All Questions
6,056 questions
0
votes
1
answer
166
views
Blow up and critical points of the projection map
Denote $Z=V(x_1, \dots, x_{n-1}) \subset \mathbb{C}^n$ and let $Bl_Z(\mathbb{C}^n)$ be the blow up of $\mathbb{C}^n$ along $Z$ together with the projection map $\pi \colon Bl_Z(\mathbb{C}^n) \to \...
- 39.3k
9
votes
3
answers
3k
views
Is the tensor product of regular rings still regular
An imprecise version of the question is that when $A$ and $B$ are regular rings, is $A \otimes B$ also regular? Please allow me to put more restrictions, here I am only interested in the case when $A$ ...
- 55
3
votes
1
answer
315
views
Intersection and tensor product of flat modules
Let $R$ be a commutative ring, $M$ be an $R$-module, and $N$ be a submodule of $M$. Assume that both $M$ and $N$ are flat, so we can identify $N\otimes_RN$, $M\otimes_RN$, and $N \otimes_RN$ as ...
- 38k
8
votes
1
answer
494
views
The removed statement from the original edition (Matsumura's "Commutative Ring Theory")
From Theorem 13.7 ii) in the original edition (written in Japanese) of Matsumura's "Commutative Ring Theory," the following statement is got rid of:
Let $A = \bigoplus_{n \geq 0}A_n$ be a ...
- 24.2k
5
votes
1
answer
421
views
Ring of continuous functions is a Jacobson ring
Let $X$ be an infinite discrete topological space. Is $$C_b(X)=\{ f \colon X \to \mathbb{R} \text{ bounded }\}$$ a Jacobson ring ?
7
votes
1
answer
578
views
What is the reason for $f_!$ not preserving discrete objects?
Let $A$ be a finitely generated $\mathbb{Z}$-algebra and let
$f: \operatorname{Spec} A \rightarrow \operatorname{Spec} \mathbb{Z}$ be the canonical map.
On pg. 53, Thm. 8.2 of https://www.math.uni-...
- 121
6
votes
0
answers
867
views
How to extend Ritt's theorem on elementary invertible bijective elementary functions?
The elementary functions according to Liouville and Ritt are the functions of a complex variable built up by applying exponentiation, logarithms and/or algebraic operations finitely often. That means, ...
- 1,053
2
votes
0
answers
464
views
Minimal number of generators of kernel of a matrix of polynomials over $\mathbb Q$ and $\mathbb Z$
$\DeclareMathOperator\im{im}$I have asked some related questions before on math.SE here and on MathOverflow here (answered here). This post is self-contained.
Let $R' = \mathbb Q[x_1,\dotsc,x_n]$, and ...
- 277
3
votes
1
answer
111
views
Projectivity of some module
Let $k$ be a algebraically closed field and suppose that $A$ and $B$ are finite dimensional $k$-algebras. If we assume that $A$ is a symmetric $k$-algebra and $A\otimes_k I$ is a projective $A\...
- 1,313
12
votes
2
answers
941
views
An analogue of the Bass-Quillen conjecture with power or Laurent series
The famous Quillen-Suslin theorem (formerly known as Serre's problem/conjecture) states that every projective module over $k[x_1,\dots, x_n]$ is free for $k$ a field. Replacing $k$ by a more general ...
- 1,313
1
vote
1
answer
133
views
Height of truncated system of parameter
Let $R$ be a Noetherian local ring of dimension $d$, and $a_1,\dots,a_d$ is a system of parameters. I am wondering whether the following statement is true:
$\mathrm{ht}(a_1,\dots,a_i)=i$ for all $i$, ...
- 1,313
1
vote
1
answer
315
views
Constructing a free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-module using a related free resolution of a $\mathbb Q[x_1,\dotsc,x_n]$-module
I have asked a related question on math.SE here, but the notation is a bit different.
As the title says, I am interested in constructing a finite free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-...
- 173
6
votes
1
answer
201
views
Does the Peskine–Szpiro intersection theorem imply Krull's ideal height theorem?
I came across these notes from a talk by Hoschter which talks about superheight of an ideal and it mentions Krull's ideal height theorem on P2-P3 in terms of superheight. Here are the notes: http://...
- 173
1
vote
0
answers
72
views
Factorizable partition polynomials
Let $p(n)$ denote the number of (unrestricted) integer partition of $n$ which has the product generating function
$$\sum_{n\geq0}p(n)\,x^n=\prod_{j\geq1}\frac1{1-x^j}.$$
On the other hand, for the ...
- 43.2k
2
votes
1
answer
195
views
Reference request: example of pointwise free module which is not projective
I am looking for a reference for examples showing the following phenomena:
Let $A$ be a commutative noetherian ring, and let $F$ be an $A$-module such that for all $p \in Spec(A)$ it holds that $F_p$ ...
3
votes
0
answers
213
views
A question related to the polynomial ring $R[x]$ over some principal ideal domain $R$?
