All Questions
6,053 questions
1
vote
0
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142
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Explicit computation of Čech-cohomology of coherent sheaves on $\mathbb{P}^n_A$
$\newcommand{\proj}[1]{\operatorname{proj}(#1)}
\newcommand{\PSP}{\mathbb{P}}$These days I noticed the following result of (constructive) commutative algebra, which I think is probably well known ...
- 12.9k
3
votes
1
answer
260
views
Cancellation in polynomial composition
Let $k$ be a field. Suppose $P,Q,R\in k[x]$ satisfy $P\circ Q=P\circ R$. What can we conclude about $Q$ and $R$?
It may not be the case that $Q=R$; for example, if $P=x^2$, any polynomials $Q,R$ with $...
- 63.9k
5
votes
2
answers
199
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Determining the multiplication via addition and some unary operation
It is known that the addition operation in a skew-field $F$ (more generally, in a quasifield) is uniquely determined by the multiplication operation and the unary involutive operation $1_{-}:F\to F$, ...
- 14.6k
2
votes
0
answers
75
views
Sum of Betti numbers and certain short exact sequence of modules of finite length over regular local ring
Let $N$ be a module of finite length over a regular local ring $R$ of characteristic $0$. Let $M$ be an $R$-module which fits into a short exact sequence $0\to N^{\oplus a}\to M \to N^{\oplus b}\to 0$ ...
- 480
1
vote
2
answers
831
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Books one can read for 2nd course in Commutative Algebra ( Self Study)
I am a student who has completed master's but couldn't take admission to a PhD program due to some unfortunate reasons.
I have done 1 course in Commutative Algebra where I followed the book " ...
- 7,127
0
votes
0
answers
97
views
Algebraic independence and substitution for quadratics
Let $f_{1},...,f_{n-1} \in \mathbb{F}[x_1,...,x_n]$ such that $\{ f_1,..., f_{n-1},x_n \}$ is algebraically independent over $\mathbb{F}$. Let $G \in \mathbb{F}[x_1,...,x_n,y_1,...,y_{n-1}]\...
- 341
5
votes
2
answers
3k
views
Zariski topology and compact \paracompact space?
Does the Zariski topology on a ring (not commutative in common) form a compact or paracompact space and why?
- 4,713
10
votes
0
answers
248
views
What is the tiling semigroup for an einstein "hat" tiling?
My undergraduate dissertation was on inverse semigroups and the key text I used for it was Lawson's, "Inverse Semigroups: The Theory of Partial Symmetries". In said book, Lawson describes ...
- 2,858
5
votes
0
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288
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Picard group of almost module category
I am very new to the world of almost mathematics and I am curious about the following:
Fix an almost mathematics situation $(R,I)$ throughout. Very generally, the almost module category comes with a ...
- 51
2
votes
1
answer
181
views
Is the derived support $\{x\in X\:|\: \mathsf{L}x^* M\neq 0\}$ closed?
Let $X$ be a scheme and consider an object $M$ of its derived category $\mathsf{D}_\text{qc}(X)$, defined as the full subcategory of $\mathsf{D}(\textsf{Mod}(\mathcal{O}_X))$ consisting of the ...
- 6,829
8
votes
3
answers
691
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Are open $\mathbb{G}_m$-invariant subschemes of an affine scheme precisely the homogeneous radical ideals of its coordinate ring?
This question is certainly not research level and in fact quite elementary which is why I asked it on math.stackexchange before: math.stackexchange. However it doesn't seem to get much attention there ...
- 101
3
votes
0
answers
93
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When can $RHom^\bullet_{A}(B/I^k\overset{L}{\otimes}_B A, A)$ be computed using formal completions?
Let $\varphi:B\to A$ be a ring homomorphism between Noetherian rings. Let $I\subset B$ be an ideal. Let $B^{\wedge}=\varprojlim_n B/I^n$ be the $I$-adic completion of $B$, and $A^{\wedge}=\...
