Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
-3
votes
0answers
47 views
If there is a finite R-module of finite inj dim, then R is Cohen [on hold]
I have a question about Exercise 3.1.24 of bruns herzog book,Cohen Macaulay rings,:If there is a finite R-module of finite injective dimension, M, then R is Cohen Macaulay ring. This is more general ...
1
vote
1answer
39 views
Does the going-up theorem hold between flat algebras?
Let $R$ be a commutative Noetherian ring with unit and $S$ a flat $R$-algebra. Does the going-up theorem hold between $R$ and $S$?
3
votes
1answer
84 views
geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings
R is a local Noeth.
What is geometric interpretation and of
1- Gorenstein rings
2- Complete intersections
3- regular rings?
how can I realize differences by geometric interpretation
1
vote
0answers
53 views
Algebraic set given by sequence of polynomials
When working on some problem, I have end up with a following situation. Suppose
$P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...
-1
votes
0answers
42 views
emb $\dim (R) = \text{depth}(R) + 1$ then $R$ is a Gorenstein [migrated]
$(R,m)$ is a Noetherian local ring such that emb $\dim(R) = \text{depth}(R) + 1$. Show that $R$ is a Gorenstein ring.(Here emb $\dim R$ means "number of minimal system of generators of m").
3
votes
0answers
90 views
operations on ideals in a subring of number field
For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$,
does this equality hold in general?
$(I+J) \cap K = (I \cap K) + (J \cap K)$
I have no counterexample yet but I couldn't prove ...
2
votes
0answers
105 views
K-theory and completion
I asked this question also on math.stackexchange. But maybe it's better to ask the Mathoverflow community.
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of ...
5
votes
0answers
48 views
K-Theory and completion [duplicate]
I vaguely remember the existence of a statement that relates the $K$-theory (in the sense of Quillen) of a noetherian local ring $A$ with maximal ideal $\mathfrak{m}$ with the $K$-theory of the ...
3
votes
2answers
394 views
An identity in an arbitrary commutative ring
This fact might be either trivial, wrong, or well known.
Let $R$ be a commutative ring. Let $u_1,\dots,u_{s-1},u_s\in R$ and $m,M\in R$. Let us assume that $m,M$ satisfy
$$(m-u_1) \dots ...
-7
votes
0answers
90 views
A simple extension problem [on hold]
Let $E$ be a finite extension of $k$ and let $p^r=[E:k]_i$. We assume that $\mathrm{char}(k)=p>0$. Assume there is no exponent $p^s$ with value of $s$ strictly less than $r$ such that $E^{p^s}$ is ...
4
votes
2answers
131 views
stability of linear systems of quadrics
A pencil of quadrics in $\mathbb{P}^n$ is a line in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of pencil of quadrics is the Grassmannian $Gr(2,N+1)$. The group $SL_{n+1}(\mathbb{C})$ ...
3
votes
3answers
233 views
Dimension of a ring after localization
Let $R$ be a Noetherian domain of dimension $\ge 1$. Let $\mathfrak{p}_i$, $i = 1, 2, ...$ be prime ideals of height one. Let $T = R[[X]]$ with $X$ is a indeterminate. For each $i \ge 1$ we set ...
-2
votes
0answers
26 views
combinatorial commutative algebra [migrated]
Is there anyone who can help me with this problem? Any hint to the solution would be appreciated!
Let $\Delta$ be a $(d-1)$-dimensional simplicial complex. Show that the h- and
f-vectors of $\Delta$ ...
3
votes
1answer
165 views
Tor dimension in polynomial rings over Artin rings
I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
0
votes
0answers
117 views
wedge product in integral ring extensions
Let $A \subseteq B$ be a finite extension of noetherian rings and let $I$ be an ideal of $B$. We consider $I$ and $B$ as finitely generated $A$-modules. The inclusion map $I \hookrightarrow B$ ...
3
votes
1answer
216 views
When is the completion of an integral domain still integral?
I have a ring $R=k[S]$ which defines an affine toric variety $X_\sigma$, where $S=M\cap \sigma^\vee$ is the semigroup from a rational polyhedral cone $\sigma$. Let $I$ be the ideal for the toric ...
