All Questions
145 questions
8
votes
1
answer
359
views
Global to local principle for f.g. $\mathbb{Z}[x]$ modules
In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know ...
- 7,893
0
votes
0
answers
448
views
Behavior of Ext under base change
Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...
- 1,713
13
votes
1
answer
697
views
Commutative algebraic version of algebraic geometric object
In my work, I have to understand certain objects in commutative algebra (for example Gorenstein rings, Cohen–Macaulay rings e.t.c). I have a reasonable background in commutative algebra (I suppose!) ...
- 30.5k
7
votes
1
answer
553
views
Relationship between Hilbert-Samuel multiplicity and polar multiplicity
Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows:
$$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \...
- 30.5k
1
vote
0
answers
180
views
Skew-symmetric multi-derivations of $k[x_1,…,x_n]/I$
Let $I = \langle f_1, \ldots f_r \rangle$ be an ideal in $R=k[x_1,\ldots,x_n]$ where $k$ is a field, and put $A = R/I$.
(If $I$ is prime then $A$ is the coordinate ring of an irreducible affine ...
- 832
8
votes
1
answer
257
views
Minimal resolution of local cohomology module
Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$
Question Can we say anything about Betti numbers ...
- 30.5k
5
votes
0
answers
92
views
Question concerning the representation dimension of a special algebra
I would like to know, if the following problem is still open:
Let $k$ denote an algebraically closed field of characteristic 3.
Determine the representation dimension of $k(C_3\times C_3)$, where $...
- 1,755
4
votes
0
answers
213
views
Reference request: Formal Existence for stacks
Is there a formal existence Theorem for coherent sheaves on locally Noetherian Artin Stacks, in the spirit of Grothendieck's Formal GAGA?
Is it available for more general stacks?
CommunityBot
- 1
2
votes
1
answer
168
views
Approximating finite type algebras over a formal power series ring
Let $k$ be a ring, let $A := k[x_{1},\dotsc,x_{d}]$ be the polynomial ring and let $A^{\wedge} := k[[x_{1},\dotsc,x_{d}]]$ be the formal power series ring. For a $d$-tuple $\mathbf{e} = (e_{1},\dotsc,...
- 28.7k
8
votes
1
answer
683
views
Is the strict henselization isomorphic to the filtered colimit of finite etale algebras?
Let $(A,\mathfrak{m})$ be a local ring, and let $A^{\mathrm{sh}}$ be the strict henselization of $A$ at $\mathfrak{m}$. Let me denote $A^{\mathrm{sh},\mathrm{fin}}$ for the filtered colimit of finite ...
- 76
1
vote
1
answer
338
views
when a family of curve is an affine morphism
Let $f: X\to B$ be a family of curves, i.e. $f$ is flat, surjective and of relative dimension 1. If each fiber is an affine curve, can we conclude that $f$ is an affine morphism? If it is not true, ...
- 28.7k
1
vote
0
answers
75
views
Formula for the index of regularity of a generic Hilbert function
Is there an explicit formula for the index of regularity of a generic Hilbert function in two variables? (i.e., the Hilbert function of an ideal of $k[X,Y]$ generated by $r$ generic forms $f_{i}$ of ...
- 11
4
votes
0
answers
472
views
formal completion of smooth morphism
Let $f: X\to S$ be a smooth morphism of schemes. Let $p$ be a section. Let $\hat{\mathscr{O}}_{X, p(S)}$ be the completion $X$ along $p(S)$. Then I think for every point $s\in S$, there exists an ...
- 4,746
3
votes
0
answers
123
views
Frobenius stratification of imperfect fields
Suppose $k$ is a (non-perfect) field of characteristic $p$, and $Fr$, its Frobenius map. I’d appreciate comments and references on the structure of the fitration/stratification of $k$ given by the ...
CommunityBot
- 1
11
votes
2
answers
2k
views
Idea behind Grothendieck's proof that formally smooth implies flat?
From this answer I learned that Grothendieck proved the following result.
Theorem. Every formally smooth morphism between locally noetherian schemes is flat.
The book Smoothness, Regularity, and ...
- 23.3k
16
votes
1
answer
733
views
Where was $I_x/I_x^2$ first introduced? (DG or AG)
Cotangent space appears in both differential geometry (DG) and algebraic geometry (AG).
In DG, given a smooth manifold $M$ and $x\in M$ one has an isomorphism $I_x/I_x^2 \cong T^*_xM$, where $I_x$ is ...
- 50.6k
1
vote
1
answer
382
views
singular locus of semi-normal variety
Let X be a seminormal variety and S be the singular locus of X. Is Blow$_S X=$ the normalisation of X?
Is the singular locus given by the conductor ideal?
- 20.5k
2
votes
0
answers
329
views
Endomorphism algebra of a coherent sheaf is locally free
What is an example of a Noetherian ring $A$ and a finitely generated $A$-module $M$ such that the endomorphism algebra $\mathrm{Hom}_{A}(M,M)$ is flat as an $A$-module but $M$ is not flat?
- 428
3
votes
0
answers
432
views
When is every submodule of a module a direct sum of indecomposable submodules?
Is there any reference for modules over a commutative ring with identity such that every submodule of them is a direct sum of indecomposable submodules? Or is there any characterization of such ...
- 6,051
1
vote
0
answers
113
views
Reference request. The adjunction $\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$
We have the adjunction
$$\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$$
where $CDGA$ is the category of commutative diffferential graded algebras and $CA$ is the category of ...
