All Questions
6,055 questions
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votes
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134
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Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
0
votes
2
answers
299
views
A question about localization of commutative rings
Given a commutative ring $ R $ and a multiplicatively closed subset $ S $ of $ R $, there are two ways to consturct $ S^{-1}R $:
define an equivalence relation $ \sim $ on $ R\times S $ and then take ...
- 33
0
votes
2
answers
284
views
Motivation and reference for Brauer algebras
I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.
- 141
0
votes
1
answer
164
views
Naive question on tensor product
Let $A, B$ be $\mathbb{C}$-algebras, which are also integral domains. Suppose there is an injective ring homomorphism $f:A \to B$. Assume further than $f$ is a finite morphism in the sense that $f$ ...
- 2,323
0
votes
1
answer
270
views
Which theorems in commutative algebra describe the closed property of curves (i.e. algebraic varieties) in algebraic geometry?
In $R^2$, we have those solutions of $x^2+y^2-1=0$ describe a closed unit disk and $y^2 = x^3 − x + 1$ describes an unclosed elliptic curve. When we consider corresponding algebras e.g. ${R^2}/ \...
0
votes
1
answer
314
views
Localization and containment in commutative ring
Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...
- 11
0
votes
1
answer
178
views
Ideal in ring of power series
Let $K$ be a field of characteristic $p$ and $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring over $K$ such that $n, p \geq 3$.
Consider the ideal $I$ defined by
\begin{...
- 563
0
votes
2
answers
167
views
Tropical version of exchange relations in cluster algebras
The exchange relation in a cluster algebra is
\begin{align}
x_k' = \frac{1}{x_k} (\prod_{j \to k}x_j + \prod_{k \to j} x_j).
\end{align}
Do we have some tropical version of this relation? Are there ...
- 6,201
0
votes
1
answer
262
views
What are the cluster algebra structures on $Gr(3,5)$?
In the paper, cluster algebra structures on $Gr(2,n)$, $Gr(3,6)$, $Gr(3,7)$, $Gr(3,8)$, $Gr(4,6)$ are described. But what are the cluster algebra structures on $Gr(3,5)$ (and $Gr(3,4)$)? Do we have ...
- 6,201
0
votes
1
answer
102
views
Elements of a ring invertible in a faithfully flat algebra
Let $R\to S$ be a commutative algebra with $S\neq 0$ free as an $R$-module.
Is it true that for the units of these rings we have $U(R)=U(S)\cap R$ ?
0
votes
2
answers
408
views
Regularity of a tensor product
Let $A \subseteq B$ and $A \subseteq C$ be commutative noetherian domains.
Assume that $A$ and $C$ are regular rings (=every localization at a maximal ideal is a regular local ring).
Assume that $B$ ...
- 2,837
0
votes
1
answer
187
views
question about valuation ring
$k$ algebraically field, $A$ $k$ algebra and valuation ring of $K$ ($K$ field fraction of $A$) and we have the transcendence degree of $K$ over $k$ is one.
i want to ask if $A$ is noetherian ring?
- 1,066
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votes
1
answer
374
views
If an algebraic set in affine n-space has a prime ideal then it is irreducible. (Hartshorne's Algebraic Geometry, Cor. 1.4)
I am confused about a step in Hartshorne's proof (final part of Corollary 1.4) that an algebraic set $Y$ in affine n-space $\mathbb{A}^n$ having a prime ideal $I(Y)$ in the polynomial ring over $n$ ...
- 19
0
votes
2
answers
1k
views
Generic rank of a coherent sheaf on a projective variety vs. generic rank on the cone
Let $S:=k[X_1,\ldots,X_n]$ be a polynomial ring over a field $k$, with its natural grading, and let $\mathfrak{p}$ be a homogeneous prime ideal of $S$. Also, let $M:=\bigoplus_{i} M_i$ be a finitely ...
- 3,101
0
votes
1
answer
327
views
Examples of complexes of modules for wich homomorphisms "homological" implies "homotopic"
Let $V$ and $V^\prime$ - complexes of modules over ring $A$, and $f, g$ - homomorphisms $V\rightarrow V^\prime$.
I am interested in various conditions on $A, V, V^\prime$: ($f$ and $g$ are ...
- 101
0
votes
1
answer
267
views
Embedding commutative associative rings in non associative rings [closed]
Let $R$ be a commutative and associative ring with unit. Can $R$ be embedded in a ring $\hat{R}$ wich is both non commutative and non associative ?
Thanks guys !
- 1
0
votes
1
answer
329
views
What is correct name of the following construction?
Consider an ideal $I=\langle f_1,f_2,\ldots,f_s\rangle$ in the polynomial ring $\mathbb{Q}[x_1,x_2,\ldots,x_n].$ Build the following set
$$
\{ g_1 f_1+g_1 f_2+\cdots+g_n f_n \},
$$
where $g_i$ ...
