All Questions
Tagged with commutative-algebra or ac.commutative-algebra
5,492 questions
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Finite generativity of algebra with valuation
Let $C$ be a commutative finitely generated algebra with no zero divisors. If necessary, we can assume it to be graded and a unique factorization domain. Let $a\in C$ be a prime element.
Let's also ...
- 4,985
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Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$
Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring.
There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
- 156k
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47
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Derived tensor products and regular sequences
Let $R \to A$ be a homomorphism of commutative rings, and let $x\in R$ be an element (or a sequence of elements in $R$, if you prefer) that is both $R$-regular and $A$-regular. Then we have
$$
A\...
- 105
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What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$, $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free commutative $\mathbb{K}$-algebra on $...
- 156k
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When do algebraic elements form a subalgebra?
If $R$ is a commutative ring and $A$ is a commutative $R$-algebra, we say that an element $x\in A$ is algebraic over $R$ if $x$ is a root of a nonzero polynomial $f \in R[X]$, or equivalently, if the &...
- 844
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Tensor product of two transcendental flat algebras is not a field?
I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
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320
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Non-negative coefficients polynomials
Let $n \in \mathbb N$ and $P,Q \in \mathbb R_+[x]$.
Is it true that $(x+1)^n\neq (x-2)^2 \times P(x)+(x-4)^2 \times Q(x)$ ?
I have asked, this question here (*), two weeks ago, but no answers.
(*) ...
- 22.5k
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Relation between Tor amplitude and $p$-complete Tor amplitude for a ring of characteristic $p$
Fix a prime number $p$. Let $A$ be a commutative ring, and consider an $A$-algbera $B$ of characteristic $p$. So we have a sequence of ring homomorphisms
$$
A \to A/pA \to B.
$$
Assume that we want to ...
- 2,806
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Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
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Application of the adjoint functor theorem to get the right adjoint of the forgetful functor from $\delta$-rings to rings (the Witt vectors)
I am studying $\delta$-rings and Witt vectors from [K] (the definition of $\delta$-ring is [K, 2.1.1]), and I am having trouble verifying that everything in Kedlaya's definition for the Witt vectors ...
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Generalization of Koszul cohomology for wedge product with a fixed q-form: literature and references?
Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $1 \leq p \leq q$, consider the cochain complex:
$0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \cdots \...
- 63.9k
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Integral domains equal to intersection of their height one localizations
Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$?
It is a standard fact that Krull domains, and thus noetherian ...
- 4,985
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Set-theoretic generation by circuit polynomials
Let $P$ be a prime ideal in $S=\mathbb{C}[x_1,\ldots , x_n],$ and write $[n] = \{ 1, \ldots , n \}.$ The algebraic matroid of $P$ can be defined according to circuit axioms as follows: $C\subset [n]$ ...
- 1,046
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Dirichlet series with integer coefficients as a UFD
I recall the following question from Ulam's book "Unsolved math problems": show that the ring of Dirichlet series with integer coefficients is a factorial ring. I believe that soon after Ulam wrote ...
- 4,985
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Colimits in commutative Banach algebras?
Let $K$ be a complete non-Archimedean field. It is known that the category $\mathrm{Ban}_K$ of $K$-Banach spaces with bounded linear maps does not have infinite colimits. The usual argument for $\...
- 63.1k
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UFD property for power series in characteristic 0
Samuel famously produced an example of a UFD, namely $S = R_{(x,y,z)}$, where $R = K[x,y,z]/(x^2+y^3+z^7)$ and $K$ has characteristic 2, such that the power series extension $S[[ x ]]$ is not a UFD. ...
- 4,985
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272
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What is known about vector subspaces of polynomial rings closed under factors?
Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is:
If $a,b \in F$, then $a+b \in F$, and
If $a,b \in R$ with $a\in R$ ...
- 1,802
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Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution
Examples of infinite dimensional involutions
Edit 2/25/23, as suggested by YCOR below: (Start)
The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
CommunityBot
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Literature Request: The derived category is Krull-Schmidt
I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question
Literature request: $K^b(\text{...
- 867
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Descent of $G$-invariant formal system of parameters using GAGF
Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \...
