# Questions tagged [ac.commutative-algebra]

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

3,791
questions

**-2**

votes

**0**answers

101 views

### Annihilator of an element and Jacobson radical

Let $R$ be a commutative ring with 1. Is there any characterization for an element $a$ of $R$ such that $\operatorname{ann}(1-a)\subseteq J(R)$ and $a\in J(R)$, where $\operatorname{ann}(x):=\{r\in R\...

**9**

votes

**4**answers

4k views

### Definition of étale for rings

Let $A \to B$ be a ring extension.
What is the definition of $B/A$ étale ?
When $A$ is a field, do we get a nice characterization ?

**7**

votes

**0**answers

194 views

### Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?

Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...

**88**

votes

**27**answers

18k views

### Errata for Atiyah-Macdonald

Is there a good errata for Atiyah-Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists inaccuracies ...

**1**

vote

**1**answer

451 views

### Classification of rings between a PID and its field of fractions?

Let $D$ be a PID and let $\mathrm{Frac}(D)$ be its field of fractions. I want to classify the intermediate rings $D\subseteq R\subseteq \mathrm{Frac}(D)$.
Theorem: Every such ring $R$ is a ...

**-3**

votes

**0**answers

138 views

### Glue DVR to itself, get a separated non-affine scheme

Here is what seems to be a fun little exercise in algebraic geometry. Take a DVR $R$ and automorphism of $Frac(R)$; glue $Spec(R)$ to itself via this automorphism. Can the glued scheme be separated ...

**1**

vote

**1**answer

72 views

### Monomials in products in power series ring on several variables

Let $A \colon= K[[X_1,\ldots,X_m,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $m + n$ variables and ${\frak m}$ be the unique maximal ideal of $A$.
For arbitrary two elements $\alpha ...

**6**

votes

**1**answer

505 views

### Does there exist a discrete valuation subring $R$ of $K((t))$ ($K$ a number field) of residue characteristic $p$ with $\mathrm{Frac}(R) = K((t))$?

Let $K$ be a number field, and let $K((t))$ be the field of formal Laurent series. Let $p > 0$ be a prime.
I have two questions:
Does there exist a discrete valuation subring $R$ of $K((t))$ of ...

**4**

votes

**2**answers

532 views

### A DVR algebra with weird automorphisms

Denote by $k$ an algebraically closed field. Can one produce a DVR $A$ over $k$ such that
the fraction field of $A$ has an automomorphism not preserving $A$
no non-trivial field extension of $k$ maps,...

**0**

votes

**1**answer

133 views

### Localization and containment in commutative ring

Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...

**1**

vote

**0**answers

93 views

### Question about Local Henselian Rings

I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces":
Here the relevant excerpt:
Remark: ...

**4**

votes

**2**answers

354 views

### When does glueing affine schemes produce affine/separated schemes?

Let $X$ be an affine scheme with an open affine subscheme $U\subset X$. Given an automorphism of $U$, we can glue $X$ with itself along $U$ to get a new scheme. Is there a description in terms of ...

**-1**

votes

**0**answers

49 views

### Infinitely generated PID algebra with infinitely many prime ideals

Given a field, is there a functorial construction of a PID algebra over it that has infinitely many prime ideals and is not finitely generated? This excludes the ring of univariate polynomials and the ...

**6**

votes

**2**answers

191 views

### What is the difference between total integral closure and integral closure?

I was advised here to make this a new question:
What is the difference between total integral closure and integral closure (geometrically, in the context of rigid analytic geometry)? I have read in ...

**1**

vote

**2**answers

202 views

### Equivalence between categories of affine schemes over $X$ and representable functors $\operatorname{Points}(x) \to \operatorname{Sets}$

I am reading Strickland - Formal schemes and formal groups. For a functor $X\colon \operatorname{Rings}\to \operatorname{Sets}$, he defines (2.14) the category of $\operatorname{Points}(X)$ in the ...

**-1**

votes

**0**answers

90 views

### Are Noetherian algebras over a field catenary?

A commutative unital ring that can be given the structure of a finitely generated algebra over a field is catenary. Is a Noetherian integral domain that can be given the structure of a (possibly ...

**2**

votes

**1**answer

255 views

### Simple object of $k[X,Y]/(Y^2)$

Let $k$ be a field. Let $A=k[X,Y]/(Y^2)$ be the quotient of polynomial ring $k[X,Y]$. Let $\mathcal{C}$ be the category of finite-dimensional $A$-modules $M$ with the action of $X$ nilpotent, and of ...

