**1**

vote

**1**answer

104 views

### Affine open subsets for algebraic group actions

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero.
Assume that $G$ acts on an affine variety $X$. Assume that $X$ contains an open orbit $U$ (so $\...

**0**

votes

**0**answers

87 views

### Quotient of Cohen-Macaulay ring

Let $A$ be a Cohen-Macaulay ring, and $I$ an ideal of $A$.
What can we say about $\operatorname{depth}(A/I)$?
I know that $\operatorname{depth}(A/I)\le \dim(A/I)$.

**0**

votes

**1**answer

91 views

### Lifting sections to completion of closed subschemes

Let $R$ be a reduced finite type $\bar{k}$-algebra, a projective morphism $\pi \colon V \rightarrow \mathrm{Spec}(R)$ and ideals $I, J \subseteq R$. Assume there is a split $s_{IJ} \colon \mathrm{Spec}...

**59**

votes

**24**answers

12k 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 ...

**5**

votes

**2**answers

193 views

### Criterion for being reflexive via Ext

In this question it was claimed that if a module $M$ over a noetherian domain $R$ satisfies $\rm{Ext}^i(M,R)=0$ for $i=1,2$, then $M$ is reflexive. Is this true? Does someone know a reference or a ...

**-2**

votes

**0**answers

35 views

### In algebraic system, what consequences brings having two elements whose powers greater than 1 coincide? [on hold]

In algebraic system, what consequences brings having two elements (say, $w_1$ and $w_2$) whose powers greater than 1 coincide?
Will the exponentiation or taking root become ambiguiuos given that $w_1$...

**1**

vote

**0**answers

65 views

### Question on Hochschild cohomology

Let given ring $A$ without zero divisors and subring $\mathbf{k}\subset Z(A)$. Let also given $M~-$ $A~-$ bimodule, such that $xm = mx, \forall x\in \mathbf{k}, m\in M$.
Is it true that if for $\...

**4**

votes

**0**answers

58 views

### A right adjoint to the truncated Witt functor?

For any ring $A$, let $\mathrm{wEt}_A$ be the category of weakly etale $A$-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor
$$
W_r : \mathrm{...

**3**

votes

**1**answer

126 views

### non-Noetherian r-Noetherian ring with Noetherian total quotient ring

A commutative ring is said to be r-Noetherian if every regular ideal is finitely generated, where an ideal is said to be regular if it contains a non-zerodivisor. Does there exist a non-Noetherian r-...

**1**

vote

**0**answers

63 views

### Monomial algebras and depth

Let $R:=k[x_1,\ldots, x_n]$ be the standard polynomial ring. Let $I\subseteq R$ be a monomial ideal of height $\ge 2,$ and $\{\ell_1, \ldots, \ell_{t}\}\subseteq R_1$ an $R$-regular sequence.
Assume $...

**7**

votes

**2**answers

1k views

### Ideals in the ring of single-variable Laurent polynomials with integer coefficients

I'm writing some software to automatically compute things like Alexander modules, Alexander ideals, Milnor signatures and Farber-Levine pairings for 1 and 2-knot complements. So among other things I'...

**3**

votes

**1**answer

121 views

### Number of free summands of finite local extensions

Suppose that $(R, m) \subseteq (S,n)$ is a finite extension of normal local domains that is:
etale on the punctured spectrum
not flat / etale at the origin
and such that the residue fields $R/m = S/...

**2**

votes

**1**answer

356 views

### Matlis' dual of injective modules

Let $(R, \mathfrak{m})$ be a commutative Noetherian complete local rings ($R$ can be regular, if you need). Let $E(R/\mathfrak{p})$ be injective hull of $R/\mathfrak{p}$, if $\mathfrak{p}= \mathfrak{m}...

**0**

votes

**1**answer

91 views

### Does the integral closure of a normal local ring in a finite extension of its fraction field have finite projective dimension?

Let $A$ be a normal local domain with field of fractions $K$. Let $L$ be a finite separable extension of $K$ (if relevant, I'm happy to assume all possible ramification is tame), and let $B$ be the ...

**1**

vote

**1**answer

67 views

### Cut ideal of two graphs?

Consider a connected graph $G$ and a connected graph $H$. Their graph ideals are their path ideals. The Alexander duality of the graph ideals give the cut ideals. $G$ and $H$ are not connected to each ...

