# Questions tagged [ac.commutative-algebra]

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

4,979
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### Blow-ups of $ F $-regular varieties at points in general position and finite generation of the Cox ring

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...

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### Invariant ring of the subvariety

Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $...

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### completed tensor products and filtered limit

Let's start with two inverse systems $\{A_p\}_{p \in \mathbb{Z}}$, and $\{B_p \}_{p \in \mathbb{Z}}$ of $C$ modules. Give each $A_p$, $B_p$, and $C$ discrete topology. Consider inverse limit topology ...

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### Spectral theorem for unital $C^{*}$-algebras

Let $A$ be a unital $C^{*}$-algebra and $a \in A$ be normal, with spectrum $\sigma(a)$. Let $B = C^{*}(a)$ be the $C^{*}$-algebra generated by $1$ and $a$, which is abelian. Let $\hat{B}$ be the space ...

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### When does a local Cohen–Macaulay ring admit a non-zero finitely generated maximal Cohen–Macaulay module of finite injective dimension?

Let $(R,\mathfrak m)$ be a local Cohen–Macaulay ring. Then, it is well- nown that there exists a non-zero finitely generated $R$-module of finite injective dimension; for instance $\operatorname{Hom}...

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### On inverse limits of $\pi$-adically complete algebras

Consider the following situation, let $\mathcal{R}$ be a discrete valuation ring with uniformizer $\pi$ (say the valuation ring of a finite extension $K$ of $\mathbb{Q}_{p}$. Let $\{ A_{n}\}_{n\in\...

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### Depth of almost complete intersection rings

Let $R$ be a regular local ring and let $I \subset R$ be an almost complete intersection ideal, that is, $\mu(I)=\text{ht}(I)+1$ where $\mu(I)$ is the number of minimal generators of $I$ and $ht(I)=\...

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### Definition of presheaf, given in Basic Algebraic geomerty 2 Shafarevich [migrated]

I was reading Basic Alegbraic geomerty 2 by Shafarevich,
In the definition of presheaf, $\mathscr F(\emptyset)=1$
To illustrate this definition they have given an following example:
suppose first ...

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### Definitions of torch ring

Well, I find myself in a tangled web of references, definitions and doubt. How do I get myself into these things? Get ready for a rollercoaster of definitions.
An FGC ring is a commutative ring whose ...

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### If $ n \in \mathbb{N} $, then does the Reynolds operator of $ \mathbb{G}_{m}^{n} $ commute with the Frobenius endomorphism?

If $ n \in \mathbb{N} $, then $ \mathbb{G}_{m}^{n} $ is linearly reductive. Let $ \beta: \mathbb{G}_{m}^{n} \to \operatorname{GL}(\mathbf{V}) $ where $ \mathbf{V} $ is a vector space over an ...

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### Analogue of Kock-Lawvere axiom for power series rings?

The Kock-Lawvere axiom for a topos $\mathcal{E}$ states that given a specified commutative ring object $R \in \mathcal{E}$, for all local Artinian $R$-algebra objects $A \in \mathcal{E}$, the morphism
...

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### $M^\wedge_I \to N^\wedge_I$ an isomorphism if $S_P^{-1}M^\wedge_P \to S_P^{-1}N^\wedge_P$ is an isomorphism for all primes $P$ containing $I$

Let $R$ be a Noetherian ring, $I \subseteq R$ an ideal, and $S \subseteq R$ a multiplicative set.
Lemma 2.3 of Adam, Haeberly, Jackowski, and May's paper A generalisation of the Segal conjecture ...

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### Rings of weak dimension ≤ 1 vs. semihereditary rings?

Rings in this question are assumed to be commutative. I am asking "natural" examples of rings of weak dimension $\le1$ which are not semihereditary. It would be better if there are integral ...

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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 ...

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### On the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties"

I am trying to understand section (3) of the Erratum to P. Scholze's "$p$-adic Hodge theory for rigid-analytic varieties" in detail. In particular, there is the following sentence on page ...

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### Intersection theory on schemes with Gorenstein singularities

Is there a good reference/book on Intersection theory on schemes with Gorenstein singularities? Does the construction of the intersection of cycles discussed in Fulton's book also hold for schemes ...

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### A smooth algebra over a $K_i$-regular ring is $K_i$-regular?

Let $A$ be a smooth algebra over a commutative ring $R$. If we assume that $R$ is $K_i$-regular for any $i\geq 0$, then $A$ is also $K_i$-regular?