- 367
20
votes
5
answers
2k
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Constructively, is the unit of the “free abelian group” monad on sets injective?
Classically, we can explicitly construct the free Abelian group $\newcommand{\Z}{\mathbb{Z}}\Z[X]$ on a set $X$ as the set of finitely-supported functions $X \to \Z$, and so easily see that the unit ...
- 426
2
votes
0
answers
117
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A very specific quotient of a determinantal variety
I'm interesting in knowing whether a certain variety defined by maximal minors is irreducible. The specific construction is as follows: let $n \geq 2$ and let $R = \mathbb{C}[a_1,b_1,c_1,d_1,e_1,f_1,...
- 185
1
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0
answers
37
views
Bounding the length of an R-module of matrices
Loosely related to this: Bounding the length in a module of evaluated skew polynomials
Let $C$ be an $\mathbb{F}_q$-vector subspace of $m \times n$ matrices over $\mathbb{F}_q$. Assume WLOG that $m \...
- 63.9k
1
vote
0
answers
120
views
Normalization and ordinary double points
Let $X$ and $Y$ be two integral projective complex varieties and $f:X\to Y$ be a finite morphism. I assume that
(1) $X$ is smooth,
(2) $f$ is the normalization morphism of $Y$, and
(3) each fiber of $...
- 389
1
vote
2
answers
164
views
General and translational Birkhoff lattices. Equational classes
By lattice I'll mean Birkhoff lattice.
The two classical equational classes of lattices are modular lattices and distributive lattices. The old problem used to be:
Is there an equational class ...
- 14.6k
0
votes
0
answers
113
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Relation between minimality and algebraic independence for binomials?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
- 16.2k
2
votes
2
answers
410
views
Dimension of the associated graded module at an ideal
Let $I$ be an ideal of a Noetherian local ring $(R, \mathfrak m)$. Define the associated graded ring $G_I(R):=\bigoplus_{n=0}^\infty I^n/I^{n+1}$. Then $G_I(R)$ is a Noetherian ring of the same ...
- 135
1
vote
1
answer
364
views
Proj construction and nilpotent homogenous elements in graded ring
Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined ...
- 6,028
0
votes
0
answers
76
views
Largest set of monomials whose span is "co-prime" to a given polynomial
Let $K$ be a number field, and let $F \in K[x_1, \cdots, x_n]$ be a polynomial. For a positive integer $d \geq 3$, define $M(F;d)$ to be the largest positive integer such that there exists a set $S$ ...
- 26.9k
5
votes
2
answers
280
views
Freeness of a quotient module over a regular local ring
Let $R$ be a regular local ring with maximal ideal $m$. Let $t\in m\setminus m^2$. Let $N$ be a submodule of a finitely generated free $R$-module $M$ satisfying
$$ tM \subseteq N \subseteq M.$$
...
- 921
4
votes
0
answers
174
views
Centers and conjugacy classes of groups relative to a pair of group homomorphisms
$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by
\begin{align*}
\mathrm{Z}(G) &\...
- 11.8k
54
votes
10
answers
16k
views
Rings in which every non-unit is a zero divisor
Is there a special name for the class of (commutative) rings in which every non-unit is a zero divisor? The main example is $\mathbf{Z}/(n)$. Are there other natural or interesting examples?
- 11.1k
2
votes
1
answer
210
views
Minimality implies algebraic independence?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
- 63.9k
2
votes
0
answers
148
views
Nilpotent polynomial matrices over $F_q$ - polynomial count variety ? ( Nilpotent cone for Hitchin-Gaudin like integrable system)
Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...
- 24.9k
7
votes
0
answers
131
views
When is a degree-$n$ homogeneous polynomial in $\mathbb{C}[x_1, x_2, \ldots , x_m ]$ the product of $n$ one-forms?
