Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
7
votes
2
answers
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Global dimension and localization
Is there any condition on a commutative ring $R$ so that the global dimension of $R$ coincides with the supremum of the global dimensions of the localizations $R_{\mathfrak{m}}$ at all maximal ideals $...
- 15.2k
7
votes
0
answers
228
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Terminology for vanishing of Hochschild homology with symmetric coefficients?
In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then
it should hopefully be understood by most readers as saying $H^n(A,M)=0$...
- 25.8k
1
vote
0
answers
35
views
Grobner basis of the toric ideal $I_{A_P}$ with respect to $<_{rev}$ consists of those binomials $t_αt_β − t_{α\cap β} t_{α\cup β}$
I try to understand the proof of the Theorem. 10.1.3.(page 185) from ''Monomial Ideals'' by Herzog & Hibi.
The reduced Grobner basis of the toric ideal $I_{A_P}$ with respect
to $<_{rev}$ ...
- 163
6
votes
1
answer
724
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Is there a relative version of Artin's approximation theorem?
I've been thinking about the following situation. I have schemes $X$ and $Y$, smooth of dimension $n$ over a base scheme $S$, together with sections of the structure maps, which are closed embeddings ...
- 61
0
votes
1
answer
175
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Codimension in zero and positive characteristic
Let $F_0,\ldots,F_m\in\mathbb{Z}[x_0,\ldots,x_n]$ be polynomials with integer coefficients and let $p$ be a prime integer. Consider the two ideals: $$I_0:=(F_0,\ldots,F_m)\subset \mathbb{Q}[x_0,\ldots,...
- 1,159
8
votes
1
answer
313
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When two determinantal ideals together generate a power of the maximal ideal?
(A somewhat technical question, but maybe it is well known.)
Consider matrices over the ring $k[[x_1,\dots,x_n]]$, whose entries vanish at the origin (i.e. belong to the maximal ideal $\mathfrak{m}$...
- 2,232
0
votes
1
answer
253
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Connected curve
Assume we have a normal,connected quasi projective scheme $Y:=X\backslash D$ where $X$ is a quasi projective scheme over field $k$, not necessarily char zero and also $D$ is a simple divisor, not ...
- 315
3
votes
0
answers
206
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Integer-matrix representation of a commutative ring
Consider a commutative ring $x_ix_j = N_{ij}^k x_k$, where $N_{ij}^k \in\{0,1,2,3,\cdots\}$, and $\{x_i\}$ is a finite set. (This is actually a fusion ring and $x_i$ are simple objects.)
How to find ...
- 4,826
2
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1
answer
1k
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For which rings does there exist an invertible Vandermonde matrix?
Suppose $R$ is a commutative ring, and $S \subset R^{n\times n}$ is an $R$-module. We are given $H_0,\dots,H_n \in R^{n\times n}$, and we know that for all $r \in R$,
$$H_0 + r H_1 + \dots r^n H_n \in ...
- 427
8
votes
1
answer
472
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Nonnegative additive functions on coherent sheaves
Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well ...
- 3,101
3
votes
0
answers
320
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a generalization of the annihilator of cokernel ideal (some new invariants of modules?) [closed]
Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $E\stackrel{A}{\rightarrow}F$. Say $rank(F)=m$.
The basic invariants of $A$ ...
- 2,232
5
votes
3
answers
985
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Isomorphism of the function field of the projective line with $\mathbf{C}(s)$
Suppose I chose two rational functions, say,
$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad
v = \frac{t^5(4+t)}{(1+4t)}.$$
Then I know that $K(X) = \mathbf{C}(u,v)$ is
the function field of the projective ...
- 53
3
votes
1
answer
221
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Extending descent data from the special fiber of an extension of DVR's
My question is about the proof of Lemma D.3 on p. 147 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud. Namely, towards the end of that proof there is the sentence "That $\varphi$ ...
- 1,103
12
votes
1
answer
949
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Discrete version of Nullstellensatz?
Hi. I was reading the paper "On the foundations of combinatorial theory (VI): The idea of a generating function" by Doubilet, Rota and Stanley, and there is a relation treated which is very ...
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9
votes
2
answers
1k
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Modules over Laurent series rings
Let $k[x]$ be the ring of polynomials over a field k in one variable x. A $k[x]$-module is a k-vector space together with a linear endomorphism (the action of x).
The field $k(x)$ of rational ...
- 66.8k
2
votes
1
answer
689
views
Finite-index free subgroups in lattices and matrix rings
It is a theorem of Selberg that a lattice $\Gamma$ in a linear group has a torsion-free subgroup of finite index. Page 64 in 'Introduction to Arithmetic Groups' by Dave Morris asserts these can be ...
