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

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45 views

Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...
-2
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0answers
47 views

Are universally catenary equidimensional local rings Cohen-Macaulay? [on hold]

Cohen-Macaulay rings are universally catenary, I do not choose catenary rings because we can find catenary but not universally catenary rings at wiki Catenary ring. Cohen-Macaulay local rings are ...
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1answer
240 views

Reference for a lemma on étale maps

The Stacks Project has the following really nice Lemma concerning étale maps of rings: Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation $$ ...
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1answer
92 views

ideals generated by two elements in the polynomial ring of two variables over a field

Let $k$ be a field. For example, $k=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime. Let $k[x,y]$ be the polynomial ring. Let $f,g\in k[x,y]$. Let the ideal $I=(f,g)$ be the ideal of $k[x,y]$ generqated ...
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0answers
23 views

Property of free submodules for a module over a PID [migrated]

This question was asked here and remains without solution. It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is ...
3
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2answers
282 views

Can there be a non-trivial epimorphism (of rings) from a field? [closed]

I apologize if this question is trivial, but I just cant figure it out. Let $K$ be a field and let $K\longrightarrow A$ be an epimorphism of rings. Is it necessary that $A=K$?
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0answers
74 views

Global dimension of graded Lie algebra

The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm ...
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0answers
71 views

a question about minimal reduction of an ideal [closed]

let $I=(x^2,y^2,xyz)$ be an ideal of $R=k[[x,y,z]]$, where $k$ is field. Then $(x^2,y^2)$ is a minimal reduction of $I$.
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72 views

a question about basic ideal [closed]

An ideal that has no reduction other than itself is called basic. If $(R,m)$ is Noetherian local ring and $I$ is a basic ideal of $R$, then $\overline{{m}{I^n}}\cap I^n={m}I^n$
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0answers
40 views

Minimal free resolution of sum of ideals

Let $S$ be the polynomial ring in $n$ variables, and let $I_1$ and $I_2$ be ideals in $S$. What can be said about the $\mathbb{Z}$-graded minimal free resolution of $I_1+ I_2$ in terms of the ...
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1answer
333 views

Does this $\mathbb{Z}_p$-algebra morphism induce a closed immersion on the generic fiber?

Let $R$ be a local and smooth $\mathbb{Z}_p$-algebra and $B$ an $R$-algebra of finite type which is an integral domain with $\operatorname{dim}B\leq \operatorname{dim}R$ such that $B/(p)$ is ...
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1answer
90 views

behavior of multiplicity in exact sequences

Bruns-Herzog define multiplicity When the ring and module are not necessarily graded as $e(M)=e(gr_m(M))$, see B-H 4.6. I have two questions: Question1. Many concepts in commutative algebra have ...
2
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1answer
203 views

Are schemes which agree on open set and its complement equal? - w/ applications to initial ideals/tropical basis

I appreciate the comments so far and am modifying based on something closer to the problem I'm interested in. I started out with something far too general. This is probably easy, but I have been ...
4
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0answers
157 views

Does integral closure commute with pushforward

Suppose that $\pi : Y \to X$ is a proper birational morphism between normal varieties (schemes, whatever). Suppose that $I$ is an ideal sheaf on $Y$. One can form $\pi_* I$ and construct an ideal ...
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0answers
37 views

Projective dimension of modules over non-discrete valuation rings

Let $\mathbb{C}_p$ be $p$-adic completion of an algebraic closure of $\mathbb{Q}_p$, $\mathcal{O}_{\mathbb{C}_p}$ be its valuation ring and $\mathfrak{m}$ its maximal ideal. Does anyone know the ...
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1answer
195 views

Integral closure of an ideal [closed]

Let $r^n+a_1r^{n-1}+\cdots+a_n=0$ be an equation of integral dependence of $r$ over an ideal $I$. Does exist a finitely generated ideal $J$, such that $J\subset I$ and $a_i\in J^i$ for all ...
2
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1answer
97 views

Families of ideals with a given initial ideal

Assume a fintie set of monomials is given. Is there a way to find the family of ideals whose initial ideal (say w.r.t revlex order) is generated by that finite set? I'll appreciate any partial answer, ...
4
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1answer
200 views

