All Questions
6,055 questions
1
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138
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Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
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0
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236
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Terminology question - "Chern number"
I have seen the term Chern number used to refer to the first Hilbert-Samuel coefficient, $e_{1}(I)$, of an ideal $I$ in a local ring $(R, m)$. (Where the Hilbert-Samuel polynomial agrees with $\...
- 113
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0
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451
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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
1
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0
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276
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Generalizations of divided-power algebras over finite fields
In Andrews, Askey, and Roy's Special Functions, the authors state that Gauß sums are finite field analogs of the $\Gamma$-function as Jacobi sums are to B-function. The $\Gamma$-function is well-...
- 1,049
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0
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351
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Regularity and limits of smooth rational curves.
Fix integers $2 < d \leq n$.
Suppose that $T$ is a smooth complex curve with marked point $0 \in T$, and $X$ is a closed subscheme of $\mathbb{P}^n_T$, flat over $T$ such that each fiber has ...
- 1,990
1
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0
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169
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Sum of two free o-submodules in a vector space over a local field
Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...
- 107
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0
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538
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Functoriality of a standard integral domain construction.
The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...
- 21
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0
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198
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Seek for good methods of computing the Krull dimension of a module?
Hi, everyone. Recently I am interested in computing the Krull dimensions of modules without using any software. However, it is not an easy job for me to do so only by its definition. Therefore, I ...
- 1,207
1
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0
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238
views
relative flatness and torsion freeness
Hi.
Question 1: let $f:X\rightarrow S$ be a proper and surjective morphism of complex reduced spaces with $X$ pure dimensional. Let $F$ be a $S$-flat coherent sheaf on $X$. Is it true that the two ...
- 435
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0
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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
1
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0
answers
534
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Integral element in the quotient of a polynomial ring
Hello,
I'm writting a "report" (to learn) on algebraic geometry, and was looking to write a proof for the following statement :
Theorem : Let $K$ be an algebraically closed field and $C_1, C_2 \...
- 11
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0
answers
268
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Rational map defined over K leads to algebra question
Hello,
Concrete algebraic question : Let $K$ be a perfect field, $\bar{K}$ a fixed algebraic closure and let $f \in \bar{K}[x_1,\ldots,x_n]$. I was wondering when there exists another polynomial (non-...
- 11
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0
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2k
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Formal power series ring & completion
I encountered the following passage in Matsumura's Commutative Ring Theory :
A a Noetherian ring, $B=A[[x]]$ a formal power series ring. $M\subset B$ a maximal ideal, $\mathfrak{m}=M\cap A$. Then $(...
- 2,857
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1
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274
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Coefficients of the monomials appearing in a Schubert polynomial
It is known that the coefficients of the monomials appearing in a Schubert polynomial are always positive. My question is: Is it always true that at least one such coefficient must be $1$? If that is ...
user111492
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3
answers
1k
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non-associative but commutative algebra [closed]
Is it possible(or may be easier) to give an example of non associative algebra but commutative?
At first sight, it seems possible to prove associativity from commutativity but later realised it may no ...
- 629
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3
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565
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Homology of koszul complex is finitely generated?
$A$ a local ring and $a_{1}$, ..., $a_{n}$ elements in its maximal ideal, $M$ a finitely generated $A$-module. In this case apparently the homologies from the Koszul complex are finitely generated as $...
- 2,857
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2
answers
232
views
Commutation of $GL_{n}$ with projective limits
Let $A$ denote a unital commutative ring. Given a system of ideals $(I_p)_{p\in P}$ indexed by a partially ordered set $P$ such that if $p \leq q$, then $I_p$ is contained in $I_q$, when is
$$GL_n ...
- 91
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1
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630
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Useless question on rank
What is the rank of $A^{n}$ if A is the zero ring? It's clearly not $n$ as many careless authors claim, since it's not even invariant. I don't think it's 0 either because it does have a linearly ...
- 2,857
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2
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754
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On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients
Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ ...
