All Questions
Tagged with commutative-algebra or ac.commutative-algebra
5,492 questions
13
votes
3
answers
1k
views
Reference for combinatorics of cell decomposition of the Hilbert scheme of points in the plane
It is known from either Morse theory or Bialynicki-Birula decomposition that the fixed points of a ${\mathbb{C}}^*$ action on a smooth algebraic variety over $\mathbb{C}$ determine a cell ...
- 156k
2
votes
2
answers
389
views
Related to fractional ideals
$K$ a field, $A\subset K$ a subring, $M\subset K$ an $A$-submodule. Define
$$(A:_{K}M):= \lbrace s\in K|sM\subset A\rbrace$$
Then it is easy to see that
$$M\subset A\Longleftrightarrow A\subset (A:_{...
- 11
2
votes
1
answer
184
views
Are there known quantitative descriptions of the fact that the common zero set of some polynomials is empty besides Nullstellensatz?
Working in a polynomial ring, if some polynomials has no common zeros, Nullstellensatz tells us a qualitative description of the propertity of these polynomials, are there known quantitative ...
- 47.4k
11
votes
2
answers
1k
views
Geometric motivation for the Stanley-Reisner correspondence
The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X_1,...,X_n]/I_\Delta
$$
where $I_\Delta$ is the ideal generated by the $X_{...
- 1,500
4
votes
1
answer
276
views
I am interested in collecting different methods of proofs that a subalgebra coincides with whole algebra.
Let $A \subset \mathbb{C}[x_1,x_2,\ldots,x_n]$ - be finitely generated graded algebra and $B$ be its subalgebra. How to prove that $A=B.?$
Unfortunalelly I know only one method to do it - to ...
- 33.8k
4
votes
1
answer
382
views
"extend a functor"
Hi,
I have probably a basic question. I have a functor $F: Sch \rightarrow Set$, an algebraic stack $M$ with a "universal family" $G\rightarrow M$ and a representability property like this: for every ...
- 11
6
votes
1
answer
640
views
The Jacobian ideal generates the socle of a complete intersection
This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here:
http://tinyurl.com/2967eov
I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ ...
- 5,846
11
votes
3
answers
972
views
Is Krull dimension non-increasing along ring epimorphisms?
Let $f \colon R \to S$ be an epimorphism of commutative rings, where $R$ and $S$ are integral domains. Suppose that $\mathfrak{p} \subset S$ is a prime such that $f^{-1}(\mathfrak{p}) = 0$. Does it ...
- 23k
4
votes
1
answer
579
views
Is there a clean definition of the residue map in Milnor K-theory?
If K is a field, v a discrete valuation, and k the residue field, there is a residue map $\partial: K^M_n(K) \to K^M_{n - 1}(k)$. All the definitions I have seen for this map involve two pages of ...
- 15.2k
3
votes
1
answer
475
views
Can the factorisation of (p) in a number field K be described by the minimal polynomial of a primitive element?
Let $K$ be a number field, with ring of integers $O_K$, and let $\alpha\in O_K$ be a primitive element for the extension $K/Q$, with minimal monic polynomial $f(x)\in Z[x]$. If $p$ is a prime number, ...
CommunityBot
- 1
1
vote
2
answers
364
views
Rig of fractions, including zero denominators
For some integral domain $R$, one forms the field of fractions $R^*$ by considering (equivalence classes of) formal pairs {$r/s : r \in R, s\in R\backslash 0$} and defining $+$ and $*$ as you'd expect ...
- 18.7k
3
votes
0
answers
591
views
Algebraic description of double vector bundles.
It is well known, by Serre-Swan theorem, that given a compact manifold M there is an equivalence of categories between the category of vector bundles over M and the category of finitely generated ...
- 51
5
votes
0
answers
210
views
Kahler differentials and the m-adic filtration
Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies $...
- 1,038
6
votes
1
answer
1k
views
Nagata's bizzare examples
Hi,
due to Nagata and his clever and bizzare examples I'm unsure in this:
1) Is there a regular ring of infinite Krull dimension?
2) Is it true that: Regular ring of finite Krull dimension = ...
- 11.6k
22
votes
4
answers
2k
views
Two questions about finiteness of ideal classes in abstract number rings
Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...
CommunityBot
- 1
7
votes
4
answers
2k
views
commuting matrices
I have a set $\{A_1, A_2, .. A_k\}$ of $n$ by $n$ real matrices and I know that they are 'perturbated versions' of a set of commuting matrices : $\{P_1,..,P_k\}$, by perturbated versions I mean that I ...
- 28.6k
18
votes
2
answers
4k
views
How to show a set of polynomials is algebraically independent?