When is a degree-$n$ homogeneous polynomial in $\mathbb{C}[x_1, x_2, \ldots , x_m ]$ the product of $n$ one-forms?
Is there any simple algorithm or criterion to check it?
I have chosen the complex ...
- 63.9k
4
votes
2
answers
227
views
Maximal subgroups of finite abelian $2$-groups
Suppose $G$ is a finite abelian $2$-group, and $S$ is a subset of $G$, $\langle S\rangle=G$,$S^{-1}=S$,$e\notin S$. How to determine whether there exists a maximal subgroup $M$ of $G$, such that $S$ ...
- 44.4k
6
votes
1
answer
272
views
Ideals of functions whose zero locus is a submanifold
Let $M$ be a smooth $m$ dimensional manifold. Suppose that $f_1,\dots,f_k\in C^\infty(M)$ are smooth functions such that the zero locus $$N:=Z_f=\lbrace p\in M:\ f_i(p)=0,\ \forall i=1,2,\dots,k\...
- 3,669
1
vote
0
answers
77
views
$M^2=0$ defines a Koszul algebra ? What if $M$ is Manin's endomorphism's of Koszul $A$ ? (Here $M^2=\sum_k M_{ik}M_{kj}$ - resembles $d^2=0$).)
Consider a matrix $M$ which elements $M_{ij}$ are generators of some algebra $K$,
impose new relations: $M^2=0$ and get a new algebra $K_{2}$.
Question 1: Is it true that $K_2$ is Koszul algebra when ...
- 24.9k
1
vote
0
answers
60
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Bounding the length in a module of evaluated skew polynomials
Let $R$ be a finite principal ideal ring, $S$ a Galois extension of $R$ of degree $m$ (so in particular $S$ is a free $R$-module of rank $m$, and we have an $R$-module isomorphism $S^n \cong \...
- 223
2
votes
1
answer
423
views
Conjecture about semigroups
Let $G$ be a finite semigroup with order $n$ odd. Let $S_i \in G, i=1,\ldots,\binom{n}{(n+1)/2}$ be all the subsets of $G$ of size $(n+1)/2$.
Let $E(S_i)$ be the set obtained "expanding" $...
- 14.6k
4
votes
1
answer
267
views
A particular morphism being zero in the singularity category
Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $...
- 35.2k
6
votes
0
answers
151
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On dual notions of morphisms of algebraic structures obtained by replacing equaliser with coequalisers
This question is based on this discussion from the Category Theory Zulip. See also the earlier question Natural cotransformations and "dual" co/limits.
Let $G$ and $H$ be groups. We define ...
- 11.8k
13
votes
2
answers
875
views
Given an irreducible polynomial over $\mathbb{Z}$, how often is it irreducible modulo a prime?
Given a monic irreducible polynomial $f\in\mathbb{Z}[x]$, I'd like to know for how many primes p we have that $f \bmod p$ is irreducible.
In the link: How many primes stay inert in a finite (non-...
- 156k
10
votes
0
answers
190
views
Is every UFD a filtered colimit of Noetherian UFDs?
I'm wondering how one could prove or disprove that any non-Noetherian UFD is a filtered colimit of Noetherian UFDs. This would allow for some absolute Noetherian approximation to be applied for ...
- 804
0
votes
0
answers
82
views
Integer valued polynomials and divided power algebra
Let $T\subset \mathbb Q[x]$ be the ring of integer valued polynomials, i.e. the polynomials $f$ with $f(\mathbb Z)\subset \mathbb Z$. In his wonderful book ”Commutative algebra with a view toward ...
- 1,907
72
votes
14
answers
22k
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Elementary / Interesting proofs of the Nullstellensatz
Is there an easy proof of the Nullstellensatz that avoids the standard Noether-normalization techniques?
One proof I know proves first the 'weak' Nullstellensatz which ensures that maximal ideals ...
- 4,713
2
votes
0
answers
75
views
When is a finitely generated commutative algebra a projective module over its invariant subalgebra?
