Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
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
views
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
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
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
views
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
views
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
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
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
20
votes
10
answers
7k
views
Resources on invariant theory
What are resources on invariant theory? Basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd prefer online / freeish ...
- 365
4
votes
0
answers
119
views
Adjoining new factors for primes in UFDs
It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...
- 18.7k
2
votes
1
answer
158
views
How to decompose a given polynomial by ideal generators
Given a finite set of polynomials $f_1, f_2,..., f_n$ of variables $x_1,...,x_m$, generating the ideal $I$, suppose that we have one more polynomial $g\in I$.
What is the algorythm for decomposing $g$ ...
- 34.3k
1
vote
0
answers
119
views
Monomorphism which is locally of finite presentation
$\DeclareMathOperator\Spec{Spec}$Let $X$ be an affine scheme of finite type over a field. Let $Y\to X$ be a monomorphism of schemes, which is locally of finite presentation. Is it true that $f$ is ...
- 63.9k
1
vote
1
answer
410
views
Morphisms of a simple sheaf over an algebra to its double dual
Given a smooth and projective surface $S$ over an algebraically closed field $k$ and a sheaf of Azumaya algebras $R$, i.e. $R$ is a locally free $O_S$-module of finite rank. Let $M$ be a coherent and ...
- 18.4k
0
votes
0
answers
57
views
Reference for packing property and König property
Can someone please suggest reference material to study about the packing property and König property of ideals and some examples?
- 869
0
votes
0
answers
100
views
Shedding faces and decomposability in simplicial complexes
Definition:
A pure d-dimensional complex
$\Delta$ is $k$-decomposable if either $\Delta$ is a $d-$simplex or $\Delta$ contains a face $F$ such that
$\dim(F) \leq k$
both $\Delta \setminus F$ and $\...
- 63.9k
5
votes
2
answers
754
views
A version of Hilbert's Nullstellensatz for real zeros
$\newcommand\R{\Bbb R}$Let $Q(x_1,\dots,x_n)\in\R[x_1,\dots,x_n]$ be an irreducible polynomial such that the dimension of the set $Z:=\{(x_1,\dots,x_n)\in\R^n\colon Q(x_1,\dots,x_n)=0\}$ (defined, say,...
- 3,599
2
votes
0
answers
100
views
Koszul cohomology associated with a regular sequence
Let $A$ be a local Noetherian ring and $M$ be an $A$-module. Let $\mathfrak{a}$ be an ideal of $A$ generated by a regular $M$-sequence $s_1,\cdots,s_r$. Let $K_\bullet(s_1,\cdots,s_r;M)$ be the Koszul ...
- 439
1
vote
0
answers
88
views
When does sum of algebraically independent polynomial become dependent?
Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a ...
- 341
7
votes
0
answers
225
views
Decomposing an endomorphism as a tensor product
$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
- 11.1k
5
votes
1
answer
349
views
Commutator subgroup of $\mathrm{GL}_n(R)$ when $R$ is a PID with unity
Hua and Reiner in their paper titled "Automorphisms of the unimodular group" have established what will be the commutator subgroup of $\mathrm{GL}_n(\mathbb{Z})$, $\forall n\geq 0$. I have ...
- 7,893
2
votes
1
answer
181
views
Hilbert–Samuel multiplicity under hypersurface sections
Let $\newcommand{\frakm}{\mathfrak{m}}(R,\frakm)$ be a reduced Noetherian local ring of dimension $d$ and $f\in\frakm^\alpha\setminus\frakm^{\alpha+1}$ a parameter of $R$, i.e. $\dim R/(f)=d-1$. Let $...
- 5,339
6
votes
0
answers
235
views
A standard name for the algebraic structure on a projective line?
Question: Is there any name for the natural algebraic structure of the projective line?
Algebraically, a projective line over a field is a set $L$ endowed with two binary operations $+$ and $\cdot$ ...
- 41.8k
101
votes
31
answers
29k
views
Errata for Atiyah–Macdonald
Is there a good list of errata for Atiyah–Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists ...