**2**

votes

**1**answer

164 views

### Intersection of two curves is not Cohen Macaulay

Let be $R=\mathbb{C} \lbrace x,y,z \rbrace$ the formal series ring and let $f_{1},f_{2},f_{3} \in R$ be nonzero elements of $R$.
(a) Consider the varieties $M:=V(f_{1},f_{2})$ and $N:=V(f_{2},f_{3})$ ...

**2**

votes

**1**answer

108 views

### Criterion for a polynomial ideal to be radical

Let $K$ be an algebraically closed field, and let $I$ be an ideal of $K[x_1,\dots,x_n]$ ($n \geq 2$). We suppose that for all $i$, and for all $a_1,\dots,a_{i-1},a_{i+1},\dots,a_n \in K^{n-1}$, the ...

**13**

votes

**4**answers

1k views

### Serre's theorem about regularity and homological dimension

One of the nicest results I know of is (Auslander-Buchsbaum-)Serre's theorem asserting that a (commutative!) local ring is regular iff it has finite global dimensional.
I'd like to ask a somewhat ...

**22**

votes

**4**answers

1k views

### Do there exist non-PIDs in which every countably generated ideal is principal?

The title pretty much says it all: suppose R is a commutative integral domain such that every countably generated ideal is principal. Must R be a principal ideal domain?
More generally: for which ...

**1**

vote

**1**answer

77 views

### How to compute graph ideal or cut ideal of a graph?

Graph ideals are a special case of Stanley-Reisner ideal, explained in Combinatorial Commutative Algebra book by Sturmfels, and graph ideals here. Graph ideals are generated by the minimal paths while ...

**1**

vote

**1**answer

109 views

### Cohen-Macaulayness of the scheme of centralizer

Let $G$ be a simply connected group over an algebraically closed field $k$, and
$I:=\{(g,\gamma)\in G\times G\vert~ g\gamma=\gamma g\}$
the scheme of centralizer.
Is $I$ a Cohen-Macaulay scheme ...

**3**

votes

**0**answers

107 views

### Quotient of Dedekind domains

Is there any characterization for a commutative ring to be a quotient of a Dedekind domain?

**4**

votes

**1**answer

187 views

### A property of minimal prime ideals in commutative reduced ring

Let $\{\mathfrak{p}_i\}_{i\in I}$ ($I$ is an infinite set) be a family of minimal prime ideals in a commutative reduced ring $R$ with identity, and let $a, b \in R$. If the ideal $\langle a, b\rangle$...

**2**

votes

**0**answers

114 views

### Torsion ideal in symmetric algebra

Let $D$ be a commutative domain, $M$ a $D$-module without torsion, and $S(M)$ its symmetric algebra.
Is the $D$-torsion ideal of $S(M)$ the prime ideal of $S(M)$?

**3**

votes

**0**answers

74 views

### Structure of valuations on $\mathbb{F}_q(X,Y)$?

I'm looking to construct all valuations on $\mathbb{Q}(X,Y)$ extending the p-adic valuation on $\mathbb{Q}$ and understand their structural properties. In doing this, to obtain 3 dimensional valuation ...

**8**

votes

**1**answer

628 views

### Valuation of an ideal in a two-dimensional regular local ring

Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field.
What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we ...

**0**

votes

**0**answers

56 views

### when: $depth\ R/I\ge depth\ R/J$ then $depth\ (R/I)_p\ge depth\ (R/J)_p$

Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite R-modules, $p$ a prime ideal, and $I$ and $J$ ideals of $R$. Here, Count Dracula proves that in general assuming $depth\ R/I\ge depth\ R/...

**4**

votes

**0**answers

101 views

### A sufficient condition for a morphism to be a closed immersion?

Let $R$ be an integral $\bar{\mathbb{F}}_p$-algebra of finite type, let $V$ be an $R$-algebra. Consider a morphism $f \colon \mathrm{Spec}(V) \rightarrow \mathbb{A}^n_R$ that has the following ...

**2**

votes

**0**answers

92 views

### Reversing the arrows-dual theorems

When one studies homological algebra one learns some basic stuff-diagram chasing, long exact sequences associated to short exact sequence of complexes and so on. Usually one works out the details with ...

**14**

votes

**4**answers

2k views

### Why are finitely generated modules over principal artin local rings direct sums of cyclic modules?

I am looking for a proof of the following fact:
If $R$ is a principal artin local ring and $M$ a finitely generated $R$-module, then $R$ is a direct sum of cyclic $R$-modules.
(Apparently such rings ...

**0**

votes

**0**answers

87 views

### is the “reduction” of an effective cartier divisor on a relative curve still a cartier divisor?

Let $C\rightarrow S$ be a smooth proper morphism of relative dimension 1, where $S$ is a Noetherian normal scheme. Let $D\hookrightarrow C$ be a relative effective Cartier divisor finite over $S$. Let ...

