All Questions
6,056 questions
1
vote
0
answers
76
views
Determine whether a set generates a residue field of an invariant ring
Fix two positive integers $m>n$.
Let $(A|Y)$ be an $m\times (n+1)$ augmented matrix consisted of $m\times (n+1)$ indeterminates, where $Y$ is a column symbolic vector of length $m$.
Denote $R=\...
- 331
1
vote
0
answers
165
views
spectral sequence Ext(R/I,H^g(M)) => Ext^{p+q}(R/I,M)
I am reading papers of Local cohomology and came across some spectral sequences. I then started reading about spectral sequences from Rotman's book. I havent finished reading the chapters on spectrals ...
- 183
1
vote
1
answer
181
views
Non-separable $\mathbb{A}^2$-form is trivial
Suppose $A$ is a finitely generated $\mathbb{Q}$-algebra and $A \otimes_{\mathbb{Q}} \mathbb{R} \cong \mathbb{R}[X, Y]$. Then is $A \cong \mathbb{Q}[X, Y]$? Here $\mathbb{R}[X, Y]$ is a two variable ...
- 4,111
1
vote
2
answers
193
views
Annihilators of sum of two ideals
Let $R$ be a commutative Noetherian ring and $I$, $J$ be two ideal of $R$.
If $x\in R$, then is $((I+J):x)=(I:x)+(J:x)$?
I would be very grateful if someone comment me.
- 113
24
votes
5
answers
2k
views
Lie groups vs Lie monoids
Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence ...
- 2,369
4
votes
0
answers
352
views
What does the cotangent complex tell you when it takes animated inputs?
These two links: What is the cotangent complex good for? and Intuition about the cotangent complex? are quite helpful in giving intution for the cotangent complex in terms of deformations but I don't ...
- 301
4
votes
1
answer
564
views
Nondegenerate pairings versus perfect pairings for finitely generated projective modules
Let $R$ be a (not necessarily commutative) ring, $M$ a left $R$-module, and $N$ a right $R$-module. We say that a pairing
$$
\langle -,-\rangle:M \otimes_R N \to R
$$
is non-degenerate if, for all $n \...
- 145
3
votes
1
answer
172
views
On the linearizability of the action of a finite group on a formal polydisc
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gl{Gl}$Let $\mathcal{D}=\mathbb{C}[[t_{1},\dotsc , t_{n}]]$ be a formal polydisc over $\mathbb{C}$, and $G$ be a finite group. On Lemma 7.8 of ...
- 149k
6
votes
2
answers
388
views
Abelian groups such that $A \cong \mathrm{End}(A)$ and "complete rings"
Motivation: for any ring $R$ there is the natural monomorphism $\mathrm{in} \colon R \to \mathrm{End}(R_{add}): r \mapsto (x \mapsto rx)$, where $R_{add}$ is an additive abelian group ( rings are ...
- 3,041
3
votes
0
answers
345
views
On a conjecture of Hartshorne
Hartshorne has a conjecture in his book Ample Subvarieties of Algebraic Varieties. It's in page 126 Conjecture 5.16 where he writes that if $B$ is a finitely generated flat module over a regular local ...
- 2,275
3
votes
0
answers
91
views
On the descent of noetherianess along completion
Let $A$ be a commutative local ring with maximal ideal $m$ and $\hat{A}$ be its $m$-adic completion. Are there any non-trivial conditions on $A$, under which $\hat{A}$ noetherian implies $A$ ...
- 541
5
votes
1
answer
524
views
Is there a non-split algebraic torus (over a finite field) satisfying the following properties?
Is there a non-split algebraic torus $T$ (over a finite field $\mathbb{F}_{\!q}$) satisfying the following properties?
$T$ is not $\mathbb{F}_{\!q}$-isomorphic to the direct product of algebraic tori ...
- 12.9k
1
vote
0
answers
41
views
A bi-variate polynomial interpolation question
Let $R$ be a commutative unital ring, and $R^{m\times k}$ denote the set of $m\times k$ matrices with entries from $R$. A matrix $U\in R^{m\times m}$ is elementary if $U$ is obtained from the identity ...
CommunityBot
- 1
2
votes
0
answers
108
views
Left-elements of a numerical semigroup generated by two elements
A numerical semigroup $S$ is a semigroup in $\mathbb{N}$ such that $\mathbb{N}\backslash S$ is finite. It is known that there exists always a set $M$ such that an element in $S$ can be expressed as a ...
- 115
3
votes
0
answers
71
views
Sums of powers in regular local rings
The following theorem is proven in Jensen-Lenzig's "Algebra, Logic, and Applications, vol 2" Sublemma 3.34.1:
Suppose that $R$ is a regular local Henselian ring with maximal ideal $\mathfrak{...
- 1,689
3
votes
0
answers
324
views
Roots of polynomials over $\mathbb{Z}/p^k\mathbb{Z}$
Over a finite field, such as $\mathbb{Z}/p\mathbb{Z}$, the number of roots of a polynomial is no larger than the degree. I'm interested in how does this generalize to $\mathbb{Z}/p^k\mathbb{Z}$.