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### Under what conditions is an open subscheme of an affine scheme affine and what ring corresponds to it?

It is well known that an open subscheme of an affine scheme is not necessarily an affine one. But what are (if possible the most general) sufficient conditions for its affinity? And is it known how, ...

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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 $...

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### Construction of a symmetric polynomial in the roots that acts like the discriminant

The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...

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### Homomorphism group of the additive group of rational numbers $\mathbb{Q}$ into quasi-cyclic group $\mathbb{Z}(p^\infty)$ [duplicate]

I read somewhere that the group of homomorphisms from the additive group of rational numbers $\mathbb{Q}$ into the quasi-cyclic group $\mathbb{Z}(p^\infty)$ is isomorphic to the additive group of ...

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### What is the name for algebras generated by elements, all of whose cubes vanish?

Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x_1, \dots, x_k$ satisfying $x_i^2 = 0$ for each index $...

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### Computing the automorphism scheme of projective space

$\newcommand{\Spec}{\operatorname{Spec}}$I'm trying to understand why $PGL_{n}$ is the automorphism scheme of $\mathbb{P}^{n-1}_{\mathbb{Z}}$.
In Conrad's Reductive Group Schemes, the following ...

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### Extending automorphisms from $\mathbb{A}^1$ to $\mathbb{P}^1$ over general rings

Over an algebraically closed field any automorphism of the affine line will extend uniquely to an automorphism of the projective line. Will that still be true if we work over a general (potentially ...

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### Condition for equality of modules generated by columns of matrices

Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...

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### Why is Fontaine's infinitesimal period ring $A_{\text{inf}}$ complete?

Fix a perfectoid field $K$ in mixed characteristic with ring of integers $\mathcal{O}$ and pseudo-uniformizer $\varpi$. Its tilt is the fraction field of $\mathcal{O}^{\flat}=\varprojlim_{x\mapsto x^{...

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### Are smooth irreducible affine varieties set theoretical complete intersection?

I'm trying to prove a lemma, so I wanted to use the next claim (which I do not know if it is true or if there is a counter example): every smooth irreducible affine variety is set theoretical complete ...

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### Example showing that $\mathbb{P}^1$ does not preserve monics

Is there an injective homomorphism of commutative rings $A \to B$ such that the induced map $\mathbb{P}^1(A) \to \mathbb{P}^1(B)$ is not injective? Here, $\mathbb{P}^1(A) = \mathrm{Hom}(\mathrm{Spec}(...

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### Is there a relevant universal property for Fitting ideals?

Let $S$ be a scheme and $\mathscr F$ a quasi-coherent sheaf on $S$, locally finitely presented. Set $S_{-1}=S$ and for all $n\geqslant 0$, let $S_n$ be the closed subscheme of $S$ defined by the $n$-...

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### Regular ring is smooth when the field is perfect

Take $A$ a (not necessarily local) commutative algebra over a field $k$ which is essentially of finite type (i.e. a localization of a finitely generated algebra). In simple words, I just want to know ...

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### How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices

Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...

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### Classfication of vector bundles on projective line over a local ring

Let $R$ be a local ring. Let $\mathbb{P}^1_R=\rm{Proj}~R[x_0, x_1]$ be the Projective line over $R$.
Is there a classification of vector bundles of rank $n$ on $\mathbb{P}^1_R$ in terms of splitting ...

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### Rings with epimorphism from a finitely generated ring

For a commutative ring with unit $R$ let's say it has property $(*)$ if there is an epimorphism in the category of rings ${\mathbb Z}[X_1,\dots,X_n]\to R$, where the former is the polynomial ring in $...

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### Adjacent reducible polynomials

Let $P[X_1, X_2, \ldots, X_N]$ be a reducible polynomial in $\mathbb{Z}[X_1, X_2, \ldots, X_N]$ such that $P[X_1, X_2, \ldots, X_N] + 1$ is also reducible. What (if anything) can we say about $P$?
One ...

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### completion and tensor product

Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers?
If $A$ is noetherian, it is clear because one has for $k$ a residue ...

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### Example of a periodic free resolution over a hypersurface

I'm reading "HOMOLOGICAL ALGEBRA ON A COMPLETE INTERSECTION,
WITH AN APPLICATION TO GROUP REPRESENTATIONS" by David Eisenbud
I'm wondering what would be a nice example illustrating Theorem 6....

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### Derivation of positively graded domain

We know that if $f\in k[X_1,{...},X_n]$ is quasi-homogeneous polynomial and $R :=k[X_1,{...},X_n]/(f)$, then any minimal generating set of $\operatorname{Der}_k(R)$ contains the Euler derivation. Is ...