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2
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316
views
Tensor powers of an algebra all isomorphic
Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism.
EDIT: Assume ...
- 4,985
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1
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327
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Can we generalise groupoids to monoid-oids? [closed]
Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories.
Groupoids correspond to small categories where every morphism is an ...
0
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2
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388
views
$\mathbb{Z}_p[\zeta]$ is Local Ring
Let consider the ring $\mathbb{Z}_p$ and $\zeta$ be a $p$-th root of unity. Especially $\zeta \not \in \mathbb{Z}_p$.
Denote with $\Phi _p(x)$ the cyclotomical polynomial in $p$. Since $p$ is a prime ...
- 6,038
0
votes
2
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768
views
Tor independence
Let $R$ be a ring. Take the polynomial ring over $R$
$$R[x_1,\dots, x_n]$$
nonzerodivisors $f,g\in R[x_1,\dots, x_n]$ such that $f$ is a polynomial in the first $k$ indeterminates, $g$ a polynomial ...
user95222
0
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1
answer
187
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Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?
Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and $...
- 17
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3
answers
186
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Behavior of duality under pull-back
I have a technical question on commutative algebra. I am not an expert in the subject, and I would like to know if there are "typical conditions" making the following possible.
Let $\varphi:R\to S$ ...
- 2,406
0
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1
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964
views
Determinants over commutative rings [closed]
Hello, I am preparing a paper on determinants in commutative rings. Someone can give me examples of applications of determinants in commutative rings to other areas of mathematics or physics. Thank ...
- 545
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1
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2k
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Are Chow groups a birational invariant?
Let us work in the category of smooth, projective varieties (say, over an algebraically closed field $k$). If $X$ and $X'$ are birational, then do they have the same Chow groups? Is there at least a ...
- 179
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2
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511
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When are seminormal rings Cohen-Macaulay?
I know that not every local seminormal ring is Cohen-Macaulay. But are 1-dimensional local seminormal rings Cohen-Macaulay?
- 179
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1
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556
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Is the multiplication between even numbers an associative algebra? [closed]
We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist?
It has been proposed as a counterexample the set of even numbers. ...
0
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3
answers
1k
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When is the radical of the extension of a prime ideal prime?
(All rings assumed to be commutative and unital)
Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial ...
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1
answer
152
views
Name for a monoid on the basis of a vector space?
Is there a name for the structure of a vector space with a monoid defined on its basis?
Given a vector space V over a field F, we can choose a basis and define a monoid on it. Now we can use each ...
- 481
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1
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162
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$\mathbb{C}(u(x,y),v(x,y),f(x)+g(y))=\mathbb{C}(x,y)$ implies $\mathbb{C}(u(x,y),v(x,y))=\mathbb{C}(x,y)$?
The following question is a direct continuation of this question:
Let $u,v \in \mathbb{C}[x,y]$.
Assume that for every $f \in \mathbb{C}[x]$ and every $g \in \mathbb{C}[y]$ (excluding the cases where $...
- 2,837
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1
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181
views
Factor $\sum_{n=1}^{N} x^n$ [closed]
I am attempting to factor an $N^{\text{th}}$ degree polynomial with coefficients strictly equal to $1$ given by the equation
$$\sum_{n=1}^{N} x^n$$
Although the Galois group for anything beyond a ...
- 547
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1
answer
426
views
Generators of $SL(n,\mathbb F_2)$? [closed]
Consider the invertible matrices in $\mathbb F_2^{n\times n}$ which are a multiplicative group structure. Is there a finite set of $2k$ (at a $k\in\mathbb Z_{\geq1}$ independent of $n$) generators for ...
- 13.9k
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1
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176
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When can one infer degrees of generators of a ring from its hilbert series
I know that for a noetherian ring, it's hilbert series can be written as $$HS(t)=\frac{P(t)}{\prod_{i=1}^d{(1-t^{d_i})}}$$ where $P(t)$ is polynomial, and there are $d$ generators of degrees $d_1,\...