- 5,998
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376
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Witt coordinates vs Joyal coordinates on the ring of Witt vectors
I am learning about Witt vectors following [K, Ch. 3], but I am having trouble with the presentation of his material (see (Q1) below).
Kedlaya's definition for ring $W(A)$ of the Witt vectors on a ...
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$\mathbb{Z}$-homomorphism and $\mathbb{Z}_p$-homomorphism
$\newcommand{\cts}{\mathrm{cts}}$Thanks for your reading. Let $A,B$ be two $\mathbb{Z}_p$-modules, where $\mathbb{Z}_p$ is the $p$-adic integer ring. I have two questions.
Is $\mathrm{Hom}_{\mathbb{Z}...
- 319
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Cohen-Macaulayness of the homogeneous coordinate ring of projective monomial curves
Let $A = \{a_0, a_1, \ldots, a_{n-1}\} \subset \mathbb{N}$ be a set of non-negative integers where we assume that $a_0 < \cdots < a_{n-1}$ and set $d := a_{n-1}$. For every $s \in \mathbb{N}$, ...
- 21
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Weak approximation in Krull domains
Suppose $R$ is a Krull domain with the field of fraction $K$. To every prime ideal $P$ of $R$ of height $1$, one can associate a $ \mathbb{Z}$-valued discrete valuation which we denote by $v_P$.
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- 4,985
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Finding prime ideals for ideal classes in arbitrary Dedekind domains
Let $R$ be an algebraic number ring with class group $C(R)$, and let $x : C(R)$. Then there exists a prime ideal $P$ in $R$ such that the ideal class $[[P]] = x$, and in fact there are infinitely many ...
- 7,893
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Derived completeness of the inverse perfection
Fix a prime number $p$, and let $R$ be a ring of positive characteristic $p$. Consider the inverse perfection of $R$, which is defined as the inverse limit
$$
R^\flat = \varprojlim(\cdots \xrightarrow{...
- 148k
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What is the fastest known algorithm for evaluating a homogeneous binary polynomial?
This question was initially posted on math.stackexchange.com, but there is no appropriate answer, hence I have the right to publish it here again.
Let $f(x,y) = \sum_{i = 0}^d f_i x^i y^{d-i}$ be a ...
- 1,833
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254
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When do (or don't) residue fields generate the derived category of a ring?
Let $R$ be a commutative ring, and $D(R)$ its derived category of unbounded chain complexes. I'm interested in when the residue fields $k(\mathfrak{p}) = \mathrm{Frac}(R/\frak p)$ for $\mathfrak p \in ...
- 3,785
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364
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Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$
Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread.
Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
CommunityBot
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Dual of finite reflexive modules
Let $A$ be a commutative ring and $M$ be a finite reflexive $A$-module, i.e. the natural map $M\to (M^{\vee})^{\vee}$ is an isomorphism. Can we deduce that the dual $M^{\vee}$ is also finite?
- 151
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Integral graded algebra of finite type is approximable
The following is the definition of approximable algebra.
An integral graded $K$-algebra $\oplus_{n\geqslant 0}B_n$ is said to be approximable if
1.$$rk_K(B_n)<+\infty,\forall n\in \mathbb{N}, $$and ...
- 1
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66
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Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
- 412
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Unpacking the plethystic substitution $h_n[n\mathbf{z}]$ in a paper by Aval, Bergeron and Garsia
I'm not familiar with the formalism of plethysm, so I need some help in unpacking the plethystic substitution $h_n[n\mathbf{z}]$ found in eqns. 5.6 and 5.9 of "Combinatorics of labelled ...
CommunityBot
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Weil restriction of cycles and norm algebra
This question is on a concrete descrption of weil restricton of an affine algebra.
Let L/K be a Galois extension. Since I only care about the quadratic case, we may assume that $\Gamma:=\operatorname{...
- 63.9k
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Alternating forms on abelian groups
Let $G$ and $H$ be abelian groups. By an alternating form, I mean a bilinear function $A\colon G\times G\to H$ such that $A(x,x)=0$ for all $x\in G$.
Question. If $A\colon G\times G\to H$ is an ...
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Question on an exercise on homological algebra?
