Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
12
votes
1answer
238 views
Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring?
I asked this question on Mathematics Stackexchange (link), but got no answer.
Let $K$ be a field, let $x_1,x_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x_1,x_2,\dots]]$.
Recall ...
2
votes
1answer
67 views
Ideal of the union of two zero loci
Let $X$ be a smooth (complex) projective variety and $\mathcal E$ a globally generated vector bundle on $X$ of rank $< dim(X)$. I would like to know, please, if there is a nice description (exact ...
2
votes
1answer
109 views
Finite maps and jacobian condition
I have asked this question in math.stackexchange but I did not have answers:
Let $k$ be an algebraically closed field and take $f_{1}, ..., f_{n} \in k[X_{1},..., X_{n}]$ with the jacobian condition: ...
5
votes
1answer
276 views
The soccer splitting problem in arbitrary commutative ring
There's a folklore problem:
Let $x_1, \cdots, x_{23} \in \mathbb{Z}$ be the weights of $23$ soccer players. Now Master Yoda want's to form two soccer teams with $11$ players each. Turns out for ...
5
votes
0answers
124 views
Reference request: A commutative variant of the Exterior Algebra
Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through
$$
p(y) = ...
0
votes
0answers
88 views
About the multiplicative group of p-adic complex
I was studying the multiplicative group of the $\mathbb{C}_p$. I'm interesed in the ring $\mathcal{O}_p$ of elements in $x\in\mathbb{C}_p$ such that $|x|_p\geq 1$. I have three questions. The first ...
7
votes
3answers
245 views
Infinite Galois descent for finitely generated commutative algebras over a field
Let $k_0$ be a field of characteristic 0, and let $k$ is a fixed algebraic closure of $k_0$.
Write $G={\rm Gal}(k/k_0)$.
Let $A_0$ be a finitely generated commutative $k_0$-algebra with a unit.
Then ...
0
votes
0answers
42 views
Can one polarize multihomogeneous polynomials?
Let for each $1 \leq i \leq l$ $V_i$ be a finite dimensional $k$-space, and
\begin{equation}
f : \times_{i=1}^l V_i \longrightarrow k
\end{equation}
a multihomogeneous polynomial map of degree $d_i$ ...
0
votes
1answer
127 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 ...
1
vote
1answer
71 views
Finding a characteristic for which the zero-locus of an ideal is not empty
I have a set of polynomials $f_1, \dots, f_m \in \mathbb{Z}[x_1, \dots, x_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x_1, \dots, x_n]$ or $\...
5
votes
1answer
176 views
Minimal resolution of local cohomology module
Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$
Question Can we say anything about Betti numbers ...
0
votes
0answers
49 views
weakly associated prime ideal
let $p$ be a prime ideal of domain $R$ and $pR_p$ be a finitely generated ideal of $R_p$. I want proof that,
If $p$ is weakly associated prime ideal of $R_-$module $M$, then $p$ is associated prime ...
4
votes
0answers
146 views
Can nonflat deformations of singularities always produce Cohen-Macaulay rings?
To make the question in the title precise, let me phrase it like this. Consider a complete local ring
$$ A := \mathbb{C}[[x_1, \dotsc, x_n]]/(f_1, \dotsc, f_m) $$
and, for definiteness, assume that $...
1
vote
2answers
236 views
A relation between an ideal and its radical
Let $I$ be an ideal of a commutative ring $R$ with $1$ such that $\sqrt{I}=I_1\cdots I_n$ where $I_i's$ are pairwise comaximal ideal of $R$. Are there ideals $J_1,...,J_n$ of $R$ such that $V(I_i)=V(...
2
votes
0answers
119 views
Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic
Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor ...
7
votes
0answers
185 views
Embedding a given affine variety as a divisor
Let $A$ be a finite type algebra over $\mathbb{C}$. Does there exist a finite type $\mathbb{C}$-algebra $B$ and a nonzero divisor $b \in B$ such that $B/b \cong A$ and $B[1/b]$ is Cohen-Macaulay (or, ...
8
votes
1answer
625 views
How to visualize the Frobenius endomorphism?
As the question title asks for, how do others "visualize" the Frobenius endomorphism? I asked my advisor and he said he didn't know and that I could go and ask on MO and possibly get miseducated. So ...
2
votes
0answers
22 views
Does the $G$-norm coincide with the ordinary norm for “quasi-$G$-Galois” extensions
Let $S$ be a commutative ring, let $G$ be a finite group acting on $S$ via automorphisms (not necessarily faithfully), and
let $R$ be a subring of $S$ consisting of elements fixed $G$.
The extension $...
3
votes
0answers
69 views
Normal set of points in the plane
When defining the normality of a scheme in the book The Geometry of Syzygies, Eisenbud says that there are just a few facts that are known for their bounds.