- 1,769
1
vote
0
answers
145
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Explicit construction of a bielliptic curve
Let $C$ be a (projective smooth complex) curve such that $K_C=2(D+p)$, with $D+p$ defining a $g_7^2$; $p$ is a base point and $D$ defines a 2-to-1 map $\varphi:C\rightarrow E\subset\mathbb{P}^2$ onto ...
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10
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0
answers
573
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Singularities arising from the Minimal Model Program (an algebraic point of view)
I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ?...
- 987
1
vote
2
answers
232
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My output of a group and inverse-closed subset in MAGMA is no longer inverse-closed when entered as input to GAP.
In MAGMA, I input the following:
G:=SmallGroup(20,3);
G;
E:=[xx:xx in G];
S:=[E[6],E[7],E[13],E[20]];
S;
S[1]^2;
S[2]^2;
S[3]*S[4];
This gives the output:
GrpPC : ...
- 13
4
votes
0
answers
202
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Shifts in the decomposition of Bott-Samelson bimodules
Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\...
- 878
2
votes
1
answer
276
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Are schemes which agree on open set and its complement equal? - w/ applications to initial ideals/tropical basis
I appreciate the comments so far and am modifying based on something closer to the problem I'm interested in. I started out with something far too general.
This is probably easy, but I have been ...
- 51
2
votes
2
answers
1k
views
Radical of annihilator of a module
I met the following problem when I studied graded ring theory. I have no idea to solve it. Please help me. Thank you very much !
Let $R$ be a commutative graded ring, $M$ is a graded R-module, $N$ is ...
- 51
1
vote
1
answer
139
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What can be said about $A$ and $B$ given the exact sequence $0 \to R^p \to A \to R^r \to R^q \to B \to 0$?
Let $A,B$ be two $R$-modules over a commutative ring $R$ (restrict to $R = \mathbb{Z}$ or $R= \mathbb{K}$ a field where appropriate). Suppose $A$ and $B$ fit into an exact sequence
$0 \to R^p \to ...
- 11
4
votes
0
answers
130
views
A right adjoint to the truncated Witt functor?
For any ring $A$, let $\mathrm{wEt}_A$ be the category of weakly etale $A$-algebras ; it is a cocomplete category. By a theorem of Van der Kallen, the truncated Witt vector functor
$$
W_r : \mathrm{...
- 7,249
3
votes
0
answers
371
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Intersection Multiplicity
Let $X$ be an hyper-surface in an affine space defined by an equation $F$. We can assume that the ground field is $\mathbb{C}$ and $X$ is normal. Take functions $f_1,\dots, f_n$ on $X$ and let $B$ ...
- 2,384
6
votes
1
answer
294
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Is there a local-global principle for integral Laurent series ?
Motivation: A real number is rational iff its decimal expansion is periodic (by "periodic" I mean periodic after some steps). Similar, a p-adic number is rational iff its p-adic expansion is periodic. ...
- 16.2k
2
votes
0
answers
120
views
Group of units of a valuation
Let K be a field. Then a subring R of K is called a valuation ring if for all $x \in K^*,$ either $x \in R$ or $x^{-1} \in R$ (or both).
It can be shown that for any valuation $v$ on $K,$ the ring $\...
- 131
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0
answers
138
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Properties of a subring of a 'completion' of k(X_1, X_2, ..., X_n)
I'm looking for a reference in commutative algebra for the properties of the ring made of polynomials in $n$ indeterminate over a field $k$ with "real exponents".
I don't even know the name of this ...
- 101
2
votes
3
answers
817
views
From reducible polynomial to an irreducible one
Is there some algebraic construction/extension to make a reducible polynomial over $\mathbb{Q}$ irreducible?
For example: consider the polynomial $x^4-x^3-x^2+1=(x-1)(x^3-x-1)\in \mathbb{Q}[x]$.
Is ...
- 151
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votes
1
answer
151
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Quadratic extension and prime ideals
Let $B/A$ be a quadratic Galois extension between local domains. Define ${\mathrm{Gal}}(B/A) = \{e,\sigma\}$.
Choose two prime ideals ${\frak P}_1, {\frak P}_2$ of $B$ such that ${\frak P}_2 = {\...
- 781
2
votes
1
answer
180
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Families of ideals with a given initial ideal
Assume a fintie set of monomials is given. Is there a way to find the family of ideals whose initial ideal (say w.r.t revlex order) is generated by that finite set? I'll appreciate any partial answer, ...
- 2,221
4
votes
1
answer
675
views
Are all (commutative) rngs ideals of (commutative) rings? [closed]
To avoid repeating it endlessly, assume all rings and rngs are commutative. I do not know if this is necessary.
The question then is exactly the title, but I think a stronger statement is true:
...
