All Questions
Tagged with commutative-algebra or ac.commutative-algebra
5,493 questions
3
votes
1
answer
502
views
Description of prime ideals in $\operatorname{Spec}(\mathbb{Z}[x_1, \dots, x_n])$
Edit: This seems to be wrong, as pointed out by Will Sawin in the comments.
The prime ideals of $\mathbb{Z}$ and $\mathbb{Z}[x]$ are well-known. It is also not too hard to compute the underlying set ...
- 35.4k
1
vote
0
answers
67
views
Conormal module of a commutative Koszul algebra
Does the conormal module $I/I^2$ of a commutative Koszul algebra $A=k[x_1,\dots,x_n]/I$ have a linear minimal free resolution? Is there a formula for the Betti numbers of $I/I^2$, or, even better, a ...
- 359
2
votes
1
answer
407
views
Finite generation of certain graded sequences of ideals
Let $U\subset\mathbb{C}^n$ be an open set containing the origin $o$ and $Y\subset U$ a complex analytic subvariety of pure codimension $c$ with ideal sheaf $\mathcal{I}_Y$. Let $\frak{a}_{\bullet}=\{{\...
CommunityBot
- 1
1
vote
0
answers
80
views
Localization of totally acyclic complex or projective modules remain totally acyclic?
Let $R$ be a commutative Noetherian ring. An acyclic complex $P$ of projective $R$-modules is called totally acyclic if for every projective $R$-module $Q$, the complex Hom$_R(P, Q)$ is also acyclic.
...
- 480
1
vote
0
answers
54
views
Algorithm for finding generating sets of projective modules
Suppose $R$ is a (Dedekind) domain and $M$ is a projective module of constant rank over $R$. We know that $M$ is finitely generated over $R$. I'm wondering is there any algorithms to produce a (...
2
votes
0
answers
182
views
Integers as polynomials in infinite variables
This question is more of a request for reference or ideas than else. Forgive (or correct) if there are imprecisions or blatant mistakes.
The main idea is that the unique factorization theorem for $\...
- 29
6
votes
2
answers
219
views
Steinitz isomorphism theorem for non-Dedekind domains
(Cross-posted from https://math.stackexchange.com/questions/4931582/steinitz-isomorphism-theorem-for-non-dedekind-domains)
Fix a Dedekind domain $R$ and fractional ideals $I, J$. It's a classical ...
- 7,893
1
vote
0
answers
181
views
Examples of semirings where the additive neutral element is not absorbing for multiplication
In the case of a non unital ring, the additive 0 must be absorbing for the multiplication because we have a⋅0 = a⋅(a − a) = a⋅a − a⋅a = 0 and similarly on the other side.
In the case of a unital ...
- 22.3k
6
votes
2
answers
1k
views
Question about the sum of odd powers equation
Quite surprisingly the following question appears while studying the billiard dynamics.
Assume we have $2n$ real numbers: $ x_1, x_2,..., x_{2n}$.
Assume also that $S_k=0$ for any odd positive integer ...
- 105k
4
votes
1
answer
685
views
Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
- 188k
1
vote
2
answers
471
views
Prime ideal of $A[X_1,...,X_d]$
Let $A$ be a UFD domain, i.e. integral and for any height one prime
${\frak p}$ of $A$, we have ${\frak p} = (u_{\frak p})$ for some $u_{\frak p} \in A$.
Once and for all, we fix the algebraic ...
- 63.9k
-3
votes
2
answers
817
views
Is there a "weak" fundamental theorem of algebra for matrices?
Let $R$ be the ring of complex $n\times n$ matrices, where $n>1$.
Does every nonconstant polynomial in $R[X]$ have a root in $R$?
Note: The "strong" fundamental theorem of algebra for ...
- 16.2k
1
vote
0
answers
108
views
Do étale coordinates give rise to a regular sequence of diagonal elements?
Fix an algebraic extension $k\subseteq K$ of fields of characteristic zero and consider a map of commutative rings $\phi\colon K\left[T_{1}^{\pm},\dots,T_{n}^{\pm}\right]\to A$ which is étale. Now ...
- 507
3
votes
2
answers
246
views
Explicit description of transfer for $K_1$
Let $R$ be a commutative regular ring and let $s \in R$ be an element such that $R / s$ is also regular. Then we have a long exact localization sequence
$$
\ldots \rightarrow K_i(R/s) \rightarrow K_i(...
- 10.8k
10
votes
1
answer
560
views
Can you constructively prove a univariate polynomial algebra over a Jacobson ring is itself Jacobson?
