# Tagged Questions

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58 views

### Quotients and radicals

Let $I, J$ be ideals in a commutative ring with identity $R$. Define the quotient ideal $(I : J)$ by $$(I : J)=\{x\in R : xJ\subseteq I\}.$$
Define the radical $r(A)$, of an ideal $A$ of $R$ by
...

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127 views

### scheme of commuting matrices

Let $k$ be any field. Let $r$ and $n$ be two positive integers.
Consider the functor $F$ from the category of $k$-schemes to the category of sets which sends a $k$-scheme $T$ to the set of matrices ...

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**2**answers

259 views

### Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?

Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...

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65 views

### About freeness of modules over the coordinate ring of an affine variety [migrated]

Let $X$ be an irreducible affine variety, $A$ be its coordinate ring, $M$ be an $A$-module. Suppose that for any maximal ideal $m$ of $A$, the localization $M_m$ is a free module of rank $n$ (finite ...

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**1**answer

266 views

### Is the regularity of finitely generated rings decidable?

Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?
I mean by regular that the localization at every prime ideal is a regular local ...

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104 views

### Thickness of the category of perfect complexes with finite length homology

Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...

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**1**answer

120 views

### Flatness of Normaliztion of regular schemes

I have a followup to the followig question: Flatness of normalization
Suppose that $X$ is a regular scheme (of finite type over a $\mathbb{C}$ if one wants) and $X'$ is the normalization of $X$ in a ...

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70 views

### Vanishing of the module of differentials of a extension of perfect fields

Let $L|F$ be a extension of perfect fields of characteristic $p$, $\phi_F:F \to F_{\phi}$, $\phi_L:L \to L_{\phi}$ the Frobenius isomorphisms ($F_{\phi}=F$ but considered as $F$-algebra via $\phi_F$). ...

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1k views

### Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...

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176 views

### Spec of an injective ring map contains minimal primes in its image?

Let $f\colon A \rightarrow B$ be an injective ring homomorphism. One knows (from EGA I, 1.2.7 or elsewhere) that the image of $\mathrm{Spec}(f)$ is dense. Does that image necessarily contain all the ...

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212 views

### Learning roadmap in Algebra [closed]

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas:
a) Commutative Algebra
b) Field Theory and Galois Theory
c) Homological Algebra
My question is ...

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61 views

### Explicit equations for conormal bundle to an affine toric variety

Let $L \subset \mathbb{Z}^n$ be a lattice and let $X_L$ be the closed toric subvariety of $\mathbb{C}^n$ cut out by the lattice ideal $I_L = \{x^{l_+} - x^{l_-} \,| \, l_+, l_- \in \mathbb{N}^n \text{ ...

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99 views

### preservation of localness among certain Krull domains

The following question essentially appeared (http://math.stackexchange.com/questions/931801/preservation-of-localness-among-certain-krull-domains) on math.SE a while ago, but nobody has done anything ...

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262 views

### Singularities arising from the Minimal Model Program (an algebraic point of view)

I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ...

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137 views

### Categorical characterization of closed imbeddings

Let $f\colon X\to Y$ be a morphism of schemes.
Let $F_X$ and $F_Y$ be the contravariant functors from the category $Sch$ of schemes to the category of sets defined via the Yoneda construction, i.e. ...

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161 views

### integral curves and differential equations on arcs

I am trying to prove a statement that is obivious in analytic setting, but makes me feel at a loss in formal algebraic setting.
Let $M$ be a smooth curve over an algebraically closed field $k$. Let ...

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71 views

### Stable analytic manifold under simple action

For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m} - 1
$$
on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...

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**1**answer

124 views

### Does projective duality preserve arithmetic-Cohen-Macaulay-ness?

Let $V$ be a vector space over $\mathbb{C}$.
Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring ...

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118 views

### Bound for the height of equations defining the singular locus of a variety

Fix positive integers $m, n, d$.
In what follows, the height of an algebraic number will mean the absolute multiplicative height.
Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...

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94 views

### $\Gamma_Z(\widetilde M)\cong\widetilde{ \Gamma_Z(M)}$

Let $R$ be a Noetherian ring and let $M$ is an $R$-module. Consider the associated affine scheme $(\text{Spec R},\mathcal{O}_{\text{Spec R}})$ and Suppose $Z\subset X$ is a closed subset of ...

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**1**answer

220 views

### Hilbert function of points in $\mathrm{P}^2$

Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If
$$
...

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159 views

### Scheme vs $A$-scheme morphisms

Let $A=S^{-1}\mathbb Z$ be a localization of a multiplicative set $S\subset \mathbb Z$.
Question 1: Let $X$ be an arbitrary $A$-scheme, and view $X_A=X\times_{\mathbb Z} A$ as an $A$-scheme via the ...

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163 views

### Zariski open set of linear forms

Let $I$ be a graded homogeneous ideal over $k[x_1, ... ,x_n]$ and $h$ a linear form, let $H$ be the corresponding hyperplane and $I_H$ the restriction of $I$ to $H$.
I am looking for a Zariski open ...

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**1**answer

214 views

### Open covering of the Hilbert functor of points

Let $R \to A$ be a homomorphism of commutative rings. Define the functor $$\mathrm{Hilb}^n_{A/R} : \mathsf{CAlg}(R) \to \mathsf{Set}$$ as follows: If $B$ is a commutative $R$-algebra, then ...

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304 views

### Canonical Sheaf of Projective Space

I am stuck on one step that occurs without explanation in several Algebraic geometry books.
Starting from the exact sequence
$$0\rightarrow \Omega_{\mathbb{P}^n}\rightarrow ...

