**0**

votes

**0**answers

71 views

### Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example).
Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...

**4**

votes

**1**answer

204 views

### Transcendence degree of the surreals over the subfield generated by the ordinals

Consider the Grothendieck ring $K[\Omega]$ of the semiring $\Omega$ of all ordinals under the operations of natural sum and product. Its quotient field $K(\Omega)$ is naturally a subfield of the ...

**1**

vote

**0**answers

76 views

### Computing the bourbaki ideals

By virtue the Griffith's paper and subsequently e.g. Goto's paper several examples of several desired class of Noetherian normal domains with specific finite length local cohomologies are constructed ...

**0**

votes

**0**answers

232 views

### Affine communication lemma and finite limits in the category of rings

Let $X$ be a scheme and $\mathrm{Spec}(B) = V \subseteq X$ be an open affine subset. When using the affine communication lemma (c.f. Theorem 6.3.2, Vakil's notes, Foundations of Algebraic Geometry), ...

**5**

votes

**2**answers

216 views

### Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ...

**1**

vote

**1**answer

137 views

### ideal of maximal minors is cohen-macaulay?

Let $k$ be an algebraically closed field.
Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of ...

**0**

votes

**1**answer

143 views

### When a proper morphism of schemes is a closed imbedding?

Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map
$$Mor_{Sch}(S,X)\to ...

**4**

votes

**1**answer

245 views

### Intersection of nonzero prime ideals is zero — does it have a name?

The Rabinowitch trick (in Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, page 132) says that $R$ (commutative unital ring) is Jacobson if and only if for every prime ideal $P ...

**1**

vote

**0**answers

81 views

### Criterion for global dimension of subring

All rings are assumed to be associative and unital.
If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...

**0**

votes

**0**answers

89 views

### Residual Intersections of a complete intersection

Let $R$ be a Cohen-Macaulay local ring and $I=(b_1,\dots,b_s)$ be a complete intersection generated by a regular sequence $\underline{b}$. Let $\mathfrak{a}\subseteq I$ such that ...

**1**

vote

**2**answers

198 views

### Computing the nonsingular projective model of a plane curve

Is there an implemented algorithm available in standard software systems (Sage, Magma, Macaulay, etc.) that will compute the nonsingular projective model of a plane curve over $\mathbb Q$?

**1**

vote

**0**answers

121 views

### Questions on prime integral ideal congruences

Suppose that we are given a fixed pair $a_1, a_2$ of non-zero irrational algebraic integers in some number field $K$ which are independent over $\mathbb{Q}$. Suppose that $\mathcal{P}$ is a prime ...

**0**

votes

**1**answer

170 views

### A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = ...

**1**

vote

**0**answers

72 views

### Does this condition imply a polynomial is a product of linear factors

Let $\Lambda$ be a lattice (i.e. $\Lambda \simeq \mathbb{Z}^n$) with a positive subcone $\Lambda^+$. Let $H: \Lambda^+ \rightarrow \mathbb{C}$ be a function such that $\forall\mu \in \Lambda^+$, ...

**4**

votes

**0**answers

145 views

### How to check whether a scheme of finite type over Spec Z is regular or not [duplicate]

Let $f_1, f_2, \ldots f_k$ be a set of polynomials in $n$ variables, with integer coefficients. These define an affine scheme $X$ of finite type over $Spec \mathbb{Z}$. (We could also consider ...

**1**

vote

**0**answers

46 views

### Name for generalization of bivariate weighted-homogeneous polynomials

A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that ...

**5**

votes

**1**answer

187 views

### Bass' stable range for Bezout rings

As discussed in this MO topic, every principal ideal domain has stable rank at most 2. The proof in the accepted answer uses the fact that PID is a unique factorization domain, but there can be no ...

**3**

votes

**0**answers

80 views

### Is there some algorithm for calculating the least number of generators of an ideal in a polynomial ring?

My question comes from the context of algebraic geometry. By Krull's principal ideal theorem, the number of generators of the defining ideal of a variety gives an upper bound of its codimension.
...

**0**

votes

**0**answers

59 views

### Isomorphy of finite $R$-algebras under special conditions

Let $R=k[x_1, \ldots, x_n]$ be the polynomial ring over a field of characteristic zero. Let $R \subseteq S$ be a finite ring extension i.e. $S$ is finitely generated as $R$-module (free if that helps) ...

**1**

vote

**0**answers

143 views

### Are there good properties of the divided power completion map?

Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed ...

**1**

vote

**0**answers

194 views

### Is this essentially of finite type algebra actually of finite type?

Let $R$ be a discrete valuation ring with a uniformizer $\pi$ and $(A, \mathfrak{m}_A)$ a local $R$-algebra that is essentially of finite type (i.e., is a localization of a finite type $R$-algebra), ...

**1**

vote

**3**answers

221 views

### Under the condition specified below, is $\mathcal{O}_X(X-V(I))=R$, where $X=\mathrm{Spec}R$?

Is it true that given a noetherian normal domain $R$ and an ideal $I$ of height $\geq 2$ we have $\mathcal{O}_X(X-V(I))=R$, where $X=\mathrm{Spec}R$?

**3**

votes

**1**answer

130 views

### “Unramified” extension of DVRs and permanence of excellence

Recall that a discrete valuation ring $R$ is excellent if the extension $\widehat{K}/K$ is separable, where $\widehat{R}$ is the completion of $R$ (with respect to the maximal ideal), $K = ...

