All Questions
Tagged with commutative-algebra or ac.commutative-algebra
5,493 questions
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
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
4
votes
1
answer
643
views
An application of Zorn's lemma.
Suppose that $R$ is a commutative ring, an $R$-module $M$ is said to be finitely embedded if $M$ has a finitely generated essential socle. Now Let $M$ be finitely embedded and not
artinian, let $S$ ...
- 21
4
votes
2
answers
1k
views
Diagrams consisting of triangles and squares
S. Lang gives a statement on page x of his 'Algebra':
Most of our diagrams are composed of triangles and squares as above, and to verify that a diagram consisting of triangles and squares is ...
- 85.6k
12
votes
0
answers
2k
views
Finding ideals of $\mathbb{Z}[x]$ generated by $n$ elements and no fewer
As the title says, I'm trying to find ideals of $\mathbb{Z}[x]$ generated by $n$ elements and no fewer. I suspect $(2^k, 2^{k-1} x, 2^{k-2} x^2, ..., x^k)$ is generated by no fewer than $n=k+1$ ...
2
votes
2
answers
983
views
Torsion in tensor products over noncommutative rings
I know that the problem of torsion in tensor products, even of torsion free modules, is a very delicate thing. Unfortunately i don't have a deeper insight into this subject, so i don't know how things ...
- 3,702
5
votes
0
answers
769
views
Looking for a reference for a generalization of the Weierstrass preparation theorem
I am looking for a reference for the following generalization of the Weierstrass preparation theorem for formal power series. Suppose that $A$ is a noetherian complete local ring with residue field $k$...
- 27k
3
votes
2
answers
344
views
Pseudo-idempotent matrix generating a free module
Let $R$ be a commutative ring with $1$. Let $n$ and $k$ be nonnegative integers, and let $A\in\mathrm{M}_n\left(R\right)$ be a matrix such that $A\cdot R^n\cong R^k$ as $R$-modules. Assume that $A^2=\...
- 6,919
4
votes
2
answers
467
views
Maximal separable extensions of residue fields
Assume that $(A,m)$ is a Noetherian normal local domain, $K = Quot(A) \subset E, F$ Galois extensions of $K$. If $B=\overline{A}^{E}$, $C=\overline{A}^F$, and $D=\overline{A}^{EF}$ and we choose ...
- 19.6k
3
votes
2
answers
804
views
A problem for finite dimensional commutative algebra
Let $(A,m)$ be a local commutative associative algebra over the field of complex numbers, $m^n\ne 0$, $m^{n+1}=0$ for some $n>0$, and
(1) $A$ is finite dimensional as vector space
(2) for any ...
- 6,262
1
vote
1
answer
307
views
A problem on Moebius transformations
We have the following result:
Let $R=\mathbb{C}[t]_f$, with $f=(t-a_1)(t-a_2)\cdots (t-a_n)$. Then the automorphism group of $R$ is isomorphic to the group of all Moebius transformations which fix (...
- 66.3k
10
votes
3
answers
2k
views
A question about an application of Molien's formula to find the generators and relations of an invariant ring
In the very beginning of the book "Introduction to Invariants and Moduli" Shigeru Mukai
proves Molien's formula for the Hilbert series of the invariant ring of a finite group action on $\mathbb C^n$. ...
- 11.6k
9
votes
3
answers
2k
views
If a polynomial f is irreducible then (f) is radical, without unique factorization?
Is there a short way to prove that for each irreducible polynomial $f$ in $k[x_1,...,x_n]$ the principal ideal $(f)$ is radical without using unique factorization of polynomials? A short proof of this ...
CommunityBot
- 1
13
votes
1
answer
3k
views
When are complex polynomial maps almost surjective?
Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
- 17.1k
0
votes
0
answers
254
views
What is Castelnuovo-Mumford regularity of this algebra?
Let $M=\mathbb{C}[f_1,f_2,\ldots,f_r]$ is finitely generated algebra, $f_i \in S:=\mathbb{C}[x_1,x_2,\ldots,x_n],$ $\deg(x_i)=1, 1<\deg(f_i)<99.$ Suppose that minimal free resolution of $...
- 301
2
votes
1
answer
996
views
Count the number of homogeneous polynomials
Is there a general way of counting the number of homogeneous polynomials of certain degree in a complex projective space or a weighted complex projective space, mod the ideal generated by some ...
