All Questions
6,055 questions
0
votes
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544
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isomorphism between vector spaces and modules - Commutative Algebra
Hi, Let $M_i$ be A modules. Then we know that $Ass (\oplus M_i) = \bigcup Ass(M_i) $. We consider here isomorphisms between modules.
Now consider a stanley ...
- 287
0
votes
0
answers
254
views
What is Castelnuovo-Mumford regularity of this algebra?
Let $M=\mathbb{C}[f_1,f_2,\ldots,f_r]$ is finitely generated algebra, $f_i \in S:=\mathbb{C}[x_1,x_2,\ldots,x_n],$ $\deg(x_i)=1, 1<\deg(f_i)<99.$ Suppose that minimal free resolution of $...
- 301
0
votes
0
answers
198
views
why a reduced ring can be embedded into a sum of integral rings?
Hi,
the question is exactly
"why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?"
Is this simply because in the normalization process we can have many irreducible ...
- 141
0
votes
0
answers
197
views
Existence of flat models of a smooth finite type algebra over $R((t))$
Let $k$ be a field, $R$ a $k$-algebra (of finite type if necessary),
$B$ an algebra of finite type over ring of the formal Laurent series $R((t))$, which is smooth.
Up to this generality, can one ...
- 51
0
votes
0
answers
183
views
Standard system of parameters and an example
Let $(R,m)$ be a local Noetherian ring. A system of parameters $\bf{x}$$:=x_{1}, \dots, x_{d}$ is a standard system of parameters if $(\bf{x})H^{i}_{m}(R/(x_{1}, \dots, x_{j}))=0$ holds for all non-...
- 113
0
votes
1
answer
164
views
How to design or create or generate a bijective ring map? [closed]
How to design or create or generate a bijective ring map?
- 11
0
votes
0
answers
165
views
Support sets along a ring homomorphism.
Let $(R,m)$ and $(S,n)$ be commutative noetherian local rings, and $f: R\rightarrow S$ be a local homomorphism (i.e., $f(m) \subseteq n$) with $S$ flat as $R$-module. If $M$ is a finite generated $R$-...
- 1,207
0
votes
1
answer
502
views
Finiteness of injective hull of residue field for Artin local ring
$(A,\mathfrak{m})$ an Artin local ring, $E(A/\mathfrak{m})$ the injective hull of $A/\mathfrak{m}$. How do I see that $E(A/\mathfrak{m})$ is a finite $A$-module?
- 2,857
0
votes
1
answer
214
views
number of representations by sums of three squares (with coefficients)
There are formulas for counting the number of representations of a positive integer $N$ as a sum of three integer squares. What is a reference for
$$
\#\{(x,y,z)\in \mathbf{N}^3: 5^4 x^2+y^2+z^2=N\}
?$...
- 3,062
0
votes
1
answer
349
views
Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)
Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...
- 10.5k
0
votes
2
answers
331
views
finite global dimension vs integral Domain
For the quotient of polynomial rings over complex number field,
its global dimension is finite is equivalent to it is domain.
is this true?
- 77
-1
votes
1
answer
803
views
Question on an exercise on homological algebra?
Suppose $R$ has finite global dimension $n$, $N$ is a finitely generated module, $F$ is a free module, and $\operatorname{Ext}^n(N, F) \neq 0$. Then $\operatorname{Ext}^n(N, R)$ is also non-trivial.
...
- 77
-1
votes
1
answer
224
views
Group isomorphism $R^{\times} \simeq C_n\times C_2$? [closed]
Let $n$ be an positive integer and $R:=\mathbb{Z}[X]/(X^2,nX)$.
Do we have $R^{\times} \simeq C_n\times C_2$ (the group of units of $R$) as it seems to be the case for small values of $n$.
If so, do ...
- 2,829
-1
votes
1
answer
512
views
Functions of several variables over finite fields [closed]
For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial ...
- 2,583
-1
votes
2
answers
671
views
Do Gorenstein rings necessarily have a finite projective dimension (as a module over itself)? [closed]
Do Gorenstein rings necessarily have finite projective dimensions?
- 2,857
-1
votes
1
answer
919
views
coprime and strictly coprime ideals
in the book "etale cohomolgy" milne is using two notions: coprime ideals and strictly coprime ideals. It seems to me that both the notions are same.
because (f(t))+(g(t))=(f(t),g(t)).
What am i doing ...
user111251
-1
votes
1
answer
178
views
About n-tuple unimodular
Let ($\mathcal{O}$, $\mathcal{M}$, k) be an DVR and $F_{1},...,F_{n} \in \mathcal{O}[X_{1},...,X_{n}]$ such that detJF = 1 where JF is the matrix $(\frac{\partial F_{i}}{\partial X_{j}})$. Suppose ...
