Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,317
questions
6
votes
1
answer
2k
views
$f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?
Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied:
(1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$.
(2) ...
2
votes
1
answer
606
views
Generators of an ideal of $K[[X_1,X_2,X_3]]$
Let us consider an irreducible polynomial $$f \colon= \alpha_e + \alpha_{e-1}X_1 + ... + \alpha_1X_1^{e-1} + X_1^e \in K[[X_2,X_3]][X_1]$$ and set $$\iota_1 \colon K[[X_2,X_3]] \hookrightarrow K[[X_1,...
1
vote
1
answer
171
views
Morphisms preserving weak normality
I would like to find a class of morphisms for which weakly normality descends. The notion of seminormlaity is very close to the one of weakly normality and for seminormal schemes one has Theorem 5.8 ...
0
votes
1
answer
242
views
A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]
For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define
$A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...
1
vote
1
answer
155
views
An ideal and its J-radical
Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$. Now let $J=\cap_{I\subseteq m\in Max(R)}m$. Set $A:=\{p\in Spec(R): I\subseteq p\}$ and $B:=\{p\in Spec(R): J\subseteq p\}$, where $...
4
votes
0
answers
104
views
Compute the closure of graph of function from complement of hypersurface in $\mathbb{A}^n$
I'm hoping someone can give me some tips to help speed up computation on the following problem:
Suppose I have a map $G=(g_1/f,\dots,g_m/f):\mathbb{A}^n\setminus{V(f)}\to \mathbb{A}^m$. I'm ...
5
votes
2
answers
477
views
Finite maps and jacobian condition
Let $k$ be an algebraically closed field and take $f_{1}, ..., f_{n} \in k[X_{1},..., X_{n}]$ with the jacobian condition: $\det J_{f} = 1$. Let $A:= k[X_{1},...,X_{n}]/(f_{1},...,f_{n})$ and ...
7
votes
0
answers
119
views
Is there a homological interpretation for the cokernel of the kernel of a map between complexes induced by tensor product?
Let $A$ be a free abelian group of rank 2, and let $S = \mathbb{Z}[A]\cong\mathbb{Z}[a_1^{\pm1},a_2^{\pm1}]$ the group algebra for $A$.
Let $t : S\times S\rightarrow S$ be the $S$-module map given by ...
7
votes
1
answer
203
views
Injective indecomposable modules over Laurent polynomial rings
What does the injective envelope of $\mathbb C[x,x^{-1}]/(p(x,x^{-1}))$ as a $\mathbb C[x,x^{-1}]$-module look like where $p(x,x^{-1})$ is an irreducible element? I’m sure this is well known, but ...
1
vote
1
answer
571
views
Sum of initial ideals
Let $S_1=k[x_1,\ldots,x_n]$ and $S_2=k[y_1,\ldots,y_m]$ be two polynomial ringsover a field $k$ and $I\subset S_1$ and $J\subset S_2$ be two ideals. Let $S=k[x_1,\ldots,x_n,y_1,\ldots,y_m].$
Question
...
2
votes
1
answer
129
views
$(x + y + z)....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$
$$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is an nth root of unity.
The ...
0
votes
0
answers
241
views
gcd of polynomial values
Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...
24
votes
3
answers
3k
views
Automorphisms of a weighted projective space
What is the automorphisms group of the weighted projective space $\mathbb{P}(a_{0},...,a_{n})$ ?
Consider the simplest case of a weighted projective plane, take for instance $\mathbb{P}(2,3,4)$; any ...
10
votes
1
answer
815
views
Do commutative rings without unity have the IBN property?
Let $R$ be a commutative rng, i.e. a commutative ring without an identity element.
Does $R$ still have the Invariant Basis Number (IBN) property?
Recall that a ring is said to have the IBN ...
5
votes
0
answers
442
views
Fraction fields of strict henselizations of DVRs
Let $A_{1},A_{2}$ be discrete valuation rings whose fraction fields are isomorphic. Let $A_{i}^{\mathrm{sh}}$ be the strict henselization of $A_{i}$, and let $K_{i}$ be the fraction field of $A_{i}^{\...
14
votes
2
answers
776
views
Can the methods of classical algebraic geometry be made rigorous with a synthetic approach?
There are approaches to real analysis that use an axiomatization of nilpotent infinitesimals to enable rigorous synthetic reasoning about infinitesimals, which is arguably closer to the reasoning ...