- 301
0
votes
3
answers
892
views
local Artin algebras
Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A_1\oplus\ldots\oplus A_n$ into local Artin subalgebras, see for example Atiyah-McDonald, ...
0
votes
1
answer
65
views
Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 631
0
votes
1
answer
170
views
Isn't every algebraic operad equipped with a trivial weight?
In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...
- 215
0
votes
1
answer
470
views
Can every idempotent ideal be generated by an idempotent?
This problem comes from this commutative algebra problem
Let $R$ be a commutative ring with identity, $I$ is a finite generated
ideal of $R$ such that $I^2=I$, then exists $e\in R$ such that $I=Re$.
...
- 245
0
votes
1
answer
166
views
Blow up and critical points of the projection map
Denote $Z=V(x_1, \dots, x_{n-1}) \subset \mathbb{C}^n$ and let $Bl_Z(\mathbb{C}^n)$ be the blow up of $\mathbb{C}^n$ along $Z$ together with the projection map $\pi \colon Bl_Z(\mathbb{C}^n) \to \...
0
votes
1
answer
193
views
Is it true that $g-t$ is divisible by $f$?
Assume $f\in k[x_1,\ldots, x_n]$ is irreducible. Let for $g\in k[x_1,\ldots, x_n]$, $\partial(g)$ is divisible by $f$ for each derivation $\partial$ with $f\in\ker\partial$. Is it true that $g-t$ is ...
- 329
0
votes
1
answer
454
views
Conjugacy in the quaternion group
Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
- 1,703
0
votes
1
answer
495
views
Roots of unity, in the completion of the ring of integers of a number field, at a prime ideal
Let $p \in \mathbb{Z}$ be a prime and consider the number field $k = \mathbb{Q}[x]/(x^{(p^2 -1)/2} - p)$. We shall denote by $O_k$ the ring of integers of $k$. Let $\beta \in O_k$ be such that $\beta^{...
- 727
0
votes
1
answer
243
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+...
- 1,209
0
votes
2
answers
501
views
Inverse limit of finite flat morphisms
Suppose that an inverse limit of finite flat morphisms $X_k\to S$ of qcqs schemes, with affine transition maps, is $X\to S$, such that a closed fiber of $X\to S$ is finite.
Is $X\to S$ finite?
user120812
0
votes
1
answer
276
views
Invariance of the fiber-dimension of a finite map
Let $A\subseteq B$ be commutative Noetherian rings such that $A$ is a regular ring, i.e., $A_{\mathfrak{m}}$ is a regular local ring for all maximal ideals $\mathfrak{m}$ of $A$ and $B$ is a finite $A$...
0
votes
2
answers
640
views
Does there exist an Affinization or Projectivization process for Varieties?
Let us consider the classical isomorphism of real manifolds between $S^2$ and ${\mathbb CP}^1$. First strange thing we have here is that both are varieties, but $S^2$ is an affine and ${\mathbb CP}^1$ ...
0
votes
1
answer
206
views
Rings with a property similar to integral domains
For an integral domain $R$, the intersection of two non-zero ideals is also non-zero, because the product of any two non-zero elements is non-zero.
Is the converse true, i.e. if $R$ has the ...
- 11
0
votes
1
answer
207
views
The injectivity of Noetherian ring
Let $R$ be a ring with 1, $M$ be a left $R$-module. Then $M$ is fp-injective if every $R$-homomorphism from a finitely presented left ideal to $M$ extends to a homomorphism of $R$ to $M$ i.e. if $\...
- 107
0
votes
2
answers
367
views
Ring with Cohen-Macaulay canonical module
Let $(R,m)$ be Noetherian local ring which is an imagine of a Gorenstein ring $(S,n)$. Set
$$ K_R:= Ext_S^{s-d}(R,S), $$
where $d=\dim R$, $s=\dim S$.
If $K_R$ is Cohen-Macaulay (i.e. $R$ is a ...
- 89
0
votes
2
answers
200
views
Differential operators between modules, $\mathcal{D}_A(M, M)$ necessarily a filtered, almost commutative ring?
See here. Does it follow immediately that $\mathcal{D}_A(M, M)$ as defined in the link is a filtered, almost commutative ring? How can I visualize this geometrically?
0
votes
2
answers
296
views
Hochster-Roberts Theorem reciprocal
Given a Cohen-Macaulay ring $R$ over a field of characteristic zero and $G$
a reductive algebraic group acting on $R$, then the ring of ivanriants $R^G$
is also Cohen-Macaulay. This is known as ...