1
vote
1answer
158 views
Failure of Noether normalization
I was in a class recently where we were trying to roughly count the dimensions of certain spaces of rational maps from algebraic curves into closed subschemes $Z \subseteq \mathbb{A}^n$. One way to ...
2
votes
1answer
49 views
Kernel of the induced map of the wedge product
Let $A$ be a noetherian ring and let $M$ be a finitely generated $A$-module. Let $F$ be a free $A$-module and let $d: F \to M$ be a homomorphism which maps a basis of $F$ to a minimal set of ...
-1
votes
0answers
16 views
Indefinite quaternion algebra over Q [migrated]
Let $D$ be an indefinite quaternion algebra over $\Bbb Q$. We have a chosen isomorphism $\iota \colon D \otimes_{\Bbb Q} {\Bbb R} \cong {\mathrm{M}}_2({\Bbb R})$.
Q: If we choose another isomorphism ...
11
votes
2answers
262 views
Varieties where every algebra is free
I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
18
votes
5answers
1k views
(Short) Exact sequences with no commutative diagram between them
This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting.
The problem is to produce an example of the following ...
-3
votes
0answers
63 views
Descending chain condition for radical ideals of $\mathbb{Z}_p[x_1, \ldots, x_n]$ [duplicate]
Suppose $p$ is a prime. Is it true that $\mathbb{Z}_p[x_1, \ldots, x_n]$ has descending chain condition on its radical ideals?
1
vote
0answers
105 views
Descending chain condition for radical ideals
For which integral domains $R$ (not filed) the ring $R[x_1, \ldots, x_n]$ satisfies descending chain condition for radical ideals? I am not expert in Ring Theory and I need an answer to construct some ...
7
votes
0answers
268 views
name for a degree-like invariant of a power series over a commutative ring
Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$
f ...
0
votes
0answers
79 views
Family of curve singularities whose generic mebers are smooth
Let $f: (X,x)\rightarrow (\mathbb C,0)$ be a deformation of a curve singularity $(X_0,x)$, and let $f: X \rightarrow T$ be a sufficiently small representative. Assume that $(X,x)$ is reduced and pure ...
2
votes
1answer
14 views
Separations between notions of rank for modules over commutative (semi-)rings with no zero divisors.
Let $M$ be an $m$-by-$n$ matrix, here are three definitions$^5$ that we could use for rank:
$rk(M) = \min k$ such that for matrices $P$, and $Q$ with $P$ of size $m$-by-$k$ and $Q$ of size ...
1
vote
1answer
80 views
Decomposition of skew-symmetric maps
Let $A$ be a ring and let $F$ be a finitely generated, free $A$-module. Let $\alpha: F \to \textrm{Hom}_A (F, A)$ be a skew-symmetric homomorphism, i.e. $\alpha(x)(y)=-\alpha(y)(x)$ for all $x,y \in ...
3
votes
2answers
164 views
Are seminormal rings regular in codimension 1?
Let $A$ be a seminormal ring. (Assume that $A$ is a finitely generated $k$-algebra, if it helps.) Is it true that $A$ is regular in codimension 1? I know this is true for normal rings. If the ...
4
votes
1answer
135 views
Tannaka–Krein duality
First I would like to stress that maybe I don't have a necessary background from the theory of Lie groups. I met the topic of Tannaka–Krein duality while reading the book of Gracia–Bondia, Varilly and ...
3
votes
1answer
302 views
What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?
Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...
2
votes
0answers
85 views
syzygy of a generalized cohen-macaulay module
Let $R$ be a local, noetherian ring of dimension $d$ and suppose it is generalized cohen-macaulay. Is it true that For any finitely generated $ R $-module $ M $, which is maximal generalized ...
3
votes
1answer
237 views
For what varieties do we have results on the category of singularities?
Let $X$ be a singular variety. Define the (triangulated) category of singularities (as in Orlov's paper)
as the Verdier quotient of the derived category of coherent sheaves on $X$ modulo the full ...
1
vote
1answer
150 views
Extend morphism between coherent sheaves in $\mathbb{P}^n$
Let $\mathcal{F}_1, \mathcal{F}_2$ be coherent sheaves over $\mathbb{P}^n_{\mathbb{C}}$ for $n \ge 3$. Now, $\Gamma_*(\mathcal{O}_{\mathbb{P}^n})=\mathbb{C}[X_0,...X_n]$. Denote by $U_0$ the affine ...
