All Questions
6,057 questions
5
votes
0
answers
917
views
Height of maximal homogeneous ideals
Let $R= \oplus_{n \ge 0} R_n$ be a graded Noetherian commutative ring and suppose $R_0$ is Artinian.
Do all maximal homogeneous ideals of $R$ have the same height ?
Let $R_{>0}$ be the ideal ...
- 16.2k
8
votes
2
answers
1k
views
Jacobian Conjecture for unit triangular matrices
This question is about the Jacobian conjecture for a special case. I will first explain the Jacobian conjecture (since it is something every mathematician should know about).
Let $k$ be an ...
- 3,093
8
votes
1
answer
472
views
Nonnegative additive functions on coherent sheaves
Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well ...
- 3,101
2
votes
1
answer
301
views
Is there existing terminology for this technical condition on semilattices?
Given a semilattice $S$, a subset $E$, and a positive integer $n$, let $E^{[n]}$ be the set of all products of $n$-tuples in $E$. Thus $\bigcup_{n\geq 1} E^{[n]}$ is nothing but the subsemigroup of $S$...
- 25.8k
15
votes
2
answers
1k
views
Exact sequence of monoids
What is the right definition of an exact sequence of monoid homomorphisms?
I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6,
http://www.math.ucla.edu/~balmer/Pubfile/...
- 3,009
2
votes
1
answer
470
views
Nonequivalent extensions with the same terms
I just construct an exact sequence $0\to M\to M\oplus N\to N\to0$ of $\mathbb{Z}$-modules that does not split, where $M=\mathbb{Z}$, $N=(\mathbb{Z}/2\mathbb{Z})^\\mathbb{N}$, and the map from $M$ to $...
- 854
4
votes
1
answer
2k
views
(algebraic) Taylor expansion for polynomials (rational functions) with coefficients in an arbitrary field.
This a probably very easy question and I am not sure whether it has been asked before (although I searched for it). Moreover I really hope this is nothing which can be found in any standard ...
- 1,558
0
votes
1
answer
494
views
Example: Nil radical of noetherian Rings with a map to simple noetherian rings
A basic example in commutative algebra: Let $A$ $B$ be noetherian rings, with $B$ simple noetherian. Suppose that for every element $b$ in $B$, there exists a power $b^{n}$ ...
6
votes
2
answers
3k
views
Algebraic varieties and UFD
Given an affine algebraic variety $V$ such that $\Gamma(V,\mathcal{O}_V)$ is a UFD, its sheaf of ring can be determined easily since one can show that:
$$\Gamma(D(f_1) \cup \cdots \cup D(f_n),\...
- 1,829
4
votes
1
answer
550
views
"Numerical Criterion" for Flatness
Let $R$ be a Dedekind ring, let $S = \mathrm{Spec} R$, and let us suppose that $f: X \to S$ is a finite morphism. Note that $X$ is not required to be connected. Does there exist a "numerical criterion"...
- 1,199
15
votes
3
answers
3k
views
State of the art for Gersten's conjecture for K-theory?
Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence
$$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to \...
- 1,347
3
votes
0
answers
313
views
Tor over non-noetherian local ring
Let $(A,m,k)$ be a local ring and let $M$ be a finite torsion $A$-module.
Is ${\rm Tor}^A_1(M,k)$ finite over $k$?
I am aware that the conclusion holds for any finite $A$-module $M$ when $A$ in ...
- 53
4
votes
1
answer
319
views
Reference request on Leray numbers
The Leray number $L_{\Bbbk}(K)$ (relative to a field $\Bbbk$) of a simplicial complex $K$ is the least $d\geq 0$ such that $\widetilde H_n(C,\Bbbk)=0$ for all $n\geq d$ and all induced subcomplexes $C$...
- 38.6k
4
votes
1
answer
468
views
Relationship between the notions of "excellent ring" and "universally catenary Nagata ring"
Every excellent ring is both universally catenary and Nagata. How "close" is a universally catenary Nagata ring to being excellent?
Context: I have not worked very much with the notions described ...
- 7,338
2
votes
3
answers
359
views
On the comparison of linear topologies on a local ring
Let $R$ be a local ring, $a_{\lambda}$ be a decreasing net of ideals, indexed by a directed set, such that each $a_{\lambda}$ is contained in the nilradical ideal and $\bigcap a_{\lambda}=(0)$. Then ...
