**3**

votes

**1**answer

267 views

### Which commutative rings have irreducible (maximal) spectra?

Does there exist any term (or, maybe, a "description"?) for commutative unital noetherian rings such that their Jacobson ideals are prime (and so, their maximal spectra are irreducible)? What is the ...

**1**

vote

**0**answers

137 views

### A family of maximal ideals

Let $m_i $, $i \in I,$ be an infinite family of maximal ideals in a commutative ring with identity (it is not supposed to be Noetherian). When does there exist $j \in I$ such that $\cap_{i\not= j} ...

**1**

vote

**0**answers

107 views

### Algebraic independence criterion

Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...

**4**

votes

**1**answer

118 views

### Reference on the classification of (low rank) Gorenstein rings over $\mathbb{C}$

I am interested in the question of the classification of (low rank) Gorenstein rings over $\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring ...

**1**

vote

**0**answers

65 views

### Cubic, divisor of rational function $x/z$? [closed]

Let $k$ be a field, and let $a \neq 0$, $1 \in k$. Let $C = V(y^2z - x(x-z)(x - az))$. What is the divisor of the rational function $\psi([x, y, z]) = x/z \in k(C)$?

**1**

vote

**0**answers

134 views

### Local cohomology commuting with fiber

Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$).
Let $M$ be an $A[x]$-module, which is finitely generated as an ...

**1**

vote

**1**answer

89 views

### Theorem 16 , Chapter 5 of Northcott 's, Finite Free Resolutions: p.grade

Let $R$ be a commutative ring with identity. D.G. Northcott's, Finite Free Resolutions, has:
and in Theorem 16 of Chapter 5 proves that:
$p.grade(I,M) = p.grade(P,M)$ for some prime ideal $P$ ...

**9**

votes

**0**answers

156 views

### $A$-module is free if and only if equation involving Hilbert-Poincaré series holds, $M$ infinitely generated case

See my question here.
Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector ...

**19**

votes

**1**answer

661 views

### Examples of Noetherian overkill

I have read in many places that the noetherian hypothesis is often overkill - both in commutative algebra and in ($\overset?=$) algebraic geometry. In particular, I've read that coherence and finite ...

**0**

votes

**0**answers

111 views

### The limit of parametrized algebraic variety

Consider the following situation. Take our field to be complex number $\mathbb{C}$, and the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$. Suppose one has an ideal $I \subset ...

**0**

votes

**1**answer

71 views

### Generalization of a Result about degree bounds of invariant rings

A theorem of Knop states that if $G$ is semisimple and connected acting on a vector space $V$ over a field $K$ of characteristic 0, then the degree of the Hilbert series of $K[V]^G$ is less than or ...

**16**

votes

**1**answer

357 views

### Koszul complex for non-Koszul algebras

Let $A$ be a graded, connected, locally finite, quadratic algebra over a field $k$; that is, $A$ may be presented as $T(V)/I$, where $V = A_1$ is a finite dimensional $k$ vector space, and the ideal ...

**5**

votes

**1**answer

170 views

### Assuming $depth M\ge depth N$, what can one say about $depth M_p$ and $depth N_p$?

Definition. Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite R-modules, $p$ a prime ideal, and $I$ an ideal such that $IM\neq M$. Then the common length of the maximal $M$-sequences in ...

**3**

votes

**2**answers

354 views

### Application of sheaves theory in ring theory

Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?

**0**

votes

**0**answers

180 views

### Basic question about power series and complete group algebras

This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange.
Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...

**2**

votes

**2**answers

180 views

### Product of reduced affinoid spaces over a field is reduced (reference request)

Let $K$ be a field of characteristic zero complete with respect to a non-Archimedean absolute value. Suppose that $A$ and $B$ are two affinoid $K$-algebras. I'd like a reference that will answer the ...

**2**

votes

**0**answers

94 views

### Factorization of linear recurrences

For each (commutative unitary) ring $R$, let $\mathfrak{R}(R)$ be the set of all linear recurrences over $R$, that is, the set of all sequences $(a(n))_{n \geq 0}$ in $R$ such that
$$a(n+k) = r_1 ...

**5**

votes

**2**answers

403 views

### A systematic canonical construction of the Hodge star operator

I'm struggling to make sense of the Hodge star as a global canonical object. Here are my struggles so far and some questions:
Let $M$ be a finitely generated projective $R$-module (hence locally free ...

**14**

votes

**1**answer

339 views

### Joyal's construction of the spectrum of a commutative ring

I am trying to understand bits and pieces of Lawvere's article Continuously Variable Sets; Algebraic Geometry = Geometric Logic. I'm not doing very well.
I know this is a lot to ask, but basically, I ...

