All Questions
Tagged with commutative-algebra or ac.commutative-algebra
5,492 questions
4
votes
1
answer
711
views
Faltings' category of almost modules
Hi,
Let $V$ be an integral domain with an ideal $m\subset V$ and put $K = S^{-1}V$ where
$S = {1}\cup m$ (a multiplicatively closed subset). Is it true that the category of almost
$(V,m)$-modules is ...
- 57.6k
8
votes
2
answers
217
views
Flipping Hilbert series of semigroup rings
I'll first give intuition, and then give a precise statement.
For $|z|<1$, we have $\sum_{i \geq 0} z^i = 1/(1-z)$. For $|z|>1$, we have $\sum_{i<0} (-1) z^i=1/(1-z)$. Thus, the two ...
- 156k
5
votes
1
answer
220
views
When are these rings regular?
Let $R$ be a noetherian regular domain. Suppose that $a, b \in R$, with $b \neq 0$, and consider the ring $S:=R[\frac{a}{b}]=R[X]/(bX-a)$. Is $S$ regular? If this is not the case are there some ...
- 5,846
7
votes
3
answers
2k
views
Products of Ideal Sheaves and Union of irreducible Subvarieties
Assume I have a nonsingular, irreducible, algebraic variety $X$ and irreducible, nonsingular subvarieties $Z_1,\ldots,Z_k\subseteq X$. Let $\mathcal{I}_i$ be the ideal sheaf of $Z_i$ and $\mathcal{I}:=...
- 39.3k
1
vote
0
answers
138
views
Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
10
votes
1
answer
1k
views
Formally smooth morphisms, the cotangent complex, and an extension of the conormal sequence
I'm reading Daniel Quillen's paper "Homology of commutative rings," in which he proves:
A finitely presented morphism of rings $A \to B$ is
Formally etale iff $L_{B/A}$ (this denotes the cotangent ...
- 19.6k
2
votes
0
answers
245
views
Is simplicity preserved under completion of the base ring?
Let $(A,\mathfrak{m})$ be a noetherian local ring and $R$ be an $A$-algebra, which is finitely generated generated as an $A$-module (module finite $A$-algebra). Let $\widehat{A}$ be the $\mathfrak{m}$-...
- 1,391
37
votes
3
answers
3k
views
What does it mean geometrically that an element in a domain is irreducible?
Consider a domain $A$ and a non-zero element $f\in A$. That element $f$ is prime if and only if the subscheme $V(f)\subset \operatorname{Spec}(A)$ is integral and this is a completely satisfactory ...
- 20.8k
2
votes
1
answer
455
views
Generic liftings of a regular sequence on the initial ideal
Hi everyone,
I've got a question about explicitly lifting regular sequences. Let $I$ be an ideal in a polynomial ring $S$ with some term order. We'll denote the initial ideal by $in(I)$. It is ...
- 46
4
votes
1
answer
661
views
Modules with flat duals
Let $R$ be a commutative ring, $M$ an $R$-module, $M^*=Hom_R(M,R)$ its dual. What are sufficient (and possibly necessary) conditions on $M$ that ensure that $M^*$ is flat? Is there a name for such ...
CommunityBot
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3
votes
1
answer
268
views
Universal catenarity and Laurent algebras
A Noetherian (commutative) ring $A$ is called universally catenary if every $A$-algebra of finite type is catenary. If one wants to know whether $A$ is universally catenary, then this definition ...
- 19.6k
6
votes
1
answer
2k
views
Hochschild and cyclic homology of smooth varieties
Many of the standard sources which discuss the Hochschild Kostant Rosenberg theorem and cyclic homology for smooth varieties such as Loday and Weibel's paper "The Hodge Filtration and Cyclic Homology" ...
- 8,385
6
votes
1
answer
1k
views
reference for p-local and p-complete integers
Can anyone suggest a good thorough reference for $p$-localization and $p$-completion of the integers? I'm an algebraic topologist who's found himself washed up without any intuition.
In particular, ...
- 45.7k
2
votes
0
answers
506
views
Zariski's Main theorem [closed]
Sir,
I am studying Zariski's Main theorem. May i know good source of problems related to Zariski's main theorem to understand it better?
Thanks in advance.
- 161
3
votes
0
answers
474
views
Jacobson-Bourbaki correspondence
The Jacobson-Bourbaki correspondence induces the traditional, finite Galois correspondence by suitable restriction; I've been pondering two things: 1. Are there any (other) interesting applications of ...
- 52.9k
3
votes
0
answers
473
views
Infinite Galois correspondence "according to Artin"
Ever since Artin's lectures on Galois Theory one knows how to set up and derive the usual Galois correspondence in the finite(-dimensional) case using just a bit of elementary Linear Algebra, and ...
9
votes
2
answers
667
views
irreducibility of generic linear combination of polynomials?
