All Questions
Tagged with ac.commutative-algebra ag.algebraic-geometry
2,098 questions
46
votes
0
answers
1k
views
Enriched Categories: Ideals/Submodules and algebraic geometry
While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...
- 3,249
2
votes
1
answer
575
views
Can you suggest a good name for a local homomorphism φ:(R,m)->(S,n) of local rings with the property that φ(m)S is n-primary?
Can you suggest a good name for a local homomorphism $(R,\mathfrak{m})\stackrel{\varphi}{\rightarrow}(S,\mathfrak{n})$ of local rings with the property that $\varphi(m)S$ is $\mathfrak{n}$-primary?
...
- 3,101
7
votes
1
answer
2k
views
The space of valuations of a function field
Hello, I'm looking for someone who can help me to understand Zariski's theory of valuations.
First I outline the theory: we take a field $K$ which is a finitely generated transcendent extension of ...
- 1,804
4
votes
1
answer
382
views
Is the subspace of DVR's of the Zariski-Riemann space still quasi-compact?
If $S$ is the Zariski-Riemann space of a noetherian subring $k$ of a field $K$, Zariski-Samuel prove that $S$ is quasi-compact. If $S'$ is the subspace of valuations that are discrete (i.e. that ...
- 1,347
21
votes
1
answer
2k
views
When does the relative differential $df=0$ imply that $f$ comes from the base?
Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be
defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of
$B \otimes_A B \to B$, as in [Hartshorne II.8].
If $df=0$,...
- 27.8k
5
votes
0
answers
287
views
sheaves with free abelian stalks over compact space
The question concerns a sheaf $S$ of abelian groups over a compact space $X$. Suppose each stalk $S_x$ is finite rank free. Is the group of global sections free?
- 345
0
votes
2
answers
511
views
When are seminormal rings Cohen-Macaulay?
I know that not every local seminormal ring is Cohen-Macaulay. But are 1-dimensional local seminormal rings Cohen-Macaulay?
- 179
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 ...
- 387
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 ...
- 71
1
vote
0
answers
345
views
Is the ideal of denominators preserved under flat pullback?
Let $\phi \colon A \to B$ be a flat homomorphism of rings (commutative, with unit). Let $R$ be the total ring of fractions of $A$ (obtained by inverting all nonzerodivisors), and let $S$ be the total ...
- 7,328
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.
...
- 7,609
3
votes
3
answers
517
views
Cohen-Macaulay property for reducible schemes
I have the following question about certain schemes being Cohen-Macaulay.
Let $X$ be the union of all coordinate $k$-planes in
${\mathbb A}^N$. Is it CM?
Let $R$ be a collection of $k$-element ...
- 7,037
12
votes
1
answer
949
views
Discrete version of Nullstellensatz?
Hi. I was reading the paper "On the foundations of combinatorial theory (VI): The idea of a generating function" by Doubilet, Rota and Stanley, and there is a relation treated which is very ...
- 902
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 $X$?
- 95
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) ...
- 1,347
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
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
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})\...
- 3,101
0
votes
0
answers
166
views
Can the zero-degree part of $M_f \otimes_{S_f} N_f$ be identified with $M_{(f)} \otimes_{S_{(f)}} N_{(f)}$?
The isomorphism ${(M \otimes _ {S} N)} _ {f} = M _ {f} \otimes _ {S _ {f}} N _ {f}$ is well-known. Here, $S$ is a graded ring, and $M,N$ are graded $S$ modules.
Now, let $f$ be any homogeneous ...
- 945
1
vote
0
answers
169
views
Choosing generators of a submodule with divisibility properties
Looking at an open subset $U$ of the plane, containing $0 \in \mathbb{C}^2$, with coordinates $x$ and $y$.
Given a quotient sheaf $O_U^n \rightarrow T$, with $supp(T)=\lbrace0\rbrace$. Let $K$ be the ...
- 1,391
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 ...
- 683
10
votes
3
answers
3k
views
Counter-examples to Krull's intersection theorem
The more general form of Krull intersection theorem says:
Let $R$ be local and Noetherian and $I \subset R$ a proper ideal. If $M$ is finitely generated over $R$, and $N=\cap_1^{\infty} I^iM$, then ...
- 103
4
votes
1
answer
398
views
A terminology question: formally finite ??
Is there a name for a local homomorphism $\varphi:A\longrightarrow B$ of local rings $A$ and $B$, whose completion $\hat{\varphi}:\hat{A}\longrightarrow\hat{B}$ is a finite homomorphism? (that is, $\...
- 3,101
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. ...
- 149
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
7
votes
2
answers
649
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....
- 63.1k
8
votes
2
answers
1k
views
A name for "not quite saturated" graded modules
Let $M$ be a finitely generated graded module over a graded ring $R$. Let $\mathcal{F}$ be the corresponding coherent sheaf on $\operatorname{Proj} R$. There is a natural map of graded $R$-modules
$$\...
