Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
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Obstruction map for local singularities via tangent (Andre-Quillen) cohomology
Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent (Andre-...
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Projectivity of one Tate algebra over another
Let $\mathbf{Q}_p \langle X_1,\dots,X_n \rangle$ be the $n$-variable Tate algebra, i.e. the subalgebra of $\mathbf{Q}_p[[X_1,\dots ,X_n]]$ of power series which converge on the closed unit polydisk in ...
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Is the generation of rings by their units a question in K-theory?
Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...
- 13.3k
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Question about the "nullstellensatz" for projective schemes
Yeasterday I asked a question on math stackexchange which simplifies to the following:
Assume that $G$ is a graded ring and $A \subseteq G$ is a homogeneous radical ideal. Is it true that $IV(...
- 3,830
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Modules over a Gorenstein ring
$A$ a Gorenstein ring, $M\neq 0$ a finite $A$-module with finite injective dimension. According to Bruns, this implies that $M$ has finite projective dimension. How do I see that?
- 2,857
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Efficient Algorithm for Matrix Version of Waring's Problem
Given an $n \times n$ matrix $A$ with entries in a commutative and associative ring with $1$ (say $Z[x_{1},\dots,x_{n^{2}}]$), the following paper guarantees existence of seven $B_{i}$s such that $A = ...
- 13.9k
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Cohomology vanishing for formal completions of modules?
Let $A$ be a ring, Noetherian or even of finite type over a field if necessary. Let $I$ be an ideal in $A$, $\widehat{A}$ the formal completion of $A$ along $I$, $M$ an $A$-module, finitely generated ...
- 1,329
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Proof that a finitely generated projective module over a Von Neumann Regular ring is free
I'm searching for a proof that a finitely generated projective module over a Von Neumann Regular ring such that all the localizations have the same rank is free. I know that this result is true, ...
- 163
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159
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Relation between local cohomology and koszul cohomology of multigraded ring
Let $R$ be a ${\mathbb {Z}}^k$-graded Noetherian ring, $J=(x,y)$ an ideal of $R$ where $x,y\in R_{(1,\ldots,1)}.$ Is this following true $$H_{J}^i(R)=\underset{n}\varinjlim{H^i((x^n,y^n),R)},$$ where $...
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Does the first singular cohomology of an ACM surface vanish?
Hi everybody, I am interested in the following:
Let $I\subset S=\mathbb{C}[x_0,\ldots ,x_n]$ be a graded ideal such that $\operatorname{depth}(S/I)\geq 3$, and let $X^h$ denote the analytic space ...
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How to compute the ring of invariants of SO_3(k) acting on a polynomial ring
Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.
${\rm SO}_3(k)$ acts on $A$ in the following way: Given $g \in {\...
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Comparation of dimensions of rings
Let $k$ be a field, $A$ be an integral domain, $B \subset A$, and $A, B$ are both finitely generated $k$ algebra. Let $p \subset B$ be a prime ideal. Suppose there exists prime ideals $q \subset A$, ...
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Graded Betti Numbers of a Graded Ideal with Linear Quotients
Exercise 8.8 in Monomial Ideals by Herzog and Hibi:
Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators $f_{1},...,f_{m}...
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Betti Numbers and number of generators
Suppose that $R:=k[x_0,\dots,x_n]$ and $I$ is an ideal. Is there any relation between finding the minimal generators of $I$ and the graded betti numbers of the module $R/I$?
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local Artin algebras
Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A_1\oplus\ldots\oplus A_n$ into local Artin subalgebras, see for example Atiyah-McDonald, ...
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Does existence of an isolated solution imply the Jacobian determinant is non-zero?
Let $f_1,\dots,f_n$ be formal power series in $\mathbb{C}[[x_1,\dots,x_n]]$ whose constant terms are all zero (i.e. $f_1,\dots f_n$ are not units in the ring). Suppose further that the radical of the ...
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Homologue of the Inertia group and of the Frobenius theorem for the group of values of a valuation
As I said previously, I have some problems in the theory of valuations and places.
Let L/K be a finite (say) Galois extension, F a place of L, and v a valuation of L.
I denote by l and k the residue ...
