All Questions
145 questions
1
vote
1
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466
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Bertini-type theorem in positive characteristic [closed]
Let $f:X \to Y$ be a morphism of finite type of irreducible schemes over an algebraically closed field of characteristic $0$. Assume that $Y$ is non-singular. Let $x \in X$ be a closed point and $T_xf:...
- 503
3
votes
0
answers
131
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Classification of faithfully flat morphisms between formal power series
Let $\mathbb{C}[[z_1,\dots,z_n]]$ denote the algebra of formal power series.
I am interested in faithfully flat morphisms
$$Spec(\mathbb{C}[[z_1,\dots,z_m]])\to Spec(\mathbb{C}[[z_1,\dots,z_n]]),\, m\...
- 21.8k
12
votes
1
answer
551
views
Presenting $\mathbb{Q}[[t]]$ as an explicit colimit of smooth $\mathbb{Q}$-algebras: an explicit example for the Popescu's theorem
By the seminal Popescu's theorem, $R=\mathbb{Q}[[t]]$ is a filtered colimit of smooth $\mathbb{Q}$-algebras. Could you give me a hint: which $\mathbb{Q}$-algebras can yield such a colimit? My problem ...
- 16.9k
1
vote
0
answers
112
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Asymptotic stability of prime divisors
Suppose $I$ is an ideal in a formally equidimensional local ring $R.$ Let $A(I)$ and $\overline A(I)$ denote Ass$R/I^n$ and Ass$R/\overline{I^n}$ for all large $n$ respectively.
My question is
What ...
- 1,713
0
votes
1
answer
598
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Reference for a lemma on étale maps
The Stacks Project has the following really nice Lemma concerning étale maps of rings:
Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation
$$ B\...
- 1,838
3
votes
0
answers
98
views
Primary ideals as functions vanishing to a certain degree
Let $k$ be the field of complex numbers. Let $I$ be a primary ideal in $k[X_1,\ldots, X_n]$, let $\operatorname{rad}(I)=P$, and let $X$ be the algebraic variety corresponding to $P$.
I am trying to ...
- 3,897
5
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0
answers
204
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Where can I find Andre's "Cinq exposés sur la désingularisation"?
Many expositions of Popescu's desingularization theorem indicate that an other proof of this theorem can be found in
"Cinq exposés sur la désingularisation" by M. Andre, Ecole Polytechnique ...
- 7,249
2
votes
0
answers
94
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Maximal quotient ring of a commutative ring
Let $R$ be an associative ring in which an identity
element is not assumed. A right quotient ring of $R$ is
an overring $S$ such that for each $a\in S$ there
corresponds $r\in ...
- 21
4
votes
1
answer
164
views
The volume around a singular isolated root when equalities are loosened
Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a double-...
- 5,971
0
votes
1
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217
views
quasi-projective and separated as topological properties
Let $X$ be a non-reduced noetherian scheme over $\mathbb{Z}$ or $\mathbb{C}$. Assume that $X^{red}$ is quasi-projective and separated, does the same hold for $X$ ?
(By the way, projective implies a ...
- 2,384
2
votes
0
answers
270
views
Looking for a reference in commutative algebra
I need "I.G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973) 23–43." in my research, but it seems to be very old and rare. Does anyone know a site for ...
- 21
4
votes
1
answer
167
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Reference on the classification of (low rank) Gorenstein rings over $\mathbb{C}$
I am interested in the question of the classification of (low rank) Gorenstein rings over $\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring ...
- 41
0
votes
1
answer
381
views
Samuel multiplicity
Let $X$ be the hyper-surface defined by
$$f:=\sum_{i=1}^k x_i^n=0$$
in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal
$$I=(x_1^{n-1},\dots , x_k^{n-1}) $$
What is ...
- 2,384
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
3
votes
0
answers
371
views
Intersection Multiplicity
Let $X$ be an hyper-surface in an affine space defined by an equation $F$. We can assume that the ground field is $\mathbb{C}$ and $X$ is normal. Take functions $f_1,\dots, f_n$ on $X$ and let $B$ ...
- 2,384
4
votes
1
answer
258
views
Reference request for division algebras, over $\mathbb{Q}_{p}((t))$
I was looking for a possible reference that would answer the following question,
Let $\mathbb{Q}_{p}$ be the $p$-adic numbers and $\mathbb{Q}_{p}((t))$ be the field of Laurent polynomials over $\...
- 143
2
votes
1
answer
155
views
On conflicting descriptions for tor of a local cohomology group
Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1_C(\mathcal{O}_X)$ the sheaf associated to the presheaf $U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ...
- 833
2
votes
0
answers
115
views
Need a reference/proof for computing the regularity of ideal of points in $\mathbb P^d$?
In a lecture notes on 'Cohomology modules' i read the following remark:
Given a set $X$ of points in $\mathbb P^d$,using the Local Cohomology modules one can easily compute the reg$(S_X)$ where $...
- 121
2
votes
1
answer
450
views
Resolution of singularity of polynomials
Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$.
By Hironaka's desingularization theorem, there exists a birational map ...
