Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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Reference request for $R$-index
Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ ...
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How to show integrally closed implies topologically unibranch
On p.52 of Mumford's book Algebraic Geometry: Complex projective varieties, he states that
$$\mathcal{O}_{x.X} \text{is integrally closed} \ \Rightarrow X \ \text{is topologically unibranch at } \ ...
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A linear subspace of $\mathbb{R}[X_1,\cdots,X_n]$ and its generated set
Let $n$ be a positive integer and $W_n$ be the linear subspace of the real vector space $\mathbb{R}[X_1,\cdots,X_n]$ generated by the following set
$$S_n=\{X_1^{i_1}\cdots X_n^{i_n}:i_1+\cdots+i_n=n\ \...
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local complete intersection
The following is an Exercise 1.1.11 of Hartshorne's Algebraic Geometry.
Let $Y\subset \mathbb{A}^3$ be the curve given parametrically by $x=t^3, y=t^4, z=t^5$. Show that $I(Y)$ is a prime ideal of ...
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Bounded Index of Nilpotency of $R[x]$
A ring $R$ is called with bounded index (of nilpotency) $n$ if $n$ is the smallest natural number such that $a^n=0$ for all nilpotent $a \in R$.
Now let $R$ be a commutatitve ring with bounded index $...
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Ring $R$ such that $R^n$ contains unimodular elements that are not part of a free basis for all $n \geq 2$
Let $R$ be a commutative ring. A vector $(c_1,\ldots,c_n) \in R^n$ is unimodular if $Rc_1 + \cdots + Rc_n = R$. Say that a vector $\vec{v} \in R^n$ is a basis element if there exists a free basis ...
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Primes in a (commutative) Jacobson ring
Recall that a commutative ring is Jacobson if every prime ideal is the intersection of the maximal ideals that contain it.
In the exercises of a commutative algebra course I gave I asked the ...
- 3,545
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Irreducible components of reduced complete intersection
Let $Z$ be an irreducible and reduced scheme. Does there exist a reduced complete intersection $Y$ such that $Z$ is an irreducible component of $Y$?
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primary ring and primary module [closed]
I want to know that in the commutative rings, the definition of the primary module is coincide with the definition of the primary ring? And where i can find it?
The primary ring is a ring when ab=0, ...
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Finiteness of normalization of Noetherian normal domain
I have the following question:
Let $A$ be an integrally closed Noetherian domain, $K$ its field of fractions. let $L$ be a finite extension of $K$, and $B$ the integral closure of $A$ inside $L$. Is ...
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elementary exact sequence of normal sheaves
Let $Z \subset Y \subset \mathbb{A^n}$ be a smooth subvarieties of $\mathbb{A^n}$.
I'm trying to show that there is an exact sequence of normal bundles.
$0 \rightarrow N_{Z/Y} \rightarrow N_{Z} \...
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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 ...
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Base change of trace for Gorenstein or Cohen-Macaulay morphisms
This is basically a question of functoriality for base change of CM morphisms.
EDIT: $\text{ }$ Brian Conrad sent me an email explaining the that this is indeed true, and follows from his book. I'...
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Minimal length of quotient by parameter ideals
Consider a commutative noetherian local ring $R$ of dimension $d$ and define
$$c_R\colon=\min_{(x_1,\ldots,x_d)} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of ...
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Is There a maximal space that is a P-space?
we guess there is no maximal space which is also a P-space. Am I right? Do u know a counter example?
clarifications:
Maximal space is that space with topology $\tau$ which is maximal crowded topology ...
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A refined version of Krull's height theorem
A classical result in algebraic geometry states that every irreducible component of a variety defined by $r$ polynomials in affine $n$-space has dimension not less than $n-r$. This is a special case ...
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The Geometric Intuition behind Minimal Primary Decompositions versus an arbitrary Primary Decomposition
Given an ideal in a Noetherian domain, it has a primary decomposition. This decompositions may not be minimal. I am interested in the relationship of primary ideals that are not in a minimal primary ...
