Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
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Is every connected scheme path connected?
Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following.
Let's ...
- 47.5k
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2
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An elementary lemma of commutative algebra
Let $R$ be a commutative ring, $M$, $N$ $R$-modules, and $f: M\rightarrow N$ a homomorphism. It is known that $f$ is injective (surjective) if and only if $f_m$ is injective (surjective) for all ...
- 1,207
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When are two ideals in a regular local ring generated by a regular sequence?
Hello!
Let $R$ be a regular local ring, and let $I,J\subset R$ be ideals. I'd like to understand the "meaning" of the existence of a regular sequence $(x_1,...,x_n)$ in $R$ such that $I$ is generated ...
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Differential graded structures on free resolution?
Hello!
In "Homological Algebra on a Complete Intersection", Eisenbud proves the following:
Let $A$ be a commutative ring, $M$ be an $A$-module and $F^{\ast}\to M$ an $A$-free resolution. Further, ...
- 2,756
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Can we characterise affine open subschemes of ${\rm Spec}(A)$?
Let $A$ be any ring, commutative with identity, and let $I\subset A$ be an ideal $\neq A$. Let $U\subset{\rm Spec}(A)$ be the open subscheme obtained by "removing" the closed set $V(I)$ of all the ...
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Are local, Noetherian rings with principal maximal ideal PIR?
A question asked by a friend. I believe it's false, but lack a decisive counterexample.
This question shows that it is true for valuation rings, but I know too little about them.
In the wider ...
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Free commutative magma over a set
BOURBAKI, inside his book on ALGEBRA defines and provides explicit constructions concerning the concepts of free magma, free monoid (and implicitly free semi-group) and free group, and as well free ...
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Localization of a polynomial ring at a prime ideal.
If $R=\mathbb{C}[x,y]$ is the polynomial ring in two variables $x$ and $y$ then we know that the localization of R at the multiplicative set $S=[1,x,x^2,x^3,...]$ is given by $R_x=\mathbb{C}[x,x^{-1},...
- 153
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In which commutative algebras does any derivation possess a flow?
Definitions
Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a derivation of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$. If $\...
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How to tell if two random polynomials are identical
Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...
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Morphisms of a simple sheaf over an algebra to its double dual
Given a smooth and projective surface $S$ over an algebraically closed field $k$ and a sheaf of Azumaya algebras $R$, i.e. $R$ is a locally free $O_S$-module of finite rank. Let $M$ be a coherent and ...
- 1,391
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Is there a field which is the union of finitely many proper subfields?
Is there a field which is the union of finitely many proper subfields?
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Flatness on the fiber
Hi.
Let $f:A\rightarrow B$ be a local morphism of locally noetherian (reduced) rings with $B$ $A$-flat. Let $M$ be an $B$-module of finite type.
Question: Which conditions ensure the following:
$N\...
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Is there a straightforward way to solve unmixed, homogeneous systems of polynomials?
I came across this problem in my research. It might just be an easy algebraic geometry question, but I don't know much algebraic geometry.
Suppose we have a system of $k\leq n$ polynomials in $\...
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Length of a module over different rings
Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$.
Under what ...
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Discriminant and Different
First some context. In most algebraic number theory textbooks, the notion of
discriminant and different of an extension of number fields $L/K$, or rather, of the corresponding extension $B/A$ of their ...
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3
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Integral closure of a regular ring
Let $A$ be a noetherian integral local ring. Let $K$ be its fraction field, $L$ an algebraic field extension of $K$, and $B$ the integral closure of $A$ in $L$. If $A$ is supposed to be regular, is ...
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ideal transform
Let $I$ be an ideal of a commutative ring $R$. $M$ be an $R$-module. In Local cohomology: an algebraic introduction with geometric applications of Brodmann M. P., Sharp R. Y we have
$$D_I(M)=\mathop {\...
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What is lim⟶ I^n M?
Let $R$ be a commutative ring, $I$ is an ideal of $R$, $M$ is an $R$-module.
$$IM\supset I^2M\supset I^3M\supset\cdots$$
What is $\mathop {\lim }\limits_{\begin{subarray}{c}
\longrightarrow \\
\...
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2
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Why do we care whether a PID admits some crazy Euclidean norm?
An integral domain $R$ is said to be Euclidean if it admits some Euclidean norm: i.e., a function $N: R \rightarrow \mathbb{N} = \mathbb{Z}^{\geq 0}$ such that: for all $x, y \in R$ with $N(y) > 0$,...
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Why is the symmetric monoidal structure on invertible modules strict?
Let $N$ be an object in a symmetric monoidal category. Then the braid map $N\otimes N\to N\otimes N$ is almost never the identity, and this is the obstruction to making a symmetric monoidal category ...
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Dual of Zorn's Lemma? [closed]
It seems to me that the dual of Zorn's Lemma should be true: if $S$ is a non-empty partially ordered set and every chain of $S$ has a lower bound in $S$, then $S$ has at least one minimal element.
...
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6
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1
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Exotic isomorphism of matrix rings
Let R and S be commutative rings with a 1 different from zero. Let m and n be positive integers. Assume the ring of m-by-m matrices over R is isomorphic to the ring of n-by-n matrices over S. Does ...
user5810
2
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1
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Linear independence in the algebraic closure of $\mathbb{C}(z)$
Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.)
Define $w_i=(\...