For the sake of simplicity, I will work over the complex numbers.
Let $A$ be a finitely generated algebra and $G$ any finite group of algebra automorphisms. Then, by Noether's Theorem, $A^G$ is also a ...
- 7,127
6
votes
1
answer
2k
views
Why is this theorem attributed to J.-P. Serre?
Page $117$ of Atiyah, MacDonald's Introduction to Commutative Algebra text has the following theorem. Let $P(M,t)$ denote the Poincare- series of $M$.
$\textbf{Theorem.}$ $\bigl(\mathsf{Hilbert-Serre}...
- 4,713
2
votes
0
answers
134
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A pattern in factorization of certrain symmetric polynomials made of sums of squares and products
Secondary-school pupils may learn things like
$$(a+b-c)^2 = a^2+b^2+c^2 +2ab - 2ac - 2bc$$
and that no choice of plus versus minus can give us
$$
(\pm a\pm b\pm c)^2 = a^2+b^2+c^2 - 2ab - 2ac - 2bc,
$$...
1
vote
1
answer
155
views
Convergence of a product in $\mathbb Q_2[[X]]$
I thought it would be very easy to prove, but in fact, I did not manage to prove or disprove this fact:
the sequence of polynomials $$\left(\prod_{j=0}^k\big(1-2^{2^j}X\big)\right)_{k\in\mathbb N}$$
...
- 50.6k
5
votes
1
answer
248
views
On the bounded derived category of sheaves with coherent cohomology
Let $(X,\mathcal{O}_X)$ be a locally ringed space such that $\mathcal{O}_X$ is locally notherian, and let $\operatorname{Coh}(\mathcal{O}_X)$ be the category of coherent $\mathcal{O}_X$-modules. The ...
- 9,229
4
votes
2
answers
544
views
Membership problem in monoids
What is the simplest example of a monoid with undecidable membership problem? In other words, I'm looking for a concrete monoid $S$ such that there is no algorithm which takes elements $s_1,...,s_n$ ...
- 236k
13
votes
0
answers
260
views
Big list of Hochster dual concepts
Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and ...
- 11.8k
0
votes
1
answer
287
views
When is the power-bounded subring top. of finite type?
Very naive question here. Let $K$ be a complete nonarchimedean field, $A$ a reduced affinoid $K$-algebra. When is the power-bounded subring $A^\circ$ topologically of finite type, in the sense that we ...
CommunityBot
- 1
3
votes
1
answer
164
views
Minimality of the Koszul resolution
Let $R = \mathbb{C}[x,y]$ and $V = \mathbb{C}x\oplus\mathbb{C}y$. Then, the Koszul resolution of $R$ (as an $R$-bimodule) is given by
\begin{align*}
0\to R\otimes_{\mathbb{C}}\wedge^2V\otimes_{\mathbb{...
- 35.2k
5
votes
0
answers
187
views
Isbell duality for monoids and groups
Isbell Duality
$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Sets}{\mathsf{Sets}}\newcommand{\rmL}{\mathrm{L}}\newcommand{\rmR}{\mathrm{R}}\newcommand{\B}{\...
- 11.8k
2
votes
1
answer
241
views
Sheaves which are locally free on subschemes of dimension zero
Let $\mathscr{F}$ be a coherent sheaf on a scheme $X$ with reasonable assumptions.
Obviously if I restrict to any point $x \in X$, the restriction $\mathscr{F}|_x$ is free over $x$.
I am interested in ...
- 148k
17
votes
1
answer
782
views
Injective ring homomorphism from $\mathbb{Z}_p[[x,y]]$ to $\mathbb{Z}_p[[x]]$
Is there an injective $\mathbb{Z}_p$-ring homomorphism from $\mathbb{Z}_p[[x,y]]$ to $\mathbb{Z}_p[[x]]$?
CommunityBot
- 1