I would be shocked if the following were not true, but I can't seem to see a proof.
Claim:
Let $R$ be an integral domain containing an AC (uncountable if you wish) field $k$. Let $a, b \in R$, and ...
- 30.5k
3
votes
1
answer
320
views
Decision problem about the existence of solution for an integer matrix equation
Given $A,B,C \ $ integer matrices of dimensions $l \times m$, $l \times n$ and $l \times m$, we want to decide (algorithmically) about the existence of $X$ (unimodular) and $Y \ $ integer matrices ...
CommunityBot
- 1
10
votes
1
answer
3k
views
Rings with all modules projective ?
Is there a classification of the commutative rings (with unit) such that each module over the ring is projective ?
- 13.6k
12
votes
2
answers
3k
views
Vector bundles on affine scheme
I have already asked similar questions before, but now I realized that there a nice
general way to ask what I want. Namely let $X$ be a normal affine variety over
a field $k$. Assume first that $k$ is ...
- 23.3k
6
votes
2
answers
850
views
Decomposition of finite algebras over finite fields
Let $K$ be a number field, $Z_K$ its ring of integers, and $p$ a rational prime number. Then $A_p = Z_K/(p)$ is a finite ${\mathbb F}_p$-algebra. Using ideal arithmetic in $Z_K$ and the Chinese ...
9
votes
1
answer
1k
views
Fixing a mistake in "An introduction to invariants and moduli"
On page 13 of the book "An introduction to invariants and moduli" of Mukai
http://catdir.loc.gov/catdir/samples/cam033/2002023422.pdf there is a mistake, in the end of the proof of Proposition 1.9. ...
CommunityBot
- 1
5
votes
2
answers
491
views
Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
1
vote
1
answer
272
views
Criteria for Preservation of a Module Structure under Extension of Scalars.
Let $A\to B$ be a morphism of (commutative) algebras and $M$ a $B$-module. The $A$-bilinear map $B\times M\to M$ given by $(b,m)\mapsto bm$ induces a surjective homomorphism $B\otimes_{A}M\to M$.
...
- 5,846
9
votes
3
answers
1k
views
Quasi-compact maps in Number Theory
Can someone give me an example of a non-quasi-compact morphism of schemes which arises naturally in the field of Algebraic Number Theory?
- 19.6k
2
votes
2
answers
492
views
Model Theoretic Localization
This is a re-post on a previous question I asked. My first question was too vague to warrant detailed responses. Really, I have two specific questions to ask.
1) Let $\sigma = (A; \{0,1\}; +, \times)...
- 73.1k
2
votes
1
answer
521
views
Kahler differentials of a hypersurface over a non-algebraically closed field
The following was recently on my algebraic geometry homework:
Let $k$ be an algebraically closed field, $f\in B=k[x_1,\ldots,x_n]$, and $A=B/(f)$. Show that $\Omega_{A/k}$ is locally free of rank $...
- 3,968
6
votes
1
answer
970
views
Reflexive sheaves on singular surfaces
Let $S$ be a normal surface over an algebraically closed field $k$ and let
$s$ be a point of $S$. Let ${\mathcal F}$ be a reflexive sheaf on $S$ of generic rank $n$ . Consider the (derived) fiber of $...
- 156k
9
votes
2
answers
1k
views
Non-Standard Prime
Hello,
My question is about the non-standard models of the integers. If we add to the Peano's axioms $P$ of arithmetic the following axioms for a fixed constant $c$:
$c \neq 0$, $c \neq 1$, $c \neq 1+...
- 47.1k
8
votes
1
answer
3k
views
When a tensor product of two local rings is a local ring?
This is a follow-up to Is tensor product of local algebras local?.
Let $A, B$ and $C$ be local rings (commutative and noetherian). Suppose that we have local ring maps $C \to A$ and $C \to B$.
What ...
CommunityBot
- 1
3
votes
1
answer
546
views
Center of the category of $R$-algebras
Let $R$ be a ring (assumed to be commutative and with unit). What is the center of the category of $R$-algebras, i.e. $\text{Z}(R\text{-Alg})$? This is a commutative monoid. See this recent question ...
CommunityBot
- 1
4
votes
1
answer
2k
views
Unsolved problems concerning Artinian Rings and Artinian Modules
I am preparing a write-up on Topics on Artinian Rings and Modules for a project. I hope to mention some unsolved problems in the domain of these objects along the way. Till now, I have been able to ...
CommunityBot
- 1
4
votes
0
answers
1k
views
An example of a noetherian ring in which the integral closure of a finite extension of its field of fractions is not noetherian
Similar to another question I posted. Does anyone know of an example of a noetherian ring in which the integral closure of a finite extension of it's field of fractions is not noetherian.
