All Questions
5,985 questions
1
vote
0
answers
351
views
Regularity and limits of smooth rational curves.
Fix integers $2 < d \leq n$.
Suppose that $T$ is a smooth complex curve with marked point $0 \in T$, and $X$ is a closed subscheme of $\mathbb{P}^n_T$, flat over $T$ such that each fiber has ...
- 1,990
1
vote
0
answers
169
views
Sum of two free o-submodules in a vector space over a local field
Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...
- 107
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
12
votes
1
answer
480
views
Extending properties of commutative rings to schemes
I'm trying to pin down the various ways we can extend a property of commutative rings to a corresponding property for schemes. Let $P$ be a property of commutative rings. We could define a scheme $(X,\...
- 4,165
0
votes
2
answers
232
views
Commutation of $GL_{n}$ with projective limits
Let $A$ denote a unital commutative ring. Given a system of ideals $(I_p)_{p\in P}$ indexed by a partially ordered set $P$ such that if $p \leq q$, then $I_p$ is contained in $I_q$, when is
$$GL_n ...
- 12.3k
3
votes
2
answers
467
views
Chern character of Hom-sheaves
I'm reading the book about moduli spaces by Huybrechts and Lehn, and i'm stuck understanding a proof, it is Theorem 6.1.8.:
Given a K3-surface $X$ and a 2-dimensional space $M$, coherent and torsion ...
- 1,391
1
vote
1
answer
434
views
Equality of chern classes and isomorphism
Given two torsion free coherent sheaves $M$ and $N$ wit $rk(M)=rk(N)=r$ on an smooth projective surface $S$, by definition $det(M):=\Lambda^r(M)^{\*\*}$.
Is the following criterion correct?
$M\cong ...
- 2,181
6
votes
1
answer
301
views
Orbits in commutative groups.
Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization ...
- 5,654
0
votes
3
answers
892
views
local Artin algebras
Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A_1\oplus\ldots\oplus A_n$ into local Artin subalgebras, see for example Atiyah-McDonald, ...
- 163
1
vote
1
answer
146
views
Is every nontrivial morphism already injective in this case?
I'm a little bit suprised at the moment, so i'll ask here if I see this wrong:
Given a sheaf of algebras $R$ ( e.g. maximal order or Azumaya) on a smooth projective scheme $X$ with generic point $p$. ...
- 42.9k
4
votes
2
answers
610
views
Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay?
I'm not sure how long this iterative questions can go on, but let me try again. Let's say $X$ is a Cohen-Macaulay scheme with an action of $\mathbb{G}_m$ (i.e. if $X$ is affine, a grading on the ...
CommunityBot
- 1
4
votes
1
answer
375
views
Regular sequence of elements of degree 1 for a homogeneous Cohen-Macaulay ring
Assume that a positively graded ring R is generated in degree 1. Is it true that, if R is Cohen-Macaulay, then there exists a regular sequence x of elements of degree 1 so that R/x is zero dimensional?...
- 5,846
4
votes
3
answers
622
views
Examples of DVRs of residue char p and ramification e
I am looking for concrete examples of a complete discrete valuation ring $R$ of characteristic 0, residue characteristic $p$ and ramification index $e$. By residue characteristic, I mean the ...
CommunityBot
- 1
2
votes
1
answer
504
views
A question arising from the Krull intersection theorem.
Let R be a local ring, I an ideal, M a finitely generated module and $N=\cap _nI^nM$. Then the Krull intersection theorem states that $N=IN$. Now if R is a local ring of characteristic $p>0$, for ...
- 19.6k
2
votes
2
answers
369
views
vectors with entries from a finite ring
I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
- 56.6k
3
votes
2
answers
589
views
Comparing homomorphisms over different base rings
I am trying to compare some homomorphism groups over different base rings, so given a commutative local ring $(A,\mathfrak{m})$ and a finite dimensional Azumaya algebra $R$ over $A$.
If $M$ and $N$ ...
- 27k
16
votes
1
answer
2k
views
Commuting Matrices and the Weak Nullstellensatz
In the Wikipedia article on Hilbert's Nullstensatz,
http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz
the following application of the Weak Nullstensatz is mentioned:
Commuting matrices
...