I'm ...
CommunityBot
- 1
1
vote
0
answers
140
views
The first syzygy module of a binomial ideal
It is known how you compute the first syzygy module of a monomial ideal but it seems an hard work to do the same for binomial ones. I don't know any procedure to aim that, so I would like kindly if ...
- 31
1
vote
2
answers
1k
views
Extension class and cup product
Recall that the group $Ext^1(F'',F')$ parametrizes extensions $$0 \rightarrow F' \rightarrow F \rightarrow F'' \rightarrow 0$$ as follows: given one such extension, consider the long exact cohomology ...
- 4,492
9
votes
1
answer
743
views
Why is choice needed in Ellis' Lemma?
Ellis Lemma on idempotent elements asserts that:
Lemma (Ellis). Every compact semigroup has an idempotent.
The proof below is excerpted from Todorcevic's Introduction to Ramsey Spaces, Lemma 2.1.
...
- 16.5k
4
votes
1
answer
631
views
On the multiplicities of an ideal on a smooth variety
Let $X$ be a smooth variety, $\xi$ be a point of $X$ and $\mathfrak{a}$ be an ideal sheaf. If we define $mult_{\xi} \mathfrak{a}$ to be the largest integer $p$ such that $\mathfrak{a} \cdot \mathcal{O}...
- 35.5k
2
votes
1
answer
258
views
Possible "algebraic" direction in hyperplane arrangements
I have been studying hyperplane arrangements from R. Stanley's notes and so far, I have read until lecture 5. But, Lecture 6 and end of Lecture 5 seems very combinatorial. I'm more interested in the &...
- 24.2k
15
votes
1
answer
649
views
Primes that must occur in every composition series for a given module
Let $M$ be a finitely generated module over the commutative noetherian ring $R$. Let ${\cal C}(M)$ be the set of all primes $P$ in $R$ such that $R/P$ appears as a quotient in every composition ...
- 2,821
11
votes
2
answers
530
views
Undecidability of irreducibility of infinite families of integer polynomials?
A recent question, Is irreducibility of polynomials $\in\mathbb{Z}[X]$ over $\mathbb{Q}$ an undecidable problem? was quickly answered in the negative. I am wondering if there is a simple example of a ...
- 82.7k
3
votes
0
answers
149
views
What direction does the derivation of an inseparable algebraic variable point in?
I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
- 1,969
15
votes
2
answers
1k
views
Is irreducibility of polynomials $\in \mathbb{Z} [X]$ over $\mathbb{Q}$ an undecidable problem?
There are a number of criteria for determining whether a polynomial $\in \mathbb{Z} [X]$ is irreducible over $\mathbb{Q}$ (the traditional ones being Eisenstein criterion and irreducibility over a ...
- 12.9k
1
vote
1
answer
90
views
Affine semigroup generating a lattice
This is a cross-post from MSE.
Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
- 63.9k
7
votes
1
answer
350
views
Pushouts of injective monoid homomorphisms
Given a pushout square in the category of monoids
$$\begin{array}{ccc}A & \rightarrow & M \\ \downarrow && \downarrow \\ N & \rightarrow & P\end{array}$$such that $A \to M$ and ...
- 64k
36
votes
4
answers
12k
views
Flatness and local freeness
The following statement is well-known:
Let $A$ be a commutative Noetherian ring and $M$ a finitely generated $A$-module. Then $M$ is flat if and only if $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-...
- 4,219
7
votes
0
answers
365
views
Residue field of a ring does not depend upon the maximal ideal
Let $\mathbb{K}$ be a field and let $A$ be a $\mathbb{K}$-algebra. We will say that $A$ is residually $\mathbb{K}$ if for every maximal ideal $\mathfrak{m}$ we have that the structural morphism $\...
0
votes
0
answers
185
views
Exactness of $I$-adic completion in a certain non-finitely generated case
I would like the functor
$$(-\otimes_{\mathbb Z} F)\hat{}: \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}_{\mathrm{f.g.}}\longrightarrow \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}$$
to be exact, where completion is w....
- 91
8
votes
3
answers
740
views
Is there some example that nicely extends the multiplication of natural numbers?
Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The ...
- 10.5k
2
votes
1
answer
234
views
Algorithm for compact polynomial expressions
Sometimes an ugly polynomial (perhaps in several variables) can be expressed as a small sum of much simpler polynomials. Can this be done algorithmically? More precisely:
Is there a reasonable
...
- 63.9k
3
votes
0
answers
92
views
Solutions of the vector field $D=A\frac{\partial}{\partial X}+B\frac{\partial}{\partial Y}$ with $A,B\in k[[X,Y]]$
I am trying to understand the article Reduction of Singularities of the Differential Equation $Ady=Bdx$ by Arno van den Essen.
Let $A,B\in k[[X,Y]]$ be formal power series and $D$ the vector field $A\...
