# Tagged Questions

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### If $R\rightarrow S$ is faithfully flat then show that it is pure, and reference for purity

I was reading about $F$-purity and $F$-splittings, when I came across then following statement which I can't proof: Definition: Let $R$ be a commutative ring with identity, and $M,N$ be $R$-modules. ...
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### An explict description of the integral closure of $A=k[x,y]/\langle x^3-y^2\rangle$.

Let $k=\mathbb C$ and $A=k[x,y]/\langle x^3-y^2\rangle$. Denote by $X$ and $Y$ the cosets of $x$ and $y$ in $A$. Question: How do we see that the integral closure $A'$ of $A$ is $k[Y/X]$? Since ...
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### defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
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### Canonical isomorphism between Cauchy sequence completion and inverse limit

I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can ...
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### Quotient of polynomials, PID but not Euclidean domain?

While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the Euclidean domain page of Wikipedia that $$\mathbb{R}[X,Y]/(X^2+Y^2+1)$$ is ...
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### What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?

I surveyed commutative algebra texts and found Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" to be the most accessible for me. The book outlines a first course in commutative ...
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### The family of schemes $\operatorname{Spec} A[x]/(x^n)$

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for ...
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### Primary decomposition of $I = (x^2, y^2, xy)$

I want to find a primary decomposition of the ideal $$I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field. How to proceed? Are there algorithms to find such decompositions? Where can I find ...
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### Generalization of Chinese Remainder Theorem to infinite ideals

I'm looking for any (obviously weaker) generalization of this famous theorem in the special case that the family of ideals is not finite.
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### Suggest a good book or reference on graded modules over polynomial rings

I am looking for reference books or papers on graded modules over the polynomial ring $k[x_0, \ldots, x_n]$. Any good commutative algebra text like Eisenbud's Commutative Algebra already contains a ...
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### Powers generate monomials

What is a reference in the literature for the following fact? Let $A$ be a commutative $\mathbb{Q}$-algebra. Then every monomial in $A$ of degree $n$ may be written as a linear combination of $n$th ...
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### Commutative algebra with a geometric flavor

Does anybody know where can I find a book with topics similar to the ones in Atiyah's Introduction to commutative algebra, but with some sort of geometric motivation? Thanks!
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I'm studying the theory of Noether but I have only 4 pages of lecture notes with no details or examples. Are there any good lecture notes or chapters you know about? In my lectures the basics of ...
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### Universal property of de Rham differential.

Suppose $A$ is a commutative algebra over a field $k$. It is well known that there is a module that generalizes the notion of differential $1$-forms. It is denoted $\Omega^1_{k}(A)$ and is called the ...
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### a theorem on the dimension of finite algebras over a field (Hartshorne)

Robin Hartshorne in his Algebraic Geometry, Theorem 1.8A(b), p. 6, says that if $B$ is an integral domain which is a finitely generated $k$-algebra, $k$ a field, and $p$ a prime ideal of $B$, then ...
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### Regular local ring result - reference request

Reference needed for the following result: Let $R$ be a regular local ring with maximal ideal $\mathfrak m$. If $A$ is a flat $R$-algebra and $A/(\mathfrak m)$ is a domain, then $A$ is a domain. ...
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### (Finite) continued fractions over a general domain

I am looking for some literature (articles or books) where finite continued fractions over a general integral domains (that is, in a fraction field of that domain, but the "coefficients" are from the ...
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### A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
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### Die Relationen, welche zwischen den elementaren symmetrischen Functionen bestehen - Translation?

I am trying to find a translation of this paper either in English or French (preferably English). I am not very optimistic, but i thought of asking in case somebody is more resourceful :)