All Questions
1,947 questions with no upvoted or accepted answers
10
votes
0
answers
854
views
Scholze's infinite to finite type ring theory reductions?
In following essay "The Perfectoid Concept: Test Case for an Absent Theory" by Michael Harris, there is the following sentence I found to be quite striking.
The most virtuosic pages in Scholze's ...
- 119
10
votes
0
answers
722
views
Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 109k
10
votes
0
answers
729
views
Mumford's intuition for flatness
In Mumford's book Algebraic Geometry II, he writes on page 179..."In order to get at what I consider the intuitive content of "flat" we need first a deeper fact..."
After the deeper fact is proven he ...
- 431
10
votes
0
answers
409
views
Higher Adeles of a scheme and related topics
Let $X$ be a noetherian scheme. I will describe a construction of a simplicial ring which I think is called the Bellinson higher Adeles complex (or something similar).
Consider the augmented ...
- 7,789
10
votes
0
answers
1k
views
What is the etale fundamental group of Spec Z((x))?
I know the etale fundamental group of $\mathbb{Z}$ is trivial. For algebraically closed fields $K$, the etale fundamental group of $K((x))$ is $\hat{\mathbb{Z}}$, since all covers in this case are ...
- 10.7k
10
votes
0
answers
573
views
Singularities arising from the Minimal Model Program (an algebraic point of view)
I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?...
- 987
9
votes
0
answers
273
views
What is known about vector subspaces of polynomial rings closed under factors?
Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is:
If $a,b \in F$, then $a+b \in F$, and
If $a,b \in R$ with $a\in R$ ...
- 1,802
9
votes
0
answers
188
views
Surjectivity of a bilinear map $A^m\times A^n\to A$ for a polynomial ring $A$
Let $k$ be a field and $A:= k[x_1, \dots, x_d]$.
Question: Suppose $M$ is an $m\times n$ matrix over $A$. If the entries of $M$ generate the unit ideal of $A$, must there exist $a\in A^m, b\in A^n$ ...
- 191
9
votes
0
answers
180
views
How should we picture the set of monomial orders (= positive monoid orders on $\mathbb{N}^k$)?
Motivation: So apparently there's some sort of sport competition currently going on where I live, which leads to countries being given an element of $\mathbb{N}^3$ called a “medal count”, and not ...
- 32.5k
9
votes
0
answers
267
views
Can "$\exists\mathcal{X}(R\cong C(\mathcal{X}))$" be expressed in "large" infinitary second-order logic?
Originally asked and bountied at MSE without success:
Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ...
- 21.1k
9
votes
0
answers
460
views
Does the book "Algebra III" exist (within the Encyclopaedia of Mathematical Sciences series from Springer)?
Within the series "Encyclopaedia of Mathematical Sciences", as published by Springer, one finds the 8 volumes, namely,
the volumes I, II, IV, V, VI, VII, VIII, IX but zbMath has no listing ...
- 161
9
votes
0
answers
366
views
Proof of Artin–Rees / Krull intersection motivated by universal property of blowup
I was very confused by the proof of Artin–Rees / Krull intersection theorem when I was younger.
Now that I learnt about blow up— I saw the Rees algebra again and I want to now gain a better ...
- 515
9
votes
0
answers
441
views
Commutative algebra details on patching when proving $R = \mathbb{T}$ theorem (Calegari-Geraghty Paper)
I have originally posted this on math.SE and been suggested to post this here. I'm merely an undergraduate student and it is the first time for me to ask questions here. I'm sincerely sorry if these ...
- 639
9
votes
0
answers
204
views
Reverse mathematics of Noetherian rings over $\mathbb{Q}$
Take the Hilbert Basis Theorem over the rational numbers in this form in the language of Second Order Arithmetic: For every $n\in N$ every ideal of the polynomial ring $\mathbb{Q}[x_1,\dots,x_n]$ is ...
- 11.1k
9
votes
0
answers
204
views
Standard reference/name for "initial ideals $\Leftrightarrow$ associated graded rings"
Let $R$ be commutative ring with a $\mathbb Z$-grading $\deg$ and let $I\subset R$ be an ideal. On one hand, we may consider the initial ideal $\mathrm{in}_{\deg}(I)$. That is the space spanned by ...
- 3,513
9
votes
0
answers
263
views
Is $[JK:(x)][JK:(y,z)]\subseteq JK$ in $k[x,y,z]$?
Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $J,K\subseteq m$ be proper homogeneous ideals in $S$. Is this true that we always have:
$$[JK:(x)][JK:(y,z)]\subseteq JK \ ?$$
Some ...