**2**

votes

**0**answers

104 views

### Image of a Weil divisor (height 1 prime) under normalization map

Let $R$ be a noetherian integral domain and let $R'$ be the normalization of $R$ in a finite field extension of the fraction field of $R$. Let $\varphi:Spec(R') \rightarrow Spec(R)$ be the ...

**0**

votes

**0**answers

72 views

### Properties of a subring of a 'completion' of k(X_1, X_2, …, X_n)

I'm looking for a reference in commutative algebra for the properties of the ring made of polynomials in $n$ indeterminate over a field $k$ with "real exponents".
I don't even know the name of this ...

**1**

vote

**0**answers

135 views

### Spivakovski-Popescu-Neron desingularisation

For $A \colon= {\Bbb F}_p[[X_1,...,X_d]]$, by generalising Popescu-Neron's method, Spivakovski proved that $A$ is written by smooth sub-algebras. That is,
$A \cong \underset{\lambda \in \Lambda}{\...

**2**

votes

**1**answer

139 views

### Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings
$$
\begin{array}{}
R & \xrightarrow{f_2} & R_2 \\
\downarrow{f_1} & &...

**0**

votes

**0**answers

65 views

### Hochschild cohomology of smooth commutative algebra

The beautiful Hochschild-Kostant-Rosenberg theorem tells us $HH^*(A,A)=\wedge^* \text{Der}(A)$ for a smooth affine algebra $A$. I wonder what happens when we consider $HH^*(A,A\otimes A)$, where $A\...

**0**

votes

**1**answer

371 views

### Number of elements in a fiber

Let $A\subseteq B$ be normal affine domains over an algebraically closed field of characteristic 0. If it is given that the corresponding morphism of schemes Spec $B\rightarrow$ Spec $A$ is quasi-...

**3**

votes

**1**answer

92 views

### Antisymmetrization of the Hochschild cocycle

Let $A$ be a commutative (unital, complex) algebra and let $\varphi$ be a $n+1$-linear functional on $A$ (we will call it cochain). Define $$(b\varphi)(a_0,a_1,...,a_{n+1}):=\sum_{j=0}^{n}(-1)^j\...

**6**

votes

**0**answers

217 views

### On describing a sort of “well-behaved” subgroups of a free abelian group

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case.
Let $M$ be a free abelian group and $N$ a ...

**4**

votes

**0**answers

152 views

### 'Noether normalization' for finite group schemes

Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$.
Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...

**13**

votes

**2**answers

648 views

### Bass' stable range condition for principal ideal domains

In his algebraic K-Theory book Bass gives the following property on a ring $R$ and a number $n$:
For every $n$ elements $v_1, \ldots, v_n$ that generate the unit ideal there are numbers $r_1, \ldots ...

**0**

votes

**0**answers

127 views

### Localisation of the formal power series ring

Let $A \colon= K[[X_1,...,X_d]]$ be a formal power series ring of $d$-variables over a field $K$.
Let ${\frak a}$ be a height $r$ prime of $A$ given by ${\frak a} \colon= (f_1,...,f_r)$, where
$f_1 ...

**57**

votes

**2**answers

5k views

### How would you solve this tantalizing Halmos problem?

$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
...

**0**

votes

**1**answer

95 views

### Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$

I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals.
The ...

**5**

votes

**1**answer

196 views

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

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

**1**

vote

**0**answers

70 views

### Explicit construction of a bielliptic curve

Let $C$ be a (projective smooth complex) curve such that $K_C=2(D+p)$, with $D+p$ defining a $g_7^2$; $p$ is a base point and $D$ defines a 2-to-1 map $\varphi:C\rightarrow E\subset\mathbb{P}^2$ onto ...

**2**

votes

**1**answer

96 views

### Generalized height of elements in abelian groups

In the book Infinite abelian groups Vol. I by L. Fuchs, on page 154, the notion of the generalized $p$-height of an element in an abelian group is defined, as follows:
Let $A$ be an abelian group ...

**1**

vote

**1**answer

403 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 ...

**22**

votes

**2**answers

713 views

### When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?

Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...

**2**

votes

**3**answers

379 views

### Krull-dimension of local domain

Let $(R,{\frak m}_R)$ be a local domain (not necessarily Noetherian). That is, $R$ is integral and ${\frak m}_R$ is the unique maximal ideal of $R$. Suppose that ${\frak m}_R$ is finitely generated.
...