Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
18
votes
0
answers
382
views
Deforming a basis of a polynomial ring
The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions $\{S_\...
- 27.9k
3
votes
1
answer
443
views
rational singularities of a certain variety
Let $X$ be a variety in $\mathbb{C}^{10}$ defined by the ideal
$I=\left<xz'-x'z, y'(u+z)-y(u'+z'), t'(u-z)-t(u'-z')+xy'-x'y\right>$ of $\mathbb{C}[x,y,z,u,t,x',y',z',u',t']$.
Note that $I+\left&...
- 127
6
votes
1
answer
689
views
When do we have derived "Hartogs" for quasi-coherent sheaves?
If $X$ is a noetherian scheme , then for any quasi-coherent sheaf $\mathscr{F}$ on $X$ that satisfies Serre's (S2) condition and an inclusion of an open subscheme $j:U\to X$ with complement of ...
- 2,040
0
votes
1
answer
964
views
Determinants over commutative rings [closed]
Hello, I am preparing a paper on determinants in commutative rings. Someone can give me examples of applications of determinants in commutative rings to other areas of mathematics or physics. Thank ...
- 545
4
votes
0
answers
896
views
A strong form of implicit function theorem (what happens when the derivative is degenerate?)
(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
- 2,232
2
votes
2
answers
451
views
Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated?
Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth ...
- 54.6k
1
vote
0
answers
107
views
Question on Hochschild cohomology
Let given ring $A$ without zero divisors and subring $\mathbf{k}\subset Z(A)$. Let also given $M~-$ $A~-$ bimodule, such that $xm = mx, \forall x\in \mathbf{k}, m\in M$.
Is it true that if for $\...
- 11
0
votes
0
answers
197
views
Name of some commutative ring akin to $p$-adics
I need help in identifying the naming convention of some commutative ring described below.
Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ...
1
vote
1
answer
220
views
Arbitrary chains of prime ideals in $R[X]$
For a commutative ring $S$ of finite Krull dimension $d$, we have $1+d\leq \dim(S[X])\leq 2d+1$. One proof of this uses the fact that if $Q_1\subset Q_2\subset Q_3$ is a chain of prime ideals of $S[X]$...
- 415
2
votes
1
answer
709
views
Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$
Let $R=\mathbb{Z}_p[[X]]$ where $\mathbb{Z}_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R_{(p)}$ $(R$ localised at the ideal $pR)$ $?$
- 193
2
votes
0
answers
379
views
local rings with finite type maximal ideal
Let $A$ be a local ring with a maximal ideal $\mathfrak{m}$ finitely generated (not principal).
Is there a sufficient condition for $A$ to be noetherian?
For example, we know that the completion $\...
- 3,472
7
votes
2
answers
1k
views
The rank of a not necessarily finitely generated module.
This question is motivated by this one. The main point of the question (was) to try to weaken the notion of rank. After the answers and comments, it seems this is not a good way to do it, but perhaps ...
- 42.9k
8
votes
1
answer
1k
views
Torsion submodule
$A$ a commutative Noetherian domain, $M$ a finitely generated $A$-module. How can I show that the kernel of the natural map $M\rightarrow M^{**}$, where $ M^{ * *}$ is the double dual (with respect to ...
- 2,857
4
votes
1
answer
258
views
Reference request for division algebras, over $\mathbb{Q}_{p}((t))$
I was looking for a possible reference that would answer the following question,
Let $\mathbb{Q}_{p}$ be the $p$-adic numbers and $\mathbb{Q}_{p}((t))$ be the field of Laurent polynomials over $\...
- 143
11
votes
1
answer
631
views
Can flatness be specified by a natural coherent sheaf?
More precisely:
Given a finite-type morphism $f \colon X \to Y$ of nice schemes (say, both of finite type over a field), is there a "natural" coherent sheaf $\mathcal F_f$ on $X$ such that the ...
- 7,338
9
votes
0
answers
400
views
Weierstrass division theorem for henselian rings
Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...