2
votes
1
answer
138
views
Steinberg components of local deformation rings
Let $F=\mathbb Q_l$ and $E$ be a finite extension of $\mathbb Q_p$ with residue field $k \cong O_E/m_E$ ($p \neq l$). Let $\bar r: \Gamma_F=Gal(\bar F/F) \to GL_2(k)$ be a continuous representation. ...
8
votes
1
answer
1k
views
Is the sheaf of smooth functions flat?
Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?
18
votes
1
answer
4k
views
How should I think about the module of coinvariants of a $G$-module?
Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$.
...
2
votes
0
answers
161
views
Prime ideals being finitely-generated implies coherence?
Let $R$ be a non-noetherian local domain. Suppose that the following two conditions hold for $R$$\colon$
$(*)$$~\quad$An arbitrary prime ideal ${\frak P}$ of $R$ such that ${\mathrm{ht}}({\frak P}) &...
14
votes
1
answer
2k
views
How to visualize the Frobenius endomorphism?
As the question title asks for, how do others "visualize" the Frobenius endomorphism? I asked some people in real life and they said they didn't know and that I could go and ask on MO and possibly get ...
1
vote
0
answers
70
views
Persistence of saturation closure
Let $R$ be a ring. A closure operation $cl$ on a set of ideals $\mathcal{I}$ of
$R$ is a set map $cl: \mathcal{I}\longrightarrow \mathcal{I}( I\mapsto I^{cl})$ satisfying the following conditions:
(i)...
5
votes
0
answers
138
views
Cohen-Macaulay local ring and its quotient by minimal prime ideal
Assume $R$ is a Noetherian local ring which is Cohen-Macaulay, must there exist an ideal $I$ of $R$ such its radical is a minimal prime and the quotient ring $R/I$ is still Cohen-Macaulay? If the ...
3
votes
0
answers
119
views
Can one characterize power series that have polynomial inverses?
Going through my Commutative Algebra notes I found out that I don't know the answer to this question.
Let $A$ be a commutative ring with unit and let $f(T):=\sum_{k=0}^\infty a_k T^k \in A[[T]]$ be a ...
0
votes
1
answer
249
views
Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals
If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
2
votes
0
answers
166
views
Solving solutions to systems of polynomial equations over $\mathbb Z$
Macaulay Resultants help identify common root in $\mathbb P^{n-1}(\mathbb K)$ of $n$ homogeneous polynomials in $n$ variables when $\mathbb K$ is algebraically closed. Is it possible that some type of ...
0
votes
0
answers
300
views
Krull dimensions and regular sequences
I am trying to understand if a certain condition on quotient rings is sufficient for a sequence to be regular. Here is the setting:
Let $\mathbb{C}[u_1,...,u_n]$ be the ring of regular functions on $\...
6
votes
0
answers
344
views
Ideals of orders in number fields
Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{O}$ be an order of $K$ and $A$ a proper ideal of $\mathcal{O}$, by which I mean that $\mathcal{O} = \{\alpha \in K : \...
2
votes
1
answer
147
views
Transitivity of an invariant of finitely generated field extensions
For a finitely generated extension of fields $K/k$, let us define "$S_{K/k}$" to be the minimum of the degrees $[K:\ell]$ where $\ell/k$ ranges over the purely transcendental subextensions of $K$ with ...
7
votes
1
answer
368
views
An infinite dimensional local domain whose chains of primes are finite
Does there exist a local domain of infinite dimension in which every chain of prime ideals is finite?
Of course, such a ring must be neither noetherian nor catenary.
(This question arose while ...
1
vote
2
answers
299
views
Noetherian module, over Noetherian ring, which is isomorphic to its double dual [duplicate]
Let $M$ be a finitely generated module over a Noetherian ring $R$ such that $M$ is isomorphic with its double dual $M^{**}=Hom(Hom(M,R),R)$.
Then is the natural map $j:M \to M^{**}$ defined as $j(m)(...
8
votes
3
answers
504
views
How to use Hilbert series to count combinatorial objects?
In THE SLOPES DETERMINED BY n POINTS IN THE PLANE by JEREMY L. MARTIN, Page 2, Theorem 1.1, a Hilbert series is used to compute some combinatorial objects:
Let $R_n=k[m_{1,2}, \ldots, m_{n-1,n}]$, ...