**6**

votes

**3**answers

501 views

### Is this formally étale morphism of schemes an isomorphism?

In the last days I came to consider the following question which I'd be happy to see answered by the affirmative:
if $f:X\to S$ is a morphism of schemes which is formally étale, quasicompact,
...

**2**

votes

**1**answer

144 views

### Given a locally nilpotent derivation over a field of characteristic 0 and a local slice, how is the ring homomorphism below defined?

Let $K$ be a field of characteristic 0 and $A$ a $K$-domain. Let $D:A\longrightarrow A$ be a locally nilpotent K-derivation, that is, $D(k)=0$ for all $k\in K$, $D(ab)=(Da)b+a(Db)$ for all $a,b\in A$, ...

**1**

vote

**1**answer

190 views

### Self-similarity for simple algebraic structures [closed]

I'm doing this thread because I have some ideas about how to define self-similarity in algebra, but I don't know if this is known at all. Any critics, comments and references are more than welcomed. ...

**7**

votes

**1**answer

480 views

### Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?

I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them there are idealizations and valuation domains. But the first non-noetherian ring we are thinking ...

**10**

votes

**1**answer

255 views

### How do I check that this is a Frobenius algebra?

Let $f_1,f_2,\ldots,f_n\in \mathbb C[z_1,\ldots, z_n]$ be such that the quotient ring
$$A:=\mathbb C[z_1,\ldots, z_n]/(f_1,f_2,\ldots,f_n)$$
is finite dimensional (in other words, it's a ...

**0**

votes

**0**answers

59 views

### Projective dimension of a sub-ideal

Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either ...

**1**

vote

**1**answer

177 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
...

**0**

votes

**0**answers

60 views

### question about Baer sum of extensions

Let $E_1$ and $E_2$ be extensions of $\mu_p$ by $\mathbb{Z}/{p\mathbb{Z}}$. Assume that $E$ contains $E_1$ and $E_2$ both, and $E_1 \cap E_2 = \mathbb{Z}/{p\mathbb{Z}}$.
Then, does $E$ contain their ...

**3**

votes

**0**answers

116 views

### Invariant Theory over finite adeles

Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...

**6**

votes

**0**answers

159 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 ...

**1**

vote

**1**answer

280 views

### Is every polynomial ring over any field regular?

Is every polynomial ring over any field regular?
For a field that is algebraically closed, it is true as any maximal ideal of $k[x_1,...,x_n]$ corresponds to a point $(t_1,...,t_n)$ in ...

**7**

votes

**2**answers

333 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 ...

**15**

votes

**1**answer

429 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 ...

**3**

votes

**0**answers

72 views

### Integral group rings on which stably free modules are free

Let $G$ be a torsion-free group and $ZG$ the integral group rings. Recall that a projective module $P$ over $ZG$ is stably free if there is an isomorphism $P \oplus ZG^n \cong ZG^m$. Are there known ...

**0**

votes

**1**answer

210 views

### Uniform Artin-Rees

The Artin-Rees lemma states that if $R$ is a Noetherian ring, $I \subseteq R$ is an ideal and $N \subseteq M$ are finitely generated $R$-modules, then there exists an integer $k$ such that for every ...

**1**

vote

**1**answer

133 views

### Lifting a direct summand of a free module

[EDIT]: After getting a nice counter example provided by Steven Landsburg I realize that I forgot to impose an important condition...namely $R$ is supposed to be complete w.r.t. the $I$-adic topology. ...

**7**

votes

**2**answers

587 views

### Does this construction yield the surreal numbers?

There are two simple constructions for creating arbitrarily large non-Archimedean ordered field extensions of the reals.
First given such a field one may consider rational functions over that field ...

**0**

votes

**1**answer

111 views

### Extending derivations to the superposition closure

Let $X$ be a set and $\mathcal{F}\subseteq {\mathbb{R}^X}$ an arbitrary family of functions.
The superposition closure of $\mathcal{F}$ is defined as
$$
\overline{\mathcal{F}}=\{ ...

**1**

vote

**0**answers

124 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 ...

**8**

votes

**1**answer

367 views

### Deligne's exterior power

In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, ...

**6**

votes

**3**answers

249 views

### About the Dimension of a complete local ring

Let $k$ be a field, and let $A$ be a local, noetherian, complete k-algebra with residue field $k$. Suppose that there are elements $t_1,\dots,t_n$ in the maximal ideal of $A$ such that the map ...

**3**

votes

**1**answer

183 views

### Flatness of Normalization of regular schemes

I have a followup to the following 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 ...

**0**

votes

**0**answers

73 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 $\phi_F$). ...

**5**

votes

**1**answer

181 views

### computing the nonnegative part of a $\mathbb{Z}$-graded ring

Let $R = \bigoplus_{n \in \mathbb{Z}} R_n$ be a $\mathbb{Z}$-graded commutative ring with nonnegative part $R^+ = \bigoplus_{n \geq 0} R_n$ and nonpositive part $R^- = \bigoplus_{n \geq 0} R_{-n}$. By ...

**16**

votes

**3**answers

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 ...

**3**

votes

**1**answer

257 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 ...

**1**

vote

**0**answers

48 views

### a generalization of the annihilator of cokernel ideal

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $F\stackrel{A}{\rightarrow}G$. Its basic invariants are the Fitting ideals, ...