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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Simple proof of that $k[X]^G$ Cohen-Macaualy ($G$ finite)?
Let $X$ be a (EDIT: non-singular, or even $\mathbf A^n$) algebraic variety over a field $k$ (alg. closed). Suppose $G$ is a finite group acting on $X$, $|G|\neq 0$ in $k$. Then $k[X]^G$ is Cohen-...
- 31
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1
answer
127
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Degree bound for power of ideal
Let $I$ be an ideal in a commutative graded ring $R$, $M$ be a finitely generated graded $R$-module. Let $\varepsilon(M)$ be the smallest degree of a homogeneous element of $M$. An ideal $J$ is called ...
- 87
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Criteria for Preservation of a Module Structure under Extension of Scalars.
Let $A\to B$ be a morphism of (commutative) algebras and $M$ a $B$-module. The $A$-bilinear map $B\times M\to M$ given by $(b,m)\mapsto bm$ induces a surjective homomorphism $B\otimes_{A}M\to M$.
...
- 486
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79
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Saturation of a subalgebra over the Tate-algebra inside the power series ring
Let $A$ be a discrete valuation ring and $\pi$ a uniformizer.
Over $A$ we consider the Tate-algebra
$$A\langle t \rangle =\{ f=\sum_{n=0}^\infty a_nt^n \mid a_n\in A, \lim_{n\to \infty} \lvert a_n\...
5
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0
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308
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Properties of the Zariski-Riemann topology on the set of valuations
One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...
- 51
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179
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Randomized alternative to Buchberger's algorithm
Richard Lipton's blog describes a A New Way To Solve Linear Equations by Prasad Raghavendra.
Can the ideas in this algorithm be generalized to systems of polynomial equations to provide a randomized ...
- 529
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1
answer
320
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covers of complete regular local rings
It is well-known that if one assumes algebraic closedness and characteristic 0 of the residue field then finite covers of complete DVRs are all of the form $A[x]/(x^m-a)$ for some $a \in A$ (direct ...
- 4,070
3
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2
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335
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Gaps in Dimension Polynomials
There are several notions of rank/dimension defined on differential fields. However, we do not have a reasonable way to estimate these typically ordinal valued invariants. Especially, we do now know a ...
- 355
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124
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Singularity locus in terms of ideals.
Let $X$ be a smooth affine variety over a field, $Z\subset Y\subset X$ are closed (reduced) subvarieties. What are the possible ways to verify whether $Y$ is singular at $Z$ i.e. whether $Z$ is ...
- 16.9k
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Subrings of invariants in divided power algebras
I am wondering to what extent the functors "ring of invariants under a group action $G$"
and "divided power envelope with respect to a $G$-stable ideal" commute.
To be precise, let $R$ be a ...
- 1,609
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178
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The transcendence degree of the algebras of invariants
Let $V_n,V_m$ be the vector $\mathbb{C}$-spaces of the binary forms of degrees $n,m$ considered as usual $SL_2$-modules. Let $I_{n,m}=\mathbb{C}[V_n \oplus V_m]^{SL_2}$ and $C_{n,m}=\mathbb{C}[...
- 301
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1
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350
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Non-representability by a binary quadratic form
Let $k$ be an arbitrary field, $d\in k$, and $d$ is not a square in $k$.
Consider the binary quadratic form $f(x,y)=x^2-d y^2$
(it is the norm from $k(\sqrt{d})$ to $k$).
I am looking for a reference ...
- 14.2k
3
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answers
123
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Flatness over Jacboson ring
This is an elementary question which did not get answered on math.stackexchange. I would like to know the answer for expository purposes.
I need either a reference or a counter-example to the ...
- 63
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232
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Recovering syzygies of zero-dimensional schemes from those of their general linear projections.
Assume that I have a reduced zero-dimensional scheme $Z \subset \mathbb{P}^3$, not contained in any hyperplane, of degree $mn$ and having the following property:
For a general outer projection $\pi ...
- 66.3k
2
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1
answer
271
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Spectra of rings that are projective module over a subring
This question is motivated from my last question here. I wonder if one has a ring A and an over-ring of this ring say B, and if we know that B is a projective A-module can we have a particular idea of ...
