All Questions
Tagged with ac.commutative-algebra ag.algebraic-geometry
182 questions
3
votes
0
answers
334
views
Which sheaves on a projective bundle are flat over the base scheme?
Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.
Are there any ...
- 1,025
3
votes
1
answer
420
views
Automorphisms of complete discrete valuation ring
Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb F}...
- 87
3
votes
1
answer
618
views
When is an almost geometric quotient flat?
All varieties here are over $\Bbb C$. Let $G$ be a reductive algebraic group acting algebraically on affine $n$-space $\Bbb A^n$. Let $R$ be the coordinate ring of $\Bbb A^n$. Assume that the natural ...
- 3,079
3
votes
1
answer
196
views
Number of free summands of finite local extensions
Suppose that $(R, m) \subseteq (S,n)$ is a finite extension of normal local domains that is:
etale on the punctured spectrum
not flat / etale at the origin
and such that the residue fields $R/m = S/n$...
- 20.5k
3
votes
1
answer
784
views
Presentation of the tautological bundle of the Grassmannian
Consider a Grassmannian $G=Gr(r,n)$ embedded in projective space $P^n$ by its Plucker embedding. Is there a way of writing down a presentation of the tautological bundle of $G$, as a module over the ...
- 235
2
votes
3
answers
341
views
Linear homogenous polynomials that generates one quadratic polynomial
Let $P_1, \ldots, P_m$, $Q_1, \ldots, Q_k \in \mathbb{C}[x_0,\ldots,x_n]$ be linear homogenous polynomials. Let $f$ be a homogenous quadratic polynomial of degree $2$.
Assume that for every $i$ and ...
- 2,576
2
votes
1
answer
639
views
How to show that the intersection of two certain affine varieties is reduced?
$\DeclareMathOperator\codim{codim}$Let $X=V(I)$, $Y=V(J)$ be two affine varieties. I'd like to know possibile strategies to understand when their schematic intersection, i.e. $X\cap Y=V(I+J)$, is ...
- 165
2
votes
1
answer
310
views
Lüroth theorem for $k \subset k(f,g) \subseteq k(x)$
Let $k$ be a field of characteristic zero (I do not mind to assume that $k=\mathbb{C}$, if things are easier in this case).
Lüroth theorem says that a field $L$, $k \subset L \subset k(x)$ containing ...
- 2,837
2
votes
1
answer
85
views
Separability of $\mathbb{C}[x,y_1,\ldots,y_r]$ over $\mathbb{C} + (h,y_1,\ldots,y_r)$
The answer to this MO question says the following:
Lemma 1. Let $h \in \mathbf C[x]$ be a polynomial of degree $n \geq 2$. Then $\mathbf C+(h) \subseteq \mathbf C[x]$ is unramified if and only if $h$ ...
- 2,837
2
votes
1
answer
1k
views
common roots of bivariate polynomial equations
Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but different coefficients. ...
- 59
2
votes
0
answers
833
views
Applications of Jordan-Holder theorem in an abelian category
The Jordan-Holder theorem says that any chain of subobjects of a finite length object can be refined to a composition series, and that any composition series has the same length.
This theorem holds ...
- 129
2
votes
0
answers
216
views
On an application of the going-down theorem of Cohen-Seidenberg in Mumford
There is a following result in Mumford's red book of schemes (Chapter II Section 8). Here $R$ is a valuation ring with algebraically closed fraction field $k$.
Let $Z \subset \mathbb{P}^n_k$ be an ...
- 3,625
2
votes
2
answers
898
views
Number of generators of an ideal in a polynomial ring over a Noetherian ring
Let $R$ be a Noetherian ring. By the Hilbert Basis Theorem the polynomial ring $R[x_1, \ldots , x_n]$ is also a Noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[...
- 545
1
vote
1
answer
324
views
Simultaneous triangularizability over a commutative ring
Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property
(*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent.
McCoy showed that, if $...
