All Questions
6,056 questions
3
votes
2
answers
382
views
Localization at multivariate monic polynomials
Let $R$ be a ring and consider a monomial order $<$ on $R[X_1,\ldots,X_n]$. A nonzero polynomial $f \in R[X_1,\ldots,X_n]$ is said to be monic if its leading coefficient with respect to $<$ is $...
- 33.8k
6
votes
1
answer
430
views
Splitting a nilpotent into square-zeros by ring extension
Let $R$ be a commutative ring. It is well-known that if $b \in R$ and $c \in R$ are two nilpotent elements with $b^k = 0$ and $c^\ell = 0$ (where $k$ and $\ell$ are positive integers), then $b+c$ is ...
- 33.8k
7
votes
1
answer
271
views
Algebraic proof that the monoid ring of a torsion-free monoid is reduced
In what follows, I say that a monoid $M$ is torsion-free if the $n$-th power map is injective for all $n \geq 1$. I have a proof of the following result:
Claim: if $M$ is a torsion-free commutative ...
- 15.5k
1
vote
0
answers
104
views
Intuitively, what makes Bernoulli umbra so similar to the zero divisors in split-complex numbers?
Notation. Here I will denote Bernoulli umbra (its moments are Bernoulli numbers $B_n$) as $B_-$, $B_-+1$ as $B_+$ (an umbra with moments being Bernoulli numbers except $B_1=1/2$).
I will denote the ...
- 10.1k
10
votes
1
answer
265
views
Finite coverings by closed subschemes
Let $X$ be a scheme. Assume we have two closed subschemes $Y_1$, $Y_2$ that cover $X$ set-theoretically. Are there closed subschemes $Y'_1$, $Y'_2$ with the same underlying sets, such that the ...
- 2,665
1
vote
1
answer
152
views
Cohen-Macaulay quotient ring and symbolic power
Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{a} \subset R$ be an ideal. Let
$$ \mathfrak{b} = \bigcap \{R \cap \mathfrak{a} \cdot R_\mathfrak{p} \text{ } \colon \mathfrak{p} \in \...
- 878
1
vote
0
answers
87
views
When is the product of two elements in algebraic closures of rational functions a constant function?
I have one question on some interactions between sum and product of elements in algebraic clsoures of rational polynomials over algebraically closed fields.
My question is as follows:
Let E and F be ...
- 11
2
votes
2
answers
134
views
On a generating set of numerical semigroups of multiplicity three
Let $S$ be a numerical semigroup. Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the Frobenius number of $S$; let $g(S)=|\mathbb ...
- 785
2
votes
0
answers
161
views
A note on perfect ideals
I am reading an article that cites this note:
E.S. Golod, A note on perfect ideals, in: A.I. Kostrikin (Ed.), Algebra Collection, Nauka, Moscow, 1980,
pp. 37–39.
E. S. Golod, A note on perfect ideals, ...
- 43
1
vote
1
answer
119
views
Injective resolutions of the module of Kähler differentials
Let $k$ be a field, $A=k[x_1,\dots,x_n]/I$ an affine algebra and $\Omega_{A|k}$ the $A$-module of Kähler differentials.
By abstract nonsense there exists an injective resolution $\mathcal I$ of $\...
CommunityBot
- 1
6
votes
2
answers
329
views
Algebraic numbers which prescribed degree which does not belong to some fields
In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...
- 20.4k
2
votes
0
answers
189
views
Root systems and subroot systems
Given the root system $E_{6}$ with basis $\alpha_{1},\dotsc,\alpha_{6}$. How would I find all subroot systems of $E_{6}$ (up to Weyl equivalence) where I can write the basis of each subroot system in ...
- 21
2
votes
0
answers
65
views
Decidability of the solvability of quadratic systems
Let a finite collection of (complex) unknowns $\{x_1,\ldots,x_n\}$ be given, as well as an affine system $AX=B$ in the quadratic variables $X:=[x_i x_j : i\leq j]$, with entries in a computable ...
- 5,428
4
votes
0
answers
522
views
Equivariant sheaves over affine schemes
Let $k$ be a field, let $G$ be a linear algebraic group over $k$ and
let $A$ be a commutative $k$-algebra which is acted on by $G$.
We say that an $A$-module $M$ is a $(G,A)$-module if it satisfies ...
22
votes
6
answers
6k
views
When is a blow-up non-singular?
Suppose that $X$ is a non-singular variety and $Z \subset X$ is a closed subscheme. When is the
blow-up $\operatorname{Bl}_{Z}(X)$ non-singular?
The blow-up of a non-singular variety along a non-...
- 4,713
12
votes
1
answer
656
views
Does every map from a noetherian ring to a valuation ring factor through a DVR?
Let R be a noetherian ring and V a valuation ring with maximal ideal $\mathfrak{m}_V$. Does every morphism of rings $\varphi: R \rightarrow V$ factor through a discrete valuation ring?
One may ...
