All Questions
5,985 questions
1
vote
1
answer
162
views
systems of parameters vs. minimal "exhausting" systems in a Noetherian local ring
Hello,
Probably this is a very easy question.
Fix a Noetherian local ring $A$, and an $A$-module of finite type $M$.
Lets call a system $ x_1 , \ldots , x_m \in \mathfrak{m}$ $M$-exhausting, if $M / ...
- 3,137
5
votes
1
answer
2k
views
module of differentials of formal power series ring and of its field of quotiens
For any $A$-algebra $B$ ( commutative ring with 1 ), we have the existence of $\Omega_{B/A}$, the module of relative differentials of $B$ over $A$, which can be defined by an universal property. In ...
- 1,214
4
votes
4
answers
596
views
Generalization of Jordan Decomposition for Several Commuting Operators
Recently I became curious about the following question:
Let $V$ be a finite dimensional vector space over $k$ and let $A_1, \cdots, A_n: V \rightarrow V$ be a set of commuting maps. Question: ...
- 9,132
2
votes
3
answers
1k
views
General hyperplane sections and projection from a point
Let $k$ be an algebraically closed field, and consider some subscheme $X\subset \mathbb{P}_k^n$. Let $x$ be a closed point of $X$, and $H$ a general hyperplane containing $x$. There is a regular map $\...
- 42.9k
7
votes
1
answer
1k
views
Are squarefree monomial ideals on a regular system of parameters in a regular local ring radical?
Suppose $(R,m)$ is a regular, local ring. Let $x_1,x_2,...,x_n$ be a regular system of parameters. Let $I$ be an ideal generated by squarefree monomials in the $x_i$'s. Is $I$ a radical ideal? The ...
CommunityBot
- 1
6
votes
2
answers
456
views
Immerse an affine schemes into $A^n_S$
Suppose $f: X\rightarrow S$ is of finite type, S is Noetherian. Now X=Spec B is affine, but the morphism f is not an affine morphism. S is not affine (or really f does not factor through any affine ...
- 11.1k
3
votes
1
answer
398
views
Is the first part of Eisenbud's Proposition 15.15's proof o.k?
In the chapter on Gröbner bases from Eisenbud's "Commutative Algebra" the following statement appears as Proposition 15.15 (page 344):
Let $F$ be a free $S$ module with basis and monomial order ...
- 4,280
8
votes
2
answers
537
views
Prime avoidance in adjacent degrees
Let $\mathfrak{p}_1, \dotsc, \mathfrak{p}_k$ be relevant homogeneous primes ideals in the graded ring $R := \Bbbk[x_0, \dotsc, x_n]$, where $\Bbbk$ is a field. Prime avoidance (in Eisenbud's ...
- 11.1k
20
votes
1
answer
2k
views
Tropical homological algebra
Has anyone out there thought about homological algebra over the tropical semifield $\mathbb{T}$? For example, I'm interested in the Hochschild homology and cyclic homology of tropical algebras, if ...
- 24k
3
votes
0
answers
336
views
Antisymmetric functions of the roots of unity: an elementary conjecture
Let $z_1, z_2, \cdots z_N$ be $N$ variables obeying the condition $z_i^M=1$ for some positive integer $M>N$.
Let $F_N$ be the space of antisymmetric polynomials of these variables. Given a set $E = ...
- 378
20
votes
3
answers
2k
views
Is every integral epimorphism of commutative rings surjective?
That's the question. Recall that a morphism $f\colon A\to B$ of commutative rings is integral if every element in $B$ is the root of a monic polynomial with coefficients in the image of $A$ and that $...
- 5,039
3
votes
2
answers
534
views
An easy example of a (1/quasi-)Gorenstein ring with non-trival canonical divisor class.
Suppose that $R = S/I = k[x_1, \dots, x_n]/I$ is a (normal) domain of finite type over a field (or any semi-local ring $k$ with a dualizing complex). In this case, I can define $\omega_R = \textrm{...
CommunityBot
- 1
15
votes
2
answers
2k
views
prime ideals in regular local rings
Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. ...
- 30.5k
1
vote
1
answer
320
views
covers of complete regular local rings
It is well-known that if one assumes algebraic closedness and characteristic 0 of the residue field then finite covers of complete DVRs are all of the form $A[x]/(x^m-a)$ for some $a \in A$ (direct ...
- 22.6k
1
vote
1
answer
924
views
Torsion-free and torsionless abelian groups
This question is motivated by my most spectacular answer on MO (:
Let $A$ be a module over $\mathbb Z$. $A$ is said to be torsion-free if $na=0$ implies $n=0$ or $a=0$ for any $n\in \mathbb Z, a\...
