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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0answers
70 views

On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)

I'm trying to take spectral sequences as a black box for application in commutative algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
5
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0answers
42 views

Chevalley restriction theorem: group vs lie algebra version

Let $G$ be a (split) reductive group over $k$, $T$ a split maximal torus, and W its Weyl group. I sometimes see the Chevalley restriction theorem stated as (1) $k[G]^G \xrightarrow{\sim} k[T]^W$ and ...
2
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1answer
88 views

The cokernel of an irreducible monomorphism is always the third term of an AR sequence with indecomposable middle term?

I recently studied the structure of the AR quiver of Dynkin type $\mathbb{A}_n/I$, $\mathbb{D}_n/I$ with $I$ any admissable ideal, and found that the cokernel of an irreducible monomorphism is always ...
6
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1answer
248 views

When annihilator of ideal and ideal is co maximal

Let $R$ be a commutative ring with identity. It is not always true that ideal $J$ and annihilator of ideal $ann(J)$ are co maximal (ex: integral domain) Is there a sufficient (necessary) condition( ...
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0answers
73 views

Concerning $\mathbb{C}[a,b,c,d]$, with special generators $a,b,c,d$

The following is a question I have asked in MSE; hopefully, it is ok to ask it here. Let $a,b,c,d \in \mathbb{C}[x,y]$. Assume that: (1) Each two of $\{a,b,c,d\}$ are algebraically independent. In ...
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75 views

Some basic definitions on the ring $\mathbb{Z}_n[x]$ [closed]

Let $n\in\mathbb{N}^*$ be a composite number. So $\mathbb{Z}_n[x]$ is not an integral domain. Recently, we need to research the Chinese remainder theorem (CRT) on $\mathbb{Z}_n[x]$, we need to find ...
5
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0answers
163 views
+100

Is $\mathrm{End}-\{0\}=\mathrm{Aut}$ for derivation Lie algebra?

Is it true that every nonzero endomorphism of Lie $\mathbb{C}$-algebra $\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\partial_{x_n}$ is an automorphism? As I ...
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1answer
359 views

Does every prime power generate a primary ideal?

Let $R$ be a commutative ring with identity and let $p\in R$ be a prime element (i. e. $(p)$ is a prime ideal). If $R$ is an integral domain, it can be shown that $(p^k)$ is a primary ideal for every $...
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89 views

Units in group rings

Let $F$ be any field with $p$ elements and $G$ be any finite $p$-group, combining together they form a group ring $FG$. And $V(FG)$ denotes group of units of coefficient-sum equal to 1 in $FG$. We ...
5
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154 views

Open covers by ind-affine ind-schemes

Many apologies if this is totally standard! I couldn't find it in the literature. Background definitions: A presheaf $X: \textbf{Aff}^\text{op} \to \textbf{Set}$ is an ind-scheme if it is a filtered ...
4
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1answer
89 views

When is the intersection of two determinantal ideals equal to the product?

Let $S = k[x_{i,j}\mid 1\leq i\leq n, 1\leq j\leq m]$ be a polynomial ring over an arbitrary field $k$. Let $M$ be a generic $n\times m$ matrix of indeterminates in the ring $S$ where $n\leq m$. For ...
4
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1answer
458 views

Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)

Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$) $$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x ...
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0answers
117 views

On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber

This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory. Allow me to first give a minor introduction. Let $(...
37
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2answers
5k views

What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
0
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1answer
77 views

An example of a special $1$-dimensional non-Noetherian valuation domain

I am looking for a $1$-dimensional non-Noetherian valuation domain $R$ such that there exists a sequence $\{a_i\}_{i=1}^\infty$ of elements of $R$ such that $\langle a_1\rangle \subsetneqq\langle a_2\...
4
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0answers
80 views

$K$-group of category of bounded chain complexes of Projective modules with finite length homologies

For a Commutative Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free ...
2
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0answers
66 views

Reference request: Cohen-Macaulay representations

Question: Are there good references of Cohen-Macaulay (CM) representation theory with concrete calculations of examples. In particular, I look for ones which contatin the classification of CM modules ...
3
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1answer
211 views

Prime ideals of formal power series ring that are above the same prime ideal

Let $R$ denote a commutative ring with identity and let $R[[X]]$ denote the ring of formal power series over $R$ in an indeterminate $X$. If $I$ is an ideal of $R$, then $I[[X]]$, the set of power ...
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1answer
108 views

Is a ideal of direct limit of rings is itself a direct limit of ideal?

