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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(algebraic) Taylor expansion for polynomials (rational functions) with coefficients in an arbitrary field.
This a probably very easy question and I am not sure whether it has been asked before (although I searched for it). Moreover I really hope this is nothing which can be found in any standard ...
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0
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Are there good properties of the divided power completion map?
Let $Y \to X$ be a closed immersion of smooth schemes over, say, the ${\rm Spec}(\mathbb{Z}_p)$. The completion map $$X_{/Y}\to X$$ is an ind-closed immersion (sometimes called pseudo-closed immersion)...
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Free and surface groups cohomology
What is a good reference for results on cohomology of finite rank free groups and surface groups with group ring coefficients?
I am interested in the case when the group acts on its group ring via ...
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(When) is isomorphism on differentials enough to guarantee that a map is étale?
I'm sorry if this is too easy for MO.
Let $S$ be a locally noetherian scheme, flat over $\mathrm{Spec}\,\mathbb{Z}$, $X$ and $Y$ be flat $S$-schemes locally of finite presentation, and let $f:X\to Y$ ...
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Dimension of a module (which is not necessarily finite)
Let $R=\bigoplus_{ n\in\mathbb N}R_{ n}$ be a Noetherian standard ring defined over an Artinian local ring. Let $M=\bigoplus_{ n\in\mathbb N}M_{ n}$ be an $\mathbb N$-graded $R$-module (not ...
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Is there an irreducible integral polynomial in two variables which is reducible for every value of one of the variables?
Is there a polynomial $f(x,y)$ in two variables, with integer coefficients, such that $f$ is irreducible over the complex numbers (i.e., in $\mathbb{C}[x,y]$), but for every integer $n$, the ...
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flatness and derived completion
Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...
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Closure of the set of principal ideals under a certain operation
Suppose $K$ is a field, and $R$ is the polynomial ring $K[x_1, \ldots, x_n]$.
Suppose $S$ is a set of ideals of $R$ satisfying these properties:
$S$ contains all principal ideals.
If $I$, $J$, and $...
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Number of Minimal left ideals in the full matrix ring over a finite commutative local ring
Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?
user39204
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Base change and relative Ext over noncommutative rings
Given two smooth projective schemes $X$ and $Y$ over some algebraically closed field $k$, we have $X\times Y$ with the projections $p$ to $X$ and $q$ to $Y$. Furthermore we have a "nice" sheaf of ...
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2
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Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
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Injective modules and torsion functors
(This is a related question.)
Local cohomology is studied mostly over Noetherian rings. Parts of the machinery do in fact not rely on Noetherianness, but on some weaker properties, for example the ...
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2
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The resultant of two degree n and n - 1 functions in two variables of t
I'm currently studying the implicitization of bezier curves (that is, finding a function that f(x, y) = 0 for any x and y pairs of a curve p(t)) as part of an algorithm for curve intersection. The ...
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The completion of a ring R is a domain then the ring R is a domain?
Let be R a commutative ring whit unit and let I a proper ideal of R. Let R' the completion of R respect to the ideal I (see Introduction to Commutative Algebra - M. F. Atiyah, I. G. MacDonald for the ...
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Field extension of fields [closed]
Is the field of real numbers $\mathbb{R}$ a finite extension of some subfield $k\subset \mathbb{R}$?
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Does exterior product commute functor Hom?
Let $M$ be an module over the commutative ring $R$. I'd like to ask do we have the following isomorphism?
$$Hom_R(\wedge^n_RM,R)\simeq \wedge^n_R Hom_R(M,R)$$
We can obviously see it's true for the ...
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lim Ext(a^n/b^n,R)=0
Is it true that:
Let $R$ be a local ring and $\dim R= d$. If $b\subset a$ be two proper ideals of $R$ then for $ n\in {\Bbb N}$, $\varinjlim Ext^d_R(a^n/b^n,R)=0$
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Is every (one dimensional) n-bud of total degree n also a formal group law?