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Ring of Witt Vectors and Tensor product of Fields
Let $p > 2$ be a prime, and let $\textbf{F}_{p} =
\textbf{Z}/p\textbf{Z}$. Let $k_{1}$ be a finite field over
$\textbf{F}_{p}$, and let $k$ be a perfect field of characteristic
$p$. Then we have ...
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Why is Ext^n(k,M) a vector space over k?
This might be a trivial question to experts but not to me whatsoever. Suppose that $(R,m,k)$ is a Noetherian local ring, $M$ is an $R$-finite module whose depth is $n$. One then defines the type of $M$...
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$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true:
$$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$
The ...
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244
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Finite extension of a field [closed]
Is it true, that if $A$ is finitely generated commutatative algebra over a field $k$, not necessary algebraically closed, then prime ideal $p \subset A$ is maximal if and only if $k \subset Quot(A/p)$ ...
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What is the localization of Q[x]/(x) at 0
Q is a rational field. Q[x] is polynomial ring over Q 。(x) is maximal ideal of Q[x].
Take Q[x]/(x) as a module over Q[x]. Then what is Q[x]-module Q[x]/(x) localize at 0??
I think the result is
Q[x]/...
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Is this a characterization of Dedekind domain?
Let $R$ be an integral domain. Suppose that for any two nonzero ideals $I$ and $J$, we have $I \oplus J$ is isomorphic to $R \oplus IJ$ as $R$-modules. Does this implies $R$ is a Dedekind domain?
- 1,373
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purely non-algebraic extension that is not separable
Can you give an example of a field extension $k\subseteq K$ such that, every element of $K$ is transcendental over k and $K$ is not separable over $k$?
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Reflexive sheaves on singular surfaces
Let $S$ be a normal surface over an algebraically closed field $k$ and let
$s$ be a point of $S$. Let ${\mathcal F}$ be a reflexive sheaf on $S$ of generic rank $n$ . Consider the (derived) fiber of $...
- 7,037
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Artinian property of local cohomology module over graded local ring
We know that if $(R, m)$ is a local ring, $M$ is a finitely generated $R$-module, then the local cohomology module $H^{i}_{m}(M)$ is an Artinian module for every $j$.
My question is : if $(R,m)$ is a ...
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Kernel elements for the Grothendieck group map of a commutative monoid
This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^*$ be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+...
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Catenarity of monoid algebras
Let $R$ be a commutative ring, let $M$ be a commutative monoid, and let $R[M]$ denote the corresponding monoid algebra. Suppose further that $R$ is universally catenary. One may ask for conditions on $...
- 6,700
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Related to fractional ideals
$K$ a field, $A\subset K$ a subring, $M\subset K$ an $A$-submodule. Define
$$(A:_{K}M):= \lbrace s\in K|sM\subset A\rbrace$$
Then it is easy to see that
$$M\subset A\Longleftrightarrow A\subset (A:_{...
- 2,857
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Nontrivial criteria for polynomials to have no common zeros?
When we work in $C[x_1,x_2,...,x_n]$,here $C$ denotes the complex field, we know that when polynomials $f_1,f_2,...,f_k$ have no common zeros, then there exists polynomials $g_1,...,g_k$ , such that ...
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Transformations of integer polynomials under combinations of their roots
I'm wondering whether the following ideas/questions give rise to an existing body of research. (Accordingly: please suggest appropriate changes to the tags!)
Preamble
We consider polynomials f &...
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About regular local rings and Socles
Let R be a regular local ring with $ \text{dim} R = d $. If $ 0\rightarrow R\rightarrow I_0\rightarrow ...\rightarrow I_d\rightarrow 0 $. Then why for $ 0\leq i\leq d-1 $, the socle of $ I_i $ is ...
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an example of a Noetherian domain with finitely many non-principal maximal ideals [closed]
Let $F$ be an algebraically closed field, and consider the ring $F[X, Y]$
of polynomials over $F$ in two indeterminates $X$ and $Y$. Let $S$ be the multiplicatively
closed set in $F[X, Y]$ generated ...
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Dimension of ring completion wrt to a decreasing chain of ideals
Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $(I_{n})_{n\in\mathbb{N}}$ be a decreasing chain of ideals in $R$ such that $\bigcap_{n \in \mathbb{N}} I_{n} = \{ 0 \}$. Then there is a ...
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Introduction to "commutative semialgebra"?