- 1,457
4
votes
1
answer
555
views
Base change and relative Ext over noncommutative rings
Given two smooth projective schemes $X$ and $Y$ over some algebraically closed field $k$, we have $X\times Y$ with the projections $p$ to $X$ and $q$ to $Y$. Furthermore we have a "nice" sheaf of ...
- 1,391
11
votes
1
answer
675
views
When is there a deformation of a given singularity to a normal singularity
Question: Given a variety $X_0$ with a singularity (say Cohen-Macaulay), when does this exist as a special fiber of a flat family $X \to C$ mapping to a smooth curve $C$, such that the generic fiber ...
- 20.5k
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
6
votes
1
answer
670
views
Base change of trace for Gorenstein or Cohen-Macaulay morphisms
This is basically a question of functoriality for base change of CM morphisms.
EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'...
9
votes
0
answers
2k
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Jacobian ideals reference
Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...
- 20.5k
1
vote
0
answers
359
views
Thickness of the category of perfect complexes with finite length homology
Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...
- 3,101
3
votes
1
answer
280
views
Composite families of formal power series over $\mathbb C$ as algebraic variety
I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...
- 5,428
0
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0
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350
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Reference: A nowhere vanishing section of a vector bundle is locally split
Well-known fact:
If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...
- 7,328
3
votes
1
answer
422
views
Tor dimension in polynomial rings over Artin rings
I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
- 165
8
votes
2
answers
425
views
Doing explicit computations with coordinate rings
Suppose that we are given an integral $k$-algebra $A$ of finite type explicitly, by which I mean that we are given the generators of the defining ideal $J$ where $A = k[x_1,...,x_n]/J$. What kinds of ...
- 3,918
2
votes
0
answers
171
views
Irreducibility of a general fibre
Let $A\subseteq B$ be an inclusion of affine domains over an algebraically closed field $k$ of characteristic $0$. Can someone give me a reference for the following fact?
If $A$ is algebraically ...
0
votes
0
answers
548
views
Fitting ideal sheaves and determinant bundles
I am working on a problem in algebraic geometry which comes down to a fact in commutative algebra that I am hoping is well-known.
Suppose $F$ is a coherent sheaf on a smooth variety $S$, and that the ...
- 5,857
2
votes
0
answers
135
views
Lifting of Commuting Maps of Vector Bundles
Assume that we have a vector bundle $\mathcal{F}$ over $\mathbb{P}^d(\mathbb{C})$ that is generated by global sections. Let $\pi \colon \mathcal{O}^n \to \mathcal{F}$ be the associated map that is ...
- 431
1
vote
0
answers
115
views
cohomlogy of Diagonal ring
Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let $M_{\Delta}=\...
- 1,713
3
votes
0
answers
168
views
Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
- 2,384
9
votes
0
answers
644
views
Conceptual proofs for the computation of the structure sheaf
The following lemma in commutative algebra is important for the foundations of algebraic geometry:
If $A$ is a commutative ring, $U \subseteq A$ is a finite subset generating the unit ideal, then ...
- 63.1k
3
votes
4
answers
1k
views
Matrix factorization categories for ADE singularities
What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example this. For ...
- 21k
4
votes
1
answer
203
views
Flatness and intersections of Cohen-Macaulay subvarieties
There's a commutative algebra fact that I would very much like to be true but could, for all I know, be completely false. One version that would be sufficient is:
Say $A$ is a smooth projective ...
- 1,510
4
votes
1
answer
552
views
Factorization of schemes
Let $k$ be a ring (perhaps a field). Let $M$ be the "set" of isomorphism classes of $k$-algebras and regard it as a commutative monoid with multiplication $\otimes_k $ and unit element $k$. There is ...
- 63.1k
3
votes
1
answer
571
views
Reference for submultiplicativity of length of tensor product
I am looking for a reference, in the form of a textbook, that contains proofs of following statements.
NOTE: I am NOT looking for the proofs, I am looking for a reference! Proofs of these statements ...
- 3,101
1
vote
0
answers
71
views
Name for generalization of bivariate weighted-homogeneous polynomials
A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that $p\alpha_j+q\beta_j=d$...
- 456
6
votes
0
answers
881
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Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19
Fulton's "Intersection theory" book contains the following fact (example 18.3.19):
Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...
- 30.5k
3
votes
0
answers
124
views
Singularity locus in terms of ideals.
Let $X$ be a smooth affine variety over a field, $Z\subset Y\subset X$ are closed (reduced) subvarieties. What are the possible ways to verify whether $Y$ is singular at $Z$ i.e. whether $Z$ is ...
- 16.9k
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
4
votes
1
answer
358
views
Prime-ness checking for polynomial ideals over ACFs( algebraically closed fields).
Let $f_1,\ldots f_m \in k[X]$ have degrees bounded by $l$. and $I(\bar{f})$ be the ideal generated by $\bar{f}$.
If $I(\bar{f})$ is not a prime ideal then its non-primeness is witnessed by polynomials ...
- 255
2
votes
0
answers
261
views
On a characterization of the symbolic square of prime ideals in polynomial rings
If $R=k[x_1,...,x_n]$ is a polynomial ring in $n$ indeterminates over a field $k$ of characteristic $0$, there is a characterization of the symbolic square of a prime ideal $P$ (the $n$-th symbolic ...
- 805