Given $X \subset \mathbb{P}^r$, we say ...
0
votes
0answers
155 views
Are the integers a vector space or algebra over “some” field or over “some” ring?
Every vector $v$ in a finite-dimensional vector space space $V$ of dimension $n$ over a field $F$ has a unique representation in terms of a basis ${\frak B} \subseteq V$, where a basis for $V$ is a ...
1
vote
0answers
92 views
Derivations of special rings
Consider $p$-adic field $\mathbb{Q}_p$ and let $A$ be any finite dimensional $\mathbb{Q}_p$ algebra without zero divisors (and maybe without $1$). Now we can look on $A$ as on a ring $A_{\mathbb{Z}}$ (...
3
votes
0answers
139 views
Ideals and Idempotents in a commutative ring
Let $I$, $J$, and $K$ be pairwise comaximal ideals of a commutative ring $R$ with $1$ with the property that if $x^2-x\in I$, then there exists $e^2=e\in J$ such that $x-e\in I$ and if $y^2-y\in K$, ...
5
votes
1answer
145 views
about morphisms of affine formal schemes $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$
It is well known that there is a correspondence between homomorphism of rings $A\to B$ and morphism of affine schemes $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$.
Question: (1) In analogy, is there ...
3
votes
0answers
67 views
Betti numbers of a Cohen-Macaulay Module in small projective dimension
I am trying to compute the Betti numbers of some Stanley-Reisner ring $R_\Delta$, where the underlying complex $\Delta$ is shellable and the projective dimension of the $R_\Delta$ is $3\text{ or }4$. ...
7
votes
2answers
365 views
A geometric proof of Krull's Principal ideal theorem
Krull's height theorem states that in a Noetherian, local ring $(A,\mathfrak m)$, for any $f \in \mathfrak m$, the minimal prime ideal containing $(f)$ is at most height $1$.
This is a very geometric ...
0
votes
0answers
64 views
projective module over inductive limit of rings
Let $R = \varinjlim\limits_{i\in I} R_i$ be a (commutative) ring where the limit is filtered (in fact in my case it is an increasing union of rings) and let $M$ be a finite projective module over $R$.
...
3
votes
1answer
120 views
Solutions to a system of homogeneous equations (inequalities)
Let $f_1,\ldots,f_r \in \mathbb{R}[x_1,\ldots,x_n]$ be $r$ homogeneous polynomials of the same odd degree $d$, where $d \in \{3,5,7,\ldots\}$.
For which values of $r,n,d$ there exists a real ...
6
votes
1answer
147 views
intersection of free/affine submodules, comparison with vector spaces
If $W_1,W_2 \subset V$ are finite-dimensional $k$-vector spaces of dimensions $d_1, d_2 \leq d$, respectively, then $d_1 + d_2 > d$ suffices to guarantee $W_1 \cap W_2 \neq \{0\}$. There are ...
1
vote
1answer
79 views
Is this algorithm for primary decomposition correct?
I've written some code for Sage to compute radical ideals and primary decompositions over $\overline{Q}$ (the field of algebraic numbers), and I'm not sure if it's right.
Since Singular (the ...
-2
votes
2answers
305 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 ...
2
votes
1answer
77 views
Higher degree of Hilbert's irreducibility theorem
A basic form of Hilbert's irreducibility theorem can be formulated as follows:
Let $f(t,x)\in\mathbb{Q}[t,x]\setminus\mathbb{Q}[t]$ be an irreducible polynomial. There exist infinitely many linear ...
3
votes
0answers
48 views
Prime ideal generated by two quadratic polynomials
Let $q_1$ and $q_2$ be two irreducible quadratic homogeneous polynomials in $\mathbb{C}[x_0, \ldots, x_n]$.
Consider the ideal $\langle q_1, q_2 \rangle$.
When this ideal is prime?
I am ...
7
votes
2answers
269 views
Generalized Smith Theorem for the torsion of cokernels
Let $R$ be a (commutative) domain and let $Q$ be its fraction field.
Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$.
Let $I_k=(\det \...
4
votes
0answers
193 views
Is there a converse of Abhyankar-Moh-Suzuki theorem?
The following question is the same as this question; I ask it here, since I have not got any comments there (I really apologize if I should have waited some more time before asking it here; it is just ...
1
vote
0answers
92 views
Hilbert's irreducibility theorem for prime ideals
A typical formulation of Hilbert's irreducibility theorem is like this (see [1]):
Let $k=\mathbb{Q}$ and $f\in k[x_1,\ldots,x_n,y_1,\ldots,y_m]$ be an irreducible polynomial. There exists a Zariski ...
