Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,337
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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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In an abelian category with no nontrivial Serre subcategories, does every short exact sequence split?
Question: Let $\mathcal A$ be an abelian category. Suppose that the only Serre subcategories of $\mathcal A$ are the zero category and $\mathcal A$ itself. Does it follow that every short exact ...
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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, ...
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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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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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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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Generalized wreath products of commutative algebras with Hopf algebras
Fix $k$ a commutative ring (or, if more convenient, assume it’s a field or even an algebraically closed field of characteristic 0, which is the case I’m mainly interested in). Let $A$ be a unital ...
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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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Coherent sheaves, Serre’s theorem and ext groups
Let $X$ be a smooth projective variety over an algebraically closed field $k$ (if necessary we assume that $\operatorname{ch}(k)=0$).
Let $O_X(1)$ be a very ample invertible sheaf on $X$.
Then, the ...
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Reference for a clear version of multigraded Serre-Grothendieck-Deligne correspondence local cohomology
The Grothendieck-Serre-Deligne correspondence states the following. Let $ R $ be a Noetherian, graded ring and let $ T $ be $ \operatorname{Proj}(R) $. If $ \mathcal{F} $ is a coherent sheaf on $ T $...
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Is there any sufficient or equivalent condition for the invertibility of a regular map, i.e. a self map of $\mathbb{R}^m$ with polynomial components?
Let $P:\mathbb{R}^m\to \mathbb{R}^m$ be a regular map, i.e. a map whose components are polynomials. I was wondering whether we can say anything about the the component polynomials, their degrees or ...
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Does going-down theorem hold for local homomorphism of finite flat dimension?
Let $f:(A,m)\rightarrow (B,n)$ be a local homomorphism of Noetherian local rings of finite flat dimension. Does the going-down theorem hold for $f$?
If yes, then by Theorem 15.1 in Matsumura’s ...
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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, ...
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A complex with homology $=R/p$
Given a Noetherian ring $R$ .
I am looking for a bounded complex $X$ of finitel geenerated projectives over $R$ whose homology is $R/p$. Infact I just need $X$ to have $\operatorname{Supp}(H(X)) = \...
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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 ...
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Are covering families of localizations stable under pushouts?
For a commutative ring $A$, we call a finite family of localizations $A \to A_{S_i}$ (where $S_i$ are some subsets of $A$) a covering if the canonical morphism $A \to \prod A_{S_i}$ is an effective ...
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A Weierstrass product theorem for invertible formal Laurent series over local Artinian rings?
Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition:
$$
A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 ...
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Invariant factors and commuting matrices over a discrete valuation ring
$\DeclareMathOperator\Im{Im}\DeclareMathOperator\Ker{Ker}$Let $A$ be a discrete valuation ring with uniformizer $p$. Let $X, Y\in M_n(A)$ be square matrices such that $XY=YX$, and let $X^T$, $Y^T$ be ...
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Admissibility of Ulm's invariants
Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define
$$G_{\alpha}=pG_{\beta}.$$
If $\alpha$ is a limit ...
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When do limits of $R$-modules commute with direct sum?
Let $R$ be a commutative ring. Is there any good special case in which I can say that a limit of $R$-modules commutes with direct sum? This is of course true for finite direct sums. Are there other ...
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The Krull dimension of the tensor product of rings
The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...
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Existence of a hyper plane
I am very new to algebraic geometry, and self-studying varieties. I have the following question.
Suppose $Y$ is a variety of dimension $r$ and degree $d>1$ in $\mathbb{P}^n$. Let $P$ be a ...
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A conceptual proof that bounded index subgroups of a bounded torsion abelian group contain bounded index complemented subgroups
Call an abelian group $G = (G,+)$ $m$-torsion for some natural number $m$ if one has $m \cdot x = 0$ for all $x \in G$. A subgroup $H$ of $G$ is said to be complemented if one can write $G = H \oplus ...
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What are some toy models for the stable homotopy groups of spheres?
The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero.
Question: What are some "toy models" ...
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What are the epis, monos, and extensions in the Freyd Envelope of a triangulated category?
Let $\mathcal T$ be a triangulated category (or homotopy category of a stable $\infty$-category).
Recall that the Freyd envelope of $\mathcal T$ is an abelian category $\mathcal A$ which is ...
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List of equivalent conditions for the invariant subalgebra to be polynomial
Let $k$ be a field, $P_n$ the polynomial algebra in $n$ indeterminates, and $G<\operatorname{GL}_n$ a finite group whose order is coprime to the characteristic of $k$, and that acts on $P_n$ by ...
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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,$$
where $R$ is any ...
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What is the Goldie dimension of the ring of stable stems?
Let $p$ be a prime, and let $\pi_\ast^{(p)}$ be the ring of stable homotopy groups of spheres localized at the prime $p$. This is a nonnegatively-graded-commutative ring with $\mathbb Z_{(p)}$ in ...
