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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Discovery of Hilbert polynomial
Presumably it was Hilbert who discovered Hilbert polynomials - where did they first appear?
The basic theorem is that for a finitely generated graded module $M = \bigoplus_k M_k$ over the ring of ...
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2
votes
1
answer
121
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Projective dimension and subrings
$\DeclareMathOperator\pd{pd}$Suppose that $R$ is a commutative ring and $R'$ is a subring of $R$ such that $R$ is a free $R'$-module of finite rank. Assume that both $R$ and $R'$ are regular local ...
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Is it a valuation ring?
It is known that a one-dimensional Noetherian local ring is a discrete valuation ring if it is integrally closed.
Then, even if it is not Noetherian, would a one-dimensional local ring become a ...
- 328
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0
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107
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Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
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0
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Factorial surfaces and smoothness
It is well-known that normal curves are smooth. Moreover, a UFD of Krull dimension one is regular. Is there any higher-dimensional analog?
For example, given a normal projective surface $S$ over $\...
- 389
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2
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776
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Hilbert polynomials of graded algebras evaluated at negative numbers
Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...
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A question about freeness of a certain class of abelian groups
Lets call an abelian group $G$, to be semi-free (or SF) if every nonzero subgroup of $G$ is isomorphic to $\mathbb{Z}\times H$ for some abelian group $H$.
Is every semi-free group, a free group? If ...
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1
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Do graded-commutative rings satisfy the strong rank condition?
Let $R$ be a ring. Recall that $R$ is said to satisfy the strong rank condition if, whenever $R^m \to R^n$ is a monomorphism of right $R$-modules (with $m,n \in \mathbb N$), we have $m \leq n$.
It is ...
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2
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An etale cover of a semiperfect ring
Assume that $R$ is a semiperfect ring in characteristic $p$, i.e the frobenius is surjective on $R$. I think one can prove that an etale cover of $R$ should again be semiperfect by considering the ...
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height of sum of prime ideals
Suppose $R$ is a Cohen-Macaulay local ring and $P,Q$ are prime ideals in $R$. Let the height of $P$ and $Q$ be $m$ and $n$ respectively. Then is it true that the height of $P+Q$ is at most $m+n$?
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2
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A proof of $\dim(R)+1 \leq \dim(R[T]) \leq 2 \dim(R)+1$ with the Coquand-Lombardi characterization of Krull dimension
Question. Can we use the Coquand-Lombardi characterization$^1$ of Krull dimension to prove the well-known inequalities$^2$
$$\dim(R)+1 \leq \dim(R[T]) \leq 2 \cdot \dim(R)+1,$$
where $R$ is any ...
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1
vote
0
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Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians
I've came across this problem while thinking about some properties of fat schemes.
Let me give you an explicit (motivating) example:
We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
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Is $\mathrm{Hom}_R(M,R)\neq \mathfrak m \mathrm{Hom}_R(M,R)$ if $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring and $M$ be an $R$-module such that $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$. Then, is it true that $\mathrm{Hom}_R(M,R)\neq \...
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Formal étaleness along Henselian thickenings
Assume that $f:X\to Y$ is an étale map between smooth varieties and $(S,I)$ is a Henselian pair. Let $\alpha\in X(S/I)$. Can we say that the lifts of $\alpha$ to $X(S)$ are in bijection with the lifts ...
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Why is this map a split monomorphism?
I have a question regarding a lemma in the proof of Hopkins-Neeman Correspondence.
It is the beginning part of Lemma 1.2 in the The Chromatic Tower for D(R)
Let $Y$ be an object of the derived ...
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4
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2
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342
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On a generalized homotopy transfer theorem
In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...
- 215
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1
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362
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Derivations and ideals
Let $R$ be a regular local ring and $I$ and ideal of $R$. If $D$ is a derivation of $R$, let
$$\lambda_D:I/I^2\to R/I$$
be the composition of the restriction of $D$ to $I$ and the quotient map $R\to R/...
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4
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Length of dual module
It is well known that, given a commutative ring $R$ and an $R$-module $M$, the dual module $M^\vee = \operatorname{Hom}_R(M, R)$ does not always satisfy $M^\vee \cong M \ (1)$, and not even $M^{\vee \...