- 348
-1
votes
1
answer
121
views
On rank one torsion-free modules over local rings [closed]
Let $A$ be a local ring which is also an integral domain and $M$ be a rank one $A$-module. Denote by $k$ the residue field of $A$. Is $\dim M \otimes_A k \le 1$? If not, is there a known upper-bound ...
- 2,126
-1
votes
1
answer
282
views
Invertible matrices satisfying $[x,y,y]=x$ (take 2).
This is a simpler version of this question. Let $x=\left(\begin{array}{lll} 2 & 0 & 0\\\
0& 1 & 0\\\
0 & 0 & \frac12\end{array}\right)$. Is there a $3\times 3$-matrix $y$ with ...
user6976
-1
votes
1
answer
187
views
Existence of a special type of maximal ideal in $C(X)$:
Does there exist any maximal ideal $M^p$ in $C(X)$ (the ring of continuous functions on a topological space $X$) such that each element of $M^p$ is a divisor of zero but $M^p≠O^p$?
- 5
-1
votes
1
answer
294
views
Must 'special' $u,v \in \mathbb{C}[x,y]$ be symmetric polynomials?
The idea for the following question came from Joachim König's last comment appearing
here, namely, the example with $u=x+y^3,v=x^3+y$.
Let $u,v \in \mathbb{C}[x,y]-\mathbb{C}$. Denote by $\alpha$ the ...
- 2,837
-1
votes
1
answer
209
views
Flatness of certain quotient rings
Let $p,q \in \mathbb{C}[x,y]$ be two polynomials such that $p_xp_yq_xq_y \neq 0$
(namely, each partial derivative is non-zero).
Assume that the following four conditions are satisfied:
(1) $\frac{\...
- 2,837
-1
votes
1
answer
510
views
tangent bundle of $\mathbb{P}^1$
Let V be a two dimensional vector space over k. Consider the set of isomorphism classes of all rank 1 $k[x]/x^2$-submodules of $V\otimes k[x]/x^2$. Does this set has a variety structure? Is it ...
user100841
-1
votes
1
answer
365
views
When is a local subring of a number field a valuation ring?
Do we have some good examples of local subrings of number fields which are not valuation rings?
Do we have an easy criterion for determining whether a local subring of a number field is a valuation ...
- 2,824
-1
votes
1
answer
110
views
Variety of commutative semi group [closed]
V is a variety of commutative semi group satisfying the identity $x^2 = x^3$.
I need to prove that:
$|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$.
Any hints on this ?
$F_V$ is V-free algebra.
- 155
-1
votes
1
answer
185
views
Algorithm to find symmetric function given specialization
I have a symmetric function f(c1,c2,c3,c4,c5) which, when c1 < c2 < c3 < c4 < c5, has the form p1(c1)+p2(c2)+p3(c3)+p4(c4)+p5(c5), where the p_i's happen to be polynomials of degree <=5....
- 827
-1
votes
1
answer
278
views
an example of a Noetherian domain with finitely many non-principal maximal ideals [closed]
Let $F$ be an algebraically closed field, and consider the ring $F[X, Y]$
of polynomials over $F$ in two indeterminates $X$ and $Y$. Let $S$ be the multiplicatively
closed set in $F[X, Y]$ generated ...
-1
votes
1
answer
555
views
Noetherianity assumptions in Hartshorne's book
It seems that noetherian assumptions are not necessary in many results by Hartshorne, in his book "Algebraic Geometry". How much is this true? Could you please give examples?
- 1,422
-1
votes
1
answer
410
views
When Hom(M,E) is injective? [closed]
Let $R$ be a commutative Noetherian ring with non-zero identity, $M$ be an $R$-module
and $E$ be an injective $R$-module. When $Hom(M,E)$ is injective?
Thanks.
- 11
-1
votes
1
answer
1k
views
Correspondence between submodules and quotient modules
What is the (natural) bijection between the isomorphic class of sub modules and isomorphic class of quotient modules of a finitely generated torsion module over a PID.
Is there any inclusion relation ...
- 1,269
-2
votes
2
answers
764
views
Reduced ring with all non-prime ideals finitely generated
Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even ...
user111492
-2
votes
3
answers
279
views
algebra group theory [closed]
If A, B, C be abelian grops and if A isomorph with direct sum of B and C and A be isomorph with B what we can say about C?
- 1
-2
votes
1
answer
314
views
configuration space and iterated loop space
Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C_n(X)$ as in the book The geometry of iterated loop spaces, Theorem 5.2. I want to prove that the map $\alpha_n$ in ...
