Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
2
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Computing the relations in invariant algebra
Suppose we have a ring $R$ and a finite group $G$ acting on it, Is there a way to compute the invariant ring $R^G$ explicitly? Infact I am more interested in the case of affine ring and the symmetric ...
- 596
2
votes
0
answers
148
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Are these rings Cohen Macaulay?
Let $R=k[M]$ be a monoid algebra, where $M$ is a fine and saturated monoid. If $k$ is a field, it is a theorem of Hochster that $R$ is Cohen-Macaulay. What if $k$ is a Dedekind domain? Is $R$ Cohen-...
- 493
0
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1
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475
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How to use Nakayama [closed]
Hi there,
Let R be a local commutative ring. If M and N are two R-modules with the condition that their direct sum is equal to R^n. How do I use Nakayama to show that M and N are in fact free R-...
- 1
1
vote
1
answer
132
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Morphsim from F_p[[X_1,...,X_d]].
Let f: F_p[[X_1,...,X_d]] --->> R be a surjection from a power series ring. Also assume that there is another surjection g: F_p[[Y_1,...,Y_d]] --->> R to the same local ring R.
Question: Is it ...
3
votes
1
answer
1k
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Inverse for a permutation over GF(2)
Given a permutation $f: \{0,1\}^n \rightarrow \{0,1\}^n$ as $n$ polynomials over $GF(2)$ how to get formulas for the inverse permutation $f^{-1}$?
I am interested in the answer to the previous ...
13
votes
0
answers
474
views
Refinement of concept of support of a module
My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...
- 8,727
1
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1
answer
172
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Does this solution guarantee $det(A)=0$ where $A\in M(R)$? [closed]
Suppose $R$ is a commutative ring with identity $1$ and the following matrix equation holds:
$\begin{pmatrix} a_n & & \\
\vdots & \ddots & \\
a_1 &...
- 398
3
votes
1
answer
578
views
When does grading pass to (co)-homology?
Let $G$ be an abelian group and let $R$ be a $G$-graded commutative ring, i.e., $R=\oplus_{g\in G} R_g$ with $R_gR_h\subseteq R_{g+h}$. Let $M$ be a $G$-graded $R$-module
i.e. $M=\oplus_{g\in G}M_g$ ...
- 7,609
1
vote
1
answer
921
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how to prove that localisation preserves Hom's [closed]
Can anyone tell me where I can read a proof that the natural map
$Hom_{A}(M,N)[S^{-1}]\rightarrow Hom_{A[S^{-1}]}(M[S^{-1}],N[S^{-1}])$
is an isomorphism if $M$ is finitely presented?
- 2,125
4
votes
0
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295
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The support of a finite type module on an algebraic space
I'd like to ask this question to make sure I understand a very basic thing about supports. Let $X$ be an algebraic space and F a quasi-coherent sheaf on it of finite type.
In here the schematic ...
- 1,273
2
votes
1
answer
578
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An example of a noetherian N-1 ring that is not N-2 and/or a Nagata ring
Hello is there anyone that would know where I can find an example of a noetherian N-1 ring that is not a Nagata ring. (See the Wikipedia article "Nagata ring" for the definitions of N-1 ring and ...
- 113
4
votes
1
answer
550
views
"Numerical Criterion" for Flatness
Let $R$ be a Dedekind ring, let $S = \mathrm{Spec} R$, and let us suppose that $f: X \to S$ is a finite morphism. Note that $X$ is not required to be connected. Does there exist a "numerical criterion"...
- 1,199
2
votes
1
answer
139
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integral closure of parameter ideals
Let $(R, \mathfrak{m})$ be an excellent domain of dimension $d$. Let $\mathfrak{q} = (x_1,...,x_d)$ be a parameter ideal of $R$.
Question: Is it true that $(x_1,...,x_{d-1}):x_d$ is contained in the ...
- 2,773
1
vote
2
answers
722
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Do subsets of generators of a toric ideal generate a toric ideal?
Given a toric ideal, say $J$, in a polynomial ring $k[x_1,...,x_n]$ we can find a finite
generating set for $J$. Is it possible, perhaps under additional assumptions on the structure of $J$, to give ...
- 805
1
vote
2
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639
views
Almost clean module
Please give me an example of an almost clean module $M$ over a ring $S$ so that if $x$ is a $M$ regular element then $M/xM$ is not almost clean.
- 287
2
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1
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172
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Whether the result that an ultraproduct which satisfies ACCP is automatically a field generalizes to ultrafilters on larger indexing sets
Let $F$ be an ultrafilter on some set $X$, $R$ an integral domain and $R_F$ the resulting ultraproduct ring. For an element $(a)$ in the product ring of $R$ indexed by $X$, denote its equivalence ...
