Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
4
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1
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734
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Parametric polynomial solution of a single polynomial equation
Let $P$ be a polynomial in $n$ variables with rational coefficients,
$P \in {\mathbb Q}[Z_1,Z_2, \ldots ,Z_n]$, and consider the algebraic
set
$Z=\lbrace (z_1,z_2,z_3, \ldots ,z_n) \in {\mathbb Q}^n |...
- 3,595
6
votes
1
answer
641
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The Jacobian ideal generates the socle of a complete intersection
This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here:
http://tinyurl.com/2967eov
I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ ...
- 805
2
votes
1
answer
505
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graded noetherian module
Let M be a R graded module with $M= \oplus M_i$. If M is noetherian then $M_i=0 $ for i << 0. My question is this, isn't $M_i = 0$ for all i >> 0 as well? If $(M_{n_i})_{i} \neq 0, n_i > 0$ ...
- 131
0
votes
1
answer
494
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Example: Nil radical of noetherian Rings with a map to simple noetherian rings
A basic example in commutative algebra: Let $A$ $B$ be noetherian rings, with $B$ simple noetherian. Suppose that for every element $b$ in $B$, there exists a power $b^{n}$ ...
3
votes
0
answers
140
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Determining Hilbert polynomial from some values of Hilbert function
For simplicity, let $(R,m)$ be a Noetherian local ring and $I$ an $m$-primary ideal. The Hilbert function of $I$ is defined as
$$
H_I(n): \mathbb{Z}_{\ge 0} \to \operatorname{length}_{R/m} I^n / I^{n+...
- 672
0
votes
1
answer
223
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Representation dimension of a special algebra
Hi,
I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf
I've come across a piece of information, which I don't understand, and wanted to ask, if I ...
- 1,755
2
votes
1
answer
504
views
Zero-dimensional algebras of infinite vector space dimension
Consider an algebra $A$ over a field and suppose that $A$ is zero-dimensional as a ring. It is well-known that if, in addition, $A$ is finitely generated, it has a finite vector space dimension. ...
1
vote
1
answer
434
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Equality of chern classes and isomorphism
Given two torsion free coherent sheaves $M$ and $N$ wit $rk(M)=rk(N)=r$ on an smooth projective surface $S$, by definition $det(M):=\Lambda^r(M)^{\*\*}$.
Is the following criterion correct?
$M\cong ...
- 1,391
2
votes
1
answer
227
views
Typical dimension of partial derivatives
Let $V$ be the space of all homogenous polynomials over $\mathbb{C}$ in $n$ variables of degree $d$.
Let $l,k$ be two integers and $f\in V$.
Let $\partial^{=k}(f)$ be the space of all partial ...
- 2,179
1
vote
1
answer
193
views
Union of Associated Primes.
Let $R$ be a Noetherian ring. Let $I=(x_1,x_2,...,x_t)$ be a nonzero ideal of $R$. Define $I_n=(x_1^n,x_2^n,...,x_t^n)$. Are there any results about finiteness of $\cup_n Ass_R(I^n/I_n)$?
More ...
- 11
5
votes
1
answer
500
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Are any finitely generated reflexive module a 2nd syzygy?
Are any finitely generated reflexive module a second syzygy?
(I´m thinking especially in normal noetherian domains)
More general...
Are any divisorial lattice a second syzygy?
(I´m thinking ...
2
votes
1
answer
693
views
When is the restriction map on global sections an embedding
Given a scheme $X$ with generic point p and a quasi-coherent sheaf $F$ on $X$.
Viewing $X$ as a scheme over $Spec(\mathbb{Z})$, let us assume
$f: X \rightarrow Spec(\mathbb{Z})$ is a proper map.
...
- 1,391
2
votes
1
answer
412
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derivative in the ring k[e]/e², chain rule
Let $k$ be a ring and $\overline{k} = k[\epsilon]/\epsilon^2$. For every $f \in k[t]$ there is a unique $f' \in k[t]$ such that $f(t+\epsilon)=f(t)+\epsilon f'(t)$ holds in $\overline{k}[t]$. It ...
- 63.1k
0
votes
1
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172
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minimal spans of polynomial companions of co-prime polynomials.
Is there an algorithm to determine for given $P,Q$ in $\mathbb Z[x,x^{-1}]$ with $gcd(P,Q)=1$, the value of $min\lbrace Span(A)+Span(B): A,B\in \mathbb Z[x,x^{-1}],\ A\cdot P+B\cdot Q=1\rbrace$, where ...
- 2,390
2
votes
0
answers
281
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Is evaluating limits with dual numbers sound?
Let $D$ be the ring $\mathbb{C}[\epsilon]/\langle \epsilon^2\rangle$. Define the functions $dual : \mathbb{C} \to D$ and $stdPart : D \to \mathbb{C}$ by $dual(x) := x+0\cdot \epsilon$ and $stdPart(x+...
user5810
11
votes
0
answers
1k
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Reverse mathematics strength of identically zero polynomials are the zero polynomial
According to wikipedia, the statement "every polynomial over a countable field that is not the zero polynomial has only finitely many roots" is equivalent to RCA0 over RCA0* (which is called ERCA-0 in ...
user5810
1
vote
0
answers
220
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the topology of power series ring
Hi, everyone.