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0-dimensional Gorenstein local ring.
Assume the following condition for the ring T = F_p[[X,S]]/I:
Condition 1. T is NOT a zero ring.
Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements.
Then, is T a ...
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Projective Modules/Algebras: decomposition of linear functions, and the rank formula
Let $A$ be a ring, $B$ a finite projective $A$-algebra, and $C$ a finite projective $B$-algebra. We can show that $C$ is also finite and projective when regarded as an $A$-algebra (by, for instance, ...
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rings with 'flat functions'
Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat elements, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, ...
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Elementary proof that projective space is a quotient
Fix an algebraically closed † ground field $k$ of any characteristic. I want to use the classical definition of projective $n$-space $\mathbb{P}^n$ as set quotient of $\mathbb{A}^{n+1}\...
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When fitting ideals determine the module?
Let $M$ be a module over a local ring $(R,m)$, everything is finitely generated/presented. The fitting ideals, $I_j(M)$ carry a lot of information about the module. When do they actually determine the ...
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418
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Does regularity of a prime ideal in the fibre imply regularity of the prime?
Recall that a prime $\mathfrak{p}$ is called nonsingular (regular) if the localization at that prime is a regular local ring. If all primes of a ring $R$ are nonsingular, $R$ is called regular. Let $...
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Jacobian ideals reference
Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ...
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Spectrum of an algebra object and Reconstruction of Schemes
In "Au-dessous de $\text{Spec}(\mathbb{Z})$", Toen and Vaquié define schemes relative to a complete, cocomplete symmetric monoidal category $C$ using a functorial approach.
In the introduction the ...
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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 $...
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Is pushforward along a closed immersion in the fppf topology exact?
Let $i : Z \to X$ be a closed immersion of schemes. Is $i_* : Ab((Sch/Z)_{fppf}) \to Ab((Sch/X)_{fppf})$ an exact functor?
The answer is yes in the \'etale or syntomic topology. It seems likely the ...
user45795
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A nice generating set for the symmetric power of an algebra
I'm looking for a reference for the following fact.
Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic zero. Let $S^n(A)$ ...
- 618
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The number of AR sequences with indecomposable middle term in $D_n/I$ cases
In the following $A_n/I$ or $D_n/I$ cases means the AR quiver of bound quiver algebra $A_n/I$ or $D_n/I$.
I know that the number of AR sequences with indecomposable middle term in $A_n/I$ cases is ...
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Do the solutions of the Maurer--Cartan equation form a simplicial set?
The Maurer--Cartan equation is the equation:
$$d\gamma+\frac 12[\gamma,\gamma]=0$$
where $\gamma$ represents a degree one element in a differential graded Lie algebra $\mathfrak g^\ast$. Let's denote ...
- 18.7k
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Maximal size of an almost-disjoint linearly independent family in $K^{\mathbb{N}}$
Let $K$ be a field, say infinite, and denote by $L$ the $K$-vector space $K^{\mathbb{N}}$. What is the maximal cardinality of a $K$-linearly independent subset $X$ of $L$ such that any two distinct ...
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Free group actions on varieties and algebras of coinvariants
Suppose $k$ is an algebraically closed field of characteristic zero and $A$ is a finitely generated commutative associative reduced $k$-algebra.
Suppose the group $\mathbb{Z}_2$ acts on $A$ in such a ...
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Finding generators of subalgebra of polynomial algebra $K[x_1,\cdots,x_n]$ that are invariant under the action of symmetric group
Let $I =\langle f_1,\cdots,f_m\rangle \subset K[x_1,\cdots,x_n]$be an ideal,
where $f_k\in K[x_1,\cdots,x_n].$
$K[e_1,\cdots,e_n]$ the polynomial algebra generated by the elementary symmetric ...
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Whether such an algebra has to be the Group algebra
Let $\mathbb C$ be the field of the complex numbers, $\mathbb Q$ the field of the rational numbers.