Suppose that I have $n$ homogeneous polynomials $f_1, \dots, f_n \in \mathbb{C}[x_1, \dots, x_m]$ and that $n < m$. Is there a well known method or algorithm to determine if these polynomials are ...
- 151
3
votes
1
answer
374
views
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 ...
- 11.1k
1
vote
0
answers
451
views
Proof of local structure theory for unramified morphisms [closed]
In Raynaud's "Anneaux locaux henseliens," a proof is given of the
following fact: Let $R \to S$ be a finite type morphism of rings, $\mathfrak{q}
\in \mathrm{Spec} S$, $\mathfrak{p} $ the inverse ...
- 11.1k
1
vote
1
answer
518
views
Cardinality of a linear independent subset of a free module over a commutative ring which is not an integral domain
If R is a commutative ring with unity and not an integral domain and F is a free R-module with rank k,is there a linear independent set with cardinality > k?
I prooved that this is not true if R is an ...
CommunityBot
- 1
6
votes
1
answer
806
views
Radicals of binomial ideals
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. An ideal (that can be) generated by monomials is called a monomial ideal. For the monomial ideal $M=(m_1,m_2,.....
CommunityBot
- 1
6
votes
1
answer
356
views
Constructive Bezout cofactors in the ring of algebraic integers
We see the following in an answer to This Question : (mangled by me for my purposes, all errors my fault)
Dedekind stated (in 1871) that the ring of algebraic integers is a Bezout Domain. He calls it ...
CommunityBot
- 1
3
votes
0
answers
277
views
For which dg-algebras is a dg-module with bounded cohomology quasi-isomorphic to a bounded dg-module?
Hello!
Let S be a commutative local Noetherian base ring and A be a dg-S-algebra.
Suppose M is a bounded above dg-A-module with bounded cohomology. Does there exist a bounded dg-A-module isomorphic ...
- 2,756
5
votes
1
answer
2k
views
Calculating the normalization of an algebraic surface.
Suppose $X$ is a complex algebraic projective surface with one dimensional singular locus. For example consider the hypersurface $z^5=t^2(tx^2+y^3+t^3)$, whose singular locus is along the double line $...
- 57.6k
16
votes
5
answers
5k
views
An advanced exposition of Galois theory
My knowledge of Galois theory is woefully inadequate. Thus, I'd be interested in an exposition that assumes little knowledge of Galois theory, but is advanced in other respects. For instance, it would ...
CommunityBot
- 1
4
votes
4
answers
2k
views
Why are principal ideal domains and Dedekind domains prominent, but I always seem to see Noetherian rings rather than Noetherian domains?
It seems to me that these 3 algebraic systems are closely related, but it always seems to be Noetherian rings rather than Noetherian domains appearing, and conversely I rarely seem to see principal ...
CommunityBot
- 1
16
votes
3
answers
3k
views
Is being torsion a local property of module elements?
Say $R$ is a ring, not necessarily a domain, and $M$ is an $R$-module. All rings are commutative with 1. An element $m\in M$ is called torsion if $r.m=0$ for some regular element (non-zerodivisor) $r\...
CommunityBot
- 1
11
votes
0
answers
1k
views
Reverse mathematics strength of identically zero polynomials are the zero polynomial
According to wikipedia, the statement "every polynomial over a countable field that is not the zero polynomial has only finitely many roots" is equivalent to RCA0 over RCA0* (which is called ERCA-0 in ...
CommunityBot
- 1
3
votes
0
answers
240
views
Flatness of a homogeneous quasi-coherent sheaf on the formal plane from flatness on lines through the origin?
(This is a follow-up to a previous question; as often happens, I didn't include enough hypotheses, so I'm asking a new, more-likely-to-be-true question).
Consider the formal plane $\operatorname{Spec}...
CommunityBot
- 1
11
votes
1
answer
1k
views
Can ⨁_I A be isomorphic to ∏_I A for infinite I?
Suppose $A$ is a non-zero ring (say commutative unital) and $I$ is an infinite set. Can it happen that there is an isomorphism of $A$-modules $\bigoplus_{i\in I}A\cong \prod_{i\in I}A$?
The obvious ...
- 20.8k
1
vote
1
answer
268
views
Flatness on the formal plane from flatness on lines through the origin?
Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of ...
CommunityBot
- 1
3
votes
1
answer
928
views
How exotic can DVRs be in the ring of rational functions over a local field?
Suppose that $R$ is a complete DVR with field of fractions $K$, uniformiser $\pi$ and residue field $k$.
Let $B$ be a subring of the ring $K(t)$ of rational functions over $K$. Moreover assume that $...
- 11.1k
15
votes
4
answers
1k
views
What formal properties should resolution of singularities have?