- 43
1
vote
1
answer
334
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Number of generators of $\mathfrak m$-primary ideals in $k[x, y]$
Let $R = k[x, y]$ with $k$ algebraically closed, and $\mathfrak m = (x, y)$. Suppose $I$ is an $\mathfrak m$-primary ideal of $R$, i.e., $(x, y)^n \subset I \subset (x, y)$ for some $n$. If $I_{\...
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1
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185
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Algorithm to find symmetric function given specialization
I have a symmetric function f(c1,c2,c3,c4,c5) which, when c1 < c2 < c3 < c4 < c5, has the form p1(c1)+p2(c2)+p3(c3)+p4(c4)+p5(c5), where the p_i's happen to be polynomials of degree <=5....
- 827
4
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0
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311
views
Is every graded subalgebra of gr(A) equal to gr(B) for some subalgebra B of A?
Let $A$ be a finitely generated reduced (i.e. affine) $\mathbb C$-algebra, and $D:A\longrightarrow A$ a locally nilpotent derivation (i.e. $D$ is $\mathbb C$-linear, satisfies Leibniz' rule and $\...
- 337
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1
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301
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finite generation of $G$-equivariant holomorphic maps by polynomials?
Let $V$ and $W$ be two complex vector spaces with an action of a finite group $G$. The $G$-equivariant polynomial maps from $V$ to $W$ are finitely generated as a module over the ring of $G$-invariant ...
- 483
4
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0
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1k
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Grothendieck spectral sequence [duplicate]
Possible Duplicate:
Composing left and right derived functors
Hi,
probably this question is obvious. I apologize for this.
Given functors $F$ and $G$ left exact, with as good properties as you ...
- 51
0
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0
answers
63
views
Writing a module as a direct sum
Let $q_1, q_2, q_3 \in \mathbb{Z}[x,y]$ such that $q_1, q_2$ are algebraically independent and let $S$ be an algebra generated by $q_1, q_2, q_3$ over $F_p$. If writing $S$ as a module over $F_p [...
- 11
1
vote
1
answer
392
views
If all localizations of an algebra at primes are of finite type over a field
Let $k$ be a field, and $R$ a $k$-algebra. Suppose that $R_{\mathfrak{p}}$ is a finitely generated $k$-algebra for all prime ideals $\mathfrak{p}$ of $R$. Does this imply that $R$ is also a finitely ...
- 13
0
votes
1
answer
544
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associated prime ideal [duplicate]
Possible Duplicate:
minimal prime devisor(MinAss R)
Hello All,is This conclusion true?
$(R,m)$ be a local ring.if every associated prime ideal of $R$ be minimal then every associated prime ideal ...
- 418
7
votes
1
answer
730
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Example sought of an atomic domain R such that R[t] is not atomic
Recall that an integral domain $R$ is atomic if every nonzero nonunit admits at least one factorization into irreducible elements. (Indeed, hard-core factorization theorists have replaced the word "...
- 65.4k
0
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1
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267
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Embedding commutative associative rings in non associative rings [closed]
Let $R$ be a commutative and associative ring with unit. Can $R$ be embedded in a ring $\hat{R}$ wich is both non commutative and non associative ?
Thanks guys !
- 1
2
votes
2
answers
421
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Homological dimensions of module
$(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$-module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j<\mbox{depth }(M)$ and for $j>\mbox{inj. ...
- 2,857
3
votes
1
answer
347
views
Extensions of truncated Witt vectors
Let $k$ be a perfect field of characteristic $p>0$. For any positive integers $n$, let $W_n(k)$ be the truncated Witt vectors of length $n$ with coefficients in $k$. For any positive integers $a,b$,...
- 33
1
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0
answers
2k
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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 ...
- 391
3
votes
1
answer
1k
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Amazing examples in complex Algebraic Geometry
Good example teaches sometimes more than couple of theorems. I wonder what are your favourite examples in complex algebraic geometry, the ones that were astonishing for you, the simpler (at least ...
- 161
1
vote
1
answer
248
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Poset axioms of Boolean algebra [closed]
I found in Awodey's "Theory Category", second edition p34, a set of poset axioms to define Boolean algebras, whereas I don't see how they can be sufficient.
Here are the axioms:
A Boolean algebra ...
- 517
4
votes
1
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161
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A certain property of elliptic curves in a paper by Rees
In the paper "On a problem of Zariski", David Rees presents a counterexample to the following problem of Zariski.
Let $F/k$ be a f.g. field extension, $S$ a f.g. normal integral domain over $k$ ...
- 339
3
votes
1
answer
901
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Behaviour of Hilbert functions
Let $G$ be a complex simple reductive group. Then the set of isomorphy classes $Irr G$ is isomorphic to the set of dominant weights $\Lambda_+$ in the weight lattice of the maximal torus of a Borel ...