- 7,328
8
votes
1
answer
555
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 ...
- 63.1k
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, ...
- 6,700
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 ...
- 236
1
vote
0
answers
417
views
Absolute Irreducibility in Characteristic 2
Let $\mathbb F$ be a field and $\mathbb F[x_1,\dotsc,x_n]$ the ring of multivariate polynomials in $n$ variables over $\mathbb F$. A polynomial $P\in\mathbb F[x_1,\dotsc,x_n]$ is said absolutely ...
- 456
1
vote
1
answer
346
views
Cohomology of the general linear group on punctured spectra of 2-dimensional power series rings
$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Quot{Quot}\DeclareMathOperator\GL{GL}\DeclareMathOperator\char{char}$Let $(A,\mathfrak{m})=k[[x,y]]$ with $\char(k)=0$ and $K=\Quot(A)$. Set $X=\...
- 1,391
7
votes
1
answer
1k
views
How to construct log-canonical (or Calabi-Yau), non-Cohen-Macaulay singularities of low codimensions?
(EDIT 07/06/11: although the question has not been settled definitely, Sándor's excellent answer and the comments by Angelo and ulrich have highlighted many potential obstructions to the constructions ...
- 30.5k
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 ...
- 42.9k
7
votes
1
answer
2k
views
An example of a rank one projective R-Module that is not invertible
Let $R$ be a commutative noetherian ring. I know that an $R$-module is invertible iff it is finitely generated and locally free of rank one. I presume then that there are examples of non-finitely ...
- 345
3
votes
3
answers
461
views
Multiplicity of eigenvalues in 2-dim families of symmetric matrices
Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...
- 1,452
2
votes
1
answer
325
views
Gersten for homotopy invariant K-theory of non-singular varieties.
Here is the question:
if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH_n(X) \to \prod_{x \in X^{(0)}} KH_n(k(x))$ injective?
If $X$ is ...
- 1,347
12
votes
3
answers
3k
views
Can we say anything about the Krull dimension of a localization?
I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do ...
- 30.3k
29
votes
2
answers
5k
views
Regular, Gorenstein and Cohen-Macaulay
All the statements below are considered over local rings, so by regular, I mean a regular local ring and so on;
It is well-known that every regular ring is Gorenstein and every Gorenstein ring is ...
- 1,661
2
votes
2
answers
1k
views
Irreducible component of a Cohen-Macaulay variety
Is it true that an irreducible component of a Cohen-Macaulay variety is also Cohen-Macaulay? If not, then in what cases does this fact hold?
- 23
3
votes
0
answers
916
views
Unibranch rings
Let us call a Noetherian local ring $A$ unibranch if it is a domain and the normalization map is finite and induces a bijection on spectra.
My question is as follows: is this property preserved when ...
- 1,209
24
votes
3
answers
3k
views
Automorphisms of a weighted projective space
What is the automorphisms group of the weighted projective space $\mathbb{P}(a_{0},...,a_{n})$ ?
Consider the simplest case of a weighted projective plane, take for instance $\mathbb{P}(2,3,4)$; any ...
- 8,998
3
votes
0
answers
289
views
Terminal quasi-affine varieties?
Let $U$ be a normal, irreducible, quasi-affine variety over the algebraically
closed field $k$ and consider the ring $A = \mathscr{O}(U)$ of regular
functions on $U$. Write $Max(A)$ for the ...
- 31
4
votes
0
answers
811
views
$Ext$ functor, filtered complexes and spectral sequences
Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...
5
votes
0
answers
331
views
Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
5
votes
0
answers
190
views
"Unknot" algebraic set defined by two mutually dependent set of variables
Let $n$ be an integer $\geq 4$, and let $V \subseteq {\mathbb C}^{2n-1}$ be the set of all
$(a_1,a_2, \ldots ,a_n,b_1,b_2, \ldots ,b_{n-1}) \in {\mathbb C}^{2n-1}$ such that the derivative of the ...
- 3,595
6
votes
1
answer
530
views
Does a variety contain a cartesian product of two curves?
We are given an affine variety $V\subset \mathbb{A}^n\times\mathbb{A}^n$, and wish to know if it contains a product of the form $C_1\times C_2$, where $C_1$ and $C_2$ are two curves in $\mathbb{A}^n$. ...
- 7,836
0
votes
0
answers
237
views
resolution of singular points on plane curves and base change
Let $k$ be a field and $C/k$ be an affine plane curve over $k$, namely $C = \mathrm{Spec}(A)$ for some $A = k[x,y]/(f(x,y))$, here $f(x,y) \in k[x,y]$ is an irreducible polynomial. Let $B$ be the ...
- 1,109