By Ńeron-Popescu desinaglarization theorem, If R is any regular semi-local ring containing $\mathbb{Q}$. Then R is a direct limit of regular semi-local rings $R_{i}$, where each $R_{i}$ is essentially ...
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0answers
67 views

Set-theoretic generation by circuit polynomials

Let $P$ be a prime ideal in $S=\mathbb{C}[x_1,\ldots , x_n],$ and write $[n] = \{ 1, \ldots , n \}.$ The algebraic matroid of $P$ can be defined according to circuit axioms as follows: $C\subset [n]$ ...
3
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0answers
101 views

Most tensor subspaces of low dimension have rank-1 defining equations

Let $V_1,\ldots , V_k$ be vector spaces of dimensions $n_1,\ldots , n_k$ over a field of characteristic zero. Consider the rational map $ \newcommand{\PP}{\mathbb{P}} \newcommand{\bs}{\boldsymbol} \...
2
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1answer
131 views

Does $\mathcal{A}\otimes\mathbb{C}(t)\cong\mathcal{D}\otimes\mathbb{C}(t)$ imply an isomorphism of Lie algebras?

Assume that $\mathcal{A}$ is a Lie $\mathbb{C}$-algebra. Also denote the Lie $\mathbb{C}$-algebra $\mathcal{D}=\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\...
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2answers
371 views

Determining the kernel of the localization map when defining the localization by generators and relations à la Serre

All rings considered will be commutative and unitary. Let $A$ be a ring, $S \subseteq A$ a multiplicatively closed subset. The localization $\lambda_S : A \longrightarrow A[S^{-1}]$ can be ...
9
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1answer
545 views

Curious anti-commutative ring

Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context? Here $\Lambda[x_0, x_1, x_2, \ldots]$ is the (graded-)commutative algebra (either over ...
5
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1answer
149 views

Closure of the product of subfunctors

Background: Let $X: \textbf{CRing} \to \textbf{Set}$ be a presheaf on the category of affine schemes and $Z \subseteq X$ a subfunctor. One defines $Z$ to be closed if for every ring $A$ and every ...
5
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0answers
66 views

Rational functions with trivial Weil symbols at every point

Let $f, g$ be a pair of nonzero rational functions in $\mathbb{C}(t).$ For $\lambda\in \mathbb{C}$ let $a$ be multiplicity of $g(t)$ at $\lambda$ and $b$ - multiplicity of $f(t)$ at $\lambda.$ Weil ...
2
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1answer
132 views

Sufficient conditions for $\mathrm{Der}_k(A)$ to be f.g. projective

Let $k$ be a field and $A$ a commutative $k$-algebra. What are sufficient conditions for the module of derivations $\mathrm{Der}_k(A)$ to be finitely generated projective? I'm looking for conditions ...
0
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1answer
54 views

Question about Jacobian subalgebra

Assume that the algebraically independent polynomials $f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$. Is it true that $\mathbb{C}[x, ...
3
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0answers
99 views

Infinite sum in power series ring

Let $R$ be a commutative ring with $1$, $R[[x]]$ be the power series ring over $R$ and $A$ be an (prime) ideal of $R[[x]]$ with $x\not\in A$ and $\{f_i\}_{i=1}^\infty$ be a sequence of element of $A$. ...
118
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3answers
14k views

When is the tensor product of two fields a field?

Consider two extension fields $K/k, L/k$ of a field $k$. A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often justified by ...
1
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1answer
39 views

Continuations of derivations of Jacobian subring

Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\...
5
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0answers
151 views

When do the spectra of overrings glue to a proper morphism?

This question is motivated by the construction of blowups. Let $A \subset K$ be a commutative domain and its fraction field, and let $\{A_i\}$ be some finite collection of overrings in between. Let $X ...
2
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1answer
279 views

In $\mathbb{C}[x,y]$: If $\langle u,v \rangle$ is a maximal ideal, then $\langle u-\lambda,v-\mu \rangle$ is a maximal ideal?

