# 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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### Errata for Atiyah-Macdonald

Is there a good errata for Atiyah-Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists inaccuracies ...
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### Classification of rings between a PID and its field of fractions?

Let $D$ be a PID and let $\mathrm{Frac}(D)$ be its field of fractions. I want to classify the intermediate rings $D\subseteq R\subseteq \mathrm{Frac}(D)$. Theorem: Every such ring $R$ is a ...
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### Glue DVR to itself, get a separated non-affine scheme

Here is what seems to be a fun little exercise in algebraic geometry. Take a DVR $R$ and automorphism of $Frac(R)$; glue $Spec(R)$ to itself via this automorphism. Can the glued scheme be separated ...
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### What properties define open loci in families?

This question is somehow related to the question What properties define open loci in excellent schemes?. Let $f:X\to S$ be a proper (or even projective) morphism between schemes (of finite type over ...
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### Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes

Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
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### Special idempotents in a commutative ring

Let $R$ be a commutative ring with $1$ and $e$ be an idempotent element of $R$ with the property that if $e=x+y$ (where $x, y\in R$), then there exists $r\in R$ such that either $e=rx$ or $e=ry$. ...
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### Non-minimal Krull associated primes of a PF-ring

A commutative ring $R$ is said to be a PF-ring if every principal ideal of $R$ is a flat $R$-module. Also, a prime ideal $P$ of $R$ to be a Krull associated prime of $R$ if for every element $x\in P$ ,...
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### Definition of dualizing complex

Sorry for a not research level question asking for a definition but unfortunately I nowhere found a source which explains the construction presented below in a satisfactory way. This question refers ...
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### Finding a presentation matrix with low dimension

Let $R=\mathbb Z[t^{\pm}]$ and $M$ a finitely generated $R$-module. With $A$ a presentation matrix, i.e we have the following exact sequence (usually I'm working with the case where $A$ is an square ...
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### How many distinct quaternions have a given prime norm $p$?

I seem to recall that the answer is $p + 1$, but I'm not quite sure.
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### The relationship between a finite field and a quotient ring in $\mathbb{F}_p[x]$

Let $f$ be an irreducible polynomial of degree $q$ over $\mathbb{F}_p$. Let ${\bf F}=\frac{\mathbb{F}_p[x]}{f}$ be the finite field which contain $p^q$ elements. Assume $k>1$ is an integer and ...
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### Does the Polarization Theorem for $A_1(k)$ has an analogue for $k[x,y]$?

There is an interesting theorem about the first Weyl algebra $A_1(k)= k \langle x,y | yx-xy= 1 \rangle$, $k$ is a field of characteristic zero, the Polarization Theorem, Corollary 5.5 by A. Joseph. ...
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### Extensions of rings

Let $K$ be a commutative but not necessarily unital ring. Let $C$ be a unital commutative ring. An extension is the data of a unital commutative ring $R$, a homomorphism $\phi:K\rightarrow R$, a ...
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### Two-dimensional quotient singularities are rational: why?

I've read that quotient singularities (that is, spectra of invariant subrings of finite groups acting linearly on polynomial rings) have rational singularities. Is there an elementary proof of this ...
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### Completion of a local ring of an arithmetic surface

Vaguely, an arithmetic surface is an algebraic curve with coefficients in some "ring of integers". More formally it is an integral normal scheme $\mathfrak{X}$ of dimension 2 of finite type over a ...
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### Continuous extension of the derivation in positive characteristic

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Does there exist a derivation on $\Omega$ ...
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### Converse to Tannaka duality for rings

Let $k$ be a field. It's well known and easy to prove that a $k$-algebra $R$ may be recovered from the functor $F: R-Mod \to Vect_k$ as the endomorphisms of $F$. Now suppose $\mathcal{C}$ is a ...
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Suppose that $k$ is a commutative ring and that $A$ is an Azumaya $k$-algebra. Then there is a well-known morphism from $Aut_{Alg_k}(A)$, the group of algebra automorphisms, to the Picard group $Pic(k)... 0answers 114 views ### Algorithm telling when an affine curve is planar I am sorry, I asked a misguided question here: Reference request: smooth affine curves are planar, here is my attempt at a better question. Let$\mathfrak{p}$be a prime ideal of$\mathbb{C}[x, y, z]$... 3answers 604 views ### Does there exist another form of the derivative for polynomials? Let$F : \mathbb{R}[X] \rightarrow \mathbb R[X]$be a linear map and let$H \in \mathbb{R}[u,x,y,z]$be a polynomial. Suppose that $$F(P \cdot Q) = H(F(P),F(Q),P,Q)$$ for all$P, Q \in \mathbb{R}[X]...
Let $k$ be an algebraically closed field of characteristic $0$ and $A=k[X_1, \ldots, X_n]$ with the grading induced by the total degree. Let $B$ be a graded $k$-subalgebra of $A$, ie, if $(A_k)$ is ...