- 1,990
-2
votes
1
answer
329
views
Module such that every finitely generated submodule is semisimple [closed]
Is there an example of a module $M$ (over a commutative ring) that is not free, and such that each of its finitely generated submodule is semisimple (i.e. such that any submodule of any finitely ...
- 549
-2
votes
1
answer
832
views
What is the spectrum of the ring $R((x))$ of formal Laurent series over a ring $R$? [closed]
Let $R$ be a ring and $R((x))$ the ring of formal Laurent series. The elements in the ring $R((x))$ are series of the form
$$
f = \sum_{n\in\mathbb{Z}} a_n x^n,
$$
where ${\displaystyle a_{n}=0}$ for ...
- 6,201
-2
votes
1
answer
780
views
commutative monoids have binary products? [closed]
Does the category CMonoid of commutative monoids have binary products?
thanks
- 1
-2
votes
1
answer
77
views
integral ring extension implies algebraicity of their fraction fields extension?
$\DeclareMathOperator\Fr{Fr}$There is something I don't get about the following :
Start with the well known fact that if $A\subset B\subset C$ are rings with $B$ the integral closure of $A$ in $C$, ...
- 1,031
-2
votes
1
answer
634
views
Doubt in this proof of Horrocks theorem
I'm beginning to study some research papers and I need right now to understand the solution of Vaseršteĭn of Serre's theorem (simplest proof of this theorem), to do so, I'm beginning to understand ...
- 195
-2
votes
1
answer
572
views
Tensor products of simple modules over algebras [closed]
Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules.
We can regard $M\otimes_{\mathbb C}N$ as $A\otimes_{\mathbb C} B$-modules in natural ...
-2
votes
1
answer
187
views
behavior of multiplicity in exact sequences
Bruns-Herzog define multiplicity When the ring and module are not necessarily graded as $e(M)=e(gr_m(M))$, see B-H 4.6. I have two questions:
Question1. Many concepts in commutative algebra have ...
- 1,355
-2
votes
1
answer
151
views
Quadratic extension and prime ideals
Let $B/A$ be a quadratic Galois extension between local domains. Define ${\mathrm{Gal}}(B/A) = \{e,\sigma\}$.
Choose two prime ideals ${\frak P}_1, {\frak P}_2$ of $B$ such that ${\frak P}_2 = {\...
- 781
-3
votes
2
answers
818
views
Is there a "weak" fundamental theorem of algebra for matrices?
Let $R$ be the ring of complex $n\times n$ matrices, where $n>1$.
Does every nonconstant polynomial in $R[X]$ have a root in $R$?
Note: The "strong" fundamental theorem of algebra for ...
- 360
-3
votes
1
answer
273
views
Isomorphic quotient of a Module over Noetherian commutative algebra [closed]
I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...
- 3
-3
votes
3
answers
400
views
Dense section of sheaves of modules
Here is something that isn't yet very clear to me. Say, I've got a commutative ring A. I consider the affine scheme from A, so it's a sheaf of rings over Spec A.
EDIT: And additionally let's say ...
- 2,275
-3
votes
1
answer
208
views
can the square of a polynomial be written as a sum of squares of polynomials with only even degree terms?
Let $f$ be a polynomial with real coefficients in several indeterminates $x_1, \dots, x_n$. Suppose that
$$ f = g^2 $$
for some polynomial $g$.
Is it true that we can find polynomials $h_1, \dots, h_m$...
- 331
-3
votes
1
answer
964
views
On the maximal ideal m of the formal power series ring [closed]
Let $A \colon= K[[X_1,X_2,\ldots,X_{\infty}]]$ be the formal power series ring with infinitely many variables over a field $K$. We can represent it also by the following manner$\colon$
\begin{equation*...
- 781
-3
votes
1
answer
234
views
A common name for a functorial construction of Commutative Algebra?
I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name.
Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...
- 41.9k
-3
votes
1
answer
1k
views
An elementary question about the Krull dimension of modules [closed]
Let $R$ be a commutative ring. If $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence of modules, we have that $\operatorname{Supp}M=\operatorname{Supp}M'\cup \operatorname{...
- 1,207
-3
votes
1
answer
201
views
Structure of the automorphism group of an L-rig
This question is a follow-up to Are there infinitely many L-rigs? which is already pretty convoluted.
Define the $\varphi$-evaluation morphism at a complex number $s$ as $\epsilon_{\varphi,s}:F\mapsto ...
- 6,984
-3
votes
1
answer
441
views
Depth or Grade of an ideal
Let $R$$\subset$$ S$ be commutative noetherian rings,and $I$ is an ideal of $S$.
We now that $I$ is a $R-$module.
Do we have $grade_{R}(I)$ $\le$ $ grade_{S}(I)$?
Thank you!
- 359