All Questions
Tagged with reference-request ac.commutative-algebra
402 questions
4
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0
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67
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Existence of infinitely many pairwise non-associate atoms in a ring of polynomials in $k$ variables over a Dedekind-finite unital ring
The following comes as a by-product of a more abstract result, and I'm essentially looking for a reference to it (or to something more general than it).
Corollary. Let $R$ be a non-trivial Dedekind-...
- 10.5k
4
votes
0
answers
429
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Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms
I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...
- 1,755
4
votes
0
answers
202
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Shifts in the decomposition of Bott-Samelson bimodules
Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\...
- 878
4
votes
0
answers
133
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Reference request for $R$-index
Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ ...
- 3,009
4
votes
0
answers
842
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Commutative algebra books representing the edge of research
Recently I have come across the books Combinatorial Commutative Algebra by Miller and Sturmfels along with Combinatorics and Commutative Algebra by Stanley. I will soon own a copy of each. I also ...
- 437
4
votes
0
answers
130
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Is this duality operation on simplicial complexes/Stanley-Reisner rings previously known?
Let $K$ be an abstract simplicial complex on vertices $x_1,\ldots,x_n$, then there is the familiar construction of the face ideal $I_K=\langle x_{i_1}\cdots x_{i_r} | \{x_{i_1},\ldots,x_{i_{r}}\}\not\...
- 1,488
4
votes
0
answers
110
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maximal degree of generators of graded ideals
Let $A$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal. Let $R(\mathfrak{a}) = \oplus_{n\geq 0} \mathfrak{a}^n$ be the Rees algebra of $\mathfrak{a}$. Let $I$ and $J$ be two graded ideals ...
- 2,773
4
votes
0
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188
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A non-matroidal notion of dependence on a set of ideals
Assume we are given a set of ideals $I_1, \dots, I_s$ in a commutative polynomial ring. Let's define a subset indexed by $A\subseteq [s] = \{ 1,2,\dots, s\}$ as dependent if there exists an $a\in A$ ...
- 1,961
3
votes
2
answers
571
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Free Resolution of this determinantal variety.
I am looking for a free resolution of the ideal generated by $2\times 2$-minors of a $3\times 3$ -matrix. More precisely let $M$ be a matrix (sorry but I cannot write a matrix for some TeX technical ...
- 31
3
votes
2
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470
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An identity in an arbitrary commutative ring
This fact might be either trivial, wrong, or well known.
Let $R$ be a commutative ring. Let $u_1,\dots,u_{s-1},u_s\in R$ and $m,M\in R$. Let us assume that $m,M$ satisfy
$$(m-u_1) \dots (m-u_{s-1})=0,...
- 21.8k
3
votes
3
answers
714
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Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
3
votes
2
answers
524
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Product of reduced affinoid spaces over a field is reduced (reference request)
Let $K$ be a field of characteristic zero complete with respect to a non-Archimedean absolute value. Suppose that $A$ and $B$ are two affinoid $K$-algebras. I'd like a reference that will answer the ...
- 408
3
votes
1
answer
836
views
Solving multilinear equations
Let $N=\{1,2,\ldots,n\}$. Suppose we are given $n$ equations, with each equation taking the form $\sum_{A\subseteq N}\left(c_A \prod_{i\in A}x_i \right) = 0$, where each $c_A$ is a real number ...
- 239
3
votes
2
answers
488
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Application of sheaves theory in ring theory
Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?
- 49
3
votes
2
answers
519
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Counterexample to Openness of Flat Locus
Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $$U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$$ is open in $\operatorname{...
- 3,101
3
votes
1
answer
618
views
When is an almost geometric quotient flat?
All varieties here are over $\Bbb C$. Let $G$ be a reductive algebraic group acting algebraically on affine $n$-space $\Bbb A^n$. Let $R$ be the coordinate ring of $\Bbb A^n$. Assume that the natural ...
- 3,079
3
votes
1
answer
313
views
Irreducibility of family of polynomials
Consider the following family of polynomials over $\mathbb{Q}$:
$$f_n = x^n - x^{n-1} - \dots - 1$$
Notice that these polynomials satisfy the recurrence
$$ f_{n+1} = x f_n - 1 $$
I would like to ...