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220 views

### commutative algebra, diagonal morphism

can anyone help me with the following statement (it is part of a bigger proof where it is not explained).
Let $B$ be a finite type $A$-algebra and consider the kernel $I$ of the diagonal ...

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99 views

### generalization Abhyankar's lemma

This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question.
Let ...

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125 views

### Is every Noetherian *connected* ring a quotient of a Noetherian domain?

This question is a strengthening of this question (answered negatively), and arose due to David Speyer's answer here.
Geometrically, this asks if every Noetherian connected affine scheme can be ...

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258 views

### Existence of a ring with specified residue fields

Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$?
To prevent things from being too easy, I ...

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vote

**1**answer

161 views

### Classification (and automorphisms) of torsion-free modules/sheaves

I would like to know what can be said about the classification of torsion-free modules.
For my purposes, we can assume that $R$ is the function ring of a smooth affine variety over a field. How does ...

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181 views

### Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...

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139 views

### Reference: A nowhere vanishing section of a vector bundle is locally split

Well-known fact:
If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...

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117 views

### Is the completion of an arbitrary ring w.r.p. a maximal ideal a local ring? [closed]

Give an arbitrary commutative ring (not necessarily noetherian) $A$ with $\mathfrak{m}$ a maximal ideal. Is the completion $\hat{A}$ w.r.t. the ideal $\mathfrak{m}$ a local ring? If so, is the maximal ...

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240 views

### Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables

Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function
$P_i = px^i + qy^i + rz^i$
where $x,y,z$ are coordinates. I have a few related ...

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64 views

### stability of discriminant curve of nonsingular nets of quadrics

The following are from this question:
A pencil of quadrics in $\mathbb{P}^n$ is a line in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of pencil of quadrics is the Grassmannian ...

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528 views

### Descent of functions along finite birational morphisms

Let $A\to B$ be a morphism of (unitary commutative) rings such that $B$ is module-finite over $A$ and there exists $f\in A$ which is a nonzerodivisor in $A$ and in $B$, with $A[1/f]\to B[1/f]$ an ...

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**1**answer

238 views

### Reference for comparison of heart cohomology with standard cohomology

I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations).
Let A,B be two hearts of ...

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**1**answer

454 views

### A weak version of Bass' conjecture

Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...

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226 views

### Are Abelian varieties (sometimes) globally $F$-split?

As defined by Karen Smith here, beginning of section 3? If $E$ is an elliptic curve, then it is when $E$ is ordinary. I wonder about higher dimension cases. Any references would be greatly ...

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259 views

### Let $f \colon X \to Y$ be an étale morphism of schemes. If $Y$ is integral, then is $X$ integral? [closed]

Let $f \colon X \to Y$ be an étale morphism of schemes.
We know:
(1) if $Y$ is normal, then $X$ is normal.
(2) if $Y$ is regular, then $X$ is regular.
(3) if $Y$ is reduced, then $X$ is ...

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271 views

### Are quotients of affine schemes by finite groups faithfully flat?

Let $R$ be a (Noetherian) ring, and $G$ a finite group acting on $R$. Consider the subring $R^G$. Is the map $R^G\rightarrow R$ faithfully flat?
If not, does this become true if we restrict to ...

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76 views

### Degree bounds for Grobner Basis

Let $I= \langle f_1, \ldots f_n \rangle \subset K[x_1,\ldots, x_n]$ be a homogeneous ideal and $\operatorname{deg}(f_i) \leq d$ then it has Grobner Basis where degree of each generator is less than or ...

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355 views

### Powers of elements in an Artinian Ring

Let $R$ be an local Artinian ring, with maximal ideal $\mathfrak{m}$.
Let $e$ be the smallest positive integer for which $\mathfrak{m}^e=(0)$.
Let $t$ be the smallest positive integer for which ...

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68 views

### Depth of conormal sheaf of a quotient singularity in a smooth variety

Let $U= \mathbb{C}^n/ \mathbb{Z}_r$ be a cyclic quotient singularity by the finite cyclic group $\mathbb{Z}_r$ of order $r$.
Assume that the group action is free outside a closed subset $Z \subset ...

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278 views

### (When) is isomorphism on differentials enough to guarantee that a map is étale?

I'm sorry if this is too easy for MO.
Let $S$ be a locally noetherian scheme, flat over $\mathrm{Spec}\,\mathbb{Z}$, $X$ and $Y$ be flat $S$-schemes locally of finite presentation, and let $f:X\to Y$ ...

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366 views

### Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...

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**1**answer

100 views

### Simultaneous triangularizability over a commutative ring

Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property
(*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent.
McCoy showed that, if ...

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**1**answer

169 views

### Vector Spaces of Symmetric Matrices of Low Rank

Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}_n(K)$, where ...

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**1**answer

172 views

### Constructing a ring whose spectrum is given by order ideals of Z with generic point

Put the following topology on $\mathbf{Z}_{>0} \cup \infty$: the closed sets are the initial invervals $\{1,\dots,n\}$ for all $n$. If I understood Hochster's characterization of the underlying ...

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31 views

### Bigraded analogue of Ratliff-Rush closure filtration

Consider the filtration $\lbrace{I^rJ^s}\rbrace_{r,s\in\mathbb{Z}}.$
What will be the bigraded analogue of Ratliff-Rush closure filtration $\tilde{{I}^n}=\cup_{k\geq1}({I}^{n+k}:{I}^k)$?
Will it be ...