- 3,472
1
vote
1
answer
362
views
Bounded dg algebra vs unbounded dg algebras
1)Let $Cd_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd_{\geq 0}ga\to dg_{\geq 0}Mod$ be the forgethfull ...
- 1,424
1
vote
2
answers
375
views
When is a power of an indeterminate in an ideal with 2 generators?
If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...
- 87
0
votes
0
answers
165
views
on the ``generic" modules of finite length (skyscrapers)
Let $R$ be a local or graded ring. (If it helps, can assume the ring is "good", e.g. $R=k[[x_1,..,x_p]]$, where $k$ is a field of zero characteristic.)
Let $M$ be a finitely generated $R$-module ...
- 2,232
2
votes
0
answers
100
views
Degrees of generators of ideal of finite number points in the plane
Let $I_X\subseteq \mathbb{C}[x_0,\ldots,x_n]$ be the homogeneous vanishing ideal of a set $X$ of $s$ points in $\mathbb{P}^2$. Let $d_1$ resp. $d_2$ denote the minimal resp. maximal degree of elements ...
- 3,031
7
votes
2
answers
512
views
Is the reduction of a flat, finite, surjective scheme over an integral base still flat?
Is the reduction $X_{red}$ of a flat, finite, surjective scheme $X$ over an integral base $S$ still flat?
I could possibly add that I am already aware we can assume the base $S$ to be local and ...
- 1,347
6
votes
0
answers
129
views
Center Picard group non-commutative algebra
I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra.
Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by $\mathrm{Pic}(A)$...
- 7,300
2
votes
2
answers
310
views
Can normalisations of curves over a perfect field change residue fields?
Does anyone know an example of a curve $X$ over a perfect field $k$ such that if $\tilde{X}$ is its noramlisation, there exists a point $x \in X$ and a point $y \in \tilde{X}$ over $x$ such that $k(y) ...
- 1,347
2
votes
1
answer
241
views
Hochschild cohomology of commutative quotients
Notation:
Let $k$ be a commutative local ring and let $HH^{i}(A,N)$ denote the $i^{th}$ Hochschild cohomology $k$-module of a $k$-algebra A with coefficients in an $(A,A)$-bi-module $N$.
If $x:=\{...
- 5,405
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
1
vote
0
answers
365
views
Separability and smoothness
Let $A \subseteq B$ be commutative noetherian rings.
I have found the following claim: "Separability implies smoothness" with the following explanation:
"The natural thing is to prove that a separable ...
- 2,837
5
votes
3
answers
676
views
Does every compact Hausdorff ring admit a decomposition into primitive idempotents?
Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in $\mathbf{R}$)...
- 1,498
2
votes
1
answer
768
views
About the practice of Bernstein-Kushnirenko theorem
The following refers to
common roots of bivariate polynomial equations and, in particular to the quim's and auniket's comments.
The BKK theorem (cf. arXiv:0812.4688. Theorem 5.4) asserts that if we ...
- 3,741
1
vote
1
answer
490
views
Iwasawa algebra [closed]
Let $\mathbb{Z}_p$ denotes the $p$-adic integers for a prime $p$. Suppose $M$ is a finitely generated torsion $\mathbb{Z}_p[[T]]$-module such that $\mu(M)=0$. Then $M/pM$ and $M[p]$($p$-torsion ...
- 303
2
votes
1
answer
387
views
Hilbert function of points in $\mathrm{P}^2$
Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If
$$
\mathrm{gin_{rlex}}(I)=(x_1^k,x_1^{k-1}x_2^{\lambda_{k-1}},...,x_1x_2^{\lambda_1},x_2^{\...
- 73
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
1
answer
134
views
Theorem 16 , Chapter 5 of Northcott 's, Finite Free Resolutions: p.grade
Let $R$ be a commutative ring with identity. D.G. Northcott's, Finite Free Resolutions, has:
and in Theorem 16 of Chapter 5 proves that:
$p.grade(I,M) = p.grade(P,M)$ for some prime ideal $P$ ...