1
vote
0
answers
84
views
A special family of prime ideals
I am looking for a commutative ring $R$ with identity which has a family $F:=\{p_i\}_{i\in A}$ of prime ideals such that for each $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ with $n\leqslant m$ ...
0
votes
1
answer
249
views
When does a subspace of the affine space form a regular sequence in a ring of regular functions?
Let $J$ be an ideal in the ring $\mathbb{C}[u] := \mathbb{C}[u_1,...,u_n]$ of regular functions on $\mathbb{C}^n$.
Given a subspace $\mathbb{C}^k \subset \mathbb{C}^n$ and $I$ its ideal in $\mathbb{...
2
votes
1
answer
410
views
Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$?
In [1], Propisition 6.1.9(2), it said that if $R$ is a perfectoid ring such that $pR^\circ$ is closed in $R^\circ$ (this includes the case if $R$ is of character $p$, or if $p$ is invertible in $R$, ...
15
votes
0
answers
670
views
If a polynomial ring is a finite flat module over some subring, is that subring itself a polynomial ring?
A question motivated by If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring? and If a polynomial ring is finite free over a subring, is the subring ...
1
vote
1
answer
62
views
Decomposing semihereditary rings
Let $R$ be a commutative semihereditary ring (i.e. every finitely generated ideal of $R$ is projective https://en.wikipedia.org/wiki/Hereditary_ring). Then is $R$ a finite direct product of Prufer ...
1
vote
1
answer
134
views
Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains)
Let $R$ be a commutative Noetherian hereditary ring (https://en.wikipedia.org/wiki/Hereditary_ring) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?
19
votes
2
answers
531
views
Ostrowski's Theorem for topological rings?
Ostrowski's theorem classifies all absolute values on a number field $K$.
Questions:
More generally, can one classify all Hausdorff topologies on $K$ making $K$ into a topological field?
In ...
4
votes
1
answer
126
views
On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion
Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
1
vote
0
answers
50
views
What is the ring of functions on the open unit disc with polynomially bounded Maclaurin coefficients called?
Let $R$ be the the set of complex-valued (analytic) functions $f$ on the open unit disc $\mathrm D:=\{z\in\Bbb C:|z|<1\}$ for which there exist constants $a_0$, $a_1$, ... in $\Bbb C$ and $n$ in $\...
1
vote
0
answers
41
views
Numerator-cancellable Modules
I don't know why there is no investigation of the cancellability of quotients in category of modules. What I mean by cancellability of quotients in category of modules is the following :
Let $R$ be a ...
4
votes
1
answer
269
views
Resultants for compactly represented product form polynomials?
Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the ...
1
vote
0
answers
138
views
On finite dimensional commutative algebras and regular sequences
Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...
3
votes
1
answer
910
views
Local ring of infinite dimension
Short version: Let $R$ be a commutative ring such that all chains of primes of $R$ with the same extremities have the same finite cardinality. Is $R$ locally finite-dimensional?
Longer version: Let $R$...
3
votes
0
answers
197
views
derived symmetric powers of an ideal
Let $R$ be the polynomial algebra in $n$ variables over a field $F$ of characteristic $0$. Let $m$ be the ideal of the origin: $m=(x_1,...,x_n)$.
We have a canonical map $Lsym^k(m)\to m^k$ from the ...
6
votes
0
answers
202
views
Counting exceptional divisors
Suppose that I blow up an ideal sheaf $J$ in $\mathbb A^2$ via a map $\pi : X \to \mathbb A^2$. I'd like to compute, from the ideal, how many exceptional divisors there are for $\pi$, and be able to ...
1
vote
0
answers
74
views
On some finiteness properties of cohomological algebras of complex tori
Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module.
...
5
votes
0
answers
146
views
Descending chain of subalgebras of $k[x,y]$
Let $k$ be a field of characteristic zero.
Let $\{R_i\}_{i \in \mathbb{N}}$ be a descending chain of $k$-subalgebras of $k[x,y]$: $k[x,y]=:R_0 \supseteq R_1 \supseteq R_2 \supseteq \ldots$,
such that ...
2
votes
1
answer
212
views
Homomorphisms of ring extending nicely ideal intersections
Let $\varphi\!:\!S\to R$ be a homomorphisms of $K$-algebras for some field $K$.
Let $\{a_{\lambda}\}_{\lambda}$ be a family of ideals of $S$.
Is there some "natural" assumption on $\varphi$ to ...