**0**

votes

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56 views

### Non-local differentially smooth algebra

Let $A$ be a noetherian commutative algebra over a perfect field $k$.
The algebra $A$ is said to be differentially smooth over $k$ if
(1) $\Omega^1_{A/k}$ is a projective $A$-module, and
(2) the ...

**3**

votes

**1**answer

176 views

### Maximal ideals of polynomial ring containing a fixed element

We know that for a field $k $ and $f\in k [x]$, the only maximal ideals of $k [x]$ containing $f $ are the ideals generated by prime factors of $f $. Now, I want to know that if $R $ is an arbitrary ...

**3**

votes

**0**answers

277 views

### A relation between a ring with its polynomial ring

Let $\{f_i(x)\}_{i\in I}$ be a subset of $R[x]$ where $R[x]$ is the polynomial ring of $R$(a commutative ring with identity). If the ring $R/\langle f_i^2(n)-f_i(n)\rangle_{i\in I, n\in A}$, for every ...

**2**

votes

**1**answer

304 views

### Dimension of a commutative ring

Let $R$ be a commutative ring with identity having finite Krull dimension (denoted $\dim$). Let $Nil(R)$ be the set of all nilpotent elements in $R$, and let $J(R)$ the intersection of all maximal ...

**53**

votes

**1**answer

2k views

### $R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...

**2**

votes

**0**answers

76 views

### If $A$ is an integer ring such that each $P \in A_L[X]$ has a finite number of zeros in $A$, is $A$ commutative?

Let $A$ be a ring in which the product of any two nonzero elements is nonzero (we shall say that $A$ is an integral domain, even if $A$ is non commutative). It is well-known that if $A$ is commutative,...

**2**

votes

**1**answer

303 views

### maximal ideals of polynomial ring

For a maximal ideal $n$ of a polynomial ring $ R [x] $ over a commutative ring $R$ with identity, are there conditions under which $m [x]\subset n$, for some maximal ideal $m$ of $R$?
Note: $m [x] $...

**1**

vote

**0**answers

73 views

### Grobner basis for a general algebra

Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...

**1**

vote

**0**answers

68 views

### A problem related to parametrizing $\operatorname{rank}\le r$ matrices and Segre embedding

Given a field $k$. We denote $A_{mn}=k[\{X_{ij}\}_{1\le i\le m,1\le j\le n}]$ a polynomial ring of $mn$ variables. Given $m,n,r>0$, we have a natural homomorphism $\phi\colon A_{mn}\to A_{mr}\...

**6**

votes

**1**answer

147 views

### When does 'Zariski tangent space derivative' vanishes everywhere imply that a section is constant?

Consider an abelian algebra, $R$, over the field $K$ with the properties that every residue field of $R$ is (canonically) isomorphic to $K$ (I'm not sure but I think this is necessary, otherwise we ...

**4**

votes

**1**answer

234 views

### What kinds of limits does localization of commutative rings reflect?

Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ...

**2**

votes

**0**answers

214 views

### Looking for a reference in commutative algebra

I need "I.G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973) 23–43." in my research, but it seems to be very old and rare. Does anyone know a site for ...

**4**

votes

**0**answers

114 views

### Reference request for $R$-index

Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ ...

**8**

votes

**2**answers

291 views

### Spectrum of Ring of Smooth Functions on $\mathbb{R}^n$

When we define smooth manifold, we starting with topological space $M$ which localy homeomorphic to $\mathbb{R}^n$ and setting up sheaf $\mathscr{F}(M)$ of functions on it which localy isomorphic to $\...

**0**

votes

**0**answers

53 views

### Extending grading of subring to entire ring

Let $R$ be a (commutative) subring of $S$, and assume that $R$ is graded by an abelian group $G$. Is there anything known, possibly under less general circumstances, about the existence/uniqueness of ...

**12**

votes

**1**answer

418 views

### Examples and Counterexamples in Commutative Algebra

There are Counterexamples in Analysis and Counterexamples in Topology. Is there any similar book for commutative algebra? I want to see some more (counter)examples for Atiyah and MacDonald's book. Let ...

**3**

votes

**1**answer

276 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} m_i\...

**1**

vote

**0**answers

110 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

119 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

66 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 $A$-...

**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

159 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 ...

**20**

votes

**1**answer

695 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

113 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 \mathbb{C}[x_1,x_2,\...

**0**

votes

**1**answer

74 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

426 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

171 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

363 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

188 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

95 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 a(n+...

**5**

votes

**2**answers

417 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

370 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 ...

**3**

votes

**1**answer

195 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

94 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

279 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

134 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

110 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

175 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

497 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

356 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

292 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

79 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

110 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

39 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 I_s)}^...

**17**

votes

**1**answer

474 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

423 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 ...