I have asked the following question at MSE and got one answer. Any further ideas are welcome: Let $u=u(x,y), v=v(x,y) \in \mathbb{C}[x,y]$, with $\deg(u) \geq 2$ and $\deg(v) \geq 2$. Let $\lambda, \...
2
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0answers
70 views

F-vectors of simplicial complex and f-vectors of non-faces of simplicial complex

Is there any result which gives us a relation between f-vector of simplicial complex and f-vector of nonfaces of a simplicial complex? Thank you
10
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2answers
4k views

Applications of algebraic geometry/commutative algebra to biology/pharmacology

Are there applications of algebraic geometry/commutative algebra to biology/pharmacology? It might be that some Gröbner basis technique is used somewhere? I know there are some applications to ...
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0answers
213 views

Polynomial objects in any concrete category

EDIT: The original question had a trivial answer: it's just a coproduct. New question below New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...
3
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0answers
144 views

A characterization for a commutative ring with a special intersection property for prime ideals

Let $R$ be a commutative ring with $1$ with the property that for any infinite family $\{P_i\}_{i\in I}$ of distinct prime ideals of $R$ we have $\cap_{i\not= j} P_i\subseteq P_j$ for all but fnitely ...
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4answers
2k views

Formally étale at all primes does not imply formally étale?

All rings are assumed to be commutative and unital, with all homomorphisms unital as well. On last week's homework, there was a mistake in one of the questions: (2.5) Let $R\to S$ be a ...
3
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0answers
109 views

Uniqueness of $\delta$-structure on a $p$-torsion ring

I was working through Bhargav's notes on $\delta$-rings and prismatic cohomology, specifically lecture 2, page 2, point 5 where he claims that the ring $\mathbb Z[x]/(px,x^p)$ has a unique $\delta$-...
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0answers
58 views

Automorphisms of the completion of a strict henselian local ring $R$ which come from automorphisms of $R$

Let $A\rightarrow R$ be a local homomorphism of Noetherian strict henselian local rings with completions $\hat{A},\hat{R}$. Let $u\in R^\times, x\in R$ be such that there is a unique $\hat{A}$-linear ...
2
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2answers
139 views

Commuting nilpotent matrices and conjugation isomorphisms

Trying to study isomorphism classes of certain commutative Artinian $\mathbb{C}$-algebras I was lead to the following problem about matrices. Suppose you have a (non-zero) nilpotent matrix $A\in M_n(\...
5
votes
2answers
2k views

Iterated calculation of determinants

Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
3
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1answer
113 views

Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map is surjective?

Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map $x\mapsto x^2$ is surjective? This is certainly true for fields. For DVR's, you can ...
3
votes
1answer
194 views

Condition such that the fibres of a polynomial map $p :\mathbb{C}^n\rightarrow \mathbb{C}^n$ are finite

I was told that if $A$ is the subring of $\mathbb{C}[x_1,\ldots, x_n]$ generated by the polynomials $p_1(x_1,\ldots, x_n),\ldots, p_1(x_1,\ldots, x_n)$, then the preimage $p^{-1}(c)$ via the map $p = (...
1
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0answers
105 views

Non-zero divisors on an $I$-completely flat module

Let $A$ be a commutative ring (not necessarily Noetherian), $I=(f_1, f_2, \dots, f_n) \subseteq A$ a finitely generated ideal that is generated by a regular sequence. Let $M$ be an $A$-module, and ...
1
vote
1answer
48 views

Infinite uniform dimension $\Rightarrow$ infinitely many idempotents in a localization of a quotient

Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $Sup\{k \mid R$ contains a direct sum of $k$ nonzero ideals$\}=\infty. $ How can we construct an ...
4
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1answer
133 views

A $q$-analogue of a characterization of polynomials by binomial coefficients

Considering the binomial coefficient $\binom{x}{m}$ as a polynomial in $x$, the span of $\binom{x}{0}, \binom{x}{1}, \ldots, \binom{x}{d}$ is exactly the polynomials of degree $\le d$. A closely ...
2
votes
1answer
175 views

Finite generation of certain graded sequences of ideals

Let $U\subset\mathbb{C}^n$ be an open set containing the origin $o$ and $Y\subset U$ a complex analytic subvariety of pure codimension $c$ with ideal sheaf $\mathcal{I}_Y$. Let $\frak{a}_{\bullet}=\{{\...
2
votes
1answer
141 views

For an integral domain $R$ when does $a^2 \equiv b^2 \bmod 4R$ imply $a \equiv b \bmod 2R$?

Suppose we have $a^2 \equiv b^2 \bmod 4R$ where $R$ is an integral domain. Under what conditions on $R$ can we conclude that $a \equiv b \bmod 2R$? This would hold if $2 \in R$ is a prime or the ...
3
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0answers
81 views

Is the determinantal ideal of the span of a linearly independent set of rank-one matrices radical?

Let $k$ be an algebraically closed field, and let $X_1,\dots, X_n \in M_m(k)$ be rank-one matrices that are linearly independent over $k$. For a fixed integer $1 \leq r \leq m$, consider the ideal $I \...

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