- 173
3
votes
1
answer
387
views
Term for an "almost regular" sequence
Let $R$ be a ring (commutative, with unit), $M$ an $R$-module, and $x_1, \dotsc, x_n \in R$. Consider the following two conditions:
For all $i$, the homomorphism $$\frac{M}{(x_1, \dotsc, x_{i-1})M}...
- 7,328
3
votes
1
answer
436
views
Stone topological Boolean algebras
I am looking for an initial reference for a theorem which is known, namely:
Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological ...
- 774
3
votes
1
answer
336
views
Finitely generated subrings of $\mathbb{R}$ are finitely approximable
In Ivanov's Finite Approximability of Modular Teichmüller Groups, for the proof of Lemma 2, the following is stated:
Let $G$ be a finitely generated group and $\tau: G \to \operatorname{PSL}(2,\...
- 243
3
votes
1
answer
374
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Composition and intersection of residue fields
Let $A$ be a normal ring with quotient field $K$. Let $L/K$ be a finite separable extension.
Let $E_1/K$ and $E_2/K$ be extensions of $K$ contained in
$L$. Let $B_1$ (resp. $B_2$) be the normalization ...
- 1,673
3
votes
1
answer
193
views
Definability of nilradical in the model theory of rings
I am looking for a reference dealing with the first-order definability of the nilradical of a commutative ring. The only thing I have found so far is an exercise in Wilfrid Hodges' book Model Theory (...
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
3
votes
1
answer
216
views
Simple reference for valuative criterion of integrality?
I'd like to see a complete proof of the simplest version of the following rough statement: "If $f/g$ is a rational function on a reduced scheme ($g$ not a zero divisor), and $f/g$ doesn't have poles ...
- 27.8k
3
votes
1
answer
591
views
Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S$ be a Noetherian ring which is a faithfully flat $R$-algebra. If $M,N$ are finitely generated $R$-modules such that $M\otimes_R S \cong N \...
- 31
3
votes
1
answer
221
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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.
...
- 577
3
votes
4
answers
1k
views
Matrix factorization categories for ADE singularities
What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.
Background: For ADE singularities, see for example this. For ...
- 21k
3
votes
1
answer
212
views
Flatness of finitely presented algebras
Let $R$ be a commutative (noetherian, if needed) ring, let $f_1，\ldots，f_r\in R[x_1，\ldots，x_n]$ and $A=R[x_1，…，x_n]/(f_1，\ldots，f_r)$, when is $A$ flat over $R$?
I found a nice answer for the case $n=...
- 1,906
3
votes
1
answer
332
views
Algebraic vector bundles on the punctured spectrum: an exact reference for a result
Let $(R, \mathfrak m)$ be a Noetherian local ring of depth at least $2$. Let $X=Spec(R)$ denote the affine- scheme with structure sheaf $\mathcal O_X$ and $U=Spec(R)\setminus \{\mathfrak m\}$ be the ...
- 642
3
votes
1
answer
451
views
Commutative algebra for the Conway games
I was reading the book On Number And Games and I have some question. In this book Conway constructed the set of "games" with a addition and a multiplication. I understand that the surreal numbers are ...
- 527
3
votes
1
answer
173
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$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.5k
3
votes
1
answer
606
views
Prime ideals and localizations of the ring $\mathbb Z[\{\sqrt p: p \text{ prime}\}]$
I have been trying to study the prime ideals of the ring $R:=\mathbb Z [\{ \sqrt{p_n}\}_{n=1}^\infty]$, where $p_n$ denotes the $n$-th prime. This is how far I got: I could conclude, by means of the ...
- 739
3
votes
1
answer
280
views
Composite families of formal power series over $\mathbb C$ as algebraic variety
I was led to prove that the set of composite families $(f_j)_{j \leq k}$ of germs at $0\in \mathbb C^m$ of a holomorphic function (composite = sharing a common divisor belonging to the maximal ideal) ...