- 113
15
votes
4
answers
6k
views
how to determine whether an ideal is prime or not by an algorithm
Given polynomials $f_{1},\cdots,f_{n}\in \mathbb{C}[x_{1},\cdots,x_{m}]$, do we have an algorithm to determine whether the ideal $I=(f_{1},\cdots,f_{n})$ is prime ideal or not? Of course, we assume ...
- 11.6k
3
votes
2
answers
666
views
Normality and rational singularities via Hilbert series
Let $A$ be a finitely generated ${\mathbb Z}_{\geq 0}$-graded algebra over a field without zero divisors;
assume that all graded components are finite-dimensional and that $Spec(A)$ is smooth
outside ...
- 7,037
4
votes
1
answer
284
views
When is Out$(SL_n(R))$ a torsion group ?
This question is a follow up question to this question. So my question is:
For which rings $R$ (commutative, with unit) (and which integers $n$) is $Out(SL_n(R))$ a torsion group? A consequence of ...
CommunityBot
- 1
2
votes
2
answers
406
views
Extending a polynomial function from an open subset
I am a bit embarrassed to ask this question, but still: assume that I have
a finite morphism $\pi:X\to Y$ of affine algebraic varieties over a field (probably
finiteness is too strong an assumption, ...
- 481
2
votes
1
answer
504
views
Zero-dimensional algebras of infinite vector space dimension
Consider an algebra $A$ over a field and suppose that $A$ is zero-dimensional as a ring. It is well-known that if, in addition, $A$ is finitely generated, it has a finite vector space dimension. ...
- 31.2k
2
votes
1
answer
577
views
An example of a noetherian N-1 ring that is not N-2 and/or a Nagata ring
Hello is there anyone that would know where I can find an example of a noetherian N-1 ring that is not a Nagata ring. (See the Wikipedia article "Nagata ring" for the definitions of N-1 ring and ...
- 113
8
votes
1
answer
289
views
Top degree local cohomology under action by a non-zerodivisor
Let $R$ be a noetherian commutative ring of dimension $n$, and let $M$ be a faithful finite $R$-module. Let $I$ be a proper ideal of $R$, and let $x\in I$ be a non-zerodivisor on $M$.
When does ...
- 7,328
7
votes
0
answers
518
views
An elementary question in singularities
The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...
- 3,758
4
votes
3
answers
2k
views
Chevalley's valuation extension theorem and the axiom of choice
Hello,
Do we know if the axiom of choice is needed for Chevalley's valuation/place extension theorem (i.e. the theorem that states that for every valued field and a field extension, one can extend ...
CommunityBot
- 1
8
votes
0
answers
493
views
"Consecutive" irreducible polynomials
If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then
it is easy to see that for any integer $m$, at least one of the polynomials
$P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb Z}...
- 3,595
2
votes
1
answer
186
views
Behaviour of Primes under Regular Coefficient Extensions
Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x_{1},\cdots,x_{n}]\hookrightarrow K'[x_{1},\cdots,x_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring ...
- 1,645
33
votes
2
answers
7k
views
Noetherian rings of infinite Krull dimension?
Since Noetherian rings satisfy the ascending chain condition, every such ring must contain infinitely many chains of prime ideals s.t. the heights of these chains are unbounded.
The only example I ...
- 161
8
votes
1
answer
721
views
Is this a characterization of Dedekind domain?
Let $R$ be an integral domain. Suppose that for any two nonzero ideals $I$ and $J$, we have $I \oplus J$ is isomorphic to $R \oplus IJ$ as $R$-modules. Does this implies $R$ is a Dedekind domain?
- 6,262
0
votes
0
answers
544
views
isomorphism between vector spaces and modules - Commutative Algebra
Hi, Let $M_i$ be A modules. Then we know that $Ass (\oplus M_i) = \bigcup Ass(M_i) $. We consider here isomorphisms between modules.
Now consider a stanley ...
- 287
2
votes
2
answers
282
views
ring of idempotents of the integral extension of a ring
For any commutative ring $A$, the set of idempotents of $A$ will be denoted as $E(A)$. This set has a (canonical) ring structure. With addition defined by:
$$e+'f=e(1−f)+f(1−e)$$
where $+$ and $−$ are ...
- 2,275
4
votes
2
answers
817
views
What are the units of $\mathbb{Z}/4\mathbb{Z}[x]$?
What are the units of $\mathbb{Z}/4\mathbb{Z}[x]$? Anything of the form $\pm 1 + 2 x p(x)$ for $p(x) \in \mathbb{Z}_4[x]$ works, and is in fact its own inverse. It's easy to see that any unit must ...
- 63.1k
3
votes
1
answer
494
views
universal finite differential module of affinoid algebra
Let $k$ be a value field (archimedean), for example $k = \mathbb{Q}_p$, the p-adic field.
The free Tate algebra is $$ T_n := \left\{ \ \sum a_I X^I, \ a_I \in k, \ a_I \rightarrow 0 \text{ as } |I| \...
CommunityBot
- 1