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Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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Hom functor and Cohen-Macaulay modules

Let $A$ be a local Gorenstein $\mathbb{C}$-algebra (not necessarily regular). Let $M,N$ be maximal Cohen-Macaulay $A$-modules. Is Hom(M,N) a maximal Cohen-Macaulay A-module? Note that I had asked this ...
Naga Venkata's user avatar
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Action of torus on Laurent polynomials

Let $F$ be an algebraically closed field and suppose that the torus $(F^*)^n$ acts on the Laurent polynomial ring $L$ in $n$ variables $X_1, \dots, X_n$ defined by $X_i \dashrightarrow a_iX_i$ for ...
A. Gupta's user avatar
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Connectedness of equivariant Hilbert schemes of points of affine spaces (or as orbifolds)?

Let $G$ be an abelian finite group act on $\mathbb C^n$, when the equivariant Hilbert scheme $\mathrm{Hilb}^{R}(\mathbb C^n)^G=\mathrm{Hilb}^{R}([\mathbb C^n/G])$ is connected? Now $R$ is a ...
DVL-WakeUp's user avatar
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Primary invariants on MAGMA for a graded ring

I have asked this question on mathstacks, but a collegue of mine recommended me to post it here. I am trying to find an optimal system of parameters for a graded ring using Magma. Specifically, I want ...
Rustam T's user avatar
6 votes
1 answer
651 views

Is decomposability of polynomials over a field an undecidable problem?

By a decomposition of a polynomial $F(x)$ over a field $K$ we mean writing $F(x)$ as $$ F(x)=G(H(x)) \quad(G(x), H(x) \in K[x]), $$ which is nontrivial if $\operatorname{deg} G(x)>1$ and $\...
SARTHAK GUPTA's user avatar
2 votes
0 answers
126 views

Derived tensor products and regular sequences

Let $R \to A$ be a homomorphism of commutative rings, and let $x\in R$ be an element (or a sequence of elements in $R$, if you prefer) that is both $R$-regular and $A$-regular. Then we have $$ A\...
Zuka's user avatar
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2 votes
1 answer
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Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$

Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring. There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
Justin Bloom's user avatar
3 votes
1 answer
225 views

Finite generativity of algebra with valuation

Let $C$ be a commutative finitely generated algebra with no zero divisors. If necessary, we can assume it to be graded and a unique factorization domain. Let $a\in C$ be a prime element. Let's also ...
Sasha Kucherenko's user avatar
2 votes
0 answers
59 views

Tensor product of two transcendental flat algebras is not a field?

I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
Jz Pan's user avatar
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What are the points of the algebra of polynomial functions on an arbitrary vector space?

Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
Dima Roytenberg's user avatar
5 votes
1 answer
132 views

Relation between Tor amplitude and $p$-complete Tor amplitude for a ring of characteristic $p$

Fix a prime number $p$. Let $A$ be a commutative ring, and consider an $A$-algbera $B$ of characteristic $p$. So we have a sequence of ring homomorphisms $$ A \to A/pA \to B. $$ Assume that we want to ...
Zuka's user avatar
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Length of generic intersection in local ring

Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$. If $a$ that is not a zero divisor of $R/I$ we have ...
Serge the Toaster's user avatar
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1 answer
324 views

Non-negative coefficients polynomials

Let $n \in \mathbb N$ and $P,Q \in \mathbb R_+[x]$. Is it true that $(x+1)^n\neq (x-2)^2 \times P(x)+(x-4)^2 \times Q(x)$ ? I have asked, this question here (*), two weeks ago, but no answers. (*) ...
Dattier's user avatar
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Generalization of Koszul cohomology for wedge product with a fixed q-form: literature and references?

Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $1 \leq p \leq q$, consider the cochain complex: $0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \cdots \...
user47660's user avatar
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When do algebraic elements form a subalgebra?

If $R$ is a commutative ring and $A$ is a commutative $R$-algebra, we say that an element $x\in A$ is algebraic over $R$ if $x$ is a root of a nonzero polynomial $f \in R[X]$, or equivalently, if the &...
Junyan Xu's user avatar
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1 answer
267 views

UFD property for power series in characteristic 0

Samuel famously produced an example of a UFD, namely $S = R_{(x,y,z)}$, where $R = K[x,y,z]/(x^2+y^3+z^7)$ and $K$ has characteristic 2, such that the power series extension $S[[ x ]]$ is not a UFD. ...
Jason McCullough's user avatar
4 votes
1 answer
228 views

Literature Request: The derived category is Krull-Schmidt

I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question Literature request: $K^b(\text{...
Sebastian Pozo's user avatar
1 vote
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Descent of $G$-invariant formal system of parameters using GAGF

Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \...
user267839's user avatar
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1 vote
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99 views

Cohen-Macaulayness of the homogeneous coordinate ring of projective monomial curves

Let $A = \{a_0, a_1, \ldots, a_{n-1}\} \subset \mathbb{N}$ be a set of non-negative integers where we assume that $a_0 < \cdots < a_{n-1}$ and set $d := a_{n-1}$. For every $s \in \mathbb{N}$, ...
Takatoshi Kashiwara's user avatar
1 vote
1 answer
214 views