Suppose $R$ has finite global dimension $n$, $N$ is a finitely generated module, $F$ is a free module, and $\operatorname{Ext}^n(N, F) \neq 0$. Then $\operatorname{Ext}^n(N, R)$ is also non-trivial.
...
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Theorems related to Chevalley's theorem
Recently I have read Chevalley's theorem of a complete local ring which basically says that if $(R,\mathfrak{m})$ is a complete local ring and if $\{b_n\}$ be a sequence of ideals such that $b_n \...
- 63.9k
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Irreducibility under etale ring map
Let $A\rightarrow B$ be a etale ring map between finite type algebra over algebraically closed field $k$.
If $A$ is one dimensional integral domain, is $B$ direct product of finite type integral ...
- 10.7k
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802
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Explanation for Lurie's SAG Remark 25.1.3.7
I am trying to understand the theory of simplicial commutative rings or animated rings. I just find a remark in Lurie's book Spectral Algebraic Geometry:
Remark 25.3.1.7. Let $f : R[x_1,\ldots ,x_n]\...
- 56.9k
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118
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Are ideals which are close to homogeneous subspaces close to homogeneous ideals?
Given $n, m \in \mathbb N$, let $\mathcal P$ denote the ring $\mathbb R[X_1, X_2, ..., X_n] / (X^\alpha : |\alpha| = m)$ of polynomials of degree $ < m$ with multiplication $P \cdot Q$ = $J_0^m(PQ)...
- 498
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Surjectivity of a bilinear map $A^m\times A^n\to A$ for a polynomial ring $A$
Let $k$ be a field and $A:= k[x_1, \dots, x_d]$.
Question: Suppose $M$ is an $m\times n$ matrix over $A$. If the entries of $M$ generate the unit ideal of $A$, must there exist $a\in A^m, b\in A^n$ ...
- 1,388
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Computing the Second Exterior Power of Certain Ideals in $\mathbb{Z}[\sqrt{-5}]$ and $\mathbb{Z}[\sqrt{5}]$ as Modules
I'm working on a problem involving the computation of the second exterior power of certain ideals within the rings $R_1 = \mathbb{Z}[\sqrt{-5}]$ and $R_2 = \mathbb{Z}[\sqrt{5}]$. The problem is as ...
- 7,893
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Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
- 15.7k
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58
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Universal formulas for polynomials with prescribed jets
Let $A$ be a commutative ring and $f\in A[x]$ a split monic. When $f$ is separable with roots $\mathrm Z(f)= \{ a_1,\dots ,a_k \}$, the Chinese remainder theorem (CRT) ensures that evaluation is an $A$...
- 10.5k
5
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1
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248
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Integral closure in characteristic 0
Let A be a Noetherian domain of characteristic 0, K be its field of fractions. Is the integral closure of A in K always finitely generated as A-module?
- 12.9k
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211
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When does a short exact sequence of abelian groups with $B\cong A\oplus C$ split?
$\hspace{20pt}$Duplicate on stackexchange.
This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and a ...
- 171
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0
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140
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Can an ideal in the ring of holomorphic functions on the complex plane be non-finitely generated?
Let $( I )$ be an ideal in the ring $( R )$ of all holomorphic functions of a single complex variable on the complex plane. I am interested in understanding whether it is possible for $( I )$ to be ...
- 6,357
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73
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In a ring with a $p$-derivation every $p$-power-torsion element is nilpotent
Let $p$ be a prime. The definition of $p$-derivation on a ring (aka $\delta$-ring structure) can be read in [K, Definition 2.1.1]. In short, a $\delta$-ring is a commutative ring with unity $A$ plus a ...
3
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1
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163
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Finite flat maps
Let $f : A \to B$ be a finite, finitely presented, flat map of (commutative) rings. It is a known consequence of Chevalley's theorem (on constructible sets) that the induced map $Spec B \to Spec A$ is ...
- 19.6k
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1
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191
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Does Serre's condition $S_k$ depend only on codimension $\leq k$ points?
Recall a locally Noetherian scheme $X$ has Serre's condition $S_k$ if for every $x\in X$ we have $\mathrm{depth}(\mathcal{O}_{x,X})\geq \mathrm{min}(k,\mathrm{dim}(\mathcal{O}_{x,X}))$.
Let $X$ be a ...
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