views

Localisation of two rings which is an integral extension, then integral extension still holds?

Question seems simple, but I just can't find the solution. Let A/B be an integral ring extension and let P be a prime ideal of B. By going-up theorem, there is Q, a prime ideal of A, lying over P. ...
user83008's user avatar
2 votes
0 answers
280 views

Local Profinite Ring

I haven't received any substantial responses to a similar question on math.stackexchange, so let me try here. Let $R$ be a profinite ring (that is a projective limit of finite rings). Assume that ...
Joël's user avatar
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When is a power of an indeterminate in an ideal with 2 generators?

If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...
Pierre's user avatar
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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 ...
Manoj Kummini's user avatar
2 votes
1 answer
655 views

Finite-index free subgroups in lattices and matrix rings

It is a theorem of Selberg that a lattice $\Gamma$ in a linear group has a torsion-free subgroup of finite index. Page 64 in 'Introduction to Arithmetic Groups' by Dave Morris asserts these can be ...
burtonpeterj's user avatar
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6 votes
2 answers
863 views

Does regular field extension preserve regularity?

Let $k$ be an arbitrary field and suppose that $K/k$ is a regular field extension. Let $V$ be regular scheme of finite type over $\text{Spec }k$ (not necessarily smooth). Is it true that $\text{Spec }...
Tomasz Lenarcik's user avatar
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1 answer
449 views

Iwasawa theory for Mazur's deformation ring R

The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other. Let ${\Bbb Q}_{\infty}$ be the unique ...
Pierre's user avatar
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4 answers
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On the fixed point of automorphism of $\mathbb F_3[[T]]$

Consider the automorphism $\sigma$ on ${\Bbb F}_3[[T]]$ such that $T \mapsto c_1T + f(T)$ with $c_1 = 1$ or $-1$, and $f(T) \not=0$ and the non-zero leading term $c_mT^m$ of $f(T)$ satisfies $m \geq 2$...
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1 answer
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Constructively correct notion of unique factorization domain

Recall the well-known proof that a unique factorization domain is a GCD domain: Let $x, y \in R \setminus \{ 0 \}$. Factor $x$ and $y$ into pairwise non-associated irreducible elements: $$\begin{...
Ingo Blechschmidt's user avatar
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553 views

Recursive Non-standard Models of Modular Arithmetic? [closed]

Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)...
Russell Easterly's user avatar
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0 answers
517 views

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 ...
Jack Huizenga's user avatar
1 vote
1 answer
465 views

formally étale morphisms which are also universally closed

A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion. Is it possible to characterize morphisms which are formally etale and universally ...
Andrew Stout's user avatar
1 vote
1 answer
304 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 ...
alfs's user avatar
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3 answers
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Behavior of duality under pull-back

I have a technical question on commutative algebra. I am not an expert in the subject, and I would like to know if there are "typical conditions" making the following possible. Let $\varphi:R\to S$ ...
Tommaso Centeleghe's user avatar
2 votes
1 answer
118 views

Regularity of special monomial ideals

Let $R = k[x_1\ldots x_n]$, and $a,b$ are vectors with integer entries, whose all entries of $a$ are non-negative, and say sum of coordinates of $b$ is $0$. Let $I$ be a monomial ideal generated by $x^...
Thanh Vu's user avatar
2 votes
1 answer
521 views

Geometric interpretation of a (standard) commutative algebra fact

Which is your geometric interpretation (if any) of the following commutative algebra proposition? Proposition. Let $M$ be a finitely generated $A$-module, $I\subseteq A$ an ideal, and $\phi\in \...
Qfwfq's user avatar
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2 answers
651 views

Pontryagin dual

Suppose $M$ is a $Z_p[[T]]$-module and $\widehat{M}$(the Pontryagin dual of $M$) is a finitely generated torsion $Z_p[[T]]$-module. How to prove that $\widehat{M}$ has $\mu$-invariant zero $\...
Suman's user avatar
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1 vote
1 answer
171 views

Morphisms preserving weak normality

I would like to find a class of morphisms for which weakly normality descends. The notion of seminormlaity is very close to the one of weakly normality and for seminormal schemes one has Theorem 5.8 ...
micco's user avatar
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6 votes
1 answer
659 views

A construction of Kähler differentials and Illusie cotangent complex as colimit over embeddings

Let $\Bbbk$ be a field, $X$ affine scheme of finite type over $\Bbbk$. Let $\mathcal C_X$ be the category of closed embeddings of $X$ into (say affine) smooth $Y$'s of finite type over $\Bbbk$, ...
rrrrrttttttt's user avatar
14 votes
0 answers
910 views

What is the state of art in Groebner bases

How big polynomial systems can we deal with? How do you know when you don't even have to try? Motivation: Recently I tried to solve a problem posed in another MO question and ultimately I got stuck ...
Vít Tuček's user avatar
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0 votes
2 answers
285 views

0-dimensional Gorenstein local ring.

Assume the following condition for the ring T = F_p[[X,S]]/I: Condition 1. T is NOT a zero ring. Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements. Then, is T a ...
Pierre's user avatar
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Does the coordinate ring of affine variety admit a structure of infinite dimensional variety?