Of course, commutative algebra is a fundamental topic in algebraic geometry, number theory, representation theory, and so on.
However, there are some instances (most obviously tropical geometry) ...
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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 ...
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Normality for non-noetherian schemes
I am interested to know to what extent the notion of normality makes sense on a non-noetherian scheme.
Specifically, I can ask the following question: let $\pi:X\to Y$ be a formally smooth morphism of ...
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Forgetting and tensoring up for very connective maps of $E_{\infty}$-rings
Does anything happen if I forget and tensor back up along a highly connective map of $E_{\infty}$-rings?
Here's what I mean precisely: Let $f \colon A \to B$ be a $n$-connective map between ...
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Reconstructing a polynomial from resultants
I am trying to compute a monic polynomial $f(x)$ with integer coefficients and known degree $d$. I am given $n$ pairwise coprime polynomials $g_1(x),\ldots,g_n(x)$, also with integer coefficients, ...
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Algebraic characterization of points constructible by compass and straightedge
The typical characterization of points constructible by compass and straightedge is the following:
Let $S\subseteq\mathbb{C}$ with $0,1\in S$, $K_0 = \mathbb{Q}(S\cup \bar{S})$ and $a\in\mathbb{C}$.
...
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When $R/(f)$ is regular?
For R being a commutative regular excellent Noetherian ring of finite Krull dimension which conditions on $f\in R$ can ensure that the ring $R/(f)$ is regular (so, I want a sufficient condition)? I do ...
- 16.9k
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Surjectivity of $f\colon\Gamma_Z(M)\rightarrow\bigoplus_{p\in Z\backslash Z'}\Gamma_{pR_p}(M_p)$
Suppose $Z'\subseteq Z\subseteq\text{Spec} R$ such that every element in $Z\backslash Z'$ is a minimal element (with respect to inclusion as ideals) in $Z$. Assume further that both $Z$ and $Z'$ are ...
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What is your expectation of the depth?
Let $S=k[x_1,...,x_9]$ be a polynomial ring over field $k$. Set $q_1=(x_1,x_2,x_5,x_6)$, $q_2=(x_1,x_2,x_6,x_7)$, $q_3=(x_2,x_3,x_7,x_8)$, $q_4=(x_1,x_5,x_6,x_7)$, $q_5=(x_1,x_6,x_7,x_8)$, $q_6=(x_2,...
- 21
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Is there a Dirichlet Unitary Unit Theorem?
Dirichlet's unit theorem computes the group of units of the algebraic numbers of a number field. There are a few generalisations for orders available.
Assume the order has an involution. For example, ...
- 151
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292
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Symbolic powers in regular local rings
I am having trouble understanding one of the results in the following paper
http://arxiv.org/PS_cache/math/pdf/0104/0104175v1.pdf
In proposition 3.1, the author says
Let $(R,\frak{m})$ be a ...
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477
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Comparing the homogeneous defining ideals of multiple embeddings of a projective scheme
If $X$ is a projective scheme over a field $k$ (which we may assume is algebraically closed), then under an embedding $i: X \hookrightarrow \mathbb{P}^n_k$, we may write $X = Proj(R/I)$ where $R = k[...
- 11
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718
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Bloch-Kato conjecture and Wiles' numerical criterion
I already asked this question some days ago on https://math.stackexchange.com/questions/158747/bloch-kato-conjecture-and-wiles-numerical-criterion but didn't receive any response.
In the ...
- 16.2k
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Separations between notions of rank for modules over commutative (semi-)rings with no zero divisors.
Let $M$ be an $m$-by-$n$ matrix, here are three definitions$^5$ that we could use for rank:
$rk(M) = \min k$ such that for matrices $P$, and $Q$ with $P$ of size $m$-by-$k$ and $Q$ of size $k$-by-$n$ ...
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resolution of singular points on curve
After reading Fulton's book "Algebraic Curves", I know how to do resolution of singular points on curves. Given an affine equation, I can get it's non-singular affine model, i.e the normalization of ...
- 1,109
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Finiteness for separated residually finite modules
Suppose that $A$ is a commutative noetherian Jacobson ring and $M$ is an $A$-module. Suppose in addition that $M$ is $\mathfrak{m}$-adically separated for every maximal ideal $\mathfrak{m}$, and that ...
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