- 11.6k
7
votes
1
answer
757
views
Characterizing intersection of zero sets of elementary symmetric polynomials on R^n
Stated simply, the question is:
Consider two elementary symmetric polynomials $\sigma_{k}$ and $\sigma_{k+1}$ on $\mathbb{R}^{n}$ with zero sets $U_{k}$ and $U_{k+1}$. Let $V_{i_{1}i_{2}\dotsb i_{j}...
- 544
5
votes
5
answers
4k
views
Unique factorisation and the fact that $\mathbb A^2-0$ is not an affine variety?
While learning commutative algebra and basic algebraic geometry and trying to understand the structure of results (i.e. what should be proven first and what next) I came to the following question:
...
CommunityBot
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4
votes
0
answers
367
views
criteria for reduced fibres
I was wondering if it is foolish to ask if there is a criteria on a morphism $f: X \to Y$ between separated schemes of finite type over a perfect field which will assure that all the scheme theoretic ...
- 1,347
0
votes
1
answer
379
views
Is a tensor product of two dvrs semilocal?
Under what conditions is the tensor product of two dvrs semilocal?
The same question about being reduced.
Tensor product is taken over another dvr or over a field to make things simpler.
For ...
- 343
14
votes
2
answers
2k
views
Explicit ring of differential operators for polynomial algebras over the integers?
Does anyone know of a reference or have any idea for an explicit description of the ring of differential operators for polynomial algebras over the integers? I'm hoping there is something analogous to ...
- 3,702
12
votes
2
answers
1k
views
Graded or stacky Serre duality
I am considering the following situation. $A$ is a finitely generated ring over a field $K$ with non-negative grading and $A_0=K$ of Krull dimension n+1, but I don't necessarily assume A is generated ...
- 44.7k
12
votes
4
answers
940
views
Factorizing polynomials in $\mathbf{Z}[[x]]$
Let $f(x)\in\mathbf{Z}[x]$ be a non-constant, irreducible polynomial, and let $\alpha \in\mathbf{C}$ be a root of $f(x)$. Denote by $\varphi_\alpha:\mathbf{Z}[x]\rightarrow\mathbf{C}$ the ring ...
- 20.8k
14
votes
0
answers
899
views
Frobenius upper shriek/flat of a dualizing complex
Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite.
...
- 20.5k
9
votes
2
answers
1k
views
Projective resolution of modules over rings which are regular in codimension n
All rings are Noetherian and commutative, modules are finitely generated.
It is a theorem of Serre that over a regular ring $R$, every module has a finite projective resolution.
More generally, if $...
- 30.5k
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 423
8
votes
2
answers
425
views
Doing explicit computations with coordinate rings
Suppose that we are given an integral $k$-algebra $A$ of finite type explicitly, by which I mean that we are given the generators of the defining ideal $J$ where $A = k[x_1,...,x_n]/J$. What kinds of ...
- 20.5k
1
vote
1
answer
162
views
systems of parameters vs. minimal "exhausting" systems in a Noetherian local ring
Hello,
Probably this is a very easy question.
Fix a Noetherian local ring $A$, and an $A$-module of finite type $M$.
Lets call a system $ x_1 , \ldots , x_m \in \mathfrak{m}$ $M$-exhausting, if $M / ...
- 3,137
5
votes
1
answer
2k
views
module of differentials of formal power series ring and of its field of quotiens
For any $A$-algebra $B$ ( commutative ring with 1 ), we have the existence of $\Omega_{B/A}$, the module of relative differentials of $B$ over $A$, which can be defined by an universal property. In ...
- 1,214
4
votes
4
answers
596
views
Generalization of Jordan Decomposition for Several Commuting Operators
Recently I became curious about the following question:
Let $V$ be a finite dimensional vector space over $k$ and let $A_1, \cdots, A_n: V \rightarrow V$ be a set of commuting maps. Question: ...
- 9,132
2
votes
3
answers
1k
views
General hyperplane sections and projection from a point
Let $k$ be an algebraically closed field, and consider some subscheme $X\subset \mathbb{P}_k^n$. Let $x$ be a closed point of $X$, and $H$ a general hyperplane containing $x$. There is a regular map $\...
- 42.9k
7
votes
1
answer
1k
views
Are squarefree monomial ideals on a regular system of parameters in a regular local ring radical?
Suppose $(R,m)$ is a regular, local ring. Let $x_1,x_2,...,x_n$ be a regular system of parameters. Let $I$ be an ideal generated by squarefree monomials in the $x_i$'s. Is $I$ a radical ideal? The ...
