All Questions
Tagged with reference-request ac.commutative-algebra
402 questions
3
votes
0
answers
47
views
Counting the monic atoms $f$ in the semiring $\mathbf N[x]$ with $f(0)=1$, bounded coefficients, and degree $k$ (in the limit as $k \to \infty$)
Let $H$ be the multiplicative monoid of the (usual) semiring of polynomials in one variable $x$ with coefficients in $\mathbf N$. Given $\alpha, k \in \mathbf N$, denote by $\mathcal A_k(\alpha)$ the ...
- 10.5k
1
vote
1
answer
264
views
Providing a grading for the polynomial ring over a commutative unital graded ring
Let $R$ be a commutative unital $G$-graded ring , where $G$ is a monoid ; then does there exist a $G$-grading on $R[X]$ such that whenever we have a commutative unital $G$-graded ring $S$ , $a \in S$ ...
user111524
6
votes
1
answer
731
views
Thick subcategories
I hope this question is not too trivial for mathoverfolw.
Let $R$ be a commutative ring (with $0\neq 1$) and $D_{Perf}(R)$ the triangulated category of perfect complexes. Let $C$ be a thick ...
- 125
7
votes
1
answer
911
views
The Mittag-Leffler condition as necessary and sufficient
Let $A_1\leftarrow A_2\leftarrow A_3\leftarrow\dotsb$ be a projective system of abelian groups with the projection maps $p_{ij}\colon A_j\to A_i$, $j\ge i$. The derived functor of projective limit $\...
- 15.6k
3
votes
1
answer
221
views
Alternating multisymmetric functions
I am looking for a reference on certain modules of invariants. I think that the question is quite natural so that I believe there should be some results already, but I am not able to find anything.
...
- 577
2
votes
0
answers
221
views
Meaning of the statement "$a\in I$ is a general element of $I$"
Suppose $I$ is an ideal in a Noetherian local ring $(R,m)$. In some papers I have seen the following statement:
"$a\in I$ is a general element of $I$".
What is the definition of general element ...
- 1,713
1
vote
2
answers
317
views
The algebra of regular functions of a quasi-affine toric variety
Let $k$ be an algebraically closed field of characteristic zero and let $X$
be a toric variety over $k$, i.e. $X$ is a normal, irreducible $k$-variety and it admits an algebraic action of a torus with ...
- 413
1
vote
1
answer
256
views
Lattice Sieving
What are some good references for Lattice Sieving in Number Field Sieve? Could someone suggest some research papers in this area?(Theoretical and Computational Perspective)
- 63
4
votes
1
answer
385
views
Which monoids can be realized as the monoid of ideals of a commutative monoid?
Let $H$ be a commutative monoid (written multiplicatively). We say that a set $I \subseteq H$ is an ideal of $H$ if $IH = I$. The set $\mathcal I(H)$ of all ideals of $H$ is made into a (commutative) ...
- 10.5k
2
votes
0
answers
62
views
Extensions of an ideal-theoretic criterion for a monoid to be BF
Let $H$ be a multiplicatively written, commutative monoid. We denote by $H^\times$ the set of units (or invertible elements) of $H$, and by $\mathcal A(H)$ the set of atoms (or irreducible elements) ...
- 10.5k
6
votes
0
answers
171
views
Name for property of mixed characteristic DVR: admits regular local homomorphism from DVR with finite residue field
Does anybody happen to know if there is already a name in the literature for the following property of a mixed characteristic DVR: that there exists a local homomorphism that is regular into the ...
- 4,111
3
votes
0
answers
98
views
Primary ideals as functions vanishing to a certain degree
Let $k$ be the field of complex numbers. Let $I$ be a primary ideal in $k[X_1,\ldots, X_n]$, let $\operatorname{rad}(I)=P$, and let $X$ be the algebraic variety corresponding to $P$.
I am trying to ...
- 3,897
2
votes
0
answers
94
views
Maximal quotient ring of a commutative ring
Let $R$ be an associative ring in which an identity
element is not assumed. A right quotient ring of $R$ is
an overring $S$ such that for each $a\in S$ there
corresponds $r\in ...
- 21
3
votes
0
answers
131
views
Classification of faithfully flat morphisms between formal power series
Let $\mathbb{C}[[z_1,\dots,z_n]]$ denote the algebra of formal power series.