- 320
5
votes
0
answers
232
views
Coherence of the monoid algebra of a non-finitely generated monoid
Let $P$ be an integral, sharp, finitely generated commutative monoid (say even torsion-free and saturated if you like), and consider the "rational cone" $P_\mathbb{Q}\subseteq P^{gp}\otimes_\mathbb{Z}...
- 1,030
1
vote
1
answer
1k
views
On the Completion of a complete local ring
Let $(R,\mathfrak{m})$ be a complete local ring, $a_{\lambda}$ be a decreasing net of ideals in $R$, indexed by a directed set. Consider the completion under $a_{\lambda}$-topology $A=\underleftarrow{\...
- 320
4
votes
1
answer
905
views
Flat family of normal schemes over a normal base
Let $f \colon X \to Y$ be a flat morphism of schemes over $\mathbb{C}$. Suppose that $Y$ is normal and that the fibers over the closed points of $Y$ are all normal.
Can I say something about the ...
- 43
1
vote
1
answer
403
views
Prime ideals in univariate polynomial rings
I'm looking for a textbook reference of the following elementary fact (a reference for an excercise in a textbook is also welcome):
Let $R$ be a commutative ring and let $\mathfrak{p}$ be a prime ...
- 16.2k
3
votes
1
answer
459
views
Frobenius functor and length of local cohomology
Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the ...
- 3,101
132
votes
3
answers
21k
views
When is the tensor product of two fields a field?
Consider two extension fields $K/k, L/k$ of a field $k$.
A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often ...
5
votes
6
answers
5k
views
an easy example of valuation ring which is not noetherian？ [duplicate]
Is there an easy example of valuation ring which is not noetherian？
- 77
14
votes
2
answers
1k
views
About integer polynomials which are sums of squares of rational polynomials...
I have the following question for which I haven't been able to find any reference or proof.
Suppose we know that a univariate polynomial $P(X)$ with integer coefficients is the sum of squares of two ...
2
votes
1
answer
399
views
Quotient field extension for an incomplete DVR
Let $R$ be a DVR with maximal ideal $xR$, and assume that $R$ is not complete in the $xR$-adic topology. Let $\hat{R}$ be the completion of $R$ in the $xR$-adic topology. Set $K=Q(R)$, the fraction ...
4
votes
1
answer
1k
views
What does "composé direct" mean in mathematical French?
In EGA IV, Lemma 6.14.1.1, the first sentence is
Soit $R$ un anneau composé direct d'un nombre fini de corps.
I'm guessing this means
Let $R$ be a ring that is the direct product of a finite ...
- 7,338
3
votes
1
answer
492
views
Noetherian descent extension for a given ring
For a homomorphism of rings $R \to S$, the following are equivalent:
a) $(-) \otimes_R S : \mathrm{Mod}(R) \to \mathrm{Mod}(S)$ reflects isomorphisms
b) $R \to S$ satisfies effective descent with ...
- 63.1k
10
votes
2
answers
917
views
What is a twisted modular operad?
I find Getzler and Kapranov's article Modular Operads difficult to understand. Can anyone explain what a (twisted) modular operad is conceptually, or what the underlying idea behind the concept of a ...
- 101
12
votes
2
answers
978
views
Blowing up a derived scheme
Is there a sensible notion of blowing up in any of the available frameworks for derived algebraic geometry? I am happy to remain in the affine setting, where I think the right question to ask is "what ...
- 1,126
26
votes
5
answers
3k
views
Given a polynomial f, can there be more than one constant c such that every root of f(x)-c is repeated?
The question
Let $f$ be a nonconstant polynomial over $\mathbb{C}$. Let's say that a point $c \in \mathbb{C}$ is unusual for $f$ if every root $x$ of $f(x) - c$ is repeated. Can $f$ have more than ...
- 27.7k
5
votes
1
answer
541
views
Localizability of differential operators a la Grothendieck
Hello,
Maybe this question is trivial, so sorry
Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1).
Then we can define the module of differential operators $D^{\leq n} (A)$,...
- 5,562
1
vote
1
answer
253
views
Chain of ideals in a complex algebra
Suppose $\mathfrak{A}$ is an unital algebra over complex numbers and $\mathfrak{J}$ is chain of left-ideals in $\mathfrak{A}$ ordered by inclusion such that none of its elements is countably generated....