- 63.9k
4
votes
1
answer
134
views
Terminology/literature for $\forall I\leq A,\; IB\cap A=I$
I am interested in extensions $A\leq B$ of commutative rings with the property that for all ideals $I\leq A$ we have $IB\cap A=I$. Is there a standard name for this property, or a standard reference ...
- 30.5k
2
votes
0
answers
97
views
References discussing the category of ordered commutative rings
Is there a reference anywhere discussing the category of ordered commutative rings?
I'm thinking of ordered commutative rings and ring homomorphisms preserving the order, but I would also be ...
- 10.1k
7
votes
0
answers
275
views
Split epimorphism of modules - does the finite case imply the infinite case?
Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Suppose further we have an ...
- 696
2
votes
1
answer
67
views
$E$-separated semigroups
Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$.
Observe that $X$ is ...
- 42k
2
votes
2
answers
417
views
Transition maps in trivial direct limit
If $\{X_i, i\in I\}$ is a directed system of abelian groups such that we have
$$\varinjlim_{i\in I}X_i = 0$$
is it true that for every $i$ and large enough $j\ge i$ the transition map $f_{i,j} : X_i\...
- 1,424
1
vote
0
answers
66
views
Prove that $f(M)=f^2(M)$ implies $f(M)$ is a direct summand of $M$ whenever $\text{End}_R(M)$ is a reduced ring
Let $M$ be a right $R$-module with the property that every homomorphism $\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to $S\to M$. If $S$ has the property $f^2=0$ implies $f=0$ for every $f\in S$...
- 111
29
votes
2
answers
7k
views
Elementary proof of Nakayama's lemma?
Nakayama's lemma is as follows:
Let $A$ be a ring, and $\frak{a}$ an ideal such that $\frak{a}$ is contained in every maximal ideal. Let $M$ be a finitely generated $A$-module. Then if $\frak{a}$$M=M$...
- 10.5k
11
votes
2
answers
287
views
Fundamental group under Gelfand duality
Gelfand duality states that the functor of continuous functions $C(-)$ from compact Hausdorff topological to commutative $C^*$-algebras is an equivalence of categories. In other words, all topological ...
- 21.5k
18
votes
7
answers
2k
views
Superfluous definitions
It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.
For if a and b are elements of R, and writing + for the group operation then applying ...
- 4,713
1
vote
0
answers
110
views
What is some algebraic intuition behind the fact that the (real part) of the logarithm of Bernoulli umbra plus $1$, is $-\gamma$?
Bernoulli umbra is defined in classical umbral calculus as in Taylor - Difference equations via the classical umbral calculus.
Yu - Bernoulli Operator and Riemann's Zeta Function shows that $\...
- 12.9k
9
votes
1
answer
661
views
What are abelian categories enriched over themselves?
As far as I understand, an arbitrary abelian category is not enriched over itself, for example, $\mathrm{ChainComplex}(\mathrm{Ab})$ is, right? On the other hand, the categories $\mathrm{Mod}(R)$ (in ...
- 6,962
15
votes
1
answer
664
views
R-module hom a direct summand of Z-module hom?
$\DeclareMathOperator\Hom{Hom}$Fix a commutative ring $R$. For $R$-modules $M$ and $N$, there is an inclusion of abelian groups $\Hom_R(M,N) \to \Hom_{\mathbb{Z}}(M,N).$ Are there conditions on $R$ ...
- 12.9k
2
votes
0
answers
97
views
Binary unions are effective in abelian categories
Let $\mathsf{A}$ be an abelian category and $S,T\hookrightarrow M$ be two subobjects. We can naturally form the commutative square
and it's surely cartesian. (Since intersections are given by ...
- 721
1
vote
0
answers
130
views
Stein Manifold and sheaf cohomology with support in a point
Let $X$ be a Stein manifold with analytic structure sheaf $\mathscr{O}_{X}$. Let $M$ be a coherent $\mathscr{O}_{X}$-module, $x \in X$, and $U$ a Stein open containing $x$. Write $\mathfrak{m}_{x}$ ...
- 63.9k
1
vote
0
answers
173
views
Calculating multiplication in a finite dimensional algebra over $\mathbb{Q}$
Suppose $ L $ be an extension over $ \mathbb{Q} $ of degree $ n $. Let $\{e_{1},e_{2},\dots,e_{n}\} $ be a basis of this extension. Now I know the product $ e_{i}^{2} $ and $ e_{i}e_{j} $ . So we can ...
- 923
2
votes
0
answers
104
views
Does same group of units imply surjective contraction map on spectra
Let $B\subset A$ be an inclusion of Krull dimension 1 rings, and assume $A$ a Dedekind domain. I think that $\mathfrak{p}\in Spec B$ is in the image of the contraction map $\mathfrak{P} \mapsto \...
- 675
2
votes
1
answer
314
views
Algebraically characterizing morphisms of commutative rings that are a homeomorphism on the prime spectra
Let us say that a morphism $\varphi\colon A\to B$ of rings (commutative, with unit) is a homeomorphism on the (prime) spectra iff the corresponding map $\operatorname{Spec}B\to\operatorname{Spec}A$ (...
- 20.5k