Let $A$ be a complete DVR with uniformizer $t$, $R:=A[[X]]$. What is the natural topology of $R$ ?
- 43
1
vote
1
answer
330
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semilocal total quotient ring whose J(R) is not zero
I am interested in rings in which every non-unit is a zero divisor. Can you give me an example of such a ring that also has FINITELY many maximal ideals (semilocal), and whose Jacobson radical is not ...
- 21
0
votes
1
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202
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lattice of subalgebras of a finite commutative algebra
(I) Suppose A is a finite commutative local algebra. Must every lattice of local subalgebras of A be a distributive lattice ?
By a subalgebra of A we mean an algebra contained in A that shares the ...
- 103
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2
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332
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finite global dimension vs integral Domain
For the quotient of polynomial rings over complex number field,
its global dimension is finite is equivalent to it is domain.
is this true?
- 77
4
votes
1
answer
412
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F_q-structures on schemes
Let $k|\mathbb{F}_q$ be a field extension. An $\mathbb{F}_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}_q$-subalgebra $A _0$ of $A$ such that $A _0 \otimes _{\mathbb{F}_q} k \cong A$ via the ...
- 5,243
1
vote
0
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190
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regular sequence
Let $I \subset J $ be two monomial ideals in $S=k[x_1,..,x_n]$ minimally generated by $(a_1,...,a_s)$ and $(a_1,...,a_s,b_1,...,b_r)$. I want to show that depth$_S S/I \geq$ depth$_S S/J$.
Let $c = ...
- 287
0
votes
0
answers
381
views
Completion of commutative rings.
Assume that $(R,\mathfrak{m})$ is a commutative local ring of equal characteristic zero. So $R$ contains the field of rationals. The well known $\mathfrak{m}$-adic completion of $R$ provides a ...
- 591
2
votes
3
answers
294
views
Necessary and sufficient criteria for non-trivial derivations to exist?
Off hand, does anyone know of some useful conditions for checking if a ring (or more generally a semiring) has non-trivial derivations? (By non-trivial, I mean they do not squish everything down to ...
- 2,392
3
votes
2
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467
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Chern character of Hom-sheaves
I'm reading the book about moduli spaces by Huybrechts and Lehn, and i'm stuck understanding a proof, it is Theorem 6.1.8.:
Given a K3-surface $X$ and a 2-dimensional space $M$, coherent and torsion ...
- 1,391
2
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0
answers
97
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Completion of Bezout Domain a Bezout Domain?
Let $R$ be a Bezout domain, and $I$ any ideal inside of $R$. Is the $I$-adic completion
$$ \varprojlim_i R/I^i $$
necessarily a Bezout domain? If not, what conditions (on $R$ or $I$) might ensure that ...
- 53
0
votes
1
answer
170
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Tensoring with descending chain of modules
Let $A \to B$ be a ring homomorphism. Let $M_1 \supseteq M_2\supseteq \ldots$ be an infinite chain of $A$-modules ($M_i$ not necessarily finite free). Suppose that the limit $\cap_{i=1}^{\infty} M_i$ ...
- 11
1
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0
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113
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Submodul of finite ring extension
Let $R \hookrightarrow S$ be a finite extension of noetherian rings. Let $I \subseteq S$ be an $R$-submodule of $S$. Are there any sufficient criteria on $I$ such that it is in fact an ideal of $S$? ...
- 3,031
4
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1
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643
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An application of Zorn's lemma.
Suppose that $R$ is a commutative ring, an $R$-module $M$ is said to be finitely embedded if $M$ has a finitely generated essential socle. Now Let $M$ be finitely embedded and not
artinian, let $S$ ...
- 1,207
3
votes
1
answer
571
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Reference for submultiplicativity of length of tensor product
I am looking for a reference, in the form of a textbook, that contains proofs of following statements.
NOTE: I am NOT looking for the proofs, I am looking for a reference! Proofs of these statements ...
- 3,101
1
vote
1
answer
474
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Expressing fiber product of affines via an ideal
Let $X$ (resp. $Y$) be the affine $k$-scheme defined by the ideal $I$ (resp. $J$) in the polynomial ring $k[x_1,...x_n]$ (resp. $k[y_1,...,y_m]$).
Let $Z$ be the affine scheme defined by the ideal $L$...
- 23.3k
4
votes
0
answers
811
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$Ext$ functor, filtered complexes and spectral sequences
Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...
1
vote
1
answer
230
views
On the paper "On the asymptotic linearity of Castelnuovo-Mumford regularity"
I have posted this question on MSE, however it seems not to be interested by member there, so I decided to post it here. I am sorry if you feel it is not appropriate for MO.
I am now reading the ...