Let $R[x]$ be the polynomial ring over some principal ideal domain $R$. If $R[x]/I$ is free as a $R$-module for some ideal $I$, is $I$ a principal ideal which is generated by some monic polynomial in $...
- 775
7
votes
1
answer
254
views
Group-like elements in quotients of group rings
$\DeclareMathOperator\Gr{Gr}$Let $R$ be a local ring, let $A$ be a finite abelian group, and let $I$ be a Hopf ideal of the ring $R[A]$. The quotient $R[A]\twoheadrightarrow R[A]/I$ induces a map on ...
- 308
1
vote
1
answer
95
views
Hilbert - Samuel multiplicity of $B$ when there is a surjection $A \rightarrow B$
The setting is: Let $A, B$ be commutative, Noetherian, local rings, $\phi:A \rightarrow B$ a surjective homomorphism. Both rings also come with surjections $\lambda_A, \lambda_B$ to a DVR $\mathcal{O}$...
- 301
1
vote
0
answers
280
views
Algebraic independence criterion
Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
- 13.9k
0
votes
0
answers
139
views
Primary ideals and radical of an ideal
Let $R$ be a regular local ring (for example, $R=\mathbb{C}\{x_1, \dots, x_n\}$) and let $\mathfrak{p}$ be a prime ideal in $R$.
Given an ideal $\mathfrak{a} \subset R$ such that $\sqrt{\mathfrak{a}}=\...
28
votes
3
answers
3k
views
Equivalent definitions of invertible modules
Let $R$ be commutative unital ring, and $M$ an $R$-module. $M$ is called invertible (a.k.a. projective module of rank one), if it is finitely generated, and $M_{\mathfrak{p}} \cong R_{\mathfrak{p}}$ ...
- 4,713
12
votes
1
answer
1k
views
Vanishing of $\mathrm{Ext}^i_R(N, R)$
$\DeclareMathOperator\Ext{Ext}$This question is related to Are the supports of $Ext^i(M,N)$ eventually periodic? .
Let $(R, \mathfrak{m}, k)$ be a Noetherian local ring of dimension $d$. It well known ...
- 63.9k
0
votes
1
answer
137
views
Monomial order and initial ideals
Let $S=K[x_1,\ldots, x_n]$ polynomial ring. Let $I \subseteq S$ an ideal and $<$ be a (global) monomial order in $S$. If in$_<(I)$ a radical ideal, then in$_<(I)=$ in$_<(P_1) \;\cap$ in$_&...
- 177
1
vote
1
answer
154
views
Grothendieck group and faithfully flat morpshim
For regular schemes $X$ and $Y$, and a faithfully flat morphism $f:Y \to X$, there is a flat pullback map of Grothendieck groups:
$$
f^*:K^0(X) \to K^0(Y).
$$
Is this map injective?
- 6,262
8
votes
5
answers
2k
views
$\lambda$-ring structure defined for a graded ring in Fulton–Lang's book
Given a commutative ring $A$ with unity, Grothendieck used universal polynomials to define a special $\lambda$-ring structure on $\Lambda(A):=1+t\:A[[t]]$. Suppose $A$ is graded, say $A=\bigoplus_{i=0}...
- 22.3k
2
votes
0
answers
173
views
Quotient Cohen-Macaulay ring and associated primes
Does there exist such (local) Cohen-Macaulay Noetherian ring $(R,m)$ with some $p\in \operatorname{Ass}R$ so that $R/p$ is not Cohen-Macaulay? (I am especially curious about the case where $R$ is a ...
- 5,429
8
votes
3
answers
379
views
Irrationality measure of formal power series
Hi everybody. I'm looking for an analogue of irrationality measure for formal power series with integer coefficient, the elements of $\mathbb{Z}[[x]]$. For any $f \in \mathbb{Z}[[x]]$ and positive ...
- 1,446
3
votes
1
answer
263
views
On the exactness of some completed tensor products
Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/...
- 541
16
votes
5
answers
2k
views
Why is the Hochschild homology of k[t] just k[t] in degrees 0 and 1?
Background: the Hochschild homology of an associative algebra is the homology of the complex
$$ \ldots \longrightarrow A \otimes A \otimes A \longrightarrow A \otimes A \longrightarrow A$$
where ...
- 35.5k
24
votes
1
answer
4k
views
Minimal number of generators of a homogeneous ideal (exercise in Hartshorne)
In the very first chapter Hartshorne proposes the following seemingly trivial exercise (ex. I.2.17(ii)):
Show that a strict complete intersection is a set theoretic complete intersection.
Here are ...
- 1,313
10
votes
0
answers
463
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 ...
- 4,713
4
votes
1
answer
242
views
Comparing the first-order theories of different kinds of local rings of a complex variety
Let $X$ be a complex variety containing some point $x$. Then $X$ is naturally a complex-analytic space, and we have an inclusion of rings $\mathbb{C}[X]_x\hookrightarrow\mathbb{C}\{X\}_x\...