- 31
5
votes
0
answers
2k
views
Is the radical of a homogeneous ideal homogeneous?
Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: is it true that $\sqrt{I}$ is homogeneous? In this generality,...
- 1,811
12
votes
1
answer
744
views
Is the following construction of the 0-Hecke monoid (well) known?
Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If ...
- 38.6k
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
5
votes
0
answers
308
views
Properties of the Zariski-Riemann topology on the set of valuations
One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...
- 51
2
votes
0
answers
263
views
Koszul complex for monomials
Suppose I have a list of monomials $f_1, \dots, f_m \in R = K[x_1, \dots, x_n]$.
Is there a nice description of the cohomology of the Koszul complex
\begin{equation}
\cdots \rightarrow\bigwedge^{r+1}...
- 21
3
votes
0
answers
293
views
Simple proof of that $k[X]^G$ Cohen-Macaualy ($G$ finite)?
Let $X$ be a (EDIT: non-singular, or even $\mathbf A^n$) algebraic variety over a field $k$ (alg. closed). Suppose $G$ is a finite group acting on $X$, $|G|\neq 0$ in $k$. Then $k[X]^G$ is Cohen-...
- 31
4
votes
0
answers
156
views
Characterizing non-singularity of varieties through properties of their derivations
I am interested in knowing about the possible implications between the following properties of a commutative, complex algebra:
Its spectrum is non-singular.
Its derivation module is projective and ...
- 393
5
votes
0
answers
454
views
If $p=0$ and $df=0$, is $f$ a $p$th power?
This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...
- 4,487
7
votes
1
answer
722
views
How is called a semigroup...
Does anyone know, how is called a semigroup in which every equation $ax=b$ has only a finite set (maybe empty) of solutions?
- 3,110
21
votes
3
answers
3k
views
Prime ideals in the ring of germs of continuous functions
We all know that the ring of germs of continuous functions at a point on, say $\mathbb{R}$, has a unique maximal ideal- namely, those functions that vanish at that point.
Can anyone think of a single ...
- 13.5k
2
votes
1
answer
255
views
What does the d-slice of a weighted polynomial algebra look like?
This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves.
...
- 33.8k
2
votes
1
answer
376
views
Hilbert series and resolution of a surface singularity
I have a question about the following theorem in Stanley's paper "Invariants of Finite Groups and Their Applications to Combinatorics".
Suppose that the Cohen-Macaulay $N$-graded $k$-algebra $B$ is ...
- 105
8
votes
1
answer
961
views
Vanishing constant term in powers of a Laurent polynomial
This is motivated by idle curiosity. I recently learned a result of Duistermaat and Van Der Kallen in "Constant terms of powers of a Laurent polynomial" which says that:
If the constant term of $f^...
- 85.6k
3
votes
1
answer
573
views
When does the forgetful functor S-Mod -> R-Mod induce injective maps on Ext-groups?
Assume we have a complete regular local ring $R$ and an $R$-algebra $S$.
Is there a class of such algebras $S$ with the following property:
Given two $S$-modules $M,N$, then the maps induced by the ...
- 1,391
1
vote
2
answers
596
views
A small question on commutative algebra
Recently I read the book Intersection Theory by Fulton. I think one property in his book relies on this commutative algebra conclusion. I'm not sure whether it is right.
Assume all rings are of finite ...
- 205
26
votes
1
answer
4k
views
Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?
It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...
- 363
1
vote
2
answers
313
views
How many DISTINCT vectors we get from pairs v_i + v_j for some set of given vectors v_i ?
Consider some set of vectors v_i i=1...N , v_i \in Z^k.
e.g. N = 10^4; k = 10
Consider all possible sums: v_i + v_j.
Is it possible to estimate how many DISTINCT vectors we get in advance without ...
- 24.9k
3
votes
0
answers
187
views
The role of "minimal" minimal primes
Let $R$ be a commutative ring of finite Krull dimension $n$. I'm interested in results where those minimal primes $P$ of $R$ play a role that sit at the end of a prime chain of maximal length, i.e.
$$...
- 16.2k
4
votes
1
answer
734
views
Finiteness of normalization of Noetherian normal domain
I have the following question:
Let $A$ be an integrally closed Noetherian domain, $K$ its field of fractions. let $L$ be a finite extension of $K$, and $B$ the integral closure of $A$ inside $L$. Is ...
- 5,562