- 325
4
votes
1
answer
662
views
Modules with flat duals
Let $R$ be a commutative ring, $M$ an $R$-module, $M^*=Hom_R(M,R)$ its dual. What are sufficient (and possibly necessary) conditions on $M$ that ensure that $M^*$ is flat? Is there a name for such ...
- 12.3k
1
vote
0
answers
157
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Indecomposability of image transformations (pure algebra). Open questions
W-transformations -- definitions
We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
- 7,500
1
vote
0
answers
113
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Algebraic set given by sequence of polynomials
When working on some problem, I have end up with a following situation. Suppose
$P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero ...
- 11
2
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1
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456
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Generic liftings of a regular sequence on the initial ideal
Hi everyone,
I've got a question about explicitly lifting regular sequences. Let $I$ be an ideal in a polynomial ring $S$ with some term order. We'll denote the initial ideal by $in(I)$. It is ...
- 103
5
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0
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995
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Maximal ideals in polynomial rings over algebraically closed fields - when Weak Nullstellensatz does not apply
Weak nullstellensatz describes maximal ideals in polynomial rings over algebraically closed fields at least when the cardinality number of variables is finite. Lang obtained the same conclusion also ...
- 17.6k
8
votes
0
answers
210
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Smallest class of rings closed under familiar operations
Suppose I start out with the ring $\mathbb{Z}$, and call $\mathcal{C}$ the smallest collection of (commutative, unital) rings closed under the following list of operations (which I am aware has some ...
- 5,759
2
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0
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264
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2 questions on Nagata's counterexample; $k[f_1,...,f_r]=k[g_1,...,g_s]$ vs. $k(f_1,...,f_r)=k(g_1,...,g_s)$
Let $\{a_{ij}\}$ for $i=1,2,3$, and $j=1,...,16$ be algebraically independent elements over some prime field. Let $k$ be a field containing all $a_{ij}$. Then consider $k^{16}$ as $k$-vector space and ...
- 339
0
votes
0
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428
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flat morphism between regular local rings
Suppose $f: A \rightarrow B$ is a local homomorphism of local rings. Assume that $A$ and $B$ are noetherian, regular and $\mathrm{Spec} B \rightarrow \mathrm{Spec} A$ is quasi-finite. Is is necessary ...
- 153
6
votes
1
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376
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Checking locally whether a homomorphism is a localization
All rings below are commutative with $1$.
Suppose $A\subset B$ is a subring and that $A\rightarrow A'$ is a faithfully flat ring homomorphism. [You may assume the rings are actually ${\mathbb C}$-...
- 2,083
3
votes
1
answer
202
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Do the classes of cotorsion and strongly cotorsion modules coincide?
Cotorsion modules
A module M is called cotorsion if for all flat modules X, $Ext_R^1(X,M)=0$ .
Strongly cotorsion modules
M is called strongly cotorsion if for all modules X of finite flat ...
- 1,207
7
votes
0
answers
897
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Does the property (x*y)*x = x*y have a name?
The property $(xy)x = xy$ is one of the equations satisified by a directoid. Various properties have names ($xy = yx$ is commutativity, $xx=x$ is idempotency, etc). The wikipedia page for Magma has ...
- 11.8k
1
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2
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1k
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An "Elementary" Math Question Generalized (Ring Theory Perhaps)
The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics"
"Prove that if integers a_1, ..., a_n are all distinct, then the ...
- 1,801
0
votes
0
answers
235
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Power of ideals and exact sequences
Hello, I'm reading about analytic sheaves and I've a problem to understand something that's related with commutative algebra:
Let $\mathfrak{a}\subset R$ an ideal and $M$ an $R$-module. Then,
$\...
- 987
8
votes
2
answers
217
views
Flipping Hilbert series of semigroup rings
I'll first give intuition, and then give a precise statement.
For $|z|<1$, we have $\sum_{i \geq 0} z^i = 1/(1-z)$. For $|z|>1$, we have $\sum_{i<0} (-1) z^i=1/(1-z)$. Thus, the two ...
- 156k
3
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0
answers
168
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What is known about the krull dimension of an ultrapower ring?
Let $R$ be a ring, $F$ a free ultrafilter on a set $X$ which is not countably complete, and $R_F$ the ultrapower of $R$ with $R \not\cong R_F$. The following two results are from a masters thesis ...
- 6,180
4
votes
2
answers
468
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Maximal separable extensions of residue fields
Assume that $(A,m)$ is a Noetherian normal local domain, $K = Quot(A) \subset E, F$ Galois extensions of $K$. If $B=\overline{A}^{E}$, $C=\overline{A}^F$, and $D=\overline{A}^{EF}$ and we choose ...
- 43
4
votes
1
answer
138
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Pulling back roots from the Completion
Consider the following diagram of regular local rings
$\begin{matrix}
\hat{A} & \xrightarrow{\quad\hat\varphi\quad} & \hat{B} \\
\ \uparrow\scriptstyle\alpha & \circlearrowleft & \ \...
- 4,902