**1**

vote

**0**answers

94 views

### Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?

Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...

**2**

votes

**0**answers

32 views

### Concerning $(x,y) \mapsto (x^{\frac{n}{r}+1}y + A,\mu x^{-\frac{n}{r}}+B)$

Let $r \in \mathbb{N}-\{0\}$.
Commutative case:
Let $f : (x,y) \mapsto (p,q)$ be a map from $\mathbb{C}[x,y]$ to $\mathbb{C}[x^{1/r},x^{-1/r},y]$ satisfying the following two conditions:
(i) $\...

**1**

vote

**1**answer

122 views

### A property for primitive idempotents

Let $R$ be a (commutative) ring (with identity). A nonzero idempotent $e\in R$ is called primitive idempotent, whenever it has no decomposition into $a+b$ where $a$ and $b$ are nonzero orthogonal ($...

**7**

votes

**1**answer

294 views

### A ring of generalized power series

Let $\Bbbk$ be a field; I am interested in the following ring (which I suspect is a field). Its elements are formal expressions that look like
$$ \sum_{n=0}^{\infty} a_n x^{b_n} $$
where $a_n\in \...

**3**

votes

**5**answers

838 views

### What properties define open loci in families?

This question is somehow related to the question What properties define open loci in excellent schemes?.
Let $f:X\to S$ be a proper (or even projective) morphism between schemes (of finite type over ...

**3**

votes

**0**answers

149 views

### Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes

Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...

**1**

vote

**1**answer

131 views

### Special idempotents in a commutative ring

Let $R$ be a commutative ring with $1$ and $e$ be an idempotent element of $R$ with the property that if $e=x+y$ (where $x, y\in R$), then there exists $r\in R$ such that either $e=rx$ or $e=ry$. ...

**5**

votes

**0**answers

166 views

### A non-commutative analog of a result concerning a Jacobian pair

Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$.
Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$.
Similarly, define $t_y(E)$ to be the maximum among $...

**3**

votes

**0**answers

50 views

### On the milnor number of analytic germ map

If $f:(\mathbb{C}^n,0) \to (\mathbb{C},0)$ is an analytic germ with an isolated singularity, then the Milnor number of $f$, denoted by $\mu(f)$ can be defined as $\dim_{\mathbb{C}} \mathcal{O}_n/\text{...

**1**

vote

**0**answers

53 views

### Non-minimal Krull associated primes of a PF-ring

A commutative ring $R$ is said to be a PF-ring if every principal ideal of $R$ is a flat $R$-module. Also, a prime ideal $P$ of $R$ to be a Krull associated prime of $R$ if
for every element $x\in P$ ,...

**5**

votes

**1**answer

195 views

### Definition of dualizing complex

Sorry for a not research level question asking for a definition but unfortunately I nowhere found a source which explains the construction presented below in a satisfactory way.
This question refers ...

**4**

votes

**0**answers

62 views

### Finding a presentation matrix with low dimension

Let $R=\mathbb Z[t^{\pm}]$ and $M$ a finitely generated $R$-module. With $A$ a presentation matrix, i.e we have the following exact sequence (usually I'm working with the case where $A$ is an square ...

**3**

votes

**1**answer

164 views

### How many distinct quaternions have a given prime norm $p$?

I seem to recall that the answer is $p + 1$, but I'm not quite sure.

**3**

votes

**1**answer

239 views

### The relationship between a finite field and a quotient ring in $\mathbb{F}_p[x]$

Let $ f$ be an irreducible polynomial of degree $q$
over $\mathbb{F}_p$.
Let ${\bf F}=\frac{\mathbb{F}_p[x]}{f}$ be the finite field which contain $p^q$ elements.
Assume $k>1$ is an integer and ...

**1**

vote

**0**answers

76 views

### Does the Polarization Theorem for $A_1(k)$ has an analogue for $k[x,y]$?

There is an interesting theorem about the first Weyl algebra $A_1(k)= k \langle x,y | yx-xy= 1 \rangle$,
$k$ is a field of characteristic zero,
the Polarization Theorem, Corollary 5.5 by A. Joseph.
...

**0**

votes

**1**answer

61 views

### Change chain of prime ideals so that $a \in P_1$

A text I am following uses of the following (probably basic) commutative algebraic lemma, omitting its proof.
Lemma: Let $n\in\mathbb{N}_{>0}$, and let $P_0\subsetneq P_1\subsetneq\cdots\...

