All Questions
6,056 questions
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310
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Primes of the power series rings
Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection
\begin{equation*}
\psi_{n,n-1} \colon A_n \...
- 563
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0
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195
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Algebraic generalization of Pascal's identity
Let $R$, $S$ be rings with identity. A map $f: R \times R \to S$ is said to be an a $R_S$-Pascal map if, for all $r_1, r_2 \in R$, the following relations are satisfied :
$$\begin{align*}
f(r_1-1_G, &...
- 1,549
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0
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399
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Infinite-dimensional representation theory of $K[x]$
Let $K$ be an algebraically closed field. The finite-dimensional representation theory of the polynomial algebra $K[x]$ is tame and completely understood, which I shall first summarise. It's ...
- 482
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94
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Fine tuning the growth rate of the degrees of polynomials
Let $r$ be an integer with $r>1$. Suppose that if $k\geq 0$, then $p_{k}(x)$ is a polynomial with nonnegative integer coefficients with $p_{k}(0)=1$ but where $p_{k}\neq 1$.
Suppose that
$$\...
1
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0
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55
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Schemes for conditional distributions (monads)
(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.)
Suppose you have a monad ...
- 121
1
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240
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Cohomology of a chain complex over a polynomial ring
I asked this on SE but I did not get any answer; I got some progress but I hope here I can find some help to finish the problem out.
Let $R = F[x_1, \ldots, x_n]$ be a polynomial ring over a field $...
- 81
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111
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When a semigroup ideal is a determinantal ideal?
Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \...
- 113
1
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120
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Strongly isovariant (aka fixed point reflecting / stabiliser preserving) morphisms
I have some questions about what feels like basic topics in quotients of schemes by group actions. Consequently I suspect there are well-known references; I couldn't find them, though.
First ...
- 1,961
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739
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Completion of localization of completion
Let $(A,m)$ be a noetherian local ring,
and let $p \subseteq A$ be a prime ideal.
From this data, we can construct two rings:
1. We may localize $A$ at $p$, and then complete,
obtaining the $pA_p$-...
- 11
1
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54
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Primary decomposition with parameters
$\newcommand\QQ{\mathbb{Q}}$
Considering the polynomial
$$f = x^2 - a y$$
one notes, that it is irreducible in $\QQ[x,y]$ for all $a \neq 0 \in \QQ$ and factors for $a = 0$.
More generally, let $A=...
- 708
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194
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Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index
Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
- 1,209
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112
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Notions of connected components in a finite family fibration
Let $\Pi_0:\mathsf{FinFam}(\mathsf C)\to \mathsf{FinSet}$ be the fibration exhibiting the free finite coproduct completion of $\mathsf C$. Suppose $\mathsf C$ has finite limits so that the extensive $\...
- 10.5k
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78
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Relation between lifts of simple roots and lifts of idempotents (Henselian property)
Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....
- 10.5k
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122
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Module of Kahler differentials for manifolds [duplicate]
Let $A$ be a $k$-algebra and let $\mathcal{M}_A$ be the set of all $A$-modules. In $\mathcal{M}_A$, there exists a universal object $\Omega_{A/k}$, called the module of Kahler differentials, and a $k$...
- 111
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65
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What's the probability a random module element has prime annihilator?
I'm going to pose two versions of my question---an ill-defined version and a well-defined version.
Ill-Defined Question (IDQ). Let $R$ be a (commutative, unital) Noetherian ring, $M$ a finitely ...
- 3,079
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207
views
Behavior of regularity under base change
Let $(R,\mathfrak{m})$ be a noetherian local domain. Let $(A,\mathfrak{n})$ be a regular local ring contains $R$ such that $\mathfrak{n}\cap R=\mathfrak{m}$ and $A$ is essential finitely generated as ...
- 564
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136
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Generating the annihilator ideal up to finite index
Let $\Lambda_2 = \mathbb{Z}_p[[T_1,T_2]]$ be the power series ring over $\mathbb{Z}_p$ in two variables, i.e., $\Lambda_2$ is a regular local ring of dimension 3. Let $M$ be the quotient of an ...
- 11
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180
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Skew-symmetric multi-derivations of $k[x_1,…,x_n]/I$
Let $I = \langle f_1, \ldots f_r \rangle$ be an ideal in $R=k[x_1,\ldots,x_n]$ where $k$ is a field, and put $A = R/I$.
(If $I$ is prime then $A$ is the coordinate ring of an irreducible affine ...
- 832
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290
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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?...
user47305
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70
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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)...
- 383
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0
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85
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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 $n\leqslant m$ ...
- 11
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51
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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 $\...
- 2,437
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43
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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 ...
- 933
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140
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 ...
- 1,227
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74
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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.
...
- 1,227
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360
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Intersection condition for polynomial ring and maximal ideals
In ring theory, there is interest in a condition known as the intersection condition. There is a brief comment in McConnell-Robson along these lines: Consider the ring $R = k[x,y]$ where $k$ is a ...
- 595
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116
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A generalized Cauchy type functional equation
Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$.
Then is it true that $f(x+y)=f(x)...
- 1,209
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1k
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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 ...
- 180
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140
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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 $A_{\mathbb{Z}}$ (...
- 219
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137
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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 $k[f,g] \neq k[t]$, $...