- 1,969
5
votes
1
answer
419
views
Multiple root of resultant
Let us suppose that we have two polynomials $F_1(x,y)$ and $F_2(x,y)$. Generally speaking, each of them defines a curve on the plane and the system of polynomial equations defined by them computes the ...
- 66.3k
4
votes
0
answers
211
views
Diagonalization over valuation rings
Let $\mathcal{R}$ be a valuation ring, and consider an $\mathcal{R}$-linear endomorphism $L:\mathcal{R}^{n}\rightarrow \mathcal{R}^{n}$. Is there any criterion for telling when $L$ can be diagonalized?...
- 541
4
votes
0
answers
108
views
Gröbner deformations
Let $I \subseteq S=\mathbb{k}\left[x_{1}, \ldots, x_{d}\right]$ be an ideal, $<$ be a monomial order on $S$ and let $T=S[t]=\mathbb{k}\left[x_{1}, \ldots, x_{d}, t\right]$. There exists $\omega \in ...
- 177
4
votes
1
answer
714
views
Jacobian criterion for algebraic independence over a perfect field in positive characteristics
It is well known that the Jacobian criterion for algebraic independence does not hold in general for fields of positive characteristics. However, the following partial statement seems promising:
...
- 156k
2
votes
0
answers
90
views
Monomial order and prime ideals
Let $S=K[x_1,\ldots, x_n]$ polynomial ring. Let $I \subseteq S$ an ideal and $<$ be a monomial order in $S$. Is it possible to describe the minimal primes of in$_<(I)$ from the minimal primes of ...
- 177
3
votes
1
answer
255
views
How to determine the degree of a rational function field over a relatively algebraic subfield?
Let $K$ be a field and $K(x_1,\cdots,x_n)$ be the degree-$n$ purely transcendental extension of $K$. Given homogeneous polynomials $f_1,\cdots,f_n\in K[x_1,\cdots,x_n]\setminus K$ with $\deg f_i=d_i$,...
- 5,339
29
votes
1
answer
1k
views
Is the Golomb countable connected space topologically rigid?
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
- 63.9k
3
votes
1
answer
164
views
Explicit computation of D-modules pullback
Consider the $D_{\mathbb{A}^1}$-module $M:=D_{\mathbb{A}^1}/(x)$, and the map $f:z\mapsto z^k$. I want to know $f^*(M)$. I believe it only has a single non-zero cohomology, namely in degree $0$, which ...
- 149k
62
votes
5
answers
10k
views
Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)
Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?
Basically I want to believe I can ...
- 64k
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 ...
- 2,821
2
votes
0
answers
80
views
Prime elements in integrally closed extension of domains
Let $R \subseteq S$ be an extension of integral domains such that $R$ is integrally closed in $S$. Let $P$ be a prime ideal of $R$.
Is $PS$ always a prime ideal of $S$?
For classical examples of ...
- 365
10
votes
0
answers
455
views
What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?
Consider the following statement:
If $K\subseteq L$ is a Galois extension of fields with Galois group $G$ and $x \in \mathbb{P}^n(L)$ is such that $\sigma(x)=x$ for all $\sigma\in G$, then $x \in \...
- 32.5k
4
votes
0
answers
213
views
Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
16
votes
2
answers
4k
views
A geometric reference for (affine) Gorenstein varieties and singularities
I would like to ask for a reference to some text that explains in relatively down to earth (if possible geometric) terms (for dummies) what is a Gorenstein singularity and Gorenstein variety (for a ...
- 35.5k
6
votes
0
answers
516
views
Quasi-syntomic descent and prismatic F-crystals
I am reading Bhatt and Scholze's paper on F-crystals, and they seem to be using the following result in the proof of Theorem 5.6:
let $X \to Y$ be a quasisyntomic cover of formal schemes over $\...
- 461
6
votes
1
answer
260
views
Vanishing linear combinations of minors
Let $V$ be the set of $k$ by $n$ matrices ($k<n$) with entries in $\mathbb{C}$, and let $\mathbb{C}[V]$ denote the set of polynomial functions on $V$. For any subset $I \subseteq [n] = \{1,2,\dotsc,...
- 7,300
6
votes
3
answers
411
views
Problem 0.9.10 in Cohn's "Free Ideal Rings and Localization in General Rings" (CUP, 2006)
Let $S$ be a monoid. On p. xvii of P.M. Cohn's Free Ideal Rings and Localization in General Rings (CUP, 2006), one reads that
an element $u \in S$ is regular if (quote) "[...] it can be ...
- 10.5k
6
votes
1
answer
272
views
Groebner Bases for submodule over polynomial ring with integer coefficients
It is well-known that there exist Groebner bases for ideals in polynomial ring $\mathbb Q[x]$ which can be found algorithmically Moreover, I don't think it is hard to show that there exist Groebner ...
- 1,446