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Trying to decode a module functor
This is section 5.1 from, ARITHMETIC DERIVATIVES THROUGH GEOMETRY OF NUMBERS by Hector Pasten.
Let $A$ be a commutative unitary ring, let $R$ be a commutative monoid, and let $\alpha : R \to A$ be a ...
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Structure theorem for artinian modules?
Let $K$ be a field and let $A$ be a $K$-algebra which is finite dimensional as $K$-vector space. Then the nice structure theorem for artinian rings says that we can write $A$ as the direct product of ...
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derived completion and flat base change
Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings.
We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete.
For a ...
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Dimension inequality for primary groups
Let $p$ be a prime number and $G$ an abelian group.
The group $G$ is said to be $\textbf{primary}$ if every element of $G$ has order power of $p$.
For every natural number $n$, we define
$$\ker(p^n)=\{...
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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 ...
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Integral closure in the total ring of fractions of reduced ring. Is it finitely generated?
Let $R = \bigoplus_{i = 0}^\infty R_i$ be a reduced finitely generated graded $\mathbb{C}$-algebra, $R_0 = \mathbb{C}$. Let $\overline{R}$ be the integral closure of $R$ in its total ring of fractions....
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Presentation of Chevalley groups over Bezout domains
Let $\Phi$ be a root system of type $A_1$, $A_2$, $B_2$ or $G_2$. For a (commutative, unital) ring $R$, consider the group $G_{\Phi}(R)$ defined by Steinberg's presentation as in [1, Theorem 12.1.1 ...
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The eventual number of generators of modules of which $M$ is a subquotient
Let $R$ be a (commutative) ring and let $M$ be an $R$-module. Say that $M$ is subfinitely generated if $M$ is a submodule of a finitely-generated module. Write
$$\mathcal F(M) = \{ M \rightarrowtail N ...
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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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Characterisation of even characteristic quadratic system
$\DeclareMathOperator\supp{supp}$Let $f_i \in \bar{\mathbb{F}}_2[x_1,..,x_5]$ for $1 \leq i \leq 5$ be such that
$f_1(\bar{x}) = x_1 + x_5^2 + q_1$,
$f_2(\bar{x}) = x_2 + x_1^2 + q_2$,
$f_3(\bar{x}) = ...
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A problem in commutative algebra whose solution requires algebraic geometry (resp., noncommutative algebra)?
One can argue that commutative algebra is affine algebraic geometry. However, a great deal of commutative algebra generalizes to non-commutative algebra, and in that setting there is little geometry, ...
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Example of a ruled, CM, $ \mathbb{Q} $-factorial, normal, Mori dream space whose Cox ring is integral but not CM,
This question is related to one I asked here in Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM. In ...
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Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM
Does anyone know an example of a $ \mathbb{Q} $-factorial, normal, Cohen Macaulay, projective, Mori dream space $ Z $ over a field $ k $ of arbitrary characteristic such that the Cox ring of $ Z $ is ...
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For an element in the integral closure of an ideal $I$ - which power is in $I$?
Consider an ideal $I$ in a ring $R$. If $f \in R$ belongs to the integral closure of $I$, then there is $k_0 \geq 0$ such that $f^k \in I^{k-k_0}$ for all $k \geq k_0$. Are there any known upper ...
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Lengths and additive invariants which preserve positivity
The length of a module is well-known to be an additive invariant of finite-length modules. That is, if $R$ is a ring and $Art(R)$ its category of finite-length modules, then $length : Ob (Art(R)) \to \...
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Commutant of irrep of $S_n$ (over local field)
Let $k$ be a field of characteristic zero and let $(V, \rho)$ be a finite-dimensional representation over $k$ of the symmetric group $S_n$. I would like to understand the commutant $\operatorname{End}...
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A question on classification of quadratic polynomials in even characteristic
$\DeclareMathOperator\supp{supp}$Let $f_1,...,f_n \in \bar{\mathbb{F}}_2[x_1,...,x_n]$ such that $f_i = x_i + q_i$ for $1\leq i \leq n-1$ and $f_n = q_n$ where $q_1,...,q_n$ are homogenous quadratic ...
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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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Let $A, B$ be matrices with elements in $\mathbb{Z}_n$, does $\ker A = \ker B$ imply that they are row equivalent?
Let $A, B$ be matrices with elements in $\mathbb{Z}_n$. If $A x = 0$ and $B x = 0$ have the same set of solutions, where the vectors also have elements in $\mathbb{Z}_n$, does this mean that there is ...
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Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
I am posting this question on MO since I haven't received any answers on MSE.
Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...
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Multiply an integer polynomial with another integer polynomial to get a "big" coefficient
I have copied this question from StackExchange, in the hope that some experts here can provide some relevant insight. Thanks to Greg Martin for improving the question.
Given $f(x) = a_0 + a_1 x + a_2 ...
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Existence of a symmetric matrix satisfying certain irreducible conditions
Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
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