1
vote
1answer
154 views
Does $\mathbb P^1 \times \mathbb P^1$ admit an Ulrich bundle?
In an answer to a MathOverflow question on the following link
Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$, it is mentioned that $\mathbb P^1 \times \mathbb P^1$ has an Ulrich sheaf. However, ...
1
vote
0answers
95 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 ...
5
votes
1answer
146 views
Kernel of the differential in de Rham complex in positive characteristic
Roughly, I'd like to ask how does the first terms in de Rham complex behaves for singular varieties.
Let $Y$ be a potentially singular integral scheme over a perfect field $k$ of characteristic $p$ ...
1
vote
0answers
70 views
The structure of symmetric powers of finite-dimensional local rings
Fix an algebraically closed field $k$ of arbitrary characteristic $p$ and let $R$ be a finite-dimensional local $k$-algebra (so in particular $R$ is Artinian and Noetherian). Let $S_n$ be the ...
1
vote
1answer
114 views
Does the normalization morphism induce isomorphism on residue fields?
The question is basically coming from the following situation:
Let $C$ be an integral curve over a field $k$ (EDIT and assume that $k$ is not algebraically closed) and let $\phi\colon C^N\to C$ be the ...
0
votes
1answer
118 views
Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial
The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ...
-2
votes
0answers
87 views
Cohen Macaulay modules over regular rings
Can someone give an example for the following object: $R$ a noetherian regular local ring, $M$ a finitely generated Cohen-Macaulay $R$-module such that $\dim R-\dim M=1$.
3
votes
2answers
239 views
Counterexample to Openness of Flat Locus
Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $$U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$$ is open in ...
3
votes
1answer
132 views
Is the induced ring homomorphism surjective for a finite injective morphism between affine varieties?
Let $X$ and $Y$ be affine varieties over $\mathbb C$, and consider a
morphism $f:X\to Y$ and the induced homomorhism
$$ \varphi=f^*:B=\mathbb C[Y]\to A=\mathbb C[X]. $$
It is very easy to see that ...
1
vote
1answer
87 views
Morphisms between a globally generated sheaf and a coherent sheaf(Edited)
Let $X$ be a quasi-projective irreducible scheme, $\mathcal{F}_1$ a globally generated $\mathcal{O}_X$-module and $\mathcal{F}_2$ a coherent sheaf over $X$. Suppose that $\mathcal{F}_1$ is globally ...
1
vote
0answers
74 views
When is a power series of two variables formally a rational function?
If we have a (formal) power series of two variables with positive coefficients.
Is there any necessary and sufficient condition for this to be (formally) a rational function?
1
vote
1answer
234 views
Cohomology after completion
I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...
6
votes
1answer
189 views
A question on non noetherian ring
Let $R$ be a reduced commutative non noetherian ring of dimension $d$ and $a$ a non zero divisor. Can I say that Krull dimension of $R/(a)$ is at most $d - 1$?
8
votes
1answer
341 views
Commutative algebras whose bidual is commutative
Let $k$ be a commutative ring and $A$ a commutative $k$-algebra. Call $D(A) := \mathrm{Hom}_k(A,k)$ the dual of $A$ as a $k$-module, and $DD(A) := \mathrm{Hom}_k(D(A),k)$ the dual of the latter. Let ...
2
votes
0answers
123 views
Weak assassins and essential morphisms
Let $R$ be a commutative ring and let $M\rightarrow N$ be an essential morphism of $R$-modules. Then, $M$ and $N$ have the same associated primes.
Over non-noetherian rings the notion of associated ...
5
votes
3answers
310 views
Finite index free subgroups of $\mathrm{SL}(3,\mathbb{Z})$
Does $\mathrm{SL}(n,\mathbb{Z})$ have a free subgroup of finite index for some $n \geq 3$? I know that $\mathrm{SL}(3,\mathbb{Z})$ has many free subgroups and that in the case of ...
3
votes
1answer
112 views
Toric ideal of slice of a polytope?
Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...