Derived completeness of the inverse perfection

Fix a prime number $p$, and let $R$ be a ring of positive characteristic $p$. Consider the inverse perfection of $R$, which is defined as the inverse limit $$ R^\flat = \varprojlim(\cdots \xrightarrow{...
Zuka's user avatar
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4 votes
2 answers
377 views

Witt coordinates vs Joyal coordinates on the ring of Witt vectors

I am learning about Witt vectors following [K, Ch. 3], but I am having trouble with the presentation of his material (see (Q1) below). Kedlaya's definition for ring $W(A)$ of the Witt vectors on a ...
Elías Guisado Villalgordo's user avatar
9 votes
0 answers
276 views

What is known about vector subspaces of polynomial rings closed under factors?

Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is: If $a,b \in F$, then $a+b \in F$, and If $a,b \in R$ with $a\in R$ ...
Neil Epstein's user avatar
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12 votes
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257 views

When do (or don't) residue fields generate the derived category of a ring?

Let $R$ be a commutative ring, and $D(R)$ its derived category of unbounded chain complexes. I'm interested in when the residue fields $k(\mathfrak{p}) = \mathrm{Frac}(R/\frak p)$ for $\mathfrak p \in ...
Drew Heard's user avatar
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2 votes
0 answers
133 views

Dual of finite reflexive modules

Let $A$ be a commutative ring and $M$ be a finite reflexive $A$-module, i.e. the natural map $M\to (M^{\vee})^{\vee}$ is an isomorphism. Can we deduce that the dual $M^{\vee}$ is also finite?
Y.M's user avatar
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Integral graded algebra of finite type is approximable

The following is the definition of approximable algebra. An integral graded $K$-algebra $\oplus_{n\geqslant 0}B_n$ is said to be approximable if 1.$$rk_K(B_n)<+\infty,\forall n\in \mathbb{N}, $$and ...
Ying.D's user avatar
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Colimits in commutative Banach algebras?

Let $K$ be a complete non-Archimedean field. It is known that the category $\mathrm{Ban}_K$ of $K$-Banach spaces with bounded linear maps does not have infinite colimits. The usual argument for $\...
user577413's user avatar
3 votes
1 answer
220 views

Weak approximation in Krull domains

Suppose $R$ is a Krull domain with the field of fraction $K$. To every prime ideal $P$ of $R$ of height $1$, one can associate a $ \mathbb{Z}$-valued discrete valuation which we denote by $v_P$. ...
Keivan Karai's user avatar
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2 votes
0 answers
66 views

Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
uno's user avatar
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1 answer
219 views

What is the fastest known algorithm for evaluating a homogeneous binary polynomial?

This question was initially posted on math.stackexchange.com, but there is no appropriate answer, hence I have the right to publish it here again. Let $f(x,y) = \sum_{i = 0}^d f_i x^i y^{d-i}$ be a ...
Dimitri Koshelev's user avatar
7 votes
2 answers
297 views

Finding prime ideals for ideal classes in arbitrary Dedekind domains

Let $R$ be an algebraic number ring with class group $C(R)$, and let $x : C(R)$. Then there exists a prime ideal $P$ in $R$ such that the ideal class $[[P]] = x$, and in fact there are infinitely many ...
Julian's user avatar
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1 vote
0 answers
97 views

Weil restriction of cycles and norm algebra

This question is on a concrete descrption of weil restricton of an affine algebra. Let L/K be a Galois extension. Since I only care about the quadratic case, we may assume that $\Gamma:=\operatorname{...
Guangzhao Zhu's user avatar
0 votes
0 answers
169 views

Theorems related to Chevalley's theorem

Recently I have read Chevalley's theorem of a complete local ring which basically says that if $(R,\mathfrak{m})$ is a complete local ring and if $\{b_n\}$ be a sequence of ideals such that $b_n \...
Kishor Kumar's user avatar
3 votes
1 answer
190 views

Irreducibility under etale ring map

Let $A\rightarrow B$ be a etale ring map between finite type algebra over algebraically closed field $k$. If $A$ is one dimensional integral domain, is $B$ direct product of finite type integral ...
George's user avatar
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9 votes
2 answers
803 views

Explanation for Lurie's SAG Remark 25.1.3.7

I am trying to understand the theory of simplicial commutative rings or animated rings. I just find a remark in Lurie's book Spectral Algebraic Geometry: Remark 25.3.1.7. Let $f : R[x_1,\ldots ,x_n]\...
Runner's user avatar
  • 93
9 votes
0 answers
188 views

Surjectivity of a bilinear map $A^m\times A^n\to A$ for a polynomial ring $A$

Let $k$ be a field and $A:= k[x_1, \dots, x_d]$. Question: Suppose $M$ is an $m\times n$ matrix over $A$. If the entries of $M$ generate the unit ideal of $A$, must there exist $a\in A^m, b\in A^n$ ...
Benjamin Baily's user avatar
2 votes
1 answer
337 views

