All Questions
Tagged with commutative-algebra or ac.commutative-algebra
5,493 questions
19
votes
2
answers
5k
views
Are morphisms from affine schemes to arbitrary schemes affine morphisms?
To put this question in precise language, let $X$ be an affine scheme, and $Y$ be an arbitrary scheme, and $f : X \rightarrow Y$ a morphism from $X$ to $Y$. Does it follow that $f$ is an affine ...
- 47.5k
7
votes
2
answers
1k
views
Invariants and base change
Suppose $R$ is a Noetherian commutative ring, and $M$ a finite free $R$-module, with an action of a finitely generated discrete group $G$ by $R$-linear maps.
Is there any homological condition on ...
- 3,908
6
votes
2
answers
565
views
Given 2 towers of fields, when are these fields isomorphic?
Let $F_0 \subset F_1 \subset F_2 \subset \cdots$ and $K_0 \subset K_1 \subset K_2 \subset \cdots$ be two towers of fields. Also, let $F = \cup_{i=0}^\infty F_i$ and $K = \cup_{i=0}^\infty K_i$.
Now ...
CommunityBot
- 1
11
votes
2
answers
4k
views
Definition of a Grothendieck ring
I've been looking at some definitions of Grothendieck rings. However I've not found a good definition that I've understood. Any recommendations?
I'm referring to the definition in tensor categories, ...
- 35.2k
0
votes
1
answer
223
views
Equivalent functors
Let $R$ be a commutative Noetherian ring, $M$ is a finitely generated $R$-module. If $F: Mod \to Mod$ is a left exact functor and $R^iF(E)=0$ where $E$ is injective module. Assume that $F(-) \cong Hom(...
- 73.1k
2
votes
2
answers
605
views
(non-trivial) isotrivial family of elliptic curves over C^{\times}
So How does one prove (rigorously) that
$$
Frac(\mathbb{C}[x,y,t]/(y^2-x^3-t)) \not\simeq Frac(\mathbb{C}[t][x,y]/(y^2-x^3-1))?
$$
So here $Frac$ denotes the fraction field of an integral domain.
...
- 11.1k
2
votes
2
answers
974
views
Smoothness of hypersurfaces in Grassmannians
I have a general question, and then the specific version of that question I need for research. All vector spaces over $\mathbb{C}$.
Grassmanians of planes
The $(2,n)$-Grassmannian, denoted $Gr(2,n)$...
- 108k
4
votes
1
answer
138
views
Pulling back roots from the Completion
Consider the following diagram of regular local rings
$\begin{matrix}
\hat{A} & \xrightarrow{\quad\hat\varphi\quad} & \hat{B} \\
\ \uparrow\scriptstyle\alpha & \circlearrowleft & \ \...
- 11.1k
1
vote
0
answers
383
views
Size of an abelian permutation group with generators of order 2 [closed]
Let $g_1, \ldots, g_k$ be distinct permutations on a set $\Omega$. Suppose that $G = \langle g_1, \ldots, g_k \rangle$ is an abelian permutation group with only elements of order at most 2. Is it ...
- 11
3
votes
1
answer
901
views
Behaviour of Hilbert functions
Let $G$ be a complex simple reductive group. Then the set of isomorphy classes $Irr G$ is isomorphic to the set of dominant weights $\Lambda_+$ in the weight lattice of the maximal torus of a Borel ...
CommunityBot
- 1
2
votes
0
answers
363
views
A simple problem on commutative algebra related to G.I.T
Let $G$ be a geometrically reductive algebraic group over an algebraically closed field $k$. Let $X$ be an affine variety over $k$ on which $G$ acts regularly. Then $G$ acts on the coordinate ring $A$ ...
- 1,804
6
votes
3
answers
644
views
Line bundles on fibrations
Let $f:Y \to X$ be a flat morphism with positive dimensional fibers. Is it always true that line bundles that are trivial along each fiber are of type $f^*L$ for $L$ a line bundle on $X$?