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### Cyclic homology can be recovered from topological cyclic homology?

Let $R$ be a commutative ring, $S$ a $R$-algebra of finite type.
By an equivalence of ring spectra
$$
\operatorname{HH}(S/R) \simeq \operatorname{THH}(S)\otimes_{\operatorname{THH}(R)} HR,
$$
...

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### Let $g$ be regular in $A$, when do we have $(gx_i-f_i)_{A[\underline{x}]}=A[\underline{x}]\cap (gx_i-f_i)_{A_g[\underline{x}]}$

Also asked in stackexchange.
Let $A$ be a commutative unital ring. Let $g$ be a regular element of $A$. Let $A_g$ be the localization of $A$ at the set $\{g^n:n\geq 0\}$, then we have an injection $A\...

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### Frobenius and mixed characteristic valuation rings

Let $R$ be an $\mathbf{F}_p$-algebra. Kunz's theorem says that if $R$ is Noetherian, then the Frobenius of $R$ is flat iff $R$ is regular. Following the philosophy that valuation rings often behave ...

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### When an element of a ring that is divisible by a finite set of elements is necessarily divisible by their product?

In a commutative ring $R$, when does the assumption $r_i\mid r$ for $1\le i\le n$ imply $\prod_{1\le i\le n} r_i\mid r$ (when $r_i$ are fixed)?
Does there exist any criterion for this implication that ...

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### Quantum version of Kostant's basis of ℤ-form of U(𝔤)

Kostant showed that the subring of $\mathcal U(\mathfrak{sl}_2)$ generated by the divided powers $e^c/c!$ and $f^a/a!$ has a $\mathbb Z$-basis given by the elements $\frac{f^a}{a!}\binom hb \frac{e^c}{...

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### A profinite group such that any deformation has finite image

Let $p$ be a fixed prime and $n\geq 2$ a fixed integer. Suppose that $ G $ is a profinite group such that any continuous homomorphism $ \rho:G\to {\rm GL}_{n}(F) $ has finite image for all non-...

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### Testing whether a finite dimensional algebra is a complete intersection with QPA

Let $A$ be a local commutative finite dimensional $K$-algebra for a field $K$.
We can thus assume that $A$ is given by quiver and admissible relations $KQ/I$.
It is easy to check when $A$ is ...

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### For a local complete intersection ring $(R,\mathfrak m)$ with $\mathfrak m^3=0\ne \mathfrak m^2$, $\mathfrak m$ can be generated by two elements

Let $(R,\mathfrak m,k)$ be a local complete intersection ring with $\mathfrak m^3=0\ne \mathfrak m^2$. As $0\ne \mathfrak m^2 \subseteq \text{soc}(R)$ and $R$ is Gorenstein, so we get $\mathfrak m^2 =\...

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### Has anyone attempted to generalize the notion of a higher differential of $ A $ and the sheaf of differentials $ \Omega_{A/k} $?

Over a field $ k $ of characteristic zero, actions of $ \widehat{\mathbb{G}_{a}} $ on a variety $ \operatorname{Spec}(A) $ are obtained from $ A $-derivations $ \delta $ of $ A $ over $ k $ via the co-...

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### f.g. module $M$ over a complete local CM ring of dimension 1 such that $M, \text{Hom}_R(M,M), \text{Ext}^1_R(M,M)$ have finite injective dimension

Let $(R,\mathfrak m)$ be a local, $\mathfrak m$-adically complete, Cohen-Macaulay ring of dimension $1$. Assume that there exists a finitely generated $R$-module $M$ of depth $0$ such that $M$, $\text{...

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### Are algebras of smooth functions formally smooth?

Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$?
If it helps, feel free to assume that $M$ is compact.
(This is not a joke ...

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### On $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when $\text{depth}_R(R/J)=0$, and $R\to S$ is a certain flat map of local rings

Let $(R, \mathfrak m) \xrightarrow{\phi} (S,\mathfrak n) $ be a flat homomorphism of local rings such that $\mathfrak n=\mathfrak m S +xS$ for some $x\in \mathfrak n \setminus \mathfrak n^2$. Let $J$ ...

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### Commutative local rings which satisfy Krull-Remak-Schmidt

Question 1: Can the class of local (always noetherian and commutative) rings be classified for which the Krull-Remak-Schmidt theorem (KRS) holds for finitely generated modules? They contain for ...