- 2,837
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67
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Classification of Maximal Rank Skew-Symmetric Matrices with Linear forms as entries
I am interested in canonical forms/simplifying assumptions of $(2n+1)\times (2n+1)$ skew symmetric matrices with homogeneous degree 1 polynomials in $k[x,y,z]$ as entries, and whose rank is $2n$.
I ...
- 553
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99
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Special formal lifts of smooth algebras
Let $A$ be a smooth algebra over $k$ a finite field.
Say $B$ is a $p$-adically complete smooth algebra over the Witt ring $W(k)$, lifting $A$.
Assume $B$ is of the form $W(k)\langle t_1,\ldots, t_n\...
user124171
1
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0
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80
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A question on f.g. ideals of $\textrm{Hol}(\mathbb{C}^2,\mathbb{C})$
Suppose that $I$ and $J$ are finitely generated ideals of the ring $\textrm{Hol}(\mathbb{C}^2,\mathbb{C})$ of all entire functions in two complex variables.
Then is $I\cap J$ finitely generated too?
- 23
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25
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computing quotient ideals efficiently over a polynomial ring: (I:J) when J has many generators?
Can someone guide me to a reference where an algorithm for computing I:J where
J has many generators is discussed? I know the method of using one generator at a time and then taking intersections. I ...
- 11
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0
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189
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Bijective correspondence between $\mathbb G_a$ actions on affine varieties and exponential maps on affine $k$-domains
Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[...
user111524
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0
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153
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Factorially closed, finitely generated $k$-sub-algebra of $k[X_1,X_2,X_3]$ , where $k$ is algebraically closed field of positive characteristic
Let $S$ be a sub-ring of a commutative ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S\setminus \{0\} \implies a,b \in S$.
Let $k$ be an algebraically ...
user111524
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0
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299
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If $R$ is UFD , then does $R \cong R[X,Y]$ imply $R \cong R[X]$?
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$ shows that it is possible to have commutative ring $R$ with unity such that $R \cong R[X,Y]$ but $R \ncong R[X]$.
My questions are: Is it possible ...
user111524
1
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0
answers
607
views
Push-forward along closed immersion
Let $X$ be a scheme, $p : Z\to X$ a closed immersion, $\mathcal{F}$ a locally free sheaf of modules on $Z$ of finite rank.
Assume both $\mathcal{O}_Z$ and $\mathcal{O}_X$ are coherent sheaves of $\...
user120812
1
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0
answers
103
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Local quotient of arithmetical ring
A commutative ring $R$ with $1$ is said to be arithmetical if for all ideals $a , b$ of $R$, $ a ∩ ( b + c ) = ( a ∩ b ) + ( a ∩ c )$ or the localization $R_m$ is a uniserial ...
- 21
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82
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What is known about the cohomology of the matrix monoid?
When I say the cohomology of a monoid, I mean that of its classifying space (considering the monoid as a category with a single object).
Let $M_n(R)$ be the monoid of matrices with matrix ...
- 1,726
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397
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A functor on the category of commutative rings, algebras or Banach algebras
Edit: According to the comments of abx and Yemon Choi I revise the question as follows:
Let $G$ be a group and $\mathcal{A_G}$ be the category of $G$-module commutative algebras, that is the ...
- 356
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69
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Information on structure (CI-magma with (non surjective)homorphism) of chemical transformations
Thinking about the mathematical structure of chemical transformations, between all possible components (educts, products) it occurs to me, that this structure is a commutative-idempotent groupoid(=...
1
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0
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40
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Hilbert functions of graded modules generated by mapped generators
I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help ...
- 11
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0
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75
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Formula for the index of regularity of a generic Hilbert function
Is there an explicit formula for the index of regularity of a generic Hilbert function in two variables? (i.e., the Hilbert function of an ideal of $k[X,Y]$ generated by $r$ generic forms $f_{i}$ of ...
- 11
1
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0
answers
121
views
Radial similarity of Newton polytopes
Let $k$ be a field of characteristic zero, and assume that $p,q \in k[x,y]$ is a Jacobian pair, namely, $p_xq_y-p_yq_x \in k^*$
(= the determinant of the Jacobi matrix $\in k^*$).
It is known that ...
- 2,837
1
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0
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117
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Catenary Noetherian commutative rings
A commutative ring $R$ is catenary if for any pair of prime ideals $p\subset q$, any two strictly increasing chain $p=p_0 ⊂p_1 ... ⊂p_n= q$ of prime ideals
are contained in maximal strictly ...
- 19
1
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0
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75
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partially commutative like monoids [duplicate]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ (empty word) whenever $\{a,b\}...
- 1,269
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0
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256
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partially commutative monoid [closed]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$...
- 1,269
1
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0
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35
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Grobner basis of the toric ideal $I_{A_P}$ with respect to $<_{rev}$ consists of those binomials $t_αt_β − t_{α\cap β} t_{α\cup β}$
I try to understand the proof of the Theorem. 10.1.3.(page 185) from ''Monomial Ideals'' by Herzog & Hibi.
The reduced Grobner basis of the toric ideal $I_{A_P}$ with respect
to $<_{rev}$ ...
- 163
1
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0
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255
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Presentation of amalgamated sum as a quotient of the direct sum
I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf).
I'm trying to understand why the amalgamated sum of ...
- 65