This is essentially a request for counterexamples, since I know so few $n$-buds (or as some might say, formal group law $n$-chunks). One notices that the only $1$-bud of maximal degree 1 is the ...
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When is there a deformation of a given singularity to a normal singularity
Question: Given a variety $X_0$ with a singularity (say Cohen-Macaulay), when does this exist as a special fiber of a flat family $X \to C$ mapping to a smooth curve $C$, such that the generic fiber ...
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computing quotient ideals efficiently over a polynomial ring: (I:J) when J has many generators?
Can someone guide me to a reference where an algorithm for computing I:J where
J has many generators is discussed? I know the method of using one generator at a time and then taking intersections. I ...
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2
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Finitely-generated algebra over Z
Let A be an artin ring which is also a finitely generated algebra over Z.
Show that $|A|<\infty$.
If A would have been a field then I know how to prove it. I know that A is a product of local ...
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0
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Elementary characterization of Krull dimension
I was reading the following paper: "A Short Proof for the Krull Dimension of a Polynomial Ring. Thierry Coquand and Henri Lombardi"
and came across this corollary. (This is present with a better ...
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1
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Finding reducible polynomials with restricted factors
Given $f(x),g(x) \in \mathbb{Z}[x]$, two irreducible polynomials, is there a polynomial $h(x) \in \mathbb{Z}[x]$ coprime to $f(x)$ such that $f(x) + g(x)h(x)$ is reducible over $\mathbb{Z}[x]$ with ...
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Local ring of product of varieties
Let $V$, $W$ be varieties (affine or projective) over an algebraically closed field $K$.
Let $p \in V$ and $q \in W$. Is there a description of the local ring of $V\times W$ at $(p,q)$ in terms of the ...
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0
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conditions for a subfield of a rational function field to be algebraically closed
Let's start with the following general question. Let $k$ be the ground field. Let $K=k(x_1,\cdots, x_n)$ be a rational function field and let $L$ be a subfield of $K$. Is there a condition to ...
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Derivations annihilated by powers of the augmentation ideal
Consider an augmented commutative ring $R$, with augmentation ideal $\varpi$. Let $\delta$ be a derivation of $R$. The example I have in mind is $R=\mathbb F_p[x]/(x^{p^i})$ and $\delta=d/dx$, though ...
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Local-cohomology and Hom
Let $f:R\to S$ be a flat homomorphism of commutative Noetherian rings. "Flat Base Change Theorem", compares the local cohomology modules $H^i_a(M) \otimes_R S$ and $H^i_{aS} (M\otimes_R S)$ for $i ∈ ...
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Does the coordinate ring of affine variety admit a structure of infinite dimensional variety?
We work in the category of algebraic varieties over
some algebraically closed field $k$.
By infinite dimensional variety I mean a filtration:
$$
V_0\subset V_1\subset V_2\subset\ldots
$$
where each $...
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Is there a ring which is not Hermite but is coherent?
Call a commutative unital ring $R$
Hermite if for all $m, n\in \mathbb{N}$ with $m<n$, and all $f\in R^{m\times n}$ such that transpose($f$) is left invertible (with a matrix with entries from $R$ ...
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Self-similarity for simple algebraic structures [closed]
I'm doing this thread because I have some ideas about how to define self-similarity in algebra, but I don't know if this is known at all. Any critics, comments and references are more than welcomed. ...
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Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay?
I'm not sure how long this iterative questions can go on, but let me try again. Let's say $X$ is a Cohen-Macaulay scheme with an action of $\mathbb{G}_m$ (i.e. if $X$ is affine, a grading on the ...
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When is a local subring of a number field a valuation ring?
Do we have some good examples of local subrings of number fields which are not valuation rings?
Do we have an easy criterion for determining whether a local subring of a number field is a valuation ...
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How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?
One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...
- 1,554
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1
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Simultaneous triangularizability over a commutative ring
Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property
(*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent.
McCoy showed that, if $...