- 6,253
6
votes
2
answers
5k
views
Periodic matrices
A square matrix $M$ such that $M^{k+1}=M$, for some positive integer $k$, is called a periodic matrix.
Can we characterize the periodic matrices in $\mathcal{M}_n(\mathbb{Z})$?
If we replace $\mathbb{...
CommunityBot
- 1
5
votes
2
answers
3k
views
Algorithm for Weierstrass Preparation Theorem for Formal Power Series
The Weierstrass preparation theorem for formal power series rings guarantees that if a given formal series $f(z) = \sum a_k z^k \in R[[z]]$ where $R$ is a complete local ring with maximal ideal $M$ ...
- 4,193
2
votes
2
answers
310
views
Can normalisations of curves over a perfect field change residue fields?
Does anyone know an example of a curve $X$ over a perfect field $k$ such that if $\tilde{X}$ is its noramlisation, there exists a point $x \in X$ and a point $y \in \tilde{X}$ over $x$ such that $k(y) ...
- 156k
9
votes
0
answers
349
views
Computing Ext for toric divisors
Ok, I have an affine (normal) toric variety $X = \text{Spec} k[\sigma]$. Suppose that $F, G$ are two torus invariant Weil divisors on $X$. Is there any relatively straightforward way to compute
$$
\...
- 20.5k
10
votes
1
answer
606
views
Are cluster variables prime elements?
Cluster algebras introduction
A cluster algebra is a subalgebra $A$ of $k[x_1^{\pm1},...,x_n^{\pm1}]$ generated by a set of cluster variables, which are elements which can be generated from the set $\{...
CommunityBot
- 1
3
votes
1
answer
191
views
Local coordinate system under finite integral extension
Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be a local morphism of regular local $\mathbb{C}$-algebras (of the same dimension) which makes $B$ integral over $A$.
Let $\mathfrak{m}=(x_1,\ldots,...
- 4,902
3
votes
0
answers
495
views
An elementary proof of a criterion for $k$ sufficiently large that $M_k = \Gamma(\mathbb{P}^n, \widetilde{M}(k))$
It is well known that coherent sheaves on $\mathbb{P}^n$ are equivalent, as a category, to finitely generated graded modules over the polynomial ring, provided that in the latter category, morphisms ...
- 7,328
4
votes
0
answers
303
views
Connecting group ring, abelianization
For reasons arising in algebraic topology, I'm wanting to better understand the relations between two functors from groups to abelian groups, $\mathbb{Z}[\cdot]$ and $\operatorname{ab}$; group ring ...
- 1,987
5
votes
1
answer
679
views
On the functoriality of scalar extensions of local rings (edited)
Note. I have edited my question to make it more transparent, following some very good comments that I received. I am sorry if it is a bit long.
A local homomorphism of local rings $(A,\mathfrak{m})\...
- 3,908
1
vote
2
answers
1k
views
maximal ideal in local subrings
Let $A,B$ be two local rings and put $\mathfrak{m}_A, \mathfrak{m}_B$ their maximal ideals. Now suppose that we have an injection $0 \to A \to B$ and put $\mathfrak{n} := A \cap \mathfrak{m}_B $. It ...
- 3,101
5
votes
1
answer
504
views
The ring of SL_2 invariants in sums of conjugation and tautological modules
Rings of Invariants
Consider $G=SL_2(\mathbb{C})$, and let $V$ be a finite-dimensional $G$-representation. Let $\mathbb{C}[V]$ denote the ring of polynomial functions on the space $V$; it is a free ...
CommunityBot
- 1
16
votes
2
answers
3k
views
Quotients of number rings
Hi,
Here's a question that comes up every now and then. Of course, the quotient of a number ring (ring of integers of a number field) by an ideal $I$ is a finite (Artin) ring. If we take $I$ to be ...
CommunityBot
- 1
16
votes
3
answers
2k
views
Hom(A,C) ⊗ Hom(B,D) injects into Hom(A⊗B,C⊗D): when? why?