We work in the category of algebraic varieties over some algebraically closed field $k$. By infinite dimensional variety I mean a filtration: $$ V_0\subset V_1\subset V_2\subset\ldots $$ where each $...
Tomasz Lenarcik's user avatar
4 votes
1 answer
141 views

Which power of $2$ kills $W(k)$?

Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say ...
Mikhail Bondarko's user avatar
8 votes
4 answers
5k views

Jacobson ring = a ring whose nilradical and Jacobson radical coincide?

In Wikipedia it is claimed that "A ring is called a Jacobson ring if the nilradical coincides with the Jacobson radical." Here the word "ring" means a commutative ring. However, I remember that ...
user43771's user avatar
0 votes
1 answer
345 views

Iwasawa invariants

Suppose $M$ is a finitely generated torsion $Z_p[[T]]$-module; the torsion comes from the $\mu$-invariant and the $\lambda$-invariant. Consider $M/(p)$ and $M[p]$ ($p$-torsion of $M$) which are $F_p[[...
Suman's user avatar
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2 votes
1 answer
426 views

Resolution of singularity of polynomials

Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$. By Hironaka's desingularization theorem, there exists a birational map ...
JJH's user avatar
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20 votes
1 answer
1k views

How is a descent datum the same as a comodule structure?

For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong S\...
Jonathan Beardsley's user avatar
0 votes
2 answers
355 views

Rank of a $ \mathbb{Z}_{p}[[T]] $ module

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can ...
Suman's user avatar
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2 votes
1 answer
523 views

Checking flatness using radical ideals

Let $R$ be a commutative ring and $M$ a not necessary finitely presented $R$-module. I am looking for a prove or a counterexample to the following statement: $M$ is flat as an $R$-module if and only ...
user7d229955's user avatar
3 votes
2 answers
360 views

Kahler differentials on cluster varieties

On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...
user36931's user avatar
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0 votes
1 answer
714 views

Does Noether normalization hold more general? [duplicate]

Noether normalization tells us that a finitely generated $k$-algebra is an integral extension of a polynomial algebra over the field $k$. My question is whether this still holds if we replace the ...
Andreas Maurischat's user avatar
0 votes
1 answer
172 views

$I/N$ 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 $N \subseteq I$. ...
Bacem's user avatar
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25 votes
5 answers
3k views

is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...
John Salvatierrez's user avatar
1 vote
0 answers
153 views

Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber

Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...
TNAn's user avatar
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9 votes
1 answer
1k views

Well founded induction attributed to Noether

What I know as well founded induction, namely the rule $$ \big(\forall y.(\forall z.z\lt y\Rightarrow\phi z)\Rightarrow\phi y\big)\Longrightarrow\big(\forall x.\phi x\big), $$ whose validity is the ...
Paul Taylor's user avatar
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2 votes
2 answers
652 views

Does $\Gamma_*$ commute with tensor product?

Given a coherent sheaf $\mathcal{F}$ we denote by $\Gamma_*(\mathcal{F})=\oplus H^0(\mathcal{F}(d))$. Suppose, $\mathcal{F}_1$ and $\mathcal{F}_2$ are two coherent sheaves on $\mathbb{P}^n$. Denote by ...
user43198's user avatar
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6 votes
0 answers
525 views

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. $m$...
user43161's user avatar
12 votes
1 answer
533 views

Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?
lina lie's user avatar
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6 votes
1 answer
679 views

Complexity of computing the Galois group

There has been some discussion of computing the Galois group of a polynomial over the integers, but I can't seem to find any results, or even a question of what the complexity of this might be. For ...
Igor Rivin's user avatar
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6 votes
0 answers
106 views

Irreducibility testing and factoring

It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...
Igor Rivin's user avatar
  • 95.5k
7 votes
2 answers
433 views

Pre-images of unipotent elements in $\operatorname{SL}_{n}(A)$

The starting point of this question is the (presumably) well-known theorem (the proof I know is from Abelian $\ell$-adic representations and elliptic curves from J-P.Serre in which it is a lemma for $...
Olivier's user avatar
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4 votes
3 answers
575 views

Exponentials in the opposite category of finite separable algebras

Let $K$ be a field and $G=Gal(K_s/K)$ is its absolute Galois group. Then, by Galois theory, the category of finite separable algebras over $K$ (denoted by $Sep(K)$) and the category of finite ...
Fujita Tomomi's user avatar
10 votes
2 answers
2k views

Characteristic polynomial of exterior power

Suppose $f$ is a linear map, and consider $\Lambda^k f$ as the usual exterior power of $f$ (if you prefer matrices, it is a matrix whose entries are the $k\times k$ minors of $f.$) The coefficients of ...
Igor Rivin's user avatar
  • 95.5k
2 votes
1 answer
143 views

Cohen-Macaulayness of the scheme of centralizer

Let $G$ be a simply connected group over an algebraically closed field $k$, and $I:=\{(g,\gamma)\in G\times G\vert~ g\gamma=\gamma g\}$ the scheme of centralizer. Is $I$ a Cohen-Macaulay scheme ...
prochet's user avatar
  • 3,432
12 votes
2 answers
1k views

Fields aren't group objects in Ab, so what are they?

This might be a vague question, but I am troubled by the fact that fields do not admit a nifty categorical definition. An obvious attempt such a definition would be to say that fields are commutative ...
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