All Questions
6,055 questions
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429
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Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras
For commutative rings $R \subseteq S$,
recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.
...
- 2,837
0
votes
1
answer
299
views
K3 Surfaces : Derivations and automorphisms
The literature about K3 surfaces is extensive. Let us consider the fact that the zero locus of any smooth homogeneous degree 4 equation is a K3 surface.
Let $R$ the quotient ring of a homogeneous ...
- 829
0
votes
1
answer
132
views
Special elements of the Cremona group
After asking this MO question, I wish to ask about the following special case:
Let $f$ be a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$ and denote $u:=f(x),v:=f(y)$.
Is it possible to ...
- 2,837
0
votes
1
answer
139
views
Projective modules and gradings
Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M_{\ge d}:=\oplus_{\ell \ge d} M_\ell$ the sub-module of $M$. Is $M_{\ge d}$ again a projective $A$-module?
- 2,126
0
votes
2
answers
758
views
Splitting field of an intermediate field
Consider the following 'wrong' question.
Let $f(x) \in F[x]$ be an irreducible polynomial in a polynomial ring of a field $F$. Let $L$ be the splitting field of $f(x)$ over $F$. Assume that $L$ is a ...
- 103
0
votes
1
answer
307
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 ...
- 25
0
votes
1
answer
269
views
Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals
If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
- 1,209
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votes
1
answer
339
views
Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?
If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I_m $, the $m$ -...
- 325
0
votes
1
answer
655
views
How to show two semigroups are isomorphic?
I have two finite semigroups namely $$S_1=\langle a,b: R\rangle,~~~S_2=\langle a,b: T\rangle$$ How can one show they are the same isomorphically? Should I show that the relations in one, implies the ...
- 233
0
votes
1
answer
165
views
How to classify a plane complex curve?
Let $p_1, p_2, t_1, t_2, a \in \mathbb{C}$ be constants. Consider the following plane complex curve in $\mathbb{C}^2$ ($c_1, c_2$ are indetermniates)
\begin{align}
& {p_1}^2 {p_2}^2 c_1 {t_1}^2 ...
- 6,201
0
votes
1
answer
309
views
exact short sequence of divisible groups splits? [closed]
Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequences of divisible abelian groups. Does then the sequence splits?
- 195
0
votes
1
answer
153
views
C*-algebra of free monogenic inverse semigroup
Let us take the right shift operator $S$ acting on the Hilbert space $l^2(\mathbb{N})$. Consider the C*-algebra generated the operator
$
\begin{pmatrix}
S & 0 \\
0 & S^*
\end{pmatrix}
$ ...
- 493
0
votes
1
answer
343
views
Relative Bertini Theorem
Let
$A \colon= {\Bbb C}[S_1,\ldots,S_n]$ with $1 \leq n < \infty$
$B \colon= A[X_1,\ldots,X_d]$ with $2 \leq d < \infty$.
$O \colon= (0,\ldots,0)$ be the origin of ${\mathrm{Spec}}\,B$.
...
- 563
0
votes
1
answer
73
views
A question on infinite local rings which are not division ring
Is it true that if $(R,m)$ is a (not necessary commutative) local ring then $R$ and $m$ have the same cardinal ? (Exclude TWO trivial cases: when $R$ is finite and when $R$ is a division ring)
On ...
- 271
0
votes
1
answer
300
views
Behaviour of length function under faithfully flat extension
Let $(R,m)$ and $(S,n)$ be local Noetherian rings such that $S$ is a faithfully flat extension of $R$. Let $J\subsetneq I $ ideals of $R$.
Can we relate $l_R(I/J)$ and $l_S(IS/JS)$?
PS: Here $l(-)$ ...
- 1,713
0
votes
1
answer
266
views
Example of a principal ideal which is properly contained in its relative integral closure
Let $I$ be an ideal in a Noetherian quasi-unmixed local ring $(R,\mathfrak m)$ of dimension $d.$
Let $q(I)=\overline{I}\cap I^{sat}$ where $I^{sat}=\cup_{n\geq 0}I:\mathfrak m^n.$ Then $q(I)$ is ...
- 1,713
0
votes
1
answer
372
views
The growth of the Hilbert function of a graded ring
Let $A=\bigoplus A_i$ be a finitely generated commutative unital graded algebra over a field $k$. Let $d(i)=\dim A_i$.