**4**

votes

**1**answer

190 views

### Is a Laskerian ring coherent

A commutative ring $R$ with identity is said to be coherent if every f.g. ideal of $R$ is f.p. We know that any noetherian ring is coherent. A Laskerian ring is a ring in which every ideal has a ...

**2**

votes

**1**answer

89 views

### Number of cluster variables

In the paper cluster algebras and quantum affine algebras, Section 13.5, it is said that when $\mathfrak{g}$ is of type $A_2$ and $\ell=2$, then the corresponding cluster algebra $\mathscr{A}_2$ for ...

**2**

votes

**4**answers

268 views

### graded rings and modules

I want to read about graded rings and modules. First, I saw Bruns-Herzog. But it was difficult for a beginner. Then I saw notes of Tom Marley (see Tom Marley's Homepage: Graded rings and modules). It ...

**1**

vote

**2**answers

131 views

### Can powers of a maximal ideal stabilize without vanishing?

Let $A$ be a local ring with maximal ideal $m$. Suppose that there exists some positive integer $k$ such that $m^k = m^{k+1}$.
Is necessarily $m^k = 0$ ?
If $m$ is finitely generated, this follows ...

**3**

votes

**2**answers

109 views

### Covering derivations of a quotient algebra

Let $(\mathcal{A},+,·)$ an algebra and $\mathcal{I}$ an ideal of $\mathcal{A}$.
Is easy to check that if $D\in Der(\mathcal{A})$ with $D(\mathcal{I})\subseteq I$ then $D$ induces a derivation $D_I$ ...

**3**

votes

**1**answer

171 views

### Is the embedding dimension minus the dimension upper semicontinuous?

For a Noetherian local ring $R$ with maximal ideal $\mathfrak{m}$ and residue class field $K$, consider the invariant $$\operatorname{def}(R) := \operatorname{dim}_K(\mathfrak{m}/\mathfrak{m}^2) - ...

**15**

votes

**3**answers

471 views

### Alternate proofs of Hilberts Basis Theorem

I'm interested in proofs using ideas from outside commutative algebra of Hilbert's Basis Theorem.
If $R$ is a noetherian ring, then so is $R[X]$.
or its sister version
If $R$ is a noetherian ...

**7**

votes

**1**answer

337 views

### Zero scheme of global sections of vector bundles on affine varieties

I want to understand better the notion of zero scheme of a section of a vector bundle. For simplicity I will consider the case of affine varieties.
Let $\mathbb{K}$ be an algebraically closed field, ...

**3**

votes

**1**answer

291 views

### If $F(x,y)$ is a polynomial which is not a square, then how often is the specialization $F(x,a)$ a square?

Suppose $F(x,y)$ is a polynomial in two variables over a field $K$, and $F(x,y)$ is not a square. When is $F(x,a)$ a square for $a\in K$? I would guess that Hilbert's Irreducibility Theorem might help ...

**5**

votes

**0**answers

78 views

### For any $f \in B$ which is not nilpotent, the set consisting of powers of $f$ is a multiplicative set in $\mathcal{D}_A(B)$? [closed]

Let $B$ be a commutative $A$-algebra. Let $\mathcal{D}_A(B)$ be the ring of differential operators of $B$ over $A$. Does it follow that for any $f \in B$ which is not nilpotent, the set consisting of ...

**2**

votes

**1**answer

108 views

### $Hom (T,R)$ isomorphic to $R:T$?

Let $R$ be a Cohen–Macaulay local ring with maximal ideal $m$ and $dim R = 1.$ In the paper "Almost Gorenstein rings", by "Goto, Matsuoka, Phuong", with this settings:
they have
How they reach the ...

**0**

votes

**0**answers

37 views

### Primary decompositions of squares (products) of monomial ideals

Is there anything known about the relation between the primary decomposition
of a monomial ideal $I$ and the primary decomposition of $I^2$?
In other words, given the standard primary decomposition ...

**0**

votes

**0**answers

43 views

### A quick question from the paper “Generalizations of reductions and mixed multiplicities” by D. Rees

In the proof of Theorem 1.3 in the paper "Generalizations of reductions and mixed multiplicities" by Rees here, Is it necessary to consider the ring $Q'=Q/{\bigcup\limits_{q \geq 1}(0:{(I_1\dots ...

**17**

votes

**1**answer

438 views

### Almost commuting unitary matrices

Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...

**9**

votes

**1**answer

415 views

### Is Max (R) a Hausdorff space?

I asked the following question at < http://math.stackexchange.com/questions/1508168/is-max-r-a-hausdorff-space >, but I pose it here for any help.
Recall a space is totally disconnected if the ...