CommunityBot
- 1
6
votes
2
answers
456
views
Immerse an affine schemes into $A^n_S$
Suppose $f: X\rightarrow S$ is of finite type, S is Noetherian. Now X=Spec B is affine, but the morphism f is not an affine morphism. S is not affine (or really f does not factor through any affine ...
- 11.1k
3
votes
1
answer
398
views
Is the first part of Eisenbud's Proposition 15.15's proof o.k?
In the chapter on Gröbner bases from Eisenbud's "Commutative Algebra" the following statement appears as Proposition 15.15 (page 344):
Let $F$ be a free $S$ module with basis and monomial order ...
- 4,280
8
votes
2
answers
537
views
Prime avoidance in adjacent degrees
Let $\mathfrak{p}_1, \dotsc, \mathfrak{p}_k$ be relevant homogeneous primes ideals in the graded ring $R := \Bbbk[x_0, \dotsc, x_n]$, where $\Bbbk$ is a field. Prime avoidance (in Eisenbud's ...
- 11.1k
20
votes
1
answer
2k
views
Tropical homological algebra
Has anyone out there thought about homological algebra over the tropical semifield $\mathbb{T}$? For example, I'm interested in the Hochschild homology and cyclic homology of tropical algebras, if ...
- 24k
3
votes
0
answers
336
views
Antisymmetric functions of the roots of unity: an elementary conjecture
Let $z_1, z_2, \cdots z_N$ be $N$ variables obeying the condition $z_i^M=1$ for some positive integer $M>N$.
Let $F_N$ be the space of antisymmetric polynomials of these variables. Given a set $E = ...
- 378
20
votes
3
answers
2k
views
Is every integral epimorphism of commutative rings surjective?
That's the question. Recall that a morphism $f\colon A\to B$ of commutative rings is integral if every element in $B$ is the root of a monic polynomial with coefficients in the image of $A$ and that $...
- 5,039
3
votes
2
answers
534
views
An easy example of a (1/quasi-)Gorenstein ring with non-trival canonical divisor class.
Suppose that $R = S/I = k[x_1, \dots, x_n]/I$ is a (normal) domain of finite type over a field (or any semi-local ring $k$ with a dualizing complex). In this case, I can define $\omega_R = \textrm{...
CommunityBot
- 1
15
votes
2
answers
2k
views
prime ideals in regular local rings
Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. ...
- 30.5k
1
vote
1
answer
320
views
covers of complete regular local rings
It is well-known that if one assumes algebraic closedness and characteristic 0 of the residue field then finite covers of complete DVRs are all of the form $A[x]/(x^m-a)$ for some $a \in A$ (direct ...
- 22.6k
1
vote
1
answer
924
views
Torsion-free and torsionless abelian groups
This question is motivated by my most spectacular answer on MO (:
Let $A$ be a module over $\mathbb Z$. $A$ is said to be torsion-free if $na=0$ implies $n=0$ or $a=0$ for any $n\in \mathbb Z, a\...
CommunityBot
- 1
1
vote
1
answer
601
views
Unimodular column property
Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property.
I would like to know if there is a ring $R$ that doesn't ...
- 23k
0
votes
0
answers
198
views
why a reduced ring can be embedded into a sum of integral rings?
Hi,
the question is exactly
"why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?"
Is this simply because in the normalization process we can have many irreducible ...
- 141
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 91
5
votes
0
answers
994
views
Maximal ideals in polynomial rings over algebraically closed fields - when Weak Nullstellensatz does not apply
Weak nullstellensatz describes maximal ideals in polynomial rings over algebraically closed fields at least when the cardinality number of variables is finite. Lang obtained the same conclusion also ...
- 17.6k
1
vote
0
answers
236
views
Terminology question - "Chern number"
I have seen the term Chern number used to refer to the first Hilbert-Samuel coefficient, $e_{1}(I)$, of an ideal $I$ in a local ring $(R, m)$. (Where the Hilbert-Samuel polynomial agrees with $\...
- 113
52
votes
7
answers
5k
views
What does a projective resolution mean geometrically?
For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...
- 19.6k
3
votes
2
answers
547
views
less than normal
Hi,
if we could write a classification about the known regularity which is the known class of schemes that are immediately less good than normal schemes? And which properties have they?
thank you
- 20.5k
1
vote
3
answers
896
views
Stably free module not finitely generated is free
Hi. I have read that stably free modules not finitely generated are free; this is proved in
M.R. Gabel, stably free projectives over commutative rings, Thesis, Brandeis Univ., Waltham, MA 1972.
But ...
- 113