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1
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polynomial expression for counting number of integral points of a set
Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$
Can we ...
- 1,713
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1
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Depth of polynomial ring $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$
Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. My Question is about the depth of this ring.
Question: Could we ...
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$0 :_M I^n$ is finitely generated for all $i\ge 1$?
I see the remark that:
"Let $R$ be a Noetherian commutative ring, $M$ an $R$-module and $I$ an ideal of $R.$ Assume that $0 :_M I$ is finitely generated. Then $0 :_M I^n$ is finitely generated for all ...
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Is there fppf descent of locally free modules
Being locally free is a property of quasi-coherent modules which
does not descend in the fpqc topology (see Remark Tag 05VF). But what happens for fppf coverings? More precisely we ask:
Suppose $A \...
user45795
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1
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On the Completion of a complete local ring
Let $(R,\mathfrak{m})$ be a complete local ring, $a_{\lambda}$ be a decreasing net of ideals in $R$, indexed by a directed set. Consider the completion under $a_{\lambda}$-topology $A=\underleftarrow{\...
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working with local rings: "abstract" vs "geometric" proofs
Let $R$ be a local ring (commutative, Noetherian, over an algebraically closed field; if needed Henselian). Suppose one wants to prove some statement.
Suppose $R$ happens to be the ring of "functions"...
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Localisation of injectives
When working with injective modules, one bad thing is that they do not necessarily behave well with respect to localisation. Consider a commutative ring $R$ and have a look at the following properties:...
- 6,700
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3
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Examples of DVRs of residue char p and ramification e
I am looking for concrete examples of a complete discrete valuation ring $R$ of characteristic 0, residue characteristic $p$ and ramification index $e$. By residue characteristic, I mean the ...
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betti-numbers of Gin(I), generic initial ideal of $I$
here in the paper Ideals with Stable Betti Numbers there is a theorem that I can't uderstand it, both in details (which highlighted) and sketch of the proof of (b):
can you help please?
...
- 1,355
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2
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Identity on topological space but not on scheme
I have this question just out of curiosity.
If X is a scheme, then a morphism $f: X \rightarrow X$ can be the identity on the underlying topological space of X, but not the identity on the structure ...
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On factorization algorithms for $\mathcal{O}[x]$
We know that $\mathsf{LLL}$ algorithm provides factorization procedure that runs in poly time for polynomials in $\Bbb Z[x]$ that are primitive.
What other rings $\mathcal{O}$ can we use instead of $\...
user76479
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2
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maximal ideal in local subrings
Let $A,B$ be two local rings and put $\mathfrak{m}_A, \mathfrak{m}_B$ their maximal ideals. Now suppose that we have an injection $0 \to A \to B$ and put $\mathfrak{n} := A \cap \mathfrak{m}_B $. It ...
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Formally smooth morphisms, the cotangent complex, André-Quillen cohomology, and representability of nilpotent extensions as trivial extensions over a cofibrant replacement
Recall that an $R$-algebra $R\to S$ is called formally smooth (resp. formally unramified resp. formally étale) if given any lifting problem of the form
$$\begin{matrix}
R&\to &T\\
\downarrow&...
- 19.6k
5
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1
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Structure of f.g. modules over a non-commutative ring
To what extent is the structure theorem for finitely generated modules over principal ideal domains true over non-commutative domains? I'm in particular interested in non-commutative euclidean domains ...
- 271
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1
answer
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$IM=mM$. can we say that $I$ is a reduction ideal of $m$?
Question. Let $(R,m)$ be a Noetherian local ring and $M$ be a finite faithful $R$-module. Let $I$ be an ideal of $R$ such that $IM=mM$. Can we say that $I$ is a reduction ideal of $m$? Recall that $I$ ...
- 1,355
1
vote
1
answer
483
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formally étale morphisms which are also universally closed
A morphism of schemes which is formally unramified, universally closed, and a monomorphism is a closed immersion. Is it possible to characterize morphisms which are formally etale and universally ...
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