7
votes
1answer
173 views
Is being a Frobenius algebra a rare condition for local algebras?
Let $U_{r,l,q}$ be the set of finite dimensional local algebras $A$ over a finite field with $q$ elements such that $J/J^2$ is $r$-dimensional for a number $r \geq 2$ and such that $J^l=0$ for the ...
9
votes
1answer
295 views
Rings with all non-prime ideals finitely generated
Motivated by this question, I would like to ask:
If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?
Note that ...
4
votes
1answer
148 views
Embedding a finite morphism into a finite morphism of smooth varieties
Let $f\colon X\to Y$ be a finite morphism of quasiprojective varieties (I’m most interested in the case that $f$ is normalization and the varieties are over $\Bbb C$). Is the following statement true (...
6
votes
0answers
274 views
Curious anti-commutative ring
Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context?
By $\Lambda$ I mean the free anti-commutative algebra, $x_i x_j = - x_j x_i$,
either ...
5
votes
1answer
257 views
Does Fermat's last theorem hold in the Grothendieck ring of the ordinals?
Inspired by this question and its answer, I am curious whether or not Fermat's last theorem holds in the Grothendieck ring of the ordinals under Hessenberg (commutative) operations.
The excellent ...
2
votes
0answers
79 views
Intersection of all positive powers of prime ideal in an integral domain with all ideals of finite height
Let $R$ be an integral domain with every prime ideal having finite height. Then is $\bigcap_{n>1} P^n$ a prime ideal of $R$ for every prime ideal $P$ of $R$ ? If that is not true in general, then ...
6
votes
1answer
237 views
Binomial coefficients in discrete valuation rings
Let $V$ be a complete discrete valuation ring whose residue field is a finite field $k=\mathbf{F}_q$. Let $\pi\in V$ be a uniformizer.
For any integer $d,n\ge 0$, define:
$${\pi^d \choose n} := \...
1
vote
1answer
131 views
Coefficients of the monomials appearing in a Schubert polynomial
It is known that the coefficients of the monomials appearing in a Schubert polynomial are always positive. My question is: Is it always true that at least one such coefficient must be $1$? If that is ...
0
votes
0answers
73 views
Kelly's theorem for quadratic polynomials
Let $f_1, \ldots, f_m$ be homogeneous irreducible quadratic polynomials in $\mathbb{C}[x_1, \ldots, x_n]$.
Assume that these polynomials are pairwise coprime.
Denote $P:= f_1 \cdot f_2 \ldots \...
4
votes
1answer
122 views
How to calculate the Chern class of the tensor product of a torsion free sheaf with a line bundle
I'am try to work with Chern class of the coherent sheaves, in this sense. If I have a vector bundle $E$ of rank $r$ and $L$ a line bundle we have the Chern class property
$$c_{r}(E\otimes L) = \sum_{...
2
votes
0answers
108 views
Witt vectors with $p$-torsion
Let $R$ be a commutative ring, $W(R)$ is a ring of Witt vectors over $R$. Can you give an example of a ring $R$ such that $W(R)$ has $p$-torsion?
4
votes
1answer
136 views
Jordan form on an invariant vector subspace
Let $\mathbb{F}$ be a field and $V$ an $\mathbb{F}$-vector space. Let $\operatorname{T}\in\mathrm{End}(V)$ be an $\mathbb{F}$-linear operator. It is well known that if $\dim V<\infty$ then $\...
1
vote
2answers
146 views
A question arising in the distribution theory of L. Schwartz
Let $R$ be the ring of distributions $T\in \mathcal{D}'(\mathbb{R})$ with support in $[0,\infty)$ and with the operations of pointwise addition and multiplication taken as convolution, and $I$ be the ...
1
vote
0answers
83 views
What can be said about $\{\deg(f),\deg(g),\deg(h)\}$, such that $k[f,g,h]=k[t]$?
Let $f=f(t),g=g(t),h=h(t) \in k[t]$, $k$ is a field of characteristic zero.
Denote $\deg(f)=a$, $\deg(g)=b$, $\deg(h)=c$. Assume that $a \geq 2, b \geq 2, c \geq 2$.
Assume that $k[f,g] \neq k[t]$, $...
3
votes
0answers
125 views
A Jacobian pair $(p,q)$ such that $\gcd(\deg(p),\deg(q))=2P$, $P \geq 5$ is prime
Let $p=p(x,y),q=q(x,y) \in \mathbb{C}[x,y]$ be a Jacobian pair, namely,
$p_xq_y-p_yq_x \in \mathbb{C}^*$. Denote $a:= \deg(p)$ and $b:= \deg(q)$,
where $\deg()$ denotes the total degree ($(1,1)$-...