CommunityBot
- 1
1
vote
1
answer
601
views
Unimodular column property
Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property.
I would like to know if there is a ring $R$ that doesn't ...
- 23k
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
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 91
5
votes
0
answers
994
views
Maximal ideals in polynomial rings over algebraically closed fields - when Weak Nullstellensatz does not apply
Weak nullstellensatz describes maximal ideals in polynomial rings over algebraically closed fields at least when the cardinality number of variables is finite. Lang obtained the same conclusion also ...
- 17.6k
1
vote
0
answers
236
views
Terminology question - "Chern number"
I have seen the term Chern number used to refer to the first Hilbert-Samuel coefficient, $e_{1}(I)$, of an ideal $I$ in a local ring $(R, m)$. (Where the Hilbert-Samuel polynomial agrees with $\...
- 113
52
votes
7
answers
5k
views
What does a projective resolution mean geometrically?
For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...
- 19.6k
3
votes
2
answers
547
views
less than normal
Hi,
if we could write a classification about the known regularity which is the known class of schemes that are immediately less good than normal schemes? And which properties have they?
thank you
- 20.5k
1
vote
3
answers
896
views
Stably free module not finitely generated is free
Hi. I have read that stably free modules not finitely generated are free; this is proved in
M.R. Gabel, stably free projectives over commutative rings, Thesis, Brandeis Univ., Waltham, MA 1972.
But ...
- 113
5
votes
3
answers
984
views
Isomorphism of the function field of the projective line with $\mathbf{C}(s)$
Suppose I chose two rational functions, say,
$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad
v = \frac{t^5(4+t)}{(1+4t)}.$$
Then I know that $K(X) = \mathbf{C}(u,v)$ is
the function field of the projective ...
CommunityBot
- 1
13
votes
3
answers
1k
views
Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane
It is known from either Morse theory or Bialynicki-Birula decomposition that the fixed points of a ${\mathbb{C}}^*$ action on a smooth algebraic variety over $\mathbb{C}$ determine a cell ...
- 156k
2
votes
2
answers
389
views
Related to fractional ideals
$K$ a field, $A\subset K$ a subring, $M\subset K$ an $A$-submodule. Define
$$(A:_{K}M):= \lbrace s\in K|sM\subset A\rbrace$$
Then it is easy to see that
$$M\subset A\Longleftrightarrow A\subset (A:_{...
- 11
2
votes
1
answer
184
views
Are there known quantitative descriptions of the fact that the common zero set of some polynomials is empty besides Nullstellensatz?
Working in a polynomial ring, if some polynomials has no common zeros, Nullstellensatz tells us a qualitative description of the propertity of these polynomials, are there known quantitative ...
- 47.4k
11
votes
2
answers
1k
views
Geometric motivation for the Stanley-Reisner correspondence
The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X_1,...,X_n]/I_\Delta
$$
where $I_\Delta$ is the ideal generated by the $X_{...
- 1,500
4
votes
1
answer
276
views
I am interested in collecting different methods of proofs that a subalgebra coincides with whole algebra.
Let $A \subset \mathbb{C}[x_1,x_2,\ldots,x_n]$ - be finitely generated graded algebra and $B$ be its subalgebra. How to prove that $A=B.?$
Unfortunalelly I know only one method to do it - to ...
- 33.8k
4
votes
1
answer
382
views
"extend a functor"
Hi,
I have probably a basic question. I have a functor $F: Sch \rightarrow Set$, an algebraic stack $M$ with a "universal family" $G\rightarrow M$ and a representability property like this: for every ...
- 11
6
votes
1
answer
641
views
The Jacobian ideal generates the socle of a complete intersection
This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here:
http://tinyurl.com/2967eov
I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ ...
- 5,846
11
votes
3
answers
972
views
Is Krull dimension non-increasing along ring epimorphisms?
Let $f \colon R \to S$ be an epimorphism of commutative rings, where $R$ and $S$ are integral domains. Suppose that $\mathfrak{p} \subset S$ is a prime such that $f^{-1}(\mathfrak{p}) = 0$. Does it ...
- 23k
4
votes
1
answer
579
views
Is there a clean definition of the residue map in Milnor K-theory?
If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M_n(K) \to K^M_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of ...
- 15.2k
3
votes
1
answer
475
views
Can the factorisation of (p) in a number field K be described by the minimal polynomial of a primitive element?
Let $K$ be a number field, with ring of integers $O_K$, and let $\alpha\in O_K$ be a primitive element for the extension $K/Q$, with minimal monic polynomial $f(x)\in Z[x]$. If $p$ is a prime number, ...