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Fast checking that a system of polynomial equations is satisfiable over $\mathbb{F}_2$
I have a (fairly large) system of polynomial equations, of the form
$$
c_1d_1=0,\ c_1d_2+c_2d_1=0,\ldots
$$
(In case it is relevant, all the polynomials are homogeneous of degree 2, except for exactly ...
- 18.7k
2
votes
1
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Finitely generated $\mathbb{Z}$-algebra embeds into unramified $p$-adic ring
Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded ...
- 423
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0
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396
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Non-Noetherian (classical) algebraic geometry
My starting point for this question is that, in a very classical sense, algebraic geometry is the study of solution spaces of systems of polynomial equations over an algebraically closed field. It is ...
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3
votes
1
answer
239
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Vanishing of $\operatorname{Ext}_R^{1}(M,R)$ when $R$ is a Gorenstein local ring of dimension $1$ and $M$ is not finitely generated
Let $(R,\mathfrak m)$ be a Gorenstein local ring of dimension $1$. Let $M$ be an $R$-module (not finitely generated) such that $M\neq \mathfrak m M$ and there exists a non-zero-divisor $x\in \mathfrak ...
- 413
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0
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42
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If $G$ is a connected bipartite graph, then the edge ideal $I(G)$ is normally torsion free
I am studying the paper "On the Ideal Theory of Graphs" by Simis, Vasconcelos and Villarreal, Journal of Algebra 167, No. 2, 389-416 (1994), MR1283294, Zbl 0816.13003. I got stuck at theorem ...
- 21
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0
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156
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Taylor-Wiles systems for higher dimensional deformation rings
Let $R$ be a deformation ring and $M$ be a finitely generated $R$-module.
A strategy for proving the theorems $R=T$ is to associate with $(R,M)$ a Taylor-Wiles system denoted $(R_{Q},M_{Q})$. Here I'm ...
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1
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138
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Solvability of derivation Lie algebras of local finite-dimensional commutative algebras
Let $A$ be a finite-dimensional local commutative algebra (with one) over a characteristic zero field $k$. Is it true that the Lie algebra $\operatorname{Der}_k(A)$ of $k$-derivations of $A$ is ...
- 18.4k
0
votes
0
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70
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Hensel lifting of roots of a biquadratic polynomial
Let $5$ divide $p-1$. Therefore, we have $$1+x+x^2+x^3+x^4=(x-\alpha)(x-{\alpha}^2)(x-\alpha^3)(x-\alpha^4)=f_1f_2f_3f_4$$ over $F_p,$ where $\alpha$ is an element of order $5$ in ${F_p}^\times.$ We ...
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1
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1
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Perturbing pole of Laurent polynomial/series in a single summand
I am working with the ring of Laurent polynomials $\mathbb{F}[X,X^{-1}]$ over $\mathbb{F}$ for some algebraically closed field $\mathbb{F}$ of any characteristic. I encountered a problem emerging from ...
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0
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When does an algebraically independent set "satisfy Noether normalization"?
Let $k$ be a field, $A$ a finitely generated $k$-algebra.
By Noether normalization, we know that there exists a finite morphism of $k$-algebras $\varphi : k[x_1, \ldots, x_d] \hookrightarrow A$, with $...
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5
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1
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357
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Does the category of integral domains admit a symmetric monoidal structure?
Let $\mathbf{Int}$ be the category of integral domains with injective homomorphisms. Does it admit a symmetric monoidal structure? If so, can we choose $\mathbb{Z}$ as the unit object? If it helps to ...
- 63.1k
21
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1
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638
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Grothendieck group of the Fibonacci monoid
Let's denote the Fibonacci numbers by $F_0=0,F_1=1,F_{n+2}=F_{n+1}+F_n \; \forall n \ge 0$. According to Zeckendorf's theorem, every positive integer can be represented uniquely as the sum of some (at ...
- 1,543
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0
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Does the symmetric algebra functor preserve inclusions?
Theorem: For any compact abelian group $G$, the homogeneous component $%
H^{2}\left( B_{G};%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\right) $ of degree $2$ is naturally ...