If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
- 30.5k
12
votes
1
answer
5k
views
intersection of ideals in a commutative ring vs their product
This question was inspired by this one. Given two ideals $A,B$ in a finitely generated commutative ring $R$. Is it possible to decide whether $A\cap B=AB$? Here $R$ is given by generators and ...
CommunityBot
- 1
6
votes
1
answer
761
views
Nontrivial criteria for polynomials to have no common zeros?
When we work in $C[x_1,x_2,...,x_n]$,here $C$ denotes the complex field, we know that when polynomials $f_1,f_2,...,f_k$ have no common zeros, then there exists polynomials $g_1,...,g_k$ , such that ...
- 39.9k
4
votes
0
answers
1k
views
Commutative ring Notes by M. Artin
In 1966, Professor Michael Artin gave a course for first-year graduate students at MIT on commutative algebra. In that course he covered many classical topics, (the Spectrum of a commutative ring, ...
- 165
0
votes
1
answer
251
views
What is a certain cartesian product of algebras?
Suppose $F$ is a field and $A$ the $F$-algebra $F[X]\times_{(F\times F)} F$ given by the missing corner of a cartesian square in $F$-algebras
\begin{equation}
F~\xrightarrow{\Delta} ~F\times F~ \...
- 5,846
2
votes
0
answers
261
views
On a characterization of the symbolic square of prime ideals in polynomial rings
If $R=k[x_1,...,x_n]$ is a polynomial ring in $n$ indeterminates over a field $k$ of characteristic $0$, there is a characterization of the symbolic square of a prime ideal $P$ (the $n$-th symbolic ...
- 805
4
votes
0
answers
338
views
What to call the following variant of tame ramification
Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...
- 20.5k
4
votes
2
answers
354
views
"un-nil-ifying" ideals via deformation
This is perhaps a naïve question; given an irreducible scheme $X$, is there a general procedure to find a flat family $Y \to T$ such that over some point $t_0$ we have $Y \times_T {t_0} \cong X$ but ...
- 66.3k
7
votes
0
answers
249
views
Does there exist a commutative ring R such that SL_3(R) and SL_2(R) have the same finite subgroups?
This question is inspired, of course, by this question, and I don't know enough commutative algebra to know whether it's answered by silence dogood's answer to this follow-up question. If the answer ...
CommunityBot
- 1
5
votes
0
answers
517
views
Monomial-type ideals in polynomial rings
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ...
- 805
7
votes
2
answers
566
views
Rational powers of ideals in Noetherian rings
Let $R$ be a Noetherian ring, and let $I$ be an ideal of $R$. Fix a rational number $a=\frac{p}{q}$ with $p, q\in \mathbb{Z_\geq 0}$ $q\neq 0$. We define $I_a = \{x \in R: x^q\in \overline{I^p}\}$, ...
- 805
0
votes
1
answer
329
views
What is correct name of the following construction?
Consider an ideal $I=\langle f_1,f_2,\ldots,f_s\rangle$ in the polynomial ring $\mathbb{Q}[x_1,x_2,\ldots,x_n].$ Build the following set
$$
\{ g_1 f_1+g_1 f_2+\cdots+g_n f_n \},
$$
where $g_i$ ...
- 2,537
6
votes
1
answer
1k
views
Explicit injective resolutions of (Laurent) polynomial rings
Hi,
Despite being nothing but the dual notion of projective resolution, injective resolutions seem to be harder to grasp. For example, the general form of Poincaré-Lefschetz duality given in Iversen'...
- 22.6k
5
votes
1
answer
959
views
Multiplicative Structures On Free Resolutions
Hello,
this question is related to Differential graded structures on free resolution?.
Given a regular local ring $S$ and $f\in{\mathfrak m}_S\setminus\{0\}$, I am interested in studying $R$-modules ...
CommunityBot
- 1
4
votes
2
answers
670
views
term for a "faithful" module
Is there a term for an $A$-module $M$ such that $M \otimes_A -$ takes nonzero modules to nonzero modules?
Motivation: It is a standard theorem that if $B$ is faithfully flat over $A$, then $\hbox{...
- 23k
4
votes
1
answer
2k
views
Arithmetically Cohen-Macaulay varieties
What do we mean by a variety being arithmetically Cohen-Macaulay? Is every such variety also Gorenstein?
- 20.5k
2
votes
0
answers
1k
views
Decomposition group vs Galois group of completed extension for height > 1 primes
Assume
Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions.
Let $S$ be a finite $R$-algebra, $L$ its field of fractions.
$L/F$ a (finite) Galois extension
$S$ normal in $L$
...
- 21
5
votes
1
answer
327
views
When is the projective line the seminaive projective line?
Excuse the possible naivete of this question. Since reading a nice survey article by Daniel Biss a few years ago, I'm always worried about what $P^1(R)$ is, for a ring $R$.
So that I stop worrying, ...
- 13.3k