- 2,275
2
votes
2
answers
282
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ring of idempotents of the integral extension of a ring
For any commutative ring $A$, the set of idempotents of $A$ will be denoted as $E(A)$. This set has a (canonical) ring structure. With addition defined by:
$$e+'f=e(1−f)+f(1−e)$$
where $+$ and $−$ are ...
- 2,275
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answers
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Is a *complete ring complete in the graded category ?
The question concerns a definition of Bruns, Herzog: Cohen-Macaulay Rings (before Prop. 3.6.16):
The Noetherian *local ring $(R,m)$ is said to be *complete if $(R_0,m_0)$ is complete.
(a graded ...
- 16.2k
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check the equality of two ideals in Singular http://www.singular.uni-kl.de [closed]
How one can check in Singlar that two ideals are equal? For example in macaulay 2, it is enough to write I==J, and then we will get True or False...
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1
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systems of parameters vs. minimal "exhausting" systems in a Noetherian local ring
Hello,
Probably this is a very easy question.
Fix a Noetherian local ring $A$, and an $A$-module of finite type $M$.
Lets call a system $ x_1 , \ldots , x_m \in \mathfrak{m}$ $M$-exhausting, if $M / ...
- 5,562
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0
answers
196
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Extending commuting endomorphims of a complete discrete valued field to the algebraic closure?
Is it true that any two commuting endomorphisms of a complete discrete valued field extend to commuting automorphisms of the algebraic closure?
- 341
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1
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I am interested in collecting different methods of proofs that a subalgebra coincides with whole algebra.
Let $A \subset \mathbb{C}[x_1,x_2,\ldots,x_n]$ - be finitely generated graded algebra and $B$ be its subalgebra. How to prove that $A=B.?$
Unfortunalelly I know only one method to do it - to ...
- 301
5
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0
answers
517
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Monomial-type ideals in polynomial rings
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ...
- 805
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169
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Algebraic properties of the semiring of open subsets.
Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at all?...
- 3,513
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vote
1
answer
257
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Are pullbacks from a factor of a product scheme flat over the other factor?
Given two smooth projective surfaces $X$ and $Y$ over some algebraically closed field.
Given a torsion free coherent sheaf $M$ on $X$. One has the projections $\pi_X$ and $\pi_Y$ from the product $X\...
- 1,391
1
vote
2
answers
355
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Is there a relationship between the right global dimensions of R and R[1/v]?
A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't ...
- 30.3k
3
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592
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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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614
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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
2
votes
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answers
263
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Koszul complex for monomials
Suppose I have a list of monomials $f_1, \dots, f_m \in R = K[x_1, \dots, x_n]$.
Is there a nice description of the cohomology of the Koszul complex
\begin{equation}
\cdots \rightarrow\bigwedge^{r+1}...
- 21
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2
answers
349
views
Subrings of rational functions invariant under change of sign
Let $R$ be a ring generated by $k$ rational functions in the
variables $x_1,...,x_n$ over the real numbers.
Is there an algorithm that computes a set of rational functions
$f_1,...,f_l \in R$ which ...
- 21
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287
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sheaves with free abelian stalks over compact space
The question concerns a sheaf $S$ of abelian groups over a compact space $X$. Suppose each stalk $S_x$ is finite rank free. Is the group of global sections free?
- 345
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Proof of local structure theory for unramified morphisms [closed]
In Raynaud's "Anneaux locaux henseliens," a proof is given of the
following fact: Let $R \to S$ be a finite type morphism of rings, $\mathfrak{q}
\in \mathrm{Spec} S$, $\mathfrak{p} $ the inverse ...
- 25.6k
4
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The ring generated by measures
Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...
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Connecting group ring, abelianization
For reasons arising in algebraic topology, I'm wanting to better understand the relations between two functors from groups to abelian groups, $\mathbb{Z}[\cdot]$ and $\operatorname{ab}$; group ring ...
- 1,987
3
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1
answer
321
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spurious torsion under compositions of linear maps
Say we have a PID $R$, integers $1 \leq a \leq b$, and $R$-homomorphisms $R^a \stackrel f\to R^b \stackrel g\to R^a$ with $g \circ f$ of full rank.