- 1,979
2
votes
2
answers
189
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Domains $D$ for which for any prime $P$, $D_P$ is a PID
Is there any name or alternative characterization for the class of integral domains $D$ such that for any prime ideal $P$, $D_P$ is a principal ideal domain?
- 155
5
votes
1
answer
2k
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Intersections of irreducible components
Let $V$ be an algebraic variety (not irreducible) over $\mathbb{C}$, defined by an ideal $I = \{f_1,f_2,\dots, f_n\}$. $V$ is not necessarily pure dimensional. Suppose $V = R_1\cup R_2\cup\dots\cup ...
- 1,510
5
votes
3
answers
2k
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The correspondence between affine vector bundles and f.g. projective modules
The definition of a (geometric) vector bundle over a scheme $X$ can be rewritten as follows in terms of 'not-so-geometrical algebra' if $X=Spec R$ is affine and if I am not missing something.
A ...
- 2,782
8
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0
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430
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name for a degree-like invariant of a power series over a commutative ring
Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$
f =...
- 1,812
2
votes
1
answer
1k
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If the quotient of a local ring is regular, does that imply that the original ring must be regular?
Suppose $R$ is a local ring and let $I\subset R$ be some nontrivial ideal. Are there conditions that we can place on $I$ so that if $R/I$ is regular, then so is $R$?
I am aware of the result that ...
- 145
2
votes
2
answers
308
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An example ellucidating proposition 2.8 in Milne's notes on commutative algebra
I would like to know if one can weaken conditions of Proposition 2.8 in
http://www.jmilne.org/math/xnotes/CA.pdf
The proposition says that if an ideal $a$ in a ring $A$ is contained in the union of ...
- 14.3k
0
votes
1
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805
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Does Noether normalization hold more general? [duplicate]
Noether normalization tells us that a finitely generated $k$-algebra is an integral extension of a polynomial algebra over the field $k$.
My question is whether this still holds if we replace the ...
20
votes
1
answer
3k
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On a theorem of Jacobson
In a comment to an answer to a MO question, in which Bill Dubuque mentioned Jacobson's theorem stating that a ring in which $X^n=X$ is an identity is commutative (theorem which has shown up on MO ...
- 47.3k
5
votes
1
answer
368
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Proper-class sized "ring" with no maximal ideals
Suppose I have a collection of "elements" together with operations that satisfy the axioms for a commutative ring with identity --- except that these elements form not a set, but a proper class.
Must ...
- 23k
2
votes
1
answer
285
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Theorem 2.5 in "Castelnuovo-Mumford regularity of products of ideals" by Conca & Herzog
Theorem 2.5 in Conca and Herzog, Castelnuovo-Mumford regularity of products of ideals http://arxiv.org/abs/math/0210065, says that if $R$ is a polynomial ring over a field $k$, $I$ a homogeneous ideal ...
- 398
3
votes
1
answer
463
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Endomorphism Ring of Indecomposable MCM Modules
Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...
- 161
6
votes
0
answers
293
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Constructing the normal sheaf for the plucker embedding in MAGMA (or a similar programming language)
How would one construct the normal sheaf $N_{G(2,6)/\mathbb P^{14}}$ to the plucker embedding of the grassmannian $G(2,6) \rightarrow \mathbb P^{14}$ as a sheaf in MAGMA (or another programming ...
- 1,308
3
votes
1
answer
420
views
Automorphisms of complete discrete valuation ring
Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb F}...
- 87
1
vote
0
answers
139
views
A strong form of Bezout theorem
Let $X$ be a smooth projective variety of dimension $n$, $U \subset X$, non-empty open set. For any integer $k>0$, does there exist $n$-hypersurface sections $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ ...
- 2,126
3
votes
1
answer
1k
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Does totally ramified extension really exist?
The answer is certainly "Yes", but this is the problem I met in Algebraic Number Theory by Neukirch. I guess that I must be doing something wrong, since otherwise I will get the statement "There are ...
- 95
3
votes
1
answer
427
views
Classification (and automorphisms) of torsion-free modules/sheaves
I would like to know what can be said about the classification of torsion-free modules.
For my purposes, we can assume that $R$ is the function ring of a smooth affine variety over a field. How does ...
- 17.4k
8
votes
1
answer
783
views
Question about local description of the branch locus
Let $\pi:Y\to X$ be a dominant, finite morphism of nonsingular varieties over an algebraically closed field $\Bbbk$. Assume furthermore that for all $Q\in Y$, with $P=\pi(Q)$, we have
$$\mathcal O_{Y,...
- 4,902
2
votes
1
answer
279
views
Decide two indices of Ext functor
This question is from the proof of Theorem 11.34 in the book: Twenty-four Hours of Local Cohomology.
Let $R$ and $S$ be CM local ring and $R\to S$ a local homomorphism such that $S$ is a finite ...
- 403