In general $d(i)$ is not a polynomial in $i$ (even not eventually polynomial). ...
- 2,649
0
votes
1
answer
131
views
Radical of modules [closed]
Let $R$ be a local ring with the unique maximal ideal ${\frak m}_R$ and $M$ be a $R$-module. Define
$I(M) \colon= \cap ~({\mathrm{all~ proper~ maximal ~submodules~ of}}~M)$,
where proper means ...
- 781
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votes
1
answer
2k
views
injective modules and divisible modules
The following result is basic ( P.J.Hilton, U.Stammabach, a course in homological algebra ).
Let $A$ be a principal ideal domain. Then a $A$ module is injective iff it is divisible.
Now if the ...
0
votes
1
answer
185
views
Linkage is equivalence relation
Let $(R,m)$ be a Cohen-Macaulay local ring, I and J are ideals of height $r.$ Then we say $I$ is directly linked to $J$, i.e. $I \sim J$ if there exists an ideal K generated by a regular sequence $x_1,...
- 1,713
0
votes
1
answer
232
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 ...
- 591
0
votes
3
answers
439
views
Example of indecomposable self injective ring
Is there any example of an indecomposable self-injective commutative ring with 4 or more maximal ideals?$$$$$$$$
- 9
0
votes
1
answer
381
views
Maximal group image
How does one prove: if $S$ is a finitely generated Clifford semigroup its maximal group image is actually $S_{e_{n}}$?
- 1
0
votes
1
answer
168
views
integral closure of m-primary ideals
I need help with this excercise
Let $k[X_1,\ldots,X_d]$ be the polynomial ring in $X_1,\ldots,X_d$ over a field $k$, and let $F_1,\ldots,F_m$ be forms of degree $n$. Assume that $(X_1,\ldots,X_d)=\...
- 19
0
votes
1
answer
301
views
local cohomology and radical of ideal
Let $R$ be commutative ring with identity, $M$ an $R$-module, and $I$ an ideal of $R$ . One defines $I$-torsion functor $Γ_I$ as: $\Gamma_I(M)=\bigcup_{n\in N} (0:_MI^n).$ When $R$ is Noetherian, it'...
- 1,355
0
votes
1
answer
124
views
polynomial expression for counting number of integral points of a set
Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$
Can we ...
- 1,713
0
votes
1
answer
165
views
$\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$
Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are:
Is $\inf\{i\in \mathbb N \cup \{0\}\cup \{\...
- 1,355
0
votes
1
answer
229
views
Canonical module of rees algebra
[Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if $I=(f_1,\ldots,f_g)$ is an ideal generated by a regular sequence with $g\ge 2$ then the ...
- 591
0
votes
1
answer
824
views
Extension of homomorphism to place in quotient field, or of local ring to valuation ring in quotient field
Let be given a domain $R$ (that can be supposed to be integrally closed if this can help), and $\varphi$ an homomorphism of $R$ into a field $F$.
$\varphi$ extends uniquely to a homomorphism $\varphi'...
- 687
0
votes
1
answer
615
views
Uniform Artin-Rees
The Artin-Rees lemma states that if $R$ is a Noetherian ring, $I \subseteq R$ is an ideal and $N \subseteq M$ are finitely generated $R$-modules, then there exists an integer $k$ such that for every $...
- 21
0
votes
1
answer
149
views
$I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$
In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals),
an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$.
[See Comm. Rings by Kaplansky, ...
- 41
0
votes
2
answers
824
views
Computing the minimal free resolution of a coherent sheaf on projective space
Most books on commutative algebra explain Grobner bases in the non graded case and minimal free resolutions in the local case. I like projective geometry and want to compute the minimal free ...
- 3,830
0
votes
2
answers
416
views
Computing toric ideals via saturation and Groebner bases of toric ideals
About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer.
Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...
- 103
0
votes
1
answer
172
views
$I/N$ is finitely presented module
Let $R$ be a commutative ring and $N = Nil(R)$ the set of its nilpotent elements. Suppose that $N$ is a divided prime ideal, i.e. for any ideal $I$ of $R$ either $I \subseteq N$ or $N \subseteq I$.
...