- 1,595
0
votes
1
answer
175
views
Codimension in zero and positive characteristic
Let $F_0,\ldots,F_m\in\mathbb{Z}[x_0,\ldots,x_n]$ be polynomials with integer coefficients and let $p$ be a prime integer. Consider the two ideals: $$I_0:=(F_0,\ldots,F_m)\subset \mathbb{Q}[x_0,\ldots,...
- 1,159
0
votes
1
answer
295
views
Normality and fiber product
Let $A$ and $B$ be noetherian normal rings and let $f:A\rightarrow B$ be a finite but non-flat ring homomorphism. We can also assume $Spec(A)$ connected if necessary. We put on $B$ the structure of $A$...
- 1
0
votes
2
answers
299
views
0-dimensional Gorenstein local ring.
Assume the following condition for the ring T = F_p[[X,S]]/I:
Condition 1. T is NOT a zero ring.
Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements.
Then, is T a ...
- 87
0
votes
1
answer
347
views
Iwasawa invariants
Suppose $M$ is a finitely generated torsion $Z_p[[T]]$-module; the torsion comes from the $\mu$-invariant and the $\lambda$-invariant. Consider $M/(p)$ and $M[p]$ ($p$-torsion of $M$) which are $F_p[[...
- 1,209
0
votes
1
answer
371
views
Depth of polynomial ring $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$
Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. My Question is about the depth of this ring.
Question: Could we ...
- 45
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votes
2
answers
206
views
h^0 of a sheaf supported at a point
$C$ : a projective curve over an algebraically closed field, $I$ : the ideal sheaf of $\mathcal{O}_C$ defining a point $p\in C$.
Why is the following hold?
$h^0(C,(\mathcal{O}_C/I^{r})\otimes\mathcal{...
- 1
0
votes
1
answer
128
views
What are the semigroups in which congruence classes can be multplied like sets?
For a semigroup $S$ and a congruence $\rho$ on $S$, let's say that $\rho$ is good when for all $a,b\in S$ we have that $[ab]=[a][b],$ where $[x]$ denotes the congruence class of $x$ modulo $\rho$ and ...
- 1,445
0
votes
1
answer
310
views
Technical question about height of minimal associated primes
Let $A$ be a Noetherian ring, $\mathfrak{p}\subset A$ a prime ideal of height $p$, $N$ an $A_{\mathfrak{p}}$-module of finite length, $M,M'\subset N$ finitely generated $A$-submodules such that $M\...
- 2,857
0
votes
2
answers
384
views
Projectivity of one Tate algebra over another
Let $\mathbf{Q}_p \langle X_1,\dots,X_n \rangle$ be the $n$-variable Tate algebra, i.e. the subalgebra of $\mathbf{Q}_p[[X_1,\dots ,X_n]]$ of power series which converge on the closed unit polydisk in ...
- 13.1k
0
votes
1
answer
136
views
pd finite for finite module over local CM ring?
Let (R,m) be a CM ring of dimension d, and M a finite R- module. Is pd M always finite?
- 287
0
votes
1
answer
580
views
Why is Ext^n(k,M) a vector space over k?
This might be a trivial question to experts but not to me whatsoever. Suppose that $(R,m,k)$ is a Noetherian local ring, $M$ is an $R$-finite module whose depth is $n$. One then defines the type of $M$...
- 149
0
votes
1
answer
251
views
What is a certain cartesian product of algebras?
Suppose $F$ is a field and $A$ the $F$-algebra $F[X]\times_{(F\times F)} F$ given by the missing corner of a cartesian square in $F$-algebras
\begin{equation}
F~\xrightarrow{\Delta} ~F\times F~ \...
- 1
0
votes
1
answer
223
views
On the Irreducibility of Cyclotomic polynomials
Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
- 494
0
votes
1
answer
147
views
Existence of linear disjointness between an algebraic number field and $p$-cyclotomic field over $ \mathbb{Q}$
Suppose $ K $ be an algebraic number field and $ n $ be an even integer. Is it possible to find atleast one $ p $ such that $ p\equiv 1( \text{mod}~ n)$ and $ \mathbb{Q}(\eta_p) $ is linearly disjoint ...
- 923
0
votes
1
answer
147
views
Are maps into a smooth curve equivalent to relative effective Cartier divisors?
Let $X$ be a smooth curve and $S$ an affine scheme, both over an algebraically closed field $k$.
Given a morphism $f : S \to X$, is it true that the graph morphism $\Gamma_f : S \to X \times S$ is a ...
- 426
0
votes
1
answer
213
views
Grobner basis of a submodule of a free module over polynomial ring
Let $A=\mathbb{Q} [x_1,\dots,x_n]$ be the ring of polynomials with rational coefficients. Let $T$ be an $m\times n$ matrix with entries from $A$. Consider it as a morphism of $A$-modules $T\colon A^n\...
- 21.8k