Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
12
votes
4
answers
940
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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 ...
- 2,406
32
votes
6
answers
12k
views
Duals and Tensor products
Let $A$ be a commutative ring with a unit element. Let $M$ and $N$ be $A$-modules. Let $M^v$ and $N^v$ be the dual modules. In general, do we have $M^v \otimes N^v \cong (M\otimes N)^v$? It is ...
- 530
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 ...
- 3,918
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 / ...
- 5,562
26
votes
3
answers
2k
views
Invariance of $\mathbb{Z}[x]$ under a self-equivalence of the category of commutative rings with 1
Let $\mbox{Rings}$ be the category of commutative rings with $1$.
Is there an equivalence of categories $F: \mbox{Rings} \to \mbox{Rings}$ such that
$$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$
- 635
5
votes
1
answer
2k
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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 ...
- 1,109
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: ...
- 607
3
votes
1
answer
497
views
Formally smooth maps between adic rings and regular immersions
Suppose $(A,\mathfrak{a})$ and $(B,\mathfrak{b})$ are two adically complete (commutative) noetherian rings. Let $f:A \to B$ be a continuous formally smooth formally of finite type map (that is, $B/\...
- 1,214
13
votes
5
answers
3k
views
Example of a projective module which is not a direct sum of f.g. submodules?
This semester I am teaching a graduate course in commutative algebra, and I have been taking the occasion to try to look at the proofs of some the results in my basic source material (Matsumura, ...
- 65.4k
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 $\...
- 45
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 ...
- 1,160
7
votes
1
answer
1k
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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 ...
- 225
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 ...
- 486
3
votes
3
answers
681
views
on the relative conductor of curve singularity and quotient of ideals
Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...
- 2,232
14
votes
5
answers
995
views
How can I write down polynomial relations that define when a polynomial is a square?
It's easy to tell when a polynomial is squarefree (or not): that's just the question of the vanishing of the discriminant, which can be dealt with as the resultant of $f$ and $f'$. However, given a ...
- 16k
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 ...
- 7,328
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 ...
- 6,229
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
18
votes
7
answers
2k
views
Superfluous definitions
It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.
For if a and b are elements of R, and writing + for the group operation then applying ...
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{...
- 20.5k
59
votes
4
answers
12k
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Geometric meaning of Cohen-Macaulay schemes
What is the geometric meaning of Cohen-Macaulay schemes?
Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various ...
- 63.1k
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 ...
- 4,070
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. $m=(x_1,x_2,...,x_n)R$. ...
- 225
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 ...
- 113
19
votes
6
answers
2k
views
Nonfree projective module over a regular UFD?
What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least ...
- 65.4k
53
votes
9
answers
13k
views
Is there a preferable convention for defining the wedge product?
There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
- 4,306
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
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 ...
- 91
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
- 141
4
votes
2
answers
907
views
About injective hull
Let $M$ be an $A$-module. Is its injective hull affected by whether I regard $M$ as an $A$-module or $A/\mbox{Ann}(M)$-module ?
- 2,857
5
votes
0
answers
994
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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
8
votes
1
answer
1k
views
Direct sum of injective modules over non-Noetherian rings
By the Bass-Papp theorem, if every direct sum of injective $R$-modules is injective then $R$ is Noetherian. I would like to know if there exists an injective module over $R$ non-Noetherian, that ...
- 113
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
16
votes
1
answer
2k
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Questions about spectra of rings of continuous functions
I have been thinking a bit about rings of continuous functions of various kinds -- how they motivate the more modern notion of the Zariski topology on the prime spectrum as well as how they fit into a ...
- 65.4k
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 $...
- 951
33
votes
2
answers
2k
views
If a field extension gives affine space, was it already affine space?
Let $R$ be a commutative Noetherian $F$-algebra, where $F$ is a
field (perfect, say). Assume that $R \otimes_F \overline F$ is a polynomial ring over the
algebraic closure $\overline F$.
Does it ...
- 27.8k
14
votes
2
answers
3k
views
Maximal ideal and Zorn's lemma
It is known that any nonzero ring A (say commutative with 1) has a maximal ideal. The proof uses Zorn's lemma.
Now I heard some people saying that if we assume A to be noetherian, then we don't need ...
- 1,271
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
- 73
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
2
votes
2
answers
1k
views
Why are extensions so heavily emphasized in valuation theory?
Whenever I read anything about valuations or things related to them (such as local fields) extensions always occupy a prominent position and a huge amount of effort is expended to derive results about ...
- 4,351
5
votes
3
answers
984
views
Isomorphism of the function field of the projective line with $\mathbf{C}(s)$
Suppose I chose two rational functions, say,
$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad
v = \frac{t^5(4+t)}{(1+4t)}.$$
Then I know that $K(X) = \mathbf{C}(u,v)$ is
the function field of the projective ...
- 53
2
votes
2
answers
2k
views
Projective modules over semi-local rings
Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. If the localizations $M_m$ have the same rank for all maximal ideals $m$ of $R$ then $M$ is free.
- 37
8
votes
3
answers
680
views
A question about how polynomials simplify under substitution
This is a revised and more sensible version of the original question, thanks to the kind help of Anthony Quas and J. C. Ottem.
Fix polynomials $f_{1},\ldots, f_{n} \in \mathbb{C}[t]$.
Let $M_{k}$ be ...
- 6,199
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 ...
- 1,528
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 ...
- 2,964
45
votes
8
answers
6k
views
What makes a theorem *a* "nullstellensatz."
I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...
19
votes
3
answers
4k
views
Generalized Euler phi function
Let $n$ be an integer, there is a well-known formula for $\varphi(n)$ where $\varphi$ is the Euler phi function. Essentially, $\varphi(n)$ gives the number of invertible elements in $\mathbb{Z}/n\...
- 337
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 ...
- 301
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 ...
- 41
6
votes
2
answers
2k
views
Online video of some courses
Who knows online video of Riemannian Geometry and Commutative Algebra? If you know, please recommend them to me. I am really eager to learn these courses.