- 31
1
vote
0
answers
54
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Elimination theory for variables packaged in a matrix
I am wondering if the elimination theory in computational algebraic geometry can be more efficiently carried out if all variables lies within some given matrices.
For instance, consider the following:
...
- 367
3
votes
2
answers
589
views
Comparing homomorphisms over different base rings
I am trying to compare some homomorphism groups over different base rings, so given a commutative local ring $(A,\mathfrak{m})$ and a finite dimensional Azumaya algebra $R$ over $A$.
If $M$ and $N$ ...
- 1,391
3
votes
1
answer
270
views
When is a blow-up a non-trivial product?
Suppose $X$ is an algebraic variety and let $Z \subset X$ be a subvariety. Are there some useful criteria under which the blow-up $Bl_Z X$ becomes a nontrivial product $V \times W$ of the algebraic ...
- 690
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votes
1
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322
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Height unmixed ideal
Suppose $R$ is a regular local ring and $I$ is a non-zero ideal such that $I$ is a radical ideal and $I$ is height unmixed. Suppose $J$ is any radical ideal contained in $I$ and with the same height ...
- 3
3
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1
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544
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Which monomial subalgebras are direct summands of polynomial rings
Let $S=k[x_1,\dots,x_n]$ be a polynomial ring, and $A:=k[x^{u^{(1)}}, \dots x^{u^{(l)}}]$ a monomial subalgebra, generated by monomials $x^{u^{(i)}} = \prod_{j=1}^n x_j^{u^{(i)}_{j}}$ with $u^{(i)} \...
- 1,961
1
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1
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336
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Need an example of finitely generated graded algebra such that each its graded subspace has infinite dimension.
More accurately, let $\displaystyle A=\sum_{i=0}^{\infty}A_i$ be a finitely generated graded algebra over say $\mathbb{Q}$ but $\dim A_i=\infty$ for each $i.$ Is it possible?
- 301
2
votes
1
answer
281
views
Does fiberwise exactness imply exactness?
Let $R$ be a local Noetherian domain with fraction field $K$ and residue field $\Bbbk$. Let $C^{\bullet}$ be a bounded complex of free, finitely generated $R$-modules. Suppose that $C^{\bullet} \...
- 7,338
6
votes
1
answer
1k
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The inverse limit of locally free module
A is an I-adic complete Noetherian ring. M is a finitely generated A module. For any n>0, $M/I^nM$ is a finitely generated locally free A/I^n-module. Is M necessarily a locally free A-module?
- 1,091
3
votes
1
answer
497
views
Formally smooth maps between adic rings and regular immersions
Suppose $(A,\mathfrak{a})$ and $(B,\mathfrak{b})$ are two adically complete (commutative) noetherian rings. Let $f:A \to B$ be a continuous formally smooth formally of finite type map (that is, $B/\...
- 1,214
4
votes
1
answer
463
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Reference request, direct summand conjecture in dimension 2
What's the easiest (by which I mean uses the least fancy machinery) proof of the direct summand conjecture in dimension 2?
Recall that the direct summand conjecture says that:
Conjecture (Hochster): ...
- 20.5k
0
votes
0
answers
355
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abstract algebra for component wise operations on "vectors" or what it might be called
I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations:
- multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
- 254
4
votes
1
answer
201
views
Algorithm to detect if an element of a (commutative) ring is in a subring?
For rings finitely-generated over a field, the theory of Groebner bases gives us quite an efficient algorithm for determining whether an element of the ring is in a given ideal of the ring.
Is there ...
- 41
0
votes
0
answers
259
views
Ring algebraically closed in its completion.
First I would like to be clear about the definition, which I am having trouble finding.
What does: The local ring $A$ is algebraically closed in $B\supset A$. (e.g. for $B:=\hat{A}$, the completion ...
- 807
4
votes
1
answer
256
views
Heisenberg-type groups over rings with involution
Hello everyone!
In a paper "Coverings of twisted Chevalley groups over commutative rings" Eiichi Abe intoroduced the following construction:
Let $R$ be a commutative ring and $x\mapsto\overline{x}$ ...
- 3,186
1
vote
1
answer
601
views
Unimodular column property
Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property.
I would like to know if there is a ring $R$ that doesn't ...
- 113
3
votes
1
answer
201
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How to estimate a local hilbert samuel funcion
Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ...
- 403
2
votes
0
answers
337
views
Orthogonality (wrt. Ext, Tor) in commutative noetherian rings
Hi,
it is a folklore, that:
let $p$, $q$ be two primes of a commutative Gorenstein ring $R$.
$$
\operatorname{Tor}^k(E(R/p), E(R/q)) \neq 0 \iff p = q\mbox{ and }k = \operatorname{height} p.
$$
...
- 41