Reference request for division algebras, over $\mathbb{Q}_{p}((t))$

I was looking for a possible reference that would answer the following question, Let $\mathbb{Q}_{p}$ be the $p$-adic numbers and $\mathbb{Q}_{p}((t))$ be the field of Laurent polynomials over ...
3
votes
1answer
142 views

The complex of Kahler differentials and de Rham complex

Let $A$ be a unital commutative algebra (say over complex numbers). Consider the multiplication map $m:A \otimes A \to A$ and put $\Omega^1_u(A)=\ker m$ to be the space of universal differential ...
3
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0answers
67 views

Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
3
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1answer
86 views

A vector version of the Segre embedding: what is the kernel of the ring map?

TL;DR version. Given a commutative ring $\mathbf{k}$ and $n+m$ "generic" vectors $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n, \mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_m$ in $\mathbf{k}^k$ ...
2
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1answer
121 views

Decide two indices of Ext functor

This question is from the proof of Theorem 11.34 in the book: Twenty-four Hours of Local Cohomology. Let $R$ and $S$ be CM local ring and $R\to S$ a local homomorphism such that $S$ is a finite ...
3
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1answer
113 views

An integral domain of dimension one with a non-trivial infinite intersection of prime ideals

In a (necessarily non-Noetherian) integral domain $A$ of (Krull) dimension $1$, is it possible that there is an infinite collection of prime ideals $\mathfrak{p}_i$ such that $\cap_i \mathfrak{p}_i ...
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0answers
75 views

Does $k[x][[h]]$ finitely generated as $k[[h]]$-algebra? [migrated]

Does $k[x][[h]]$ finitely generated as an algebra over $k[[h]]$? As the title, where $k$ is a field, and $xh=hx$.
2
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0answers
89 views

Lifting of Commuting Maps of Vector Bundles

Assume that we have a vector bundle $\mathcal{F}$ over $\mathbb{P}^d(\mathbb{C})$ that is generated by global sections. Let $\pi \colon \mathcal{O}^n \to \mathcal{F}$ be the associated map that is ...
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1answer
186 views

Necessary and sufficient condition for $can : A^X\otimes_A A^Y\rightarrow A^{X\times Y}$ to be an embedding

The two sets are, of course, supposed infinite. This question is related to that one Commutation of tensor products with inverse limits in a specific case where it received a (partial) answer ($A$ ...
1
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1answer
129 views

Structure of $\text{Aut}_R(R[X])$

Let $R$ be a commutative ring with identity. I'd like to know how to determine the set $\text{Aut}_R(R[X])$ of all $R$-automorphisms of $R[X]$. I've proved that all $\sigma\in\text{Aut}_R(R[X])$ ...
2
votes
0answers
117 views

Irreducibility of a general fibre

Let $A\subseteq B$ be an inclusion of affine domains over an algebraically closed field $k$ of characteristic $0$. Can someone give me a reference for the following fact? If $A$ is algebraically ...
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1answer
393 views

Example of a ring $R$ such that $\dim(R[[X]])<\dim(R[X])$

Dimension refers to the Krull dimension of a commutative ring. In the paper "Prime ideals in power series rings" J. Arnold gives an example of such a ring: Let $k$ be a field and $K=k(t)$ a ...
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0answers
57 views

How to prove that the set of maximal elements of a set of prime ideals is finite

Let $A$ be a subset of ${\rm Spec}(R)$ with $R$ noetherian Are there any techniques to prove that ${\rm max}(A)$ (ie the set of maximal elements of $A$) is finite? I'm looking for equivalent ...
1
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1answer
118 views

Arbitrary chains of prime ideals in $R[X]$

For a commutative ring $S$ of finite Krull dimension $d$, we have $1+d\leq \dim(S[X])\leq 2d+1$. One proof of this uses the fact that if $Q_1\subset Q_2\subset Q_3$ is a chain of prime ideals of ...
2
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0answers
152 views

Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$?