**7**

votes

**0**answers

159 views

### Terminology for vanishing of Hochschild homology with symmetric coefficients?

In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then
it should hopefully be understood by most readers as saying ...

**1**

vote

**0**answers

27 views

### Projectivity of a faithfully balanced self-orthogonal bimodule

Let $_RT_S$ be a faithully balanced self-orthogonal bimodule over a pair of noncommutative rings $(R,S)$, if $_RT$ is projective as a left $R$-module, can we say $T_S$ is also projective as a right ...

**0**

votes

**1**answer

79 views

### When can one infer degrees of generators of a ring from its hilbert series

I know that for a noetherian ring, it's hilbert series can be written as $$HS(t)=\frac{P(t)}{\prod_{i=1}^d{(1-t^{d_i})}}$$ where $P(t)$ is polynomial, and there are $d$ generators of degrees ...

**3**

votes

**1**answer

118 views

### Simple proof that $(p_1x_1-q_1y_1,…,p_nx_n-q_ny_n)$ is a prime ideal

Is there a nice way to show that
$$(p_1x_1-q_1y_1,\ldots ,p_nx_n-q_ny_n) \subseteq \mathbb{Z}[x_1,...,x_n,y_1,...,y_n]$$
is a prime ideal for coprime non-zero integers $p_i,q_i\,(i=1,...,n)$ ?
I ...

**0**

votes

**0**answers

176 views

### Name of some commutative ring akin to $p$-adics

I need help in identifying the naming convention of some commutative ring described below.
Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ...

**1**

vote

**0**answers

37 views

### Upper bound for the minimum number of generators of the canonical module

Let $P=k[x_1...x_n]$ be a poly over a field. Suppose that $R=k[x_1...x_n]/I$. The canonical module of $R$ is $\omega_R=Ext^{n-dim(R)}_P(R,P)$.
The question is that is there any upper bound for the ...

**3**

votes

**2**answers

205 views

### Connectedness of units in finite-dimensional commutative complex algebras

In the following, an algebra will always mean a finite-dimensional associative commutative unital algebra (over some field $k$).
Let $A$ be a $\mathbb{C}$-algebra. I am trying to understand how its ...

**1**

vote

**0**answers

115 views

### Are these quaternion algebras definite or indefinite?

By investigating a different problem I have ended up looking at Quaternion algebras and have a lot to learn about them. Before I do, however, I want to see if my idea has any hope of being useful. So ...

**2**

votes

**1**answer

92 views

### projective module of rank one over notherian ring

Is finitely generated projective module M of rank one over regular commutative notherian ring free?
Bass (Illinois Math J, 1963) showed that in case M is nonfinitely generated, it is free. I am ...

**4**

votes

**0**answers

190 views

### Commutative algebraic version of algebraic geometric object

In my work, I have to understand certain objects in commutative algebra (for example Gorenstein rings, Cohen–Macaulay rings e.t.c). I have a reasonable background in commutative algebra (I suppose!) ...

**5**

votes

**1**answer

90 views

### Flat dimension of injectives over a Gorenstein ring

Let $A$ be a Gorenstein noetherian local ring, and let $M$ be an $A$-module of finite injective dimension.
If $M$ is a finite $A$-module, it is easy to show these assumptions imply that $M$ has ...

**1**

vote

**0**answers

92 views

### A question from Hilbert's Nullstellensatz [closed]

From the Hilbert's Nullstellensatz, we know that for any algebraic closed field $K$ and any prime ideal $p$ of $K[X_1,X_2,\cdots,X_n]$, the intersection of all the maximal ideals containing $p$ is ...

**6**

votes

**1**answer

353 views

### what are the finite etale covers of $\mathbb{Z}_p((x))$?

Let $R$ be the ring of integers of some $p$-adic field $K$ (finite over $\mathbb{Q}_p$) with uniformizer $\pi$ and residue field $k$. I'd like to understand the finite etale extensions of $R((x)) := ...

**5**

votes

**1**answer

111 views

### What is the criterion for a matrix containing vectors and their permutations being invertible?

Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix ...

**4**

votes

**1**answer

159 views

### number of generators of maximal ideals in an order of a number field

let $K$ be a number field of degree $d$ over $\mathbb{Q}$), Let $\mathcal{O}\subset K $ be an order (i.e. a $\mathbb{Z}$-lattice of $K$ contained in the integer ring $\mathcal{O}_K$ of $K$). If $ ...

**3**

votes

**1**answer

111 views

### A question about prime elements in a specific integral domain

Let $R,\Omega$ be two integral domains such that $R$ is Noetherian and $\Omega=R[\alpha]$ for some $\alpha\in \Omega$.
If there are infinite prime elements in $R$, can we proof that there are ...