- 16.1k
1
vote
0
answers
198
views
Seek for good methods of computing the Krull dimension of a module?
Hi, everyone. Recently I am interested in computing the Krull dimensions of modules without using any software. However, it is not an easy job for me to do so only by its definition. Therefore, I ...
- 1,207
1
vote
1
answer
375
views
Any implemented algorithm to compute the closure of an affine variety in a product of projective spaces?
Let $I$ be an ideal of $k[x_1, \ldots, x_m, y_1, \ldots, y_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-...
- 24.8k
2
votes
1
answer
1k
views
Linear Programming Cost Function [closed]
I need to add the following to my LP problem:
If the amount of workers hired in period $t$ ($H_t$) is higher than 25, the hiring cost is only 1 instead of 1.2.
Example: if 30 workers are hired in ...
- 567
5
votes
2
answers
495
views
Sub-Hopf algebras of group algebras
Let $k$ be a field and $G$ a finite group. Is every sub-Hopf algebra over $k$ of the group algebra $k[G]$ of the form $k[U]$ for a subgroup $U$ of $G$ ?
- 12.3k
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
5
votes
0
answers
817
views
morphism which is open but not universally open
In someone's note, I have seen such an example, but I can't show that it is not universally open. Here is the example:
Let $k$ be a field and $A = k[T]_{(T)}$, the discrete valuation ring obtained ...
- 1,109
17
votes
2
answers
2k
views
How badly can Krull's Hauptidealsatz fail for non-Noetherian rings?
Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can ...
- 982
1
vote
1
answer
336
views
Need an example of finitely generated graded algebra such that each its graded subspace has infinite dimension.
More accurately, let $\displaystyle A=\sum_{i=0}^{\infty}A_i$ be a finitely generated graded algebra over say $\mathbb{Q}$ but $\dim A_i=\infty$ for each $i.$ Is it possible?
- 301
6
votes
2
answers
418
views
Does regularity of a prime ideal in the fibre imply regularity of the prime?
Recall that a prime $\mathfrak{p}$ is called nonsingular (regular) if the localization at that prime is a regular local ring. If all primes of a ring $R$ are nonsingular, $R$ is called regular. Let $...
- 30.5k
14
votes
2
answers
8k
views
Choosing the algebraic independent elements in Noether's normalization lemma
Given a field $k$ and a finitely generated $k$-algebra $R$ without zero divisors, one knows that there exist $x_1, \ldots, x_n$ algebraically independent such that $R$ is integral over $k[x_1, \ldots, ...
- 6,262
1
vote
2
answers
1k
views
Inequality-constrained linear-regression, what is the covariance of the estimator?
If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$
...
- 233
4
votes
0
answers
350
views
Artin approximation theorem for analytic functions over a field of zero characteristic
Artin's approximation theorem states: "if a system of locally analytic equations in several complex variables has a formal solution then it has a locally analytic solution".
(Artin 1968, "On the ...
- 2,232
11
votes
2
answers
697
views
Differential graded structures on free resolution?
Hello!
In "Homological Algebra on a Complete Intersection", Eisenbud proves the following:
Let $A$ be a commutative ring, $M$ be an $A$-module and $F^{\ast}\to M$ an $A$-free resolution. Further, ...
- 2,756
2
votes
2
answers
451
views
Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated?
Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth ...
- 3,918
1
vote
1
answer
167
views
Why is multiplication with a scalar no global morphism?
Given a smooth projective surface $S$ over an algebraically closed field, a sheaf rings or algebras $R$ on $S$ and a simple left $R$-module $M$, i.e. $Hom_R(M,M)=k$.Then we have $Hom_R(M,M(-i))=H^{0}(...
- 66.3k
13
votes
2
answers
2k
views
Length of I/I^2 versus Ann(I)/Ann(I)^2 in Artinian rings.
Suppose that $(A,\mathfrak{m})$ is a local Artinian ring.
If $A$ is Gorenstein, then $A$ admits a dualizing functor
on finite length modules defined by $D(M):= Hom_A(M,A)$ which preserves
lengths. If ...
CommunityBot
- 1
1
vote
0
answers
238
views
relative flatness and torsion freeness
Hi.