- 928
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2
answers
240
views
Using group presentation for its corresponding semigroup?
Somewhere Colin M. Campbell noted:
If $A$ is a semigroup defined as $$A=Sg(\pi)=\langle a_1,\cdots, a_d\mid u_1=v_1,\cdots,u_e=v_e\rangle $$ then the same generators with the same relations can ...
- 233
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2
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357
views
Rank of a $ \mathbb{Z}_{p}[[T]] $ module
Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can ...
- 1,209
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2
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1k
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What is the localization of Q[x]/(x) at 0
Q is a rational field. Q[x] is polynomial ring over Q 。(x) is maximal ideal of Q[x].
Take Q[x]/(x) as a module over Q[x]. Then what is Q[x]-module Q[x]/(x) localize at 0??
I think the result is
Q[x]/...
- 25
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2
answers
389
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Torsion of modules
Given a left module $M$ over a domain $R$, I am interested in irreducible elements $r\in R$ such that $r\cdot m=0$ for some $m\in M-\{0\}.$ I think "torsors" would be perfect name for such $...
- 2,390
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1
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568
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A confusion about power series and p-adic measures
In page 16 of these notes on $p$-adic $L$-functions, it makes the following claim:
Let $\alpha$ be a $p$-adic measure on $\mathbf{Z}_p$ which corresponds to a power series $F_{\alpha}(T) \in \mathbf{...
- 1,949
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votes
1
answer
176
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Annihilator of idempotent elements
Let $R $ be a commutative ring with 1 and $s $ a nonnilpotent element of $R $ such that for each idempotent $e $ of $R $ there exists a natural number $n_e $ such that either $s^{n_e}e=0$ or $s^{n_e}(...
- 9
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2
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2k
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Tensor products of two domains
Let $R$ be an integral noetherian regular local ring. Let $S$ be a noetherian integral domain such that $S/R$ is finite.
That is, $R \subset S$ and the surjection $R^{\oplus n} \twoheadrightarrow S$ ...
- 781
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571
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Recursive Non-standard Models of Modular Arithmetic? [closed]
Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)...
- 687
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2
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681
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Pontryagin dual
Suppose $M$ is a $Z_p[[T]]$-module and $\widehat{M}$(the Pontryagin dual of $M$) is a finitely generated torsion $Z_p[[T]]$-module. How to prove that $\widehat{M}$ has $\mu$-invariant zero $\...
- 1,209
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1
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453
views
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 ...
- 3
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1
answer
403
views
When does a power semigroup have a zero, and what can the zero be?
Let $S$ be a semigroup. The power semigroup of $S$ is the set $P(S)=2^S\setminus\lbrace\varnothing\rbrace $ with the operation $$AB=\lbrace ab\ |\ a\in A,\ b\in B\rbrace.$$
This operation is ...
- 1,445
0
votes
1
answer
372
views
the algebraic closure of strict henselian DVR
Let $A$ be a strict henselian DVR, and $\hat A$ is completion of $A$,
is $K(A)^{alg} \longrightarrow {K(\hat A)}^{alg}$ a isomorphism?
where $K(A)$ and $K(\hat A)$ are quotient fields.
- 1,921
0
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2
answers
617
views
An element in the product of schemes
Let $ f : X \to S, g : Y \to S$ be the scheme morphisms, and $ X \times_S Y$ be the product of shemes. Let $ z \in X \times_S Y$ , and $ x=p(z) , y=q(z) ,$ where $ p: X \times_S Y \to X, q: X \times_S ...
- 95
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1
answer
183
views
Projectively splitting module
Is there a name for such class of modules $M$ such that $M\rightarrow N\rightarrow 0$ splits for every $N$?
- 2,857
0
votes
2
answers
563
views
Primary decomposition of zero-dimensional modules
(I removed my motivation because it may be misleading :) )
Let $A$ be a noetherian commutative ring and let $M \neq 0$ be a finitely generated zero-dimensional (i.e. $\mathrm{dim} \ \mathrm{Supp}(M) ...
- 5,243