Application of the adjoint functor theorem to get the right adjoint of the forgetful functor from $\delta$-rings to rings (the Witt vectors)

I am studying $\delta$-rings and Witt vectors from [K] (the definition of $\delta$-ring is [K, 2.1.1]), and I am having trouble verifying that everything in Kedlaya's definition for the Witt vectors ...
Elías Guisado Villalgordo's user avatar
5 votes
1 answer
202 views

Computing the Second Exterior Power of Certain Ideals in $\mathbb{Z}[\sqrt{-5}]$ and $\mathbb{Z}[\sqrt{5}]$ as Modules

I'm working on a problem involving the computation of the second exterior power of certain ideals within the rings $R_1 = \mathbb{Z}[\sqrt{-5}]$ and $R_2 = \mathbb{Z}[\sqrt{5}]$. The problem is as ...
Haze's user avatar
  • 93
4 votes
2 answers
285 views

Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?

Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
uno's user avatar
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1 vote
0 answers
59 views

Universal formulas for polynomials with prescribed jets

Let $A$ be a commutative ring and $f\in A[x]$ a split monic. When $f$ is separable with roots $\mathrm Z(f)= \{ a_1,\dots ,a_k \}$, the Chinese remainder theorem (CRT) ensures that evaluation is an $A$...
Arrow's user avatar
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5 votes
1 answer
249 views

Integral closure in characteristic 0

Let A be a Noetherian domain of characteristic 0, K be its field of fractions. Is the integral closure of A in K always finitely generated as A-module?
Jurchik's user avatar
  • 51
4 votes
0 answers
212 views

When does a short exact sequence of abelian groups with $B\cong A\oplus C$ split?

$\hspace{20pt}$Duplicate on stackexchange. This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and a ...
cnikbesku's user avatar
  • 171
4 votes
0 answers
140 views

Can an ideal in the ring of holomorphic functions on the complex plane be non-finitely generated?

Let $( I )$ be an ideal in the ring $( R )$ of all holomorphic functions of a single complex variable on the complex plane. I am interested in understanding whether it is possible for $( I )$ to be ...
Haze's user avatar
  • 93
1 vote
1 answer
73 views

In a ring with a $p$-derivation every $p$-power-torsion element is nilpotent

Let $p$ be a prime. The definition of $p$-derivation on a ring (aka $\delta$-ring structure) can be read in [K, Definition 2.1.1]. In short, a $\delta$-ring is a commutative ring with unity $A$ plus a ...
Elías Guisado Villalgordo's user avatar
3 votes
1 answer
167 views

Finite flat maps

Let $f : A \to B$ be a finite, finitely presented, flat map of (commutative) rings. It is a known consequence of Chevalley's theorem (on constructible sets) that the induced map $Spec B \to Spec A$ is ...
Jakob's user avatar
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2 votes
1 answer
192 views

Does Serre's condition $S_k$ depend only on codimension $\leq k$ points?

Recall a locally Noetherian scheme $X$ has Serre's condition $S_k$ if for every $x\in X$ we have $\mathrm{depth}(\mathcal{O}_{x,X})\geq \mathrm{min}(k,\mathrm{dim}(\mathcal{O}_{x,X}))$. Let $X$ be a ...
Junpeng Jiao's user avatar
0 votes
0 answers
95 views

Conditions for regularity in a covering

Let $V$ be a DVR of mixed characteristic, whose residue field is a finite field of characteristic $2$. Let $R$ be a flat, finitely generated algebra over $V$, which is regular. Let $a\in R^*$ be an ...
eroq's user avatar
  • 1
2 votes
0 answers
129 views

Is the deformation of a $C^{\infty}$-manifold over Artin local algebra trivial?

$\DeclareMathOperator\Spec{Spec}$Let $X$ be a compact $C^{\infty}$-manifold without boundary. Let $(A,m)$ be a Artin local $\mathbb{C}$-algebra such that $A/m\cong \mathbb{C}$. Intuitively, a ...
Zhaoting Wei's user avatar
  • 9,019
2 votes
1 answer
198 views

Maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$

After asking this question and finding this relevant paper, I would like to ask the following question: For every $a,b \in \mathbb{C}$, denote: $A_{a,b}=\mathbb{C}[(x-a)(x-b),x(x-a)(x-b),y]$ and $B_{a,...
user237522's user avatar
  • 2,837
3 votes
0 answers
181 views

Levelled trees and the homotopy transfer theorem

In section 10.3.12 of Loday-Vallette's book "Algebraic operads", given a $P_\infty$-algebra $(A,d,\alpha)$ the Homotopy Transfer Theorem applied to $H_*(A,d)$ is studied. There, because the ...
groupoid's user avatar
  • 215
1 vote
0 answers
47 views

Examples of graded subrings of $\mathbb Q(T)$

The following question came up in some discussion on some very unrelated matters. A graded algebra $A$ is an algebra $A$ with a decomposition $A = \oplus_{i \in \mathbb Z} A_i$ such that $A_i A_j \...
user5831's user avatar
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