I am interested in faithfully flat morphisms
$$Spec(\mathbb{C}[[z_1,\dots,z_m]])\to Spec(\mathbb{C}[[z_1,\dots,z_n]]),\, m\...
- 21.8k
4
votes
0
answers
98
views
$\lambda$-Decomposition for Connes' Cyclic Complex
Let $k$ be a field of characteristic zero, and $A$ be a commutative unital $k$-algebra. Then the cyclic homology of $A$ has a $\lambda$-decomposition:
$$HC_{n}(A)=HC_{n}^{(1)}(A)\oplus \cdots \oplus ...
- 661
1
vote
0
answers
112
views
Asymptotic stability of prime divisors
Suppose $I$ is an ideal in a formally equidimensional local ring $R.$ Let $A(I)$ and $\overline A(I)$ denote Ass$R/I^n$ and Ass$R/\overline{I^n}$ for all large $n$ respectively.
My question is
What ...
- 1,713
3
votes
0
answers
86
views
$\mathbb Z$-torsion for some quadratically presented Lie rings
$\newcommand{\Z}{\mathbb{Z}}$
I asked this question on MSE but no answer so far, so I'm also asking it here.
Let $L$ be a Lie ring (a Lie algebra over $\Z$) with generators $x_1,\dots,x_n$ and ...
- 8,524
4
votes
0
answers
218
views
map of Koszul cohomology
I am reading paper "Standard systems of parameters and their blowing-up rings", J. Reine Angew. Math. 344 (1983), 201--220 of Peter Schenzel. In proof of Theorem 3.9, page 209-the second diagram, he ...
- 2,773
6
votes
0
answers
224
views
Book or survey on Dedekind-finite rings
I'm seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring $R$ is Dedekind-finite ...
- 10.5k
4
votes
0
answers
67
views
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
5
votes
2
answers
369
views
Links between tight closure and deformation theory
I am looking for links between tight closure and deformation theory. As a sample question:
Question 1. Are there geometric interpretations in terms of deformation theory of
Frobenius rationality?
...
- 32.1k
5
votes
1
answer
208
views
Zariski openness of Newton non-degenerate polynomials
Suppose you are given a convex polyhedron $\Delta$ in $\mathbb{R}^n$ (i.e. a convex hull of finitely many points in $\mathbb{Z}^n$) and consider a finite dimensional vector space $V$ over $\mathbb{C}$ ...
- 53
4
votes
1
answer
396
views
A generalization of the discriminant of a polynomial
Let $\mathbb{K}$ be a field and let $f \in \mathbb{K}[x]$ be a monic polynomial of degree $n$. Suppose that $\alpha_1, \ldots, \alpha_n$ are all the roots of $f$ (in some algebraic closure of $\mathbb{...
user40023
5
votes
1
answer
339
views
Can completely multiplicative functions be extended to $\overline{\mathbb{Q}}$ or further?
I'm looking for a subject of study that handles the following question. I'm not the most familiar with algebra; I have a strong working knowledge and that's about it, but I've been considering ...
user78249
1
vote
0
answers
136
views
Representations of finite groups over commutative rings-question and reference request
In a textbook of representation theory I have encountered the following statement without proof:
Let $R$ be a commutative ring and $G$ a finite group. If $M$ is a simple $RG$-module then the ...
- 137
1
vote
1
answer
325
views
Sheaf Hom is flat
Let $T$ be a scheme (probably integral noetherian) and $X$ a smooth projective variety. Let $K,K',A,A'$ be locally free coherent sheaves on $X\times T$. There are exact sequences:
$0\rightarrow K\...
- 297
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
12
votes
1
answer
437
views
Is there a categorical notion of reduced commutative algebras?
A commutative ring $R$ is reduced if $r^2=0 \Rightarrow r=0$ holds for all $r \in R$. Commutative rings are precisely the commutative algebra objects in the symmetric monoidal category $(\mathsf{Ab},\...
- 5,482
9
votes
2
answers
974
views
Algebras whose subalgebras are finitely generated
Let $k$ be a commutative ring. Is there a name for those commutative $k$-algebras with the property that every subalgebra is finitely generated? (Equivalently, the partial order of subalgebras is ...