In [1], Propisition 6.1.9(2), it said that if $R$ is a perfectoid ring such that $pR^\circ$ is closed in $R^\circ$ (this includes the case if $R$ is of character $p$, or if $p$ is invertible in $R$, ...
5
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1answer
310 views

UFD and fundamental group

Let $C$ be the curve $x^2+y^2-1$, defined over $\mathbb R$. It is easy to see that $\mathbb R[C]$ is not a UFD, as witnessed by the identity $(1-x)(1+x)=y^2$. On the other hand, the real locus ...
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0answers
76 views

Deformations of associative algebras and Hochschild cohomology

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer: Let $(A,\mu)$ be a commutative associative algebra ...
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0answers
71 views

Extension of scalars and projective limits

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...
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58 views

J not being subset of integral closure of I

Definition. Let $I$ and $J$ be ideals in a ring $R$. An element $r \in R$ is said to be integral over $I$ if there exist an integer $n$ and elements $a_i \in I^i, i = 1, . . . , n$, such that $$r^n + ...
1
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1answer
160 views

On Q-Cartier Divisors

I have my question on Q-Cartier Weil divisor. People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of ...
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0answers
58 views

completion of non-finitely generated ideal

Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates. Then we can consider the completion ...
3
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1answer
123 views

Uncountable chain of prime ideals in an arbitrary direct product of rings

I am only considering commutative rings with $1$. Dimension refers to Krull dimension. In the paper "Products of commutative rings and zero-dimensionality", Gilmer and Heinzer give necessary and ...
3
votes
3answers
183 views

Are linearizations of involutive PDEs locally solvable?

A possibly soft question for you guys and gals. Say a system of analytic PDEs has been completed to involution (in the sense that it's geometric symbol has a Pommaret basis, or has vanishing ...
2
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1answer
107 views

complement of an open immersion

Let $A\subseteq B$ be normal affine doamins over a field $k$ with same field of fractions. If the induced morphism of schemes $i^*:Spec\ B \rightarrow Spec\ A$ is an open immersion, how to prove that ...
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0answers
61 views

Failure of little lemma in non-separable case

A nice little lemma in commutative algebra says the following (see for instance proposition 5.17 in [Atiyah-MacDonald]): If $A$ is a Noetherian integrally closed domain, $K$ its field of fractions ...
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1answer
260 views

Number of elements in a fiber

Let $A\subseteq B$ be normal affine domains over an algebraically closed field of characteristic 0. If it is given that the corresponding morphism of schemes Spec $B\rightarrow$ Spec $A$ is ...
2
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1answer
48 views

Polynomial degree comparison of Nullstellensatz and Positivstellensatz over real algebraic sets

Suppose we have a (finite) system of polynomials $P = \{ p_i \} \subseteq \mathbb{R}[x_1, \ldots, x_n]$. Then it is well known by the Nullstellensatz that either $P$ has a simultaneous zero over ...
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0answers
104 views

Cohen-Macaulay fibers

Let $Y$ be a set of points in $\mathbb{P}^n$. Then we can write a resolution $$0\rightarrow P_n \rightarrow \cdots \rightarrow P_0\rightarrow \mathcal{O}_Y$$ where each ...
3
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1answer
92 views

Existence of Factor rings of UFDs which are UFDs

Suppose that $X=Spec(A)$ is an affine variety over an algebraically closed field $k$ which is normal and such that $Cl(X)=0$. I am interested in hypersurfaces of $X$ which again satisfy this ...
0
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1answer
85 views

Samuel multiplicity

Let $X$ be the hyper-surface defined by $$f:=\sum_{i=1}^k x_i^n=0$$ in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal $$I=(x_1^{n-1},\dots , x_k^{n-1}) $$ What is ...
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38 views

t-linked extension

Let $A\subseteq B$ be an extension of commutative integral domains. the extension is t-linked if it satisfies the following property: If P is a finitely generated ideal of A such that $P^{-1}=A$ than ...
5
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1answer
344 views

Is the ring $\mathbb{Z}_p [[x]]\otimes_{\mathbb{Z}_p} \overline{\mathbb{Q}}_p$ Noetherian?

Is the ring $\mathbb{Z}_p [[x]]\otimes_{\mathbb{Z}_p} \overline{\mathbb{Q}}_p$ Noetherian?
3
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0answers
110 views

going down theorem

Typical maps that satisfy "going down theorem" are flat morphisms and integral extensions of normal rings that are integral. Let $Spec(B)\rightarrow Spec(A)$ be a finite type morphism of k-noetherian ...