- 1,355
5
votes
3
answers
2k
views
Atiyah-MacDonald: exercise 5.29 - "local ring of a valuation ring"
The exercise is the following:
Let $A$ be a valuation ring, $K$ its field of fractions. Show that every subring of $K$ which contains $A$ is a local ring of $A$.
Does anyone know what is meant by "...
- 2,399
5
votes
1
answer
476
views
Ideals of affine space curves
Given some algebraic closed curve in the affine space $\mathbb{A}^3_\mathbb{C}$, is there a way to decide whether its ideal (polynomials in $\mathbb{C}[X,Y,Z]$ vanishing on the curve) is generated by ...
- 7,722
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 ...
- 25.6k
2
votes
1
answer
706
views
Milne, Etale Cohomology, mistake in proposition 2.5?
In Milne, Etale Cohomology, Proposition 2.5 (§2) is stated as follows:
(All rings noetherian.)
Let $B$ be a flat $A$--algebra, and consider $b \in B$. If the image of $b$ in $B/\mathfrak{m} B$ is ...
- 708
0
votes
0
answers
301
views
Irreducible component of a scheme over a dvr
Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
- 155
2
votes
1
answer
601
views
Henselization of valued field
What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
- 173
9
votes
0
answers
1k
views
When UFD implies PID
The following result is too elementary, both to state and to prove, not to be known. Can someone give a reference? Is there any hope if you don't suppose UFD (i.e. move that from the hypothesis to ...
- 907
3
votes
1
answer
193
views
Example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$
Is there any example of a homogeneous (not monomial) $(x,y)$-primary ideal $I$ in $K[x,y]$ such that $I$ is complete and and there exists a minimal reduction $J$ of $I$ such that $J\overline{I^n}=\...
- 1,713
3
votes
0
answers
181
views
Factorization of linear recurrences
For each (commutative unitary) ring $R$, let $\mathfrak{R}(R)$ be the set of all linear recurrences over $R$, that is, the set of all sequences $(a(n))_{n \geq 0}$ in $R$ such that
$$a(n+k) = r_1 a(n+...
user40023
1
vote
1
answer
109
views
section of reduced structure map
Let $R$ be a commutative ring whose characteristic is either prime or $0$, such that $R/N$ is an integral domain, where $N$ is the nilradical, and $p: R \rightarrow R/N$ the canonical map. Is there a ...
- 1,812
7
votes
2
answers
931
views
What is the German translation of "catenary ring"?
I am looking for the correct technical term in German for the notion of catenary ring in commutative algebra.
Does anyone know?
For those who don't know what a catenary ring is but would like to: ...
- 2,399
6
votes
0
answers
310
views
Algebraic closedness in field of fractions
If $A\subseteq B$ are affine domains over an algebraically closed field $k$ of characteristic zero, such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...
7
votes
1
answer
300
views
Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?
We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be
$$
\text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!,
$$
where $ P_{M} $ denotes the Hilbert ...
- 211
1
vote
1
answer
224
views
M is an R-module which is not finitely generated. is it true that $\inf \{ i| H^i_I(M)\neq 0 \}\le ht_M I?$
Let $R$ be a commutative noetherian ring (with $1$), $I$ an ideal of $R$, and $M$, a finitely generated $R$-module such that $IM \neq M$. Then, by Theorem 6.2.7 of BRODMANN-SHARP's Local Cohomology ...
- 1,355
4
votes
0
answers
80
views
Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?
Let $k$ be a field and let $R=k[X_1,...,X_n]$ be a polynomial ring. Let $F \subset R$ be a finite subset generating a radical ideal $I$ with precisely $e$ solutions over an algebraic closure of $k$. ...
- 476
4
votes
1
answer
499
views
Artin approximation theorems over non-regular rings/non-Noetherian rings
In Artin1968 he considers $\underline{analytic}$ equations, but over the ring $R=k\{x_1,..,x_n\}$. In Artin1969 he works with $R=k\{x_1,..,x_n\}/I$, not necessarily regular, but considers $\underline{...
- 2,232
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
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