Sorry for asking a linear algebra question on a research forum, but this seems to be either a case of extreme blindness on my side, or a case of a result lying much deeper than it seems.
The ...
- 63.1k
4
votes
1
answer
678
views
In what generality is the natural map $\operatorname{Hom}_R(L,M)\otimes S\to\operatorname{Hom}_{R\otimes S}(L\otimes S,M\otimes S)$ an isomorphism?
Let $k$ be a commutative ring, $R$ and $S$ commutative $k$-algebras. Let $L$ and $M$ be $R$-modules. Consider the natural map
$$\operatorname{Hom}_R(L,M)\otimes_k S \to \operatorname{Hom}_{R \...
CommunityBot
- 1
5
votes
0
answers
238
views
When does the normalization have regular special fiber?
Let's say $\mathcal{O}$ is a complete DVR with fraction field $K$ and algebraically closed residue field $k$. (The case I had in mind here was with $\mathcal{O}$ of equicharacteristic $p$, so assume ...
- 4,487
5
votes
1
answer
898
views
A little help with the unmixedness theorem?
I have two smooth subvarieties $Y$ and $Z$ of a smooth variety $X$. Their intersection $Y \cap Z$ has two irreducible components, both of the expected dimension and generically reduced. I want to ...
- 4,111
0
votes
1
answer
465
views
What is lim⟶ I^n M?
Let $R$ be a commutative ring, $I$ is an ideal of $R$, $M$ is an $R$-module.
$$IM\supset I^2M\supset I^3M\supset\cdots$$
What is $\mathop {\lim }\limits_{\begin{subarray}{c}
\longrightarrow \\
\...
CommunityBot
- 1
3
votes
1
answer
382
views
Generalizing Krull's Principal Ideal Theorem to Modules
Let $R = \mathbf{C}[x_1, \ldots, x_n]$ and let $M$ be a graded $R$-module which is finite-dimensional over $\mathbf{C}$ and suppose
$
0 \leftarrow M \leftarrow R^g \leftarrow R^d \leftarrow \cdots
$
...
- 30.5k
3
votes
1
answer
387
views
Term for an "almost regular" sequence
Let $R$ be a ring (commutative, with unit), $M$ an $R$-module, and $x_1, \dotsc, x_n \in R$. Consider the following two conditions:
For all $i$, the homomorphism $$\frac{M}{(x_1, \dotsc, x_{i-1})M}...
- 546
2
votes
1
answer
505
views
graded noetherian module
Let M be a R graded module with $M= \oplus M_i$. If M is noetherian then $M_i=0 $ for i << 0. My question is this, isn't $M_i = 0$ for all i >> 0 as well? If $(M_{n_i})_{i} \neq 0, n_i > 0$ ...
CommunityBot
- 1
17
votes
2
answers
1k
views
Dimension 1 prime ideals in the intersection of two maximal ideals
This question/problem really comes from a fact in algebraic geometry, where it says that given an irreducible variety $V$ ($\dim V \geq 2$) then for any given pair of points $x,y\in V$ there is an ...
CommunityBot
- 1
4
votes
0
answers
1k
views
Grothendieck spectral sequence [duplicate]
Possible Duplicate:
Composing left and right derived functors
Hi,
probably this question is obvious. I apologize for this.
Given functors $F$ and $G$ left exact, with as good properties as you ...
CommunityBot
- 1
9
votes
2
answers
1k
views
Modules over Laurent series rings
Let $k[x]$ be the ring of polynomials over a field k in one variable x. A $k[x]$-module is a k-vector space together with a linear endomorphism (the action of x).
The field $k(x)$ of rational ...
- 45.7k
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$...
- 47.3k
2
votes
4
answers
2k
views
A proof for a statement about polynomial automorphism
I already got a proof for the fact that if a polynomial map is surjective then it is also injective. However, I used the invariant dimension of a ring and I want a simpler proof. Bravo for any try. ...
- 38.6k
-1
votes
1
answer
282
views
Invertible matrices satisfying $[x,y,y]=x$ (take 2).