- 454
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Generic Rank of R^{1/p}
Suppose $R$ is a local Noetherian domain of dimension $d$ in characteristic $p>0$. Suppose $R^{1/p}$ is a finitely generated $R$-module, and suppose $k$ is the residue field of $R$. Is the ...
- 103
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Moduli space of modules with fixed length
Let $R$ be a (commutative) local Artinian ring, with an algebraically closed residue field $k$. I am interested in the set $L_n(R)$ of isomorphism classes of $R$-modules of length $n$.
If $R$ is a $k$...
- 30.5k
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758
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Locally square implies square
Does there exist a (noetherian) commutative ring $R$ and an element $a \in R$ such that $a$ is a square in every localization of $R$ but $a$ itself is not a square?
- 3,009
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1
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How to design or create or generate a bijective ring map? [closed]
How to design or create or generate a bijective ring map?
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Are valuation rings regular?
This question is short, and to the point:
Valuation rings are certainly integrally closed, but are they regular?
The motivation is that I'm trying to understand the resolution of singularities of ...
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14
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3
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Projective dimension of zero module
Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$:
...
- 2,857
3
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If $B \subset C \subset B_g$, is $\mathrm{Spec} C \to \mathrm{Spec} B$ necessarily an open immersion?
Let $B \subset C$ be Noetherian integral domains, and $g \in B$. Thus, $\mathrm{Spec} B \to \mathrm{Spec} B_g$ is an open immersion.
If furthermore $C \subset B_g$, does it follow that $\mathrm{Spec}...
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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
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Lower bounds on the degrees of representatives of $u^n$ as $n \to \infty$
Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra, together with a specified surjective morphism $\phi \colon k[x_1, \dotsc, x_n] \to A$. For $f \in A$, define $\...
- 7,328
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Any implemented algorithm to compute the closure of an affine variety in a product of projective spaces?
Let $I$ be an ideal of $k[x_1, \ldots, x_m, y_1, \ldots, y_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-...
- 5,339
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633
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Determining if a ring satisfies Serre's condition S_{n}
Given a specific ring $R$ (eg, $R=k[x_{1}, \cdots, x_{n}]/I)$ is there a (simple) way to determine whether or not $R$ satisfies Serre's condition $S_{n}$? In particular, is there a way to do this in ...
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Is weak normality stable under completion?
I'm curious if anyone knows a reference for the following. It seems like someone must have done this somewhere, but I couldn't find a reference.
Recall that an excellent reduced noetherian ring $R$ ...
- 20.5k
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Is a valuation domain PID when its maximal ideal is principal?
It is known that a valuation domain is a principal ideal ring if and only if its prime ideals are principal. Is it a principal ideal ring when its unique maximal ideal is principal?
- 1,207
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Standard reduction to the artinian local case?
Where can I find a clear exposé of the so called "standard reduction to the local artinian (with algebraically closed residue field", a sentence I read everywhere but that is never completely unfold?
...
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Generators of ideals in polynomial rings over commutative rings.
This is my first question; I hope it worthy of this awesome forum and its members.
Let $R$ be a commutative ring, perhaps with unit, perhaps not. As usual let $R[x]$
denote the ring of polynomials ...
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Length of I/I^2 versus Ann(I)/Ann(I)^2 in Artinian rings.
Suppose that $(A,\mathfrak{m})$ is a local Artinian ring.
If $A$ is Gorenstein, then $A$ admits a dualizing functor
on finite length modules defined by $D(M):= Hom_A(M,A)$ which preserves
lengths. If ...
user631
3
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nilpotent matrices over polynomial rings
I am looking for an analogue of the Jordan normal form for nilpotent matrices over the
polynomial ring ${\mathbb Z}[x_1, \dots, x_n]$. More precisely, is there a description for the orbits of action ...
- 6,224
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542
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Abelian varieties over local fields
Let $K$ be a local field of characteristic zero, $k$ its residue field, $R$ its ring of integers and $p$ the characteristic of the residue field $k$. Let $G$ be the Galois group of $K$, $I\subset G$ ...
- 1,673
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Q-Divisor and Determinant Map on a Maximal Order
Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring.
Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^*$ be the ...
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Computational Question about finite local rings:
Let $(A,\mathfrak{m})$ be a local Artinian ring with
finite residue field, which I'm happy to assume is $\mathbf{F}_3$.
(In particular, $A$ has finitely many elements.)
I would like to do some ...
user631
23
votes
1
answer
3k
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Modules and Square Zero Extensions
Let $R$ be a commutative ring, $RMod$ its category of modules and $CRing$ the category of commutative rings.
There's an embedding $RMod \rightarrow CRing/R$ that sends an $R$-module $M$ to the ring ...
- 1,484
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When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?
Let V be a finite dimensional symplectic vector space over $\mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is
generated by symplectic reflections, i.e. by elements $g\...
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Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19
Fulton's "Intersection theory" book contains the following fact (example 18.3.19):
Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...
- 30.5k
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Basic commutative algebra question.
Suppose that A is a local ring (commutative with unit), finite over a field k. Let L be the residue field A / m where m is the unique maximal ideal of A.
Does the dimension of L (as a k-vector space) ...
- 467
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1
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Hochschild and cyclic homology of smooth varieties
Many of the standard sources which discuss the Hochschild Kostant Rosenberg theorem and cyclic homology for smooth varieties such as Loday and Weibel's paper "The Hodge Filtration and Cyclic Homology" ...
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