- 5,428
3
votes
2
answers
426
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Reference Request: Smith Normal Form for maps between free _graded_ modules
I feel like this should be easy, but I cannot quite find a literature reference for this:
We know (i.a. from the Kaplansky reference in Does Smith normal form imply PID?) that sufficient for Smith ...
- 2,537
3
votes
1
answer
496
views
Regular ring is smooth when the field is perfect
Take $A$ a (not necessarily local) commutative algebra over a field $k$ which is essentially of finite type (i.e. a localization of a finitely generated algebra). In simple words, I just want to know ...
3
votes
1
answer
178
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On the degree of the Hilbert polynomial of a graded module over the Rees algebra
If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module ...
- 279
3
votes
1
answer
153
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Artinian Tor modules (Reference request)
I am looking for a reference for the following basic fact:
Let $R$ be a noetherian ring, let $M$ be an artinian $R$-module, let $N$ be a finitely generated $R$-module, and let $i\in\mathbb{N}$. ...
- 6,700
3
votes
1
answer
422
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Tor dimension in polynomial rings over Artin rings
I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
- 165
3
votes
1
answer
571
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Reference for submultiplicativity of length of tensor product
I am looking for a reference, in the form of a textbook, that contains proofs of following statements.
NOTE: I am NOT looking for the proofs, I am looking for a reference! Proofs of these statements ...
- 3,101
3
votes
1
answer
293
views
Freeness of modules along ring homomorphisms
This question arises from my discussion with a Master student. It concerns with the following situation: let $\phi: R \to S$ be a homomorphism between Noetherian commutative rings. Suppose the $R$-...
- 30.5k
3
votes
0
answers
74
views
Locally compact rings with reciprocals
A topological field is defined to be a topological ring $F$ with reciprocals such that the reciprocal function $F\setminus\{0\} \to F\setminus\{0\}$ is continuous. Locally compact topological fields ...
- 974
3
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0
answers
151
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Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion
(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
- 10.5k
3
votes
0
answers
70
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Degree of an even/odd part of a formal power series over a polynomial ring
Let $K$ be a field with $\operatorname{char}K\ne 2$ (say, $K=\mathbb{R}$ or $\mathbb{C}$) and consider a formal power series $f=f(x)\in K[[x]]$ such that $[K[x,f]:K[x]\,]=d$. Suppose $f_e,f_o\in K[[x]]...
- 1,190
3
votes
0
answers
173
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Intersection theory on schemes with Gorenstein singularities
Is there a good reference/book on Intersection theory on schemes with Gorenstein singularities? Does the construction of the intersection of cycles discussed in Fulton's book also hold for schemes ...
user497552
3
votes
0
answers
248
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Explicit computation of hyper Ext in terms of the homologies of the input chain complexes
This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
- 301
3
votes
0
answers
68
views
Finding generators and relations for special commutative algebras with a computer
Let $K[x_1,...,x_n]$ be the polynomial ring in $n$ variables and $a_1,...,a_m$ elements in the quotient field $K(x_1,...,x_n)$.
Let $A:=K[a_1,...,a_m]$ the ring generated by the $a_i$ in $K(x_1,...,...
- 26.5k
3
votes
1
answer
177
views
Quiver and relations for ADE singularities in dimension one
Let $A$ be an ADE-hypersurface singularity in dimension one.
For example in Dynkin type $A_n$, A is given by $K[[x,y]]/(x^2+y^{n+1})$.
Then $A$ is CM-finite and let $M$ be the direct sum of all ...
- 26.5k
3
votes
0
answers
243
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Quick proof of the first part of Kaplansky's Theorem on characterization of Noetherian domains with all maximal ideals principal
I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-...
- 739
3
votes
0
answers
69
views
On Ext-duals of injective modules for commutative rings
Let $R$ be a commutative noetherian ring and $I=E(R/p)$ the injective hull of the module $R/p$ for a prime ideal $p$.
Question: Is there a (more) explicit description of the $R$-modules $Ext_R^i(I,R)$...
- 26.5k
3
votes
0
answers
98
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Hales' generalization of the stacked bases theorem (seeking a proof)
In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ...
- 2,992