- 30.5k
9
votes
0
answers
321
views
Irreducibility of the Sylvester resultant
If $r$ and $s$ are positive integers, $R$ a commutative ring and $a_0,\dots,a_r$, $b_0,\dots,b_s$ independent variables, we can consider the polynomials $f=\sum_{i=0}^ra_iX^i$ and $g=\sum_{j=0}^sb_iX^...
- 47.3k
9
votes
0
answers
165
views
When is the rank 2 free metabelian group of exponent $n$ center free?
Let $M_n$ be the rank 2 free metabelian group of exponent $n$. For which $n$ is $M_n$ center-free?
The abelianization $M_n^{ab}\cong C_n\times C_n$, so the commutator subgroup $M_n'$ is a cyclic $(\...
- 7,527
9
votes
0
answers
239
views
Which semirings have enough injectives in their category of modules?
Let $R$ be a semiring and $Mod_R$ its category of modules. That is, $R$ is a monoid in the monoidal category of commutative monoids and $Mod_R$ is its category of modules in the usual sense.
Question ...
- 63.9k
9
votes
0
answers
425
views
Is the class of commutative generalized Euclidean rings stable under quotient and localization?
Let $R$ be an associative ring with identity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. Let ...
- 7,893
9
votes
0
answers
420
views
Geometric interpretation of minimal number of generators of a module
Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, \...
- 243
9
votes
0
answers
400
views
Weierstrass division theorem for henselian rings
Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...
- 3,472
9
votes
0
answers
260
views
Is the generation of rings by their units a question in K-theory?
Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...
- 13.3k
9
votes
0
answers
520
views
Getting a bound via polynomial equations
When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$,
\begin{cases}
&\sum\limits_{j=0}^{m-1}x_jx_{2k-j}=...
- 767
9
votes
0
answers
2k
views
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
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
9
votes
0
answers
316
views
When is $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}$?
Suppose $(R,\mathfrak{m})$ is a noetherian local ring. I am interested in ideals $\mathfrak{a}$ of $R$ for which $$\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\...
- 3,101
9
votes
0
answers
1k
views
When UFD implies PID
The following result is too elementary, both to state and to prove, not to be known. Can someone give a reference? Is there any hope if you don't suppose UFD (i.e. move that from the hypothesis to ...
- 907
9
votes
0
answers
278
views
Uncountable Lüroth problem
Question. Let $F(X)$ be the field obtained by adding an uncountable collection of indeterminates (mutually transcendental elements) to a prime field $F$. Is there an example of a subfield $E$ of $F(X)$...
- 17.7k
9
votes
0
answers
349
views
Computing Ext for toric divisors
Ok, I have an affine (normal) toric variety $X = \text{Spec} k[\sigma]$. Suppose that $F, G$ are two torus invariant Weil divisors on $X$. Is there any relatively straightforward way to compute
$$
\...
- 20.5k
9
votes
0
answers
512
views
E(n) Deformations of the infinity category Qcoh(X) with it's E(n)-tensor product
Let $X$ be a smooth scheme, then an infinity enchancement of $QCoh(X)$ has an $E_\infty$ structure and in particular an $E_n$ structure for any $n$. In this paper, http://arxiv.org/abs/0805.0157 Ben-...
- 3,758
9
votes
0
answers
281
views
Krull rings and determinantal invariants
During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
- 22.1k
8
votes
0
answers
185
views
Ring of invariants for graph automorphism
$\DeclareMathOperator\Aut{Aut}$Let $G$ be a finite simple graph with nodes numbered $1$ to $n$. Attach variables $x_1, ..., x_n$ to nodes. The graph automorphism group $\Aut G$ acts on nodes by ...
- 645
8
votes
0
answers
291
views
Image of multiplication map in tensor powers of finite-dimensional ring
Let $R$ be a (commutative, unital) ring of dimension $n$ over a field $k$. Assume the characteristic of $k$ is greater than $n$.
Then $R^{\otimes n}$ has a natural ring structure, together with an $...
- 148k
8
votes
0
answers
397
views
A criterion for rational singularities in mixed characteristic
Let $R$ be a mixed characteristic discrete valuation ring with perfect residue field and $f:X \to \mathrm{Spec}(R)$ a flat proper morphism.
If the generic fibre of $f$ is smooth and the special fibre ...
- 10.5k
8
votes
0
answers
336
views
Passing to torsion of an exact sequence
If
$$
\Theta\colon\quad 0\to A\to B\to C\to 0
$$
is an exact sequence of abelian groups, and $n$ is an integer, then one obtains an exact sequence $$
0\to A[n] \to B[n] \to C[n] \stackrel{\delta_n(\...