- 1
0
votes
3
answers
2k
views
Multiplicative functions $\phi : M_n(F) \longrightarrow F$ with $\phi(I) = 1$
Let $F$ be an infinite field and let $f \in F[x_{11},x_{12},...,x_{nn}]$ be an arbitrary polynomial in $n^2$ variables. Consider the function $\phi : M_n(F)\longrightarrow F$ defined by $\phi((a_{...
- 447
0
votes
1
answer
235
views
Projective bundles
Fix $n$ and let
$0\leftarrow \mathcal{F}\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(a_i)\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(b_i)\leftarrow \cdots$
be an exact sequence.
Then we can ...
- 39
0
votes
1
answer
606
views
Number of Minimal left ideals in the full matrix ring over a finite commutative local ring
Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?
user39204
0
votes
1
answer
227
views
krull dimension [closed]
im looking for a non-noetherian ring with infinite krull dimension.would you help?
- 15
0
votes
1
answer
1k
views
Cofibrant Replacement of chain complexes
Hi,
I encountered an issue today that I can't resolve to myself:
Consider the projective model structure on chain complexes over a ring R (Ch(R)), bounded below if you like.
Projectives in Ch(R) ...
- 109
0
votes
1
answer
315
views
Generalized Picard group (reflexive fractional ideals, principal ideals)
Given $\mathcal{O}=k[[u,v]]$ with maximal ideal $\mathfrak{m}$ and an $\mathcal{O}$-algebra $A$, free as an $\mathcal{O}$-module of rank $n^2$. $A$ is genertaed by two elements $x,y$ with $x^n=u$, $y^...
- 1,391
0
votes
1
answer
379
views
Is a tensor product of two dvrs semilocal?
Under what conditions is the tensor product of two dvrs semilocal?
The same question about being reduced.
Tensor product is taken over another dvr or over a field to make things simpler.
For ...
- 1
0
votes
1
answer
285
views
A Nomenclature Issue : Imprimitive Semigroup?
The following question was asked by me on the forum sci.math.research,
“An imprimitive group is a transitive permutation group with a non-trivial
equivalence relation compatible with the action of ...
- 113
0
votes
1
answer
370
views
Proving that two local PIDs, one inside the other, with the same field of fractions are equal.
Let $R\subset S$ be two local PIDs that have the same fields of fractions. How to prove that they are equal?
- 59
0
votes
3
answers
2k
views
Equality of elements in localization via universal property
I've been studying universal objects of universal algebra in a quite general setting and try to exhibit the structure of their elements just using the universal property. A very nice example for this ...
- 63.1k
0
votes
1
answer
473
views
A subfield $R \subseteq \mathbb{C}(x,y)$ with 'many' generators $w$, $R(w)=\mathbb{C}(x,y)$
Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$.
Here $\mathbb{N}$ includes $0$.
Assume that $R$ ...
- 2,837
0
votes
1
answer
118
views
$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$
Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$.
Is it true that $I/I^2$ is $R$-...
- 412
0
votes
1
answer
186
views
Does going-down theorem hold for local homomorphism of finite flat dimension?
Let $f:(A,m)\rightarrow (B,n)$ be a local homomorphism of Noetherian local rings of finite flat dimension. Does the going-down theorem hold for $f$?
If yes, then by Theorem 15.1 in Matsumura’s ...
- 639
0
votes
1
answer
137
views
$k(F_i)_{i=1}^{n}=k(G_j)_{j=1}^{m}$ iff there exist $a_i,b_j \in k$ such that $\langle F_i-a_i \rangle_{i=1}^{n} = \langle G_j-b_j \rangle_{j=1}^{m}$
Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$ and let $F_1,\ldots,F_n,G_1,\ldots,G_m \in \mathbb{C}[x,y]$, $n,m \in \mathbb{N}-\{0\}$.
Claim:
$\mathbb{C}(...
- 2,837
0
votes
1
answer
287
views
When is the power-bounded subring top. of finite type?
Very naive question here. Let $K$ be a complete nonarchimedean field, $A$ a reduced affinoid $K$-algebra. When is the power-bounded subring $A^\circ$ topologically of finite type, in the sense that we ...
- 783
0
votes
1
answer
88
views
Proving finite presentation [closed]
Let $R$ be an integral domain, $S$ be a finitely presented $R$ algebra. Then for a flat $R$ module $M$ which is also a finitely generated $S$ module I need to show that $M \otimes_{R}T$ is a fintely ...
- 167