- 7,893
1
vote
1
answer
149
views
Cohen-Macaulyness of Milnor algebra
Denote by $R = \mathbb{C}\{x_1, \dots, x_n\}$ the ring of germs of analytics maps at the origin in $n$ variables and let $f \in R$ such that $Sing(V(f))=V(x_1, \dots, x_{n-1})$ as sets. In addition, ...
CommunityBot
- 1
0
votes
1
answer
63
views
Changing base field for sum of polynomials
Let $L/\mathbb{Q}$ be a finite extension and $f_{1},\dotsc,f_{n}\in L[x_{1},\dotsc,x_{k}]$ be degree $d$ homogeneous polynomials. Is there a way to find homogenous degree $d’$ polynomials $g_{1},\...
- 109k
11
votes
3
answers
3k
views
For which fields K is every subring of K…?
This question was inspired by
How to prove that the subrings of the rational numbers are noetherian?
which some people found too routine to be of interest. So I have decided to liven things up a bit ...
- 12.9k
0
votes
0
answers
228
views
Generalization of "Lagrange interpolation" over non-division rings
The theorem below is from pages 4 and 5 in Singmaster - On polynomial functions $\pmod m$ (Theorem 10) on polynomials in $\mathbb{Z}_m[x]$.
Let $f$ be a polynomial function $\pmod{m}$. Then $f$ has a ...
- 4,713
6
votes
4
answers
961
views
Does every projective A/I-module come from A?
Let $A$ be a Noetherian commutative ring and $I$ an ideal in $A$. It is pretty much trivial to see that every free $A/I$-Module is obtained from a free $A$-module by tensoring over $A$ with $A/I$: ...
- 12.9k
7
votes
2
answers
637
views
An algebraic proof of Mumford's smoothness criterion for surfaces?
(Disclaimer: I'm a beginner in this area, so welcome corrections.)
Let $(X,x)$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $x$ an isolated ...
- 1,446
8
votes
3
answers
2k
views
When does the group of invertible ideal quotients = the free abelian group on the prime ideals?
I haven't learned that much about primary decomposition, but from I understand about Dedekind domains, we have that all fractional ideals are invertible and all (plain old) ideals factor uniquely into ...
- 4,713
0
votes
1
answer
181
views
Factor $\sum_{n=1}^{N} x^n$ [closed]
I am attempting to factor an $N^{\text{th}}$ degree polynomial with coefficients strictly equal to $1$ given by the equation
$$\sum_{n=1}^{N} x^n$$
Although the Galois group for anything beyond a ...
- 105k
21
votes
2
answers
1k
views
What properties define open loci in excellent schemes?
Let $R$ be an excellent Noetherian ring. A property $P$ is said to be open if the set $\{q \in \operatorname{Spec}(R) \ | \ R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open ...
- 2,821
1
vote
0
answers
470
views
Source for conjectures in commutative algebra
Do you know some books/survey papers/ websites on conjectures or open problems in commutative ring theory? I want to see if
there are very famous open problems or conjectures in commutative ring ...
- 33
1
vote
0
answers
113
views
Cohomological dimension and height of ideals
Let $I$ be an ideal in a Noetherian ring $R$. We define the cohomological dimension of $I$ to be $\operatorname{cd}(I)=\operatorname{sup}\{i\in \mathbb N:\operatorname{H}_I^i(R)\neq0\}$ and it is ...
- 33
9
votes
0
answers
267
views
Can "$\exists\mathcal{X}(R\cong C(\mathcal{X}))$" be expressed in "large" infinitary second-order logic?
Originally asked and bountied at MSE without success:
Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ...
CommunityBot
- 1
1
vote
0
answers
86
views
Buchsbaum-Eisenbud-Horrocks conjecture for finitely generated modules
Someone know a version for conjecture of Buchsbaum-Eisenbud-Horrocks whithout the assumptions that
the $R$-module $M$ has length finite?
1
vote
0
answers
134
views
A question about minimal system of generators and regular sequences
Let $(R,\mathfrak{m},k)$ a Noetherian local ring and $I$ an ideal. Suppose that:
$\mu(I)=\operatorname{grade}(I,R)+1$ and $\operatorname{pd}_R(R/I)=\operatorname{grade}(I,R)$. (Some people says $I$ is ...
- 33
5
votes
1
answer
317
views
Localization of a ring and the Hom functor
Let $R=\mathbb{Z}[x,x^{-1}]$ be the ring of Laurent polynomials in $x$, $\mathfrak{p}=(1-x)$ be an ideal in $R$ and $R_\mathfrak{p}$ be the localization. I want to know what $\text{Hom}_R(R_\mathfrak{...
- 6,262
1
vote
0
answers
124
views
Some properties for height 1 prime ideals in the local ring
Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Let $R=\mathbb{K}[x_0,x_1,\dotsc,x_n]/I$ be the coordinate ring of an affine variety/projective variety. Also, assume that $I$ ...
- 839
12
votes
2
answers
658
views
Maps between K-groups induced by rings homomorphism
Let $f: R\to S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free ...