- 5,482
2
votes
1
answer
302
views
Has this notion of powers of ideals already appeared in the literature?
My question is whether the below notion of powers of ideals, which may be regarded as a weakening of the notion of symbolic power, has been already defined in the literature and, in that case, I ask ...
1
vote
1
answer
500
views
A question on Eisenbud-Green-Harris conjecture
Let $I$ be a homogeneous ideal in a polynomial ring $k[x_1,\ldots,x_n],$ $I$ contain a regular sequence $f_1,\ldots,f_n$ such that deg$(f_i)=a_i$ and $a_1\leq\ldots\leq a_n.$ Let $d$ be a non-negative ...
- 1,713
19
votes
3
answers
1k
views
How to construct a constructive proof from a non-constructive proof using prime ideals?
The sum of two nilpotent elements of a commutative ring is nilpotent. This can be checked by a direct calculation using the binomial theorem. In fact, this calculation shows the stronger statement $x^...
- 5,482
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
4
votes
1
answer
512
views
Being Cohen-Macaulay open in Hilbert scheme?
Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) ...
- 3,031
5
votes
0
answers
164
views
Traces in finite extensions of integrally closed domains
$\def\fp{\mathfrak{p}}\def\fq{\mathfrak{q}}$I'm looking for a reference for the following commutative algebra fact.
Let $A$ be an integrally closed integral domain, with field of fractions $K$. Let $...
- 156k
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
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
5
votes
0
answers
204
views
Where can I find Andre's "Cinq exposés sur la désingularisation"?
Many expositions of Popescu's desingularization theorem indicate that an other proof of this theorem can be found in
"Cinq exposés sur la désingularisation" by M. Andre, Ecole Polytechnique ...
- 7,249
0
votes
1
answer
380
views
Milnor numbers and mixed multiplicities
section 6 of the link
Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration $\{m^rJ^s\...
- 1,713
4
votes
0
answers
429
views
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
2
votes
1
answer
569
views
Field extension and nilpotent element
Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular local noetherian $k$-algebra, $B$ another local noetherian $k$-algebra, $f:A \to B$ an injective ring homomorphism of ...
- 2,126
1
vote
1
answer
436
views
Automorphisms of rings fixing all prime ideals
Let $f,g:A \to B$ be two ring homomorphisms of noetherian rings satisfying that for any prime ideal $\mathfrak{q} \subset B$, $f^{-1}(\mathfrak{q})=g^{-1}(\mathfrak{q})=:\mathfrak{p}$ and the induced ...
- 2,126
2
votes
0
answers
115
views
Need a reference/proof for computing the regularity of ideal of points in $\mathbb P^d$?
In a lecture notes on 'Cohomology modules' i read the following remark:
Given a set $X$ of points in $\mathbb P^d$,using the Local Cohomology modules one can easily compute the reg$(S_X)$ where $...
- 121
4
votes
2
answers
548
views
Irreducible algebraic sets via irreducible polynomials
There are many results about irreducible polynomials over finite fields:
we know a cardinality of all irreducible polynomials with given degree, we know explicit examples of irreducible polynomials, ...
- 2,576
2
votes
1
answer
581
views
A paper by Y. Morita
The corresponding bibliographical details are:
Yoshihito Morita, Elementary proofs of the commutativity of rings satisfying $x^{n}=x$. Mem. Defense Acad. 18 (1978), no. 1, 1–24.
Does anybody here ...
- 8,842
4
votes
1
answer
164
views
The volume around a singular isolated root when equalities are loosened
Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a double-...
- 5,971
2
votes
1
answer
509
views
Structure theorem for non-Noetherian local rings
Is there a structure theorem (like Cohen 's structure theorem) for non-Noetherian local rings?
I am adding what I am looking for as someone asked in the comment.
If $R$ is a local domain (not ...
- 271
19
votes
1
answer
2k
views
Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?
Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$
We can refer to the elements of $\mathbb{J}$ as "joiners."
The product of joiners is inherited from $\mathbb{Z}$.
The sum of joiners will be ...
- 3,793
6
votes
0
answers
672
views
Flat + locally of finite presentation + monomorphism = open immersion
It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$:
Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then $...
- 15.9k
4
votes
0
answers
202
views
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