Let $G$ be an additive subgroup of $\mathbb Q$.
$R$ is an commutative algebra over $\mathbb C$, ...
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Annihilators of elements in symmetric algebras
Let $M$ be a module over a commutative ring, and $S(M)$ its symmetric algebra. What elements of $S(M)$ annihilate a given element $m\in M$ ? ($M$ is considered as a submodule of $S(M)$.)
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Does torsion-freeness of class group localize?
Let $R$ be a local normal domain, and let $P \in Spec (R)$. It is well known that $Cl(R) \to Cl(R_P)$ is surjective. However, I do not know any example where $Cl(R)$ is torsion-free, but $Cl(R_P)$ is ...
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Embedded associated prime and non zero divisor
$M$ is a finitely generated $A$-module of dimension $d$ such that $G(M)$ is equidimensional and $M$ does not have any embedded prime.
Given $x\in I$, where $I$ is an ideal of $A$, and $\dim G(M)/x^*G(...
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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 ...
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456
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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
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Representations of $GL(n)$ containing $S^kV$
Let $V$ be a vector space of dimension $n$.
Let $S^k V$ be a representation of $GL(n)$.
I would like to know if there exists some characterization of finite dimensional $GL(n)$ modules $V_1,V_2$ such ...
- 2,179
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Free direct summand of a module
Let (A,m) be a local ring and M be a finitely generated A-module contained in a free module F of rank r with length(F/M) < $\infty$. Then I have the following question : Is the statement "M doesn't ...
- 11
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Intersections of ideals and nilpotence
Let $R$ be a polynomial ring over a field $k$, $R = k[x_1, \dots, x_n]$. Suppose $R'$ is an associative $R$-algebra and it has the property that there exists a degree $m<n$ monomial in the $x_i$'s ...
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Reduced rings, idempotents and their prime spectrum
Let $B$ be a commutative unitary reduced ring and let $A$ be a subring of it. Let $e$ be an idempotent of $B$. Then we have a natural surjective ring homomorphism $A\rightarrow Ae$ defined by $a\...
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Do group identities of quotient with radical lift?
Let $R$ be a commutative Artinian ring and $J(R)$ its radical. Assume that the quotient $R/J(R)$ is a GI-ring.
(definitions that i use: I call a ring $S$ a GI-ring if its unit group, $\mathcal{U}(S)$,...
0
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1
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Normality and fiber product
Let $A$ and $B$ be noetherian normal rings and let $f:A\rightarrow B$ be a finite but non-flat ring homomorphism. We can also assume $Spec(A)$ connected if necessary. We put on $B$ the structure of $A$...
- 1
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2
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634
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Why bi-module, bi-bundle, etc?
This is perhaps an ill-proposed question. Any way, thank you guys.
We have a lot of bi-stuffs, such as bi-module, bi-bundle, etc. They are basically two commuting actions from two sides, left and ...
- 1,271
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917
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Height of maximal homogeneous ideals
Let $R= \oplus_{n \ge 0} R_n$ be a graded Noetherian commutative ring and suppose $R_0$ is Artinian.
Do all maximal homogeneous ideals of $R$ have the same height ?
Let $R_{>0}$ be the ideal ...
- 16.2k
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602
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vanishing of local cohomology $H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$
Edited:
I guess
$$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$
We know that if $\operatorname{Supp} H^i_I(M)\subseteq V(I)\cap \operatorname{Supp}(M)$, then
$$\operatorname{Supp} H^2_{...
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3
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algebra group theory [closed]
If A, B, C be abelian grops and if A isomorph with direct sum of B and C and A be isomorph with B what we can say about C?
- 1
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353
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Failure of Noether normalization
I was in a class recently where we were trying to roughly count the dimensions of certain spaces of rational maps from algebraic curves into closed subschemes $Z \subseteq \mathbb{A}^n$. One way to ...