CommunityBot
- 1
1
vote
2
answers
364
views
Rig of fractions, including zero denominators
For some integral domain $R$, one forms the field of fractions $R^*$ by considering (equivalence classes of) formal pairs {$r/s : r \in R, s\in R\backslash 0$} and defining $+$ and $*$ as you'd expect ...
- 18.7k
3
votes
0
answers
591
views
Algebraic description of double vector bundles.
It is well known, by Serre-Swan theorem, that given a compact manifold M there is an equivalence of categories between the category of vector bundles over M and the category of finitely generated ...
- 51
5
votes
0
answers
210
views
Kahler differentials and the m-adic filtration
Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies $...
- 1,038
6
votes
1
answer
1k
views
Nagata's bizzare examples
Hi,
due to Nagata and his clever and bizzare examples I'm unsure in this:
1) Is there a regular ring of infinite Krull dimension?
2) Is it true that: Regular ring of finite Krull dimension = ...
- 11.6k
22
votes
4
answers
2k
views
Two questions about finiteness of ideal classes in abstract number rings
Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...
CommunityBot
- 1
7
votes
4
answers
2k
views
commuting matrices
I have a set $\{A_1, A_2, .. A_k\}$ of $n$ by $n$ real matrices and I know that they are 'perturbated versions' of a set of commuting matrices : $\{P_1,..,P_k\}$, by perturbated versions I mean that I ...
- 28.6k
18
votes
2
answers
4k
views
How to show a set of polynomials is algebraically independent?
Suppose that I have $n$ homogeneous polynomials $f_1, \dots, f_n \in \mathbb{C}[x_1, \dots, x_m]$ and that $n < m$. Is there a well known method or algorithm to determine if these polynomials are ...
- 151
3
votes
1
answer
374
views
Composition and intersection of residue fields
Let $A$ be a normal ring with quotient field $K$. Let $L/K$ be a finite separable extension.
Let $E_1/K$ and $E_2/K$ be extensions of $K$ contained in
$L$. Let $B_1$ (resp. $B_2$) be the normalization ...
- 11.1k
1
vote
0
answers
451
views
Proof of local structure theory for unramified morphisms [closed]
In Raynaud's "Anneaux locaux henseliens," a proof is given of the
following fact: Let $R \to S$ be a finite type morphism of rings, $\mathfrak{q}
\in \mathrm{Spec} S$, $\mathfrak{p} $ the inverse ...
- 11.1k
1
vote
1
answer
518
views
Cardinality of a linear independent subset of a free module over a commutative ring which is not an integral domain
If R is a commutative ring with unity and not an integral domain and F is a free R-module with rank k,is there a linear independent set with cardinality > k?
I prooved that this is not true if R is an ...
CommunityBot
- 1
6
votes
1
answer
806
views
Radicals of binomial ideals
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. An ideal (that can be) generated by monomials is called a monomial ideal. For the monomial ideal $M=(m_1,m_2,.....
CommunityBot
- 1
6
votes
1
answer
356
views
Constructive Bezout cofactors in the ring of algebraic integers
We see the following in an answer to This Question : (mangled by me for my purposes, all errors my fault)
Dedekind stated (in 1871) that the ring of algebraic integers is a Bezout Domain. He calls it ...
CommunityBot
- 1
3
votes
0
answers
277
views
For which dg-algebras is a dg-module with bounded cohomology quasi-isomorphic to a bounded dg-module?
Hello!
Let S be a commutative local Noetherian base ring and A be a dg-S-algebra.
Suppose M is a bounded above dg-A-module with bounded cohomology. Does there exist a bounded dg-A-module isomorphic ...
- 2,756
0
votes
2
answers
1k
views
Degenerate case of linear programming duality?
Let's say we have a maximization linear program that looks like this: maximize $\vec{c}\vec{x}$, subject to $\matrix{A}\vec{x} \leq 0$, $\vec{x} \geq 0$. If we take the dual, we have "minimize $0\vec{...
- 837
1
vote
4
answers
978
views
Maximum average value within a rectangular bounding box
The goal is to expedite detection using the sliding window approach. In other words, an object classifier is known and I need to find where the possible locations of this object are in an image. This ...
CommunityBot
- 1
5
votes
1
answer
2k
views
Calculating the normalization of an algebraic surface.
Suppose $X$ is a complex algebraic projective surface with one dimensional singular locus. For example consider the hypersurface $z^5=t^2(tx^2+y^3+t^3)$, whose singular locus is along the double line $...
- 57.6k