For $h = f, g, g \circ f$, let $c(h)$ be the ...
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Equivalent functors
Let $R$ be a commutative Noetherian ring, $M$ is a finitely generated $R$-module. If $F: Mod \to Mod$ is a left exact functor and $R^iF(E)=0$ where $E$ is injective module. Assume that $F(-) \cong Hom(...
- 259
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Example of a ring whose minimals are annihilators of idempotents?
I'm looking for examples† of rings with the property that for each
$P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$)
† other than domains!
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Is there a standard name for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial?
Is there any existing standard terminology for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial and $\alpha$ is e.g. a complex number? I haven't been able to come up with any good ...
- 1,488
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Finite generatation of Ext
If $A$ is a Noetherian ring and $M$, $N$ are finitely generated modules over $A$, it is easy to see that $\mbox{Ext}_{A}(M,N)$ is finitely generated by taking a finitely generated projective ...
- 2,857
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How many generators for rings of partially symmetric polynomials?
Let $k$ be a field, $n$ a positive integer. The group $S_n$ acts on $R_n=k[x_1,\dots,x_n]$ by permuting indices, and $\mathcal{S}_n=R_n^{S_n}=k[s_1,\dots,s_n]$ where the $s_i$'s are the usual ...
- 13.1k
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331
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Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
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How to consider a module over the ring Q[t,t^(-1)] to be a module over the polynomial ring Q[t]? [closed]
Can we view a module over the ring $\mathbb{Q}[t,t^{-1}]$ to be a module over the polynomial ring $\mathbb{Q}[t]$?
where $\mathbb{Q}$ denote any rational number coefficients.
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Decomposition of modules using computer packages
I am interested in computing direct sum decomposition of modules over some quotients of polynomial rings over a field (do not care much about the field at this point). Does any one know a package ...
- 30.5k
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Coherence of the monoid algebra of a non-finitely generated monoid
Let $P$ be an integral, sharp, finitely generated commutative monoid (say even torsion-free and saturated if you like), and consider the "rational cone" $P_\mathbb{Q}\subseteq P^{gp}\otimes_\mathbb{Z}...
- 1,030
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216
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Modules with first Betti number bigger than the second Betti number
Let $R$ be a commutative noetherian local ring (with 1) and let $M$ be a finitely generated $R$-module. Consider a minimal free presentation of $M$ as follows: $R^{\beta_2}\rightarrow R^{\beta_1}\...
- 3,101
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85
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different and discriminant for finite invariants
Let $k$ be an algebraically closed field.
Let $B$ a $k$-algebra of finite type, normal and Cohen-macaulay. Let $G$ a finite group acting on $B$. We assume that the order of $G$ is prime to the ...
- 3,472
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0
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152
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Kählerdifferentials and normal crossing divisors
Let $k$ be an algebraically closed field of arbitrary characteristic, $X$ a smooth surface over $k$, and $D_i \subset X$ be an regular, effective Divisor such that $D=\sum D_i$
has normal crossings ...
3
votes
1
answer
293
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Freeness of modules along ring homomorphisms
This question arises from my discussion with a Master student. It concerns with the following situation: let $\phi: R \to S$ be a homomorphism between Noetherian commutative rings. Suppose the $R$-...
- 30.5k
3
votes
0
answers
106
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Graded algebras: what is the name of the coeffients
Let $R=\oplus_i R_i,$ $R_0=\mathbb{C}$ be graded finitelly generaded algebra with the Krull dimension $r$ and the Hilbert series $H(A,t).$ Then the Laurent series expansion of $H(A,t)$ at $t=1$ has ...
- 115
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1
answer
154
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Example of a non-liftable morphism from a smooth algebra
This precise question grew out from the question whether a smooth commutative $k$-algebra (char($k$)=$0$) is always cofibrant as a non-positively graded commutative differential graded co-chain $k$-...
- 257
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0
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83
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lift sections on a thickened curve
Let $X$ a curve over an algebraically closed field $k$ and $D$ a divisor on X.
Fix an integer $N$ and a closed point $x$ on $X$, we assume that $\deg(D)$ is big enough such that we have a surjective ...
- 3,472