- 3,741
1
vote
1
answer
483
views
formally étale morphisms which are also universally closed
A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion. Is it possible to characterize morphisms which are formally etale and universally ...
- 761
1
vote
1
answer
218
views
stable sheaves in characteristic $0$
Let $K$ be a non-algebraically closed field of characteristic $0$ and $X_K$ a smooth, projective, geometrically connected curve defined over $K$. If $F$ is a stable locally free sheaf on $X_K$, is it ...
- 2,323
1
vote
1
answer
698
views
When are Segre- and Veronese embeddings Gorenstein?
Given a Segre product $\mathbb P^m \times \mathbb P^n$, or more generally $\mathbb P^{m_1}\times\cdots\times\mathbb P^{m_n}$, is there a characterization in terms of $m$ and $n$, or the $m_i$, for the ...
- 83
1
vote
1
answer
367
views
Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial
The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ($\...
- 413
1
vote
0
answers
106
views
Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM
Does anyone know an example of a $ \mathbb{Q} $-factorial, normal, Cohen Macaulay, projective, Mori dream space $ Z $ over a field $ k $ of arbitrary characteristic such that the Cox ring of $ Z $ is ...
- 912
1
vote
0
answers
140
views
On finite dimensional commutative algebras and regular sequences
Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...
- 1,227
1
vote
1
answer
264
views
Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$
Let $R \subseteq S$ be two Noetherian local rings, not necessarily regular, which are integral domains,
with $m_RS=m_S$, namely, the ideal in $S$ generated by $m_R$ (= the maximal ideal of $R$) is $...
- 2,837
1
vote
0
answers
234
views
Separability of a simple ring extension
Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
- 2,837
1
vote
1
answer
285
views
a problem about ideals of polynomial rings
Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions
$$
f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i.
$$
Let $(...
- 1,990
1
vote
1
answer
235
views
Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular
Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$.
Assume that:
(1) $R$ and $S$ are (Noetherian) integral domains.
(2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...
- 2,837
1
vote
0
answers
294
views
Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
0
votes
1
answer
2k
views
Are Chow groups a birational invariant?
Let us work in the category of smooth, projective varieties (say, over an algebraically closed field $k$). If $X$ and $X'$ are birational, then do they have the same Chow groups? Is there at least a ...
- 179
0
votes
1
answer
269
views
Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals
If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
- 1,209
0
votes
0
answers
213
views
For which $f \in \mathbb{C}[x,y][T]$, all irreducible elements of $\mathbb{C}[x,y]$ remain irreducible in $\mathbb{C}[x,y,T]/(f)$
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic polynomial:
$f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$,
$a_j \in \mathbb{C}[x,y]$.
Denote: $A=\mathbb{C}[x,y]$ and $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}[x,y][...
- 2,837
0
votes
0
answers
303
views
For which monic irreducible $f \in \mathbb{C}[x,y][T]$, $\mathbb{C}[x,y][T]/(f)$ is a UFD?
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic irreducible polynomial:
$f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$,
$a_j \in \mathbb{C}[x,y]$, $0 \leq j \leq n-1$.
Denote $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}...
- 2,837
0
votes
1
answer
307
views
Glue DVR to itself, get a separated non-affine scheme
Here is what seems to be a fun little exercise in algebraic geometry. Take a DVR $R$ and automorphism of $Frac(R)$; glue $Spec(R)$ to itself via this automorphism. Can the glued scheme be separated ...
- 25
0
votes
0
answers
163
views
Question about the statement of the going-down theorem of Cohen-Seidenberg in Mumford
In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows.
Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that ...
- 3,625
0
votes
0
answers
308
views
Faithfully flat etale morphism from strictly Henselian ring (from Etale Cohomology and the Weil Conjecture by Freitag/Kiehl)
I have question about a statement found in Etale Cohomology and the Weil Conjecture by Freitag, Kiehl at the end of page 15.
It starts with the Remark 1.18 : Let $A$ be a strictly Henselian ring (i.e. ...
- 6,028