This is a simpler version of this question. Let $x=\left(\begin{array}{lll} 2 & 0 & 0\\\
0& 1 & 0\\\
0 & 0 & \frac12\end{array}\right)$. Is there a $3\times 3$-matrix $y$ with ...
CommunityBot
- 1
2
votes
0
answers
152
views
Characterization of a "Jacobian pair" member
Consider the ring ${R}$ of Puiseux series in $Y$ with coefficients in the ring $\mathbb{C}((X^*))$ of Puiseux series in $X$ with coefficients in $\mathbb{C}$; $F \in R$ is said to be a member of a ...
- 96
1
vote
2
answers
194
views
Counting hyperplane cuts vs. projections. Combinatorial identity
I have checked the following combinatorial identity for several cases and it seems to work. I would like to know if this is known or if there is a counter-example. Note, i is a given constant.
$$(i+d)...
- 85.6k
3
votes
1
answer
2k
views
Multiplicity of a singular point
Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z_T:=T\cap Z$. Is ...
- 11.1k
7
votes
2
answers
647
views
Characterization of locally free modules via exterior powers
Let $X$ be a scheme and $\mathcal{F}$ be quasi-coherent module on $X$. It is clear that if $\mathcal{F}$ is locally free of rank $n$, then $\det(\mathcal{F}) := \wedge^n \mathcal{F}$ is invertible, i....
- 22.6k
7
votes
1
answer
735
views
Can we make Buchberger's algorithm faster for a given ideal if we are allowed to vary the monomial order?
Suppose we have a finite set of generators for an ideal $I \subset R := \Bbbk[x_1,\dotsc, x_n]$, where $\Bbbk$ is a field. If we choose a monomial ordering, then Buchberger's algorithm allows us to ...
- 151k
8
votes
1
answer
553
views
Spectrum of an algebra object and Reconstruction of Schemes
In "Au-dessous de $\text{Spec}(\mathbb{Z})$", Toen and Vaquié define schemes relative to a complete, cocomplete symmetric monoidal category $C$ using a functorial approach.
In the introduction the ...
- 13.6k
6
votes
1
answer
825
views
Rings with finitely generated nilradical
Let $\mathfrak{a}$ be a monomial ideal in a polynomial algebra over some commutative ring $R$. If $R$ is reduced, then the radical $\sqrt{\mathfrak{a}}$ of $\mathfrak{a}$ is again a monomial ideal, ...
- 47.5k
0
votes
1
answer
315
views
Generalized Picard group (reflexive fractional ideals, principal ideals)
Given $\mathcal{O}=k[[u,v]]$ with maximal ideal $\mathfrak{m}$ and an $\mathcal{O}$-algebra $A$, free as an $\mathcal{O}$-module of rank $n^2$. $A$ is genertaed by two elements $x,y$ with $x^n=u$, $y^...
- 4,111
9
votes
2
answers
971
views
Simple object in derived category or stable model category?
Exist any common definition of simple objects in derived categories, or even better, in stable model categories?
I was only able to find definition for abelian categories.
Thanks.
- 39.3k
1
vote
3
answers
467
views
$\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$
Let $R$ be a commutative ring and $A$ and $B$ two $R$-module. Suppose that $A$ is free of rank $n$ with basis $a_1,\dots,a_n$. Then there is an isomorphism $\Phi: Hom_R(A,B) \to Hom_R(A,R)\otimes_R B$ ...
- 15.2k
1
vote
0
answers
263
views
In a Noetherian local ring $(R,m)$ with a prime ideal $P\neq m$, $P^{(n)}=P^n:m^{\infty}$
I had asked this question on math.stackexchange.com, but I have not received a response. Hoping to get some help here.
I am trying to prove this result and I am stuck at one step.
Let $(R,m)$ be a ...
7
votes
2
answers
1k
views
The rank of a not necessarily finitely generated module.
This question is motivated by this one. The main point of the question (was) to try to weaken the notion of rank. After the answers and comments, it seems this is not a good way to do it, but perhaps ...
CommunityBot
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