1
vote
0
answers
186
views
Is it true that monomorphisms of local Artinian $\mathbb{R}$-algebras are regular?
A Weil algebra is a finite-dimensional real algebra, in which each element is the uniquely sum of a scalar and a nilpotent (so nilpotents constitute the only maximal ideal of codimension 1). In other ...
- 6,962
1
vote
0
answers
205
views
Finding if an ideal is the radical of another one
Let's suppose we have, in the ring $\mathbb{Z} [x,y,z,w,v]$, the following polynomials:
$f=xw-yz$,
$g=x^2z-y^3$,
$h=yw^2-z^3$,
$k=xz^2-y^2w$.
The question is to prove that $I=(f,g,h,k)$ is the radical ...
- 1,561
12
votes
4
answers
3k
views
Atiyah-MacDonald, exercise 7.19 - "decomposition using irreducible ideals"
An ideal $\mathfrak{a}$ is called irreducible if $\mathfrak{a} = \mathfrak{b} \cap \mathfrak{c}$ implies $\mathfrak{a} = \mathfrak{b}$ or $\mathfrak{a} = \mathfrak{c}$. Atiyah-MacDonald Lemma 7.11 ...
2
votes
1
answer
206
views
Noetherian local ring with non-lci formal fibers
I am looking for an example of a noetherian local ring $A$ such that its formal fibers are not complete intersection rings in the sense of https://stacks.math.columbia.edu/tag/09PY (i.e., there is a ...
- 1,813
2
votes
1
answer
307
views
A question about mapping cone and resolutions
I am studying this papper
https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-44/issue-3/Betti-numbers-of-almost-complete-intersections/10.1215/ijm/1256060413.full
By Daniel ...
- 10.7k
1
vote
0
answers
122
views
What can we say when a module of differential is free?
Let $\mathbb{C}$ complex number.
$R=\mathbb{C}[x,y]/(f)(f\in \mathbb{C}[x,y])$
If the module of differential $\Omega_{R/\mathbb{C}}$ is free $R$ module of rank one,
what can we say about $R$.
How far ...
- 328
2
votes
1
answer
202
views
Artin-Schreier theorem for rings (a little different)
Motivation:
Let me recall the well-known Artin-Schreier theorem (AST) for fields in a non-formal way; if $L$ is an algebraically closed field, and $K \subset L$ a subfield not 'much smaller' than $L$, ...
- 5,407
0
votes
0
answers
87
views
Relation between nullspace and row-equivalence of matrices over $\mathbb{Z}$ and $\frac{\mathbb{Z}}{n \mathbb{Z}}$?
Two matrices $D$ and $E$ over a field have the same nullspace if only if they are row-equivalent. Is the same true if those matrices are over the ring of integers ($\mathbb{Z}$) or integers mod a ...
- 219
2
votes
0
answers
122
views
Quasi-isomorphisms of P-algebras
In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...
- 215
3
votes
1
answer
227
views
Van der Waerden's classical paper on Hilbert polynomials and Bezout's theorem
Is it possible to obtain an electronic copy of Van der Waerden's "On Hilbert series of composition of ideals and generalisation of the Theorem of Bezout", Proceedings of the Koninklijke ...
- 5,339
3
votes
1
answer
200
views
Kernel of a map of Tate algebras
Let $A$ and $B$ be a pair of noetherian Tate algebras over $\mathbb{Q}_p$, and assume $\text{dim}_{\text{Krull}}(B) > \text{dim}_{\text{Krull}}(A)$. Is it true that any morphism $B \longrightarrow ...
- 2,907
4
votes
0
answers
116
views
The criterion for dimensional conjecture for universal Galois deformation rings
I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-...
- 863
1
vote
1
answer
110
views
Particular example of a quadratic extension of a nonunital ring
I want to construct a concrete non-unital ring $R$ with the following properties:
$R$ is a noncommutative non-unital ring with a right unite $r$ i.e $t.r=t$ for any $t\in R$.
$S\subset R$ is a ...
- 55.9k