Recall the Jacobson radical of a commutative ring $\mathrm J(A)=\lbrace a\in A\mid \forall b\in A:1-ba\in A^\times\rbrace$. The Jacobson radical of a quotient by an ideal $I\vartriangleleft A$ is ...
- 35.4k
6
votes
1
answer
414
views
Constructive treatment of Jacobson rings
Which result is closest to the classical
General Hilbert's Nullstellensatz: Finite type algebras over Jacobson rings are Jacobson.
and constructively true at the same time? And where can I find a ...
- 76
0
votes
0
answers
47
views
Relationship between equation of integral dependence of an element and its inverse
Let $A$ be a reduced, Noetherian ring. Let $B$ be its integral closure. Let $b\in B$ and let $v\in B$ be its inverse. Let $b^n+\ldots a_0=0$ be an equation of integral dependence for $b$. Is there any ...
- 71
2
votes
0
answers
274
views
Can the completion of a local domain which is not a field be a field?
I would like to prove/disprove the following claim:
Let $A$ be an equicharacteristic local domain, and denote by $\widehat{A}$ its completion with respect to its maximal ideal. If $\widehat{A}$ is ...
- 293
3
votes
1
answer
413
views
Do there exist these real polynomials?
Do there exist real polynomials $P(x)$ and $Q(x)$ with nonnegative coefficients, and $n>20$ a natural number such that
$$\left(\sum\limits_{k=0}^{n} x^k\right)^2=(x-3)^2P(x)+(x+4)^2 Q(x)?$$
I have ...
- 22.5k
84
votes
31
answers
70k
views
Applications of the Chinese remainder theorem
As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
- 60.5k
1
vote
0
answers
107
views
A certain subfield of $\mathbb{C}(x,y)$
Let $A=\mathbb{C}(x+y,xy)$, the subfield of symmetric polynomials with respect to the involution $\alpha: (x,y) \mapsto (y,x)$.
Denote $G_A=\{w \in \mathbb{C}(x,y) \ | \ \mathbb{C}(x+y,xy,w)=A(w)=\...
- 2,837
11
votes
0
answers
615
views
Monotonicity of ratio of symmetric polynomials
The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
4
votes
0
answers
112
views
The $K_1$-group of integer valued polynomials
Let $R=$ Int$(\mathbb{Z}) = \{f \in \mathbb{Q}[x]| f(\mathbb{Z}) \subset \mathbb{Z}\}$. I am interested to find $K_1(R)$. I list my trials below:
Let us construct a Milnor square $$\matrix{R&\...
- 141
2
votes
1
answer
152
views
A commutative ring with unity which does not have relatively pseudo-injective ideals with zero intersection
Let $R$ be a ring with $1$ and $M$ and $N$ be any right $R$-modules. We say that $M$ is pseudo-$N$-injective if every $R$-monomorphism $f:X \to M$ from a submodule $X_R$ of $N_R$ can be extended to $N$...
- 14.6k
20
votes
4
answers
4k
views
What is interesting/useful about Castelnuovo-Mumford regularity?
What is interesting/useful about Castelnuovo-Mumford regularity?
- 708
3
votes
0
answers
114
views
English translation of Borel-Serre's "Théorèmes de finitude en cohomologie galoisienne"?
Is there an English translation of this text, or at least some English language paper that proves the same results?
I especially need a proof of the following fact which is in this paper: Say $k$ is a ...
- 31
2
votes
2
answers
416
views
Tensor product over $\mathbb{Z}$ and p-adic integer ring $\mathbb{Z}_p$
Thanks for your reading. Suppose we have two $\mathbb{Z}_p$-modules $A,B$. Do we always have $A \otimes_{\mathbb{Z}} B \simeq A \otimes_{\mathbb{Z}_p} B$, as abelian groups or $\mathbb{Z}_p$-modules? ...
CommunityBot
- 1
13
votes
2
answers
2k
views
Basics of classification of trilinear forms (when is it non-discrete)
Consider tri-linear forms, $\{A_{ijk}\}$ where $i=1,\dotsc,n_1$, $j=1,,n_2$, $k=1,\dotsc n_3$, over a field of zero characteristic, up to the equivalence $A\to (U_1,U_2,U_3)(A)$, by three matrices.
...
- 12.9k
1
vote
0
answers
141
views
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
views
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
829
views
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
5
votes
0
answers
288
views
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
views
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
views
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
views
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
views
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
vote
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
views
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
409
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
363
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 ...
- 5,998
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
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