- 2,821
1
vote
1
answer
186
views
On the trivialization of the sheaf of kahler differentials on the G-invariant topology
Let $X$ be a connected, smooth affine algebraic variety over an algebraically closed field $K$ of characteristic zero. Assume we have a finite group $G$ acting on $X$ by morphisms of $K$-schemes. ...
- 44.7k
17
votes
4
answers
2k
views
Are units of rings of functions on algebraic varieties finitely generated (mod. constants)?
Hello,
Consider the following question. Let $A$ be a finitely generated reduced algebra over an algebraically closed field $k$. Consider the group of units of $A$, modulo the group $k^*$. Is this ...
- 14.2k
6
votes
1
answer
894
views
Brauer group of rational numbers
Reading about the calculation of the Brauer group of rational numbers, the calculations of the group are extremely lengthy and technical. First of all, it will be very helpful to me if someone can ...
- 50.6k
1
vote
0
answers
74
views
Characterise set of polynomials which are zero over an ideal
This is not a specific question, but rather a question about possible techniques approaching a problem. Although this question came from research, it might not fit this forum; in which case I will ...
- 11
2
votes
0
answers
82
views
Let $R$ be a non-catenary, and $f: R \to S$ be a finite monomorphism. Can $S$ be catenary?
Let $R$ be a commutative ring with identity. Then $R$ is $\textit{catenary}$ if for each pair of prime ideal $p \subsetneq q$, all maximal chains of prime ideals $p = p_0 \subsetneq p_1 \subsetneq \...
- 1,355
19
votes
5
answers
7k
views
When a formal power series is a rational function in disguise
Given a formal power series $f \in k[[X]]$, where $k$ is a commutative field, is there any good way to tell whether or not $f\in k(X)$?
Edit: To clarify, "good way to tell" means "computable ...
- 1,446
4
votes
0
answers
510
views
Does a torsion-free coherent sheaf embed into a locally free sheaf?
Let $ X $ be a Noetherian integral regular scheme and $ \mathcal{F} $ be a torsion-free coherent sheaf. (One definition of torsion-free is that the natural map $ \mathcal{F} \rightarrow \mathcal{F} \...
- 936
4
votes
0
answers
82
views
The index of an order defined by a binary form
In his well-known paper, Nakagawa generalized a construction due to Birch and Merriman to arbitrary binary forms and orders. In particular, his construction gives a canonical algebraic order $\mathcal{...
- 26.9k
2
votes
1
answer
389
views
Relation between free resolutions and minimal free resolutions
Let $(R,\mathfrak{m},k)$ a Noetherian local ring and $M$ a finitely generated $R$-module. In this case, given $F_{\bullet}$ a free resolution of $M$, what is the relation between $F_{\bullet}$ and a ...
- 334
4
votes
1
answer
188
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Faithful module cancellation with maximal ideal
Let $k$ be a field of characteristic $0$ and $R = k[[x_1, \dotsc, x_n]]$. Suppose that $M$ is a faithful, finitely generated $R$-module and $\mathfrak{a} < R$ is an ideal such that $\mathfrak{a} M =...
- 5,039
17
votes
2
answers
2k
views
How much theory works out for "almost commutative" rings?
I've been reading about D-modules, and have seen a proof that D_X, the ring of differential operators on a variety, is "almost commutative", that is, that its associated graded ring is commutative. ...
- 1,446
3
votes
0
answers
235
views
Algebras which admit tensor calculus and (pseudo-)Riemannian geometry
It's an often observed fact that the basic notions of analysis on manifolds and (pseudo-)Riemannian geometry, such as tensors, connections and curvature, can be defined in purely algebraic terms. The ...
- 6,406
10
votes
1
answer
294
views
Rational even polynomials maximally tangent to the unit circle
This question is motivated by College Mathematics Journal problem 1196, proposed by Ferenc Beleznay and Daniel Hwang. My solution to this problem (pre-publication version here) uses Chebyshev ...
- 109k
3
votes
1
answer
201
views
Is this a true weakening of the quasi-coherence property?
Let $R$ be a commutative Noetherian ring, and $X=$Spec$(R)$ the associated affine scheme. Let $F$ be a sheaf of $O_X$-modules. Consider the following condition
(#) For all containments $V \subseteq ...
- 19.6k
15
votes
2
answers
655
views
Indecomposable contracting maps on the integers
$\def\ZZ{\mathbb{Z}}$Call a function $f : \ZZ \to \ZZ$ "contracting" if
$$|f(j) - f(i)| \leq |j-i|$$
for all $i$, $j \in \ZZ$. The contracting functions form a monoid under composition; call ...
- 149k
1
vote
0
answers
332
views
Meaning of "cut out (scheme-theoretically)"
Let $V$ be a projectively normal closed subvariety of some projective space over an algebraically closed field $\mathbb{K}$. Let $R$ be the local ring at the vertex of the affine cone over $V$ ($R$ is ...
- 839
2
votes
0
answers
244
views
Hypermodulus and what mathematical objects have it
When researching divergent integrals, I decided to introduce a concept of "modulus" or "determinant" of divergent integral (and series). Basically, it is the exponent of the real ...
- 10.1k