Question 1: let $f:X\rightarrow S$ be a proper and surjective morphism of complex reduced spaces with $X$ pure dimensional. Let $F$ be a $S$-flat coherent sheaf on $X$. Is it true that the two ...
- 435
10
votes
1
answer
1k
views
maximal ideals of $k[x_1,x_2,...]$
What can be said about the structure of maximal ideals of $R=k[\{x_i\}_{i \in I}]$, or geometric properties of $\text{Spm } k[\{x_i\}_{i \in I}]$? Here $k$ is an arbitrary field and $I$ is an infinite ...
- 63.1k
3
votes
1
answer
270
views
When is a blow-up a non-trivial product?
Suppose $X$ is an algebraic variety and let $Z \subset X$ be a subvariety. Are there some useful criteria under which the blow-up $Bl_Z X$ becomes a nontrivial product $V \times W$ of the algebraic ...
- 5,137
1
vote
1
answer
9k
views
what is the difference between the revised simplex method andthe full tableu?
No to sound naive but they look like they include the same steps to me, one's just the algorithmical representation of the other. Thanks in advance.
- 3,283
4
votes
1
answer
463
views
Reference request, direct summand conjecture in dimension 2
What's the easiest (by which I mean uses the least fancy machinery) proof of the direct summand conjecture in dimension 2?
Recall that the direct summand conjecture says that:
Conjecture (Hochster): ...
- 20.5k
2
votes
1
answer
341
views
A weaker form of Zariski's connectedness principle
Let $A$ be a complete regular local noetherian ring of dimension $d>1$ and $B$ an $A$-algebra, finite and free as $A$-module. Assume moreover that there exists an open subset $U$ of $\textrm{Spec}\ ...
CommunityBot
- 1
4
votes
1
answer
2k
views
How to find which subset of bitfields xor to another bitfield?
I have a somewhat coding-oriented problem. I have a bunch of bitfields and would like to calculate what subset of them to xor together to achieve a certain other bitfield, or if there isn't a way to ...
CommunityBot
- 1
2
votes
2
answers
827
views
Reduced varieties with no regular points?
Let $k$ be a field. Let $X$ be a reduced $k$-scheme of finite type. If $X$ is geometrically reduced, then it is a basic result that $X$ has a regular point (i.e. the local ring at that point is ...
- 1,673
6
votes
1
answer
434
views
When are two ideals in a regular local ring generated by a regular sequence?
Hello!
Let $R$ be a regular local ring, and let $I,J\subset R$ be ideals. I'd like to understand the "meaning" of the existence of a regular sequence $(x_1,...,x_n)$ in $R$ such that $I$ is generated ...
- 30.5k
1
vote
1
answer
221
views
1
vote
1
answer
815
views
Can we characterise affine open subschemes of ${\rm Spec}(A)$?
Let $A$ be any ring, commutative with identity, and let $I\subset A$ be an ideal $\neq A$. Let $U\subset{\rm Spec}(A)$ be the open subscheme obtained by "removing" the closed set $V(I)$ of all the ...
- 30.5k
3
votes
7
answers
4k
views
How to tell if two random polynomials are identical
Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...
- 32.1k
4
votes
1
answer
5k
views
Localization of a polynomial ring at a prime ideal.
If $R=\mathbb{C}[x,y]$ is the polynomial ring in two variables $x$ and $y$ then we know that the localization of R at the multiplicative set $S=[1,x,x^2,x^3,...]$ is given by $R_x=\mathbb{C}[x,x^{-1},...
- 63.1k
7
votes
3
answers
2k
views
Is there a field which is the union of finitely many proper subfields?
Is there a field which is the union of finitely many proper subfields?
- 39.9k
2
votes
1
answer
326
views
Flatness on the fiber
Hi.
Let $f:A\rightarrow B$ be a local morphism of locally noetherian (reduced) rings with $B$ $A$-flat. Let $M$ be an $B$-module of finite type.
Question: Which conditions ensure the following:
$N\...
- 30.5k
6
votes
0
answers
267
views
Is there a straightforward way to solve unmixed, homogeneous systems of polynomials?
I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry.
Suppose we have a system of $k\leq n$ polynomials in $\...
5
votes
1
answer
2k
views
Length of a module over different rings
Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$.
Under what ...
- 5,407