- 1,615
3
votes
0
answers
119
views
Finite generation of the module of invariant vector fields
Let $G$ be a linear algebraic group (not necessarily reductive) and let
$X$ be an affine variety with a regular $G$-action (everything defined over the field of complex numbers $\mathbb{C}$). Denote ...
- 413
6
votes
1
answer
731
views
Thick subcategories
I hope this question is not too trivial for mathoverfolw.
Let $R$ be a commutative ring (with $0\neq 1$) and $D_{Perf}(R)$ the triangulated category of perfect complexes. Let $C$ be a thick ...
- 9,229
11
votes
3
answers
2k
views
When is a blow-up Cohen-Macaulay?
Let $X$ be a smooth projective variety and $Z$ a closed subscheme. Let $X'$ be the blow-up of $X$ with center $Z$.
Under what conditions on $Z$ is $X'$
Cohen-Macaulay?
In the case $Z$ is non-...
- 42.9k
3
votes
1
answer
221
views
Alternating multisymmetric functions
I am looking for a reference on certain modules of invariants. I think that the question is quite natural so that I believe there should be some results already, but I am not able to find anything.
...
- 3,271
2
votes
0
answers
221
views
Meaning of the statement "$a\in I$ is a general element of $I$"
Suppose $I$ is an ideal in a Noetherian local ring $(R,m)$. In some papers I have seen the following statement:
"$a\in I$ is a general element of $I$".
What is the definition of general element ...
- 1,713
1
vote
2
answers
317
views
The algebra of regular functions of a quasi-affine toric variety
Let $k$ be an algebraically closed field of characteristic zero and let $X$
be a toric variety over $k$, i.e. $X$ is a normal, irreducible $k$-variety and it admits an algebraic action of a torus with ...
- 15.5k
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 ...
- 1,423
30
votes
6
answers
8k
views
Algebraic stacks from scratch [closed]
I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...
- 1,980
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
2
votes
0
answers
94
views
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
3
votes
0
answers
131
views
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
1
vote
0
answers
112
views
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
1
vote
1
answer
500
views
A question on Eisenbud-Green-Harris conjecture
Let $I$ be a homogeneous ideal in a polynomial ring $k[x_1,\ldots,x_n],$ $I$ contain a regular sequence $f_1,\ldots,f_n$ such that deg$(f_i)=a_i$ and $a_1\leq\ldots\leq a_n.$ Let $d$ be a non-negative ...
CommunityBot
- 1
0
votes
1
answer
380
views
Milnor numbers and mixed multiplicities
section 6 of the link
Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration $\{m^rJ^s\...
CommunityBot
- 1
5
votes
1
answer
208
views
Zariski openness of Newton non-degenerate polynomials
Suppose you are given a convex polyhedron $\Delta$ in $\mathbb{R}^n$ (i.e. a convex hull of finitely many points in $\mathbb{Z}^n$) and consider a finite dimensional vector space $V$ over $\mathbb{C}$ ...
- 156k
1
vote
1
answer
325
views
Sheaf Hom is flat
Let $T$ be a scheme (probably integral noetherian) and $X$ a smooth projective variety. Let $K,K',A,A'$ be locally free coherent sheaves on $X\times T$. There are exact sequences:
$0\rightarrow K\...
- 28.7k
2
votes
1
answer
574
views
Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.
I would like to know a reference of the following statement (or counter example).
Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...
CommunityBot
- 1
12
votes
1
answer
437
views
Is there a categorical notion of reduced commutative algebras?
A commutative ring $R$ is reduced if $r^2=0 \Rightarrow r=0$ holds for all $r \in R$. Commutative rings are precisely the commutative algebra objects in the symmetric monoidal category $(\mathsf{Ab},\...
CommunityBot
- 1
10
votes
2
answers
1k
views
Algebraic independence of exponentials
First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution ...
CommunityBot
- 1
2
votes
1
answer
302
views
Has this notion of powers of ideals already appeared in the literature?
My question is whether the below notion of powers of ideals, which may be regarded as a weakening of the notion of symbolic power, has been already defined in the literature and, in that case, I ask ...
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 ...
CommunityBot
- 1
4
votes
1
answer
512
views
Being Cohen-Macaulay open in Hilbert scheme?
Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) ...
- 4,111
5
votes
0
answers
204
views
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 ...
- 63.9k
4
votes
2
answers
548
views
Irreducible algebraic sets via irreducible polynomials
There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...
CommunityBot
- 1
2
votes
1
answer
569
views
Field extension and nilpotent element
Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular local noetherian $k$-algebra, $B$ another local noetherian $k$-algebra, $f:A \to B$ an injective ring homomorphism of ...
- 28.7k
1
vote
1
answer
436
views
Automorphisms of rings fixing all prime ideals
Let $f,g:A \to B$ be two ring homomorphisms of noetherian rings satisfying that for any prime ideal $\mathfrak{q} \subset B$, $f^{-1}(\mathfrak{q})=g^{-1}(\mathfrak{q})=:\mathfrak{p}$ and the induced ...
- 19.6k
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 $...
- 5,169
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-...
CommunityBot
- 1
4
votes
1
answer
167
views
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 ...
- 6,237
6
votes
0
answers
671
views
Flat + locally of finite presentation + monomorphism = open immersion
It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$:
Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then $...
- 15.9k