- 1,367
3
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0
answers
161
views
Amalgamation of commutative subrings
Let $A$ and $B$ be commutative subrings of a non-commutative ring $X$.
Is there always a commutative ring $Y$ containing $A$ and $B$ preserving their intersection?
This is equivalent to ask if in the ...
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0
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329
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Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
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3
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1
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228
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Van der Waerden's classical paper on Hilbert polynomials and Bezout's theorem
Is it possible to obtain an electronic copy of Van der Waerden's "On Hilbert series of composition of ideals and generalisation of the Theorem of Bezout", Proceedings of the Koninklijke ...
- 5,339
4
votes
1
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GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials
This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$.
Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
- 63.1k
4
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1
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If $\mathbb{C}(u(x,y),v(x,y),f(x))=\mathbb{C}(x,y)$, for every $f(x) \in \mathbb{C}[x]-\mathbb{C}$, then already $\mathbb{C}(u,v)=\mathbb{C}(x,y)$?
The following question is a direct continuation of this elaborate question; it is mentioned there at the end:
Let $u,v \in \mathbb{C}(x,y)$ or $u,v \in \mathbb{C}[x,y]$, if it is easier to answer in ...
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1
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0
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Algorithms to decompose a graded module over $R[x]$, where $R$ is a PID
There is a certain class of objects, which can be thought of either as modules over a ring $R[x]$ or as functors. A few equivalent definitions are given below. The question is what computer algorithms ...
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3
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1
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Question on the extension theorem from Humphreys' book on linear algebraic groups
On page 2 of Humphreys' book "Linear algebraic groups" he presents the "extension theorem". I will copy it below:
Extension theorem. Let $R/S$ be an integral extension, $K$ an ...
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3
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1
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372
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Subalgebras of quadratic algebras that are not quadratic
Suppose $A=k\oplus A_1 \oplus A_2\oplus \cdots$ is a quadratic algebra over a field $k$. Let $B$ be the subalgebra generated by a subspace $V\subseteq A_1$. What are the examples of such subalgebras $...
5
votes
1
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Generalizations of Chevalley–Shephard–Todd's Theorem?
Major Edit
I will reformulate my question signicantly, given Anton Geraschenko's comment. The old version of the question is bellow.
For simplicity, my base field is $\mathbb{C}$. If $G<\...
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1
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Homological conjectures for finite dimensional commutative algebras
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $...
- 26.5k
4
votes
1
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168
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Does local homomorphism of finite flat dimension preserve Krull dimension?
Let $f:A\rightarrow B$ be a local homomorphism of Noetherian local rings, such that the $A$-module $B$ has finite flat dimension. Is it true that the Krull dimensions of $A$ and $B$ agree? If yes, ...
- 639
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0
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85
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Regarding the common zeros of the system of equations
Consider the following two systems of n homogeneous polynomials in n variables of degree $d$ with complex coefficients:
System 1 ($S_1$):
$f_1(x_1,\dots,x_n) = 0$,
$f_2(x_1,\dots,x_n) = 0$,
$\vdots$
$...
- 1,269
3
votes
1
answer
1k
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irreducibility of the polynomial $ x^4 +1 $
Let $K$ be a field. We consider the polynomial $f(x) = x^4 + 1$. It is known that $f(x)$ is irreducible over $\mathbb{Q}$ but reducible over any finite field. Thus $ f(x)$ is reducible over any field $...
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votes
2
answers
863
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$A^2$ is isomorphic to $A^{(\omega)}$, but not $A$
Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)?
Edited to add: As no answers are forthcoming, does anyone ...
- 18.7k
0
votes
1
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118
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$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$
Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$.
Is it true that $I/I^2$ is $R$-...
- 412
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0
answers
165
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A direct proof that every projectivity between parallel lines is affine
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
- 42k
1
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0
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166
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Reference request showing that a very general Abelian variety $ A $ of genus $ g>1 $ has cyclic class group with ample generator
In Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM I asked for an example of a Cohen Macaulay, normal, ...
- 912
2
votes
1
answer
365
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Correspondence between fundamental group and geometric properties of $X$
At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...
- 613
78
votes
9
answers
26k
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Irreducibility of polynomials in two variables
Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in ...
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