**14**

votes

**5**answers

3k views

### Fast computation of a Groebner basis. What is possible?

I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or ...

**0**

votes

**0**answers

111 views

### Locus of trivialization of an extension of a vector bundle

Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.
We assume that $j_*\mathcal{E}$ is a vector bundle. In ...

**5**

votes

**1**answer

184 views

### Absolute value on tensor product of fields

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2:=\mathbb F_p((Y))$.
Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{...

**4**

votes

**0**answers

131 views

### Extensions of rings

Let $K$ be a commutative but not necessarily unital ring. Let $C$ be a unital commutative ring.
An extension is the data of a unital commutative ring $R$, a homomorphism $\phi:K\rightarrow R$, a ...

**11**

votes

**5**answers

2k views

### Two-dimensional quotient singularities are rational: why?

I've read that quotient singularities (that is, spectra of invariant subrings of finite groups acting linearly on polynomial rings) have rational singularities. Is there an elementary proof of this ...

**2**

votes

**0**answers

230 views

### Concerning certain Keller maps of $k[x,y]$

Let $k$ be a field of characteristic zero.
Let $(x,y) \mapsto (p,q) \in k[x,y]$ be a Keller map, namely, a $k$-algebra endomorphism of $k[x,y]$ with $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in k-\{0\}...

**6**

votes

**0**answers

90 views

### Relation between $\mathrm{Proj}(A[tI])$ and $\mathrm{Proj}(A[tJ])$ for ideals $I$ and $J$ of $A$ with $I^2 \subset J \subset I$

Let $k$ be a field and $A$ a noetherian local $k$-algebra.
Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$.
Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \...

**10**

votes

**1**answer

225 views

### Derivative of an algebraic power series in positive characteristic

Let $K$ be a field. It is easy to see that if the characteristic of $K$ is $0$ and $f(T)=\sum_{n\ge0}a_nT^n$ is a power series algebraic over $K(T)$, then $f'$ belongs to $K(T)(f)$.
Indeed let $P(X)=\...

**3**

votes

**3**answers

484 views

### Castelnuovo-Mumford Regularity of Ideals of Maximal Minors

I have an $m \times 2m$ matrix of linear forms over $\mathbb{C}[x,y,z,w]$. It is of the form $$M = ( x I - A z -B w \mid y I - C z - D w).$$ Here $A,B,C$ and $D$ are $m \times m$ scalar matrices. Let $...

**5**

votes

**0**answers

421 views

### Completion of a local ring of an arithmetic surface

Vaguely, an arithmetic surface is an algebraic curve with coefficients in some "ring of integers".
More formally it is an integral normal scheme $\mathfrak{X}$ of dimension 2 of finite type over a ...

**2**

votes

**0**answers

70 views

### Continuous extension of the derivation in positive characteristic

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Does there exist a derivation on $\Omega$ ...

**2**

votes

**0**answers

103 views

### Converse to Tannaka duality for rings

Let $k$ be a field. It's well known and easy to prove that a $k$-algebra $R$ may be recovered from the functor $F: R-Mod \to Vect_k$ as the endomorphisms of $F$. Now suppose $\mathcal{C}$ is a ...

**9**

votes

**2**answers

361 views

### On a morphism from the Brauer group to the Picard group

Suppose that $k$ is a commutative ring and that $A$ is an Azumaya
$k$-algebra. Then there is a well-known morphism from $Aut_{Alg_k}(A)$, the group of algebra automorphisms, to the Picard group $Pic(k)...

**3**

votes

**0**answers

114 views

### Algorithm telling when an affine curve is planar

I am sorry, I asked a misguided question here: Reference request: smooth affine curves are planar, here is my attempt at a better question.
Let $\mathfrak{p}$ be a prime ideal of $\mathbb{C}[x, y, z]$...

**3**

votes

**3**answers

604 views

### Does there exist another form of the derivative for polynomials?

Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that
$$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$
for all $P, Q \in \mathbb{R}[X]...

**5**

votes

**3**answers

627 views

### Radical generation of ideals in Noetherian rings

It is well-known that any ideal in a Dedekind domain can be generated by at most two elements. However, already for Noetherian domains of dimension 2, it is easy to construct examples of ideals that ...

**0**

votes

**0**answers

50 views

### Homogeneous basis on a polynomial subalgebra

Let $k$ be an algebraically closed field of characteristic $0$ and $A=k[X_1, \ldots, X_n]$ with the grading induced by the total degree. Let $B$ be a graded $k$-subalgebra of $A$, ie, if $(A_k)$ is ...