- 13k
8
votes
0
answers
179
views
Rlim versus tensor product
Let $R$ be a coherent ring, and let $(M_n)_{n\geq 1}$ and $(N_n)_{n\geq 1}$ be two inverse systems of finitely generated flat $R$-modules. If $R^1 \lim M_n=R^1 \lim N_n = 0$, is it true also that $R^1 ...
- 13.1k
8
votes
0
answers
313
views
Cohomology of the complement of the resonance hyperplane arrangement
Here was a question about resonance arrangement. It is defined as follows.
Let $x_i$ be the standard coordinates on $\mathbb{C}^n$. For each nonempty $I\subseteq\{1,\dots,n\}$, define the hyperplane $...
- 743
8
votes
0
answers
265
views
Chevalley restriction theorem: group vs lie algebra version
Let $G$ be a (split) reductive group over $k$, $T$ a split maximal torus, and W its Weyl group. I sometimes see the Chevalley restriction theorem stated as
(1) $k[G]^G \xrightarrow{\sim} k[T]^W$
and ...
- 689
8
votes
0
answers
220
views
Finitely generated commutative rings with the same profinite completion
Let $R_1$ and $R_2$ be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: $\widehat{R_1}\cong \widehat{R_2}$.
Suppose that $R_1$ is a domain. Does ...
- 1,418
8
votes
0
answers
119
views
Catenarity and epimorphisms of rings
Let $R$ be a commutative ring. The following are well-known:
If $R$ is catenary and $\mathfrak{a}\subseteq R$ is an ideal, then $R/\mathfrak{a}$ is catenary.
If $R$ is catenary and $S\subseteq R$ is ...
- 6,700
8
votes
0
answers
548
views
Foundational Questions on Adic Spaces
There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...
- 2,923
8
votes
0
answers
500
views
Curvilinear locus in the Hilbert scheme of points
Let $X$ be a smooth complex projective variety of dimension $d$. Then the Hilbert scheme of $n$ points $X^{[n]}$ is not irreducible in general, but it has always the main component $X^{[n]}_{sm}$ of ...
- 577
8
votes
0
answers
438
views
If $A$ is normal and $\Omega^1_{B/A}=0$ then $B$ is normal
Let $A\subseteq B$ be two noetherian domains with fraction fields $k$ and $L$, respectively. Assume that $A$ is normal and $B$ is finite as $A$-module. I'm asking myself if $B$ is also normal if $\...
- 1,252
8
votes
0
answers
127
views
universally open and connected fibers
Let $A$ be a coherent ring, and consider the map:
$Spec(A[[t]])\rightarrow Spec(A)$,
in particular, we know that it's flat. Is it universally open? Does it have connected fibers?
N.B: Easy if A is ...
- 3,472
8
votes
0
answers
213
views
"Rings" with partially defined addition in Algebra or Algebraic Geometry?
Were there ever considered sets $P$ with associative multiplication and a partially defined commutative, associative addition, $+: U\to P,$ $U\subset P\times P$, such that $x(y+z)=xy+xz$ when the left ...
- 2,390
8
votes
0
answers
575
views
Polynomial maps over $\mathbb{Z}$
It is know that an injective polynomial map $f:\overline{\mathbb{Q}}^{n} \longrightarrow \overline{\mathbb{Q}}^{n}$ is an bijection with inverse regular (Cynk-Rusek theorem). My question is following:
...
- 348
8
votes
0
answers
480
views
Connections and curvature in commutative algebra
Since on any commutative algebra $R$ over ring $S$ we have module of Kahler differentials $(\Omega_{R/S},d)$ which extends to the algebraic de-Rham complex $(\Omega^\bullet,d),$ it is natural to ...
- 1,615
8
votes
0
answers
350
views
What does the characteristic polynomial of an element in a finite flat $R$-algebra tell you?
Let $R$ be a noetherian ring, and $B$ a finite locally free $R$-algebra. Since $B$ is locally free, for every $b\in B$, multiplication by $b$ gives an $R$-linear endomorphism of $B$ as a locally free $...
- 7,527
8
votes
0
answers
439
views
Involutions on power series $\mathbb{C}[[X_1,\ldots,X_n]]$
For the ring of formal power series $\mathbb{C}[[X_1,\ldots,X_n]]$ over complex numbers,
is it true that any automorphism of order $2$ is after change of co-ordinates
given by $X_i\mapsto \pm X_i$?
- 161