# Tagged Questions

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### Category of schemes with flat morphisms

Consider the category whose objects are schemes, and for every two schemes $X$ and $Y$, morphisms $\operatorname{Hom}(X,Y)$ consist of flat morphisms $X\rightarrow Y$, only. Does this category have a ...
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### Is the maximal ideal of a localization at a prime ideal principal?

Let $X$ be a closed subvariety of $\mathbf P^{n}_{k}$ which is nonsingular in codimension one. Let $Y$ be a subvariety of $X$ of codimension one, let $\eta$ be its generic point. First question: is ...
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### Relation between elements of a ring and their annihilators

let $(R.m)$ be a local ring and $x,y$ two elements of $R$ and for ideal $I$ of $R$, we have $x$ is in $I$, $ann(x)=ann(y)$ and $x$ is uniqu minimal ideal of $R$, is there any conditions that implies, ...
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### Intersection of two polynomial ideals

In the 4-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$X'=\{x=y=0\}$$ and $$X''=\{z=x-t=0\}$$ (I'm working on a algebraically ...
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### The greatest common divisor of homogeneous polynomials

Let a matrix $$M=\begin{pmatrix} a_{01}&a_{02}&a_{03}\\a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}$$ with $a_{ij}\in k[x,y,z]$ ...
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### Question regarding Vakil's algebraic geometry notes

Exercise 1.3 D of Vakil's lecture notes on algebraic geometry asks: "Verify that $A \to S^{−1}A$ satisfies the following universal property: $S^{−1}A$ is initial among $A$-algebras $B$ where every ...
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### Plane curves isomorphic to the affine line

Let $C$ be a plane curve parametrized by $x=f(t),y=g(t)$ where $f(t),g(t)\in k[t]$. We can easily see that the coordinate ring of $C$ is isomorphic to $k[f(t),g(t)]\subset k[t]$. So $C$ is isomorphic ...
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### Interpretation of $\Omega_{A/k} \simeq A \otimes_k I/I^2$ for affine group schemes

I'm learning some group scheme stuff and there's the following result: If $A$ is Hopf $k$- algebra, then $\Omega_{A/k} \simeq A \otimes_k I/I^2$, where $I$ is the augmentation ideal. I know the ...
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### A question about locally free differential sheaf and regular local ring

Let $B$ be a local ring containing a field $k$ isomorphic to its residue field. Assume furthermore that $B$ is a localisation of a finitely generated $k$-algebra. Then $Ω_{B/k}$ is a free $B$-module ...
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### A nonfree module which is locally free

The general context is trying to understand the Picard groups of various schemes, but this question focuses on affine schemes. Let $X=Spec A$ an affine scheme. What conditions does $A$ need to ...
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### Same number of generators and relations in a complete intersection, when?

I make this question a bit more general because i think as i put it, it will have no answer because there are too many maybe irrelevant details: Given $B$ an $A$-algebra, local, of finite type (that ...
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### Local complete intersection ring

Suppose $R$ is a local Noetherian complete intersection ring that is a finite $A$-algebra, where $A$ is a DVR. If the module of differentials of $R$ is free as an $R/\mathfrak a$-module for some ...
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### Computing the Length of a finite length module.

How we can compute the length (length of a composition series) of the Artinian local ring $R=K[x,y]/(x^3,y^3)$ ? Does the following chain is a saturated chain of ideals of $K[x,y]$ ? ...
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I was reading Mumford's 'Red book on varieties and schemes', when I came across the following paragraph: I am confused about meaning of the phrase "We let $k(X)$ be the zeroth graded piece of the ...
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### use Noether normalization theorem to integrate differential forms over singular subvarieties

Let $X \subset \mathbb C^n$ be an analytic subset. I would like to show that locally around any point $x \in X$ the regular part $X_\text{reg}$ has finite volume, perhaps using the theorem below. ...
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### Tensor product of injective ring homomorphisms

What is an example of two injective homomorphisms $R \to A$, $R \to B$ of commutative rings such that $R \to A \otimes_R B$ is not injective? Of course neither $R \to A$ nor $R \to B$ can be flat in ...
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### Thinking About Fractional Ideals Geometrically

So algebraic geometry gives one a way of thinking about about rings geometrically. Like prime ideals correspond to points in the spectrum of a ring, maximal ideals are closed points and so on. This ...
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### How to define a smooth subvariety as the vanishing of local coordinates

I keep stumbling upon this fact, and would like to see or get an idea for the proof: An ideal of a smooth subvariety at a point of a smooth variety can be generated by a subset of a suitably chosen ...
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### Ideal generated by a regular sequence

I need to prove that the ideal $$I = (xz -y^2, x^2t^2 -yz^3, x^2yt^2 -xz^4) \subset R = \mathbb{K}[x,y,z,t]$$ is generated by a $R$-regular sequence. How can I do it? I don't know if this can ...
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### Why is $W_n(k)$ the unique flat lifting of a perfect field $k$ over $\mathbf{Z}/p^n$?

Let $k$ be a perfect field of characteristic $p>0$ and denote by $W_n(k)$ the ring of Witt vectors over $k$ of length $n$. In their article on the decomposition of the de Rham complex, Deligne and ...
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### Characterization of ideals generated by homogeneous polynomials in terms of $f^{(d)}$ in Gathmann's notes.

On pg. 37 of Gathmann's Algebraic Geometry notes, the following is mentioned: For every $f\in k[x_0,x_1,\dots,x_n]$ be an ideal. The following are equivalent: I can be generated by ...
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### Resultants of two polynomials over a ring

Let $k$ be a field $f,g\in k[x,y]$ be two polynomials. The resultant $R\in k[x]$ is a polynomial function of the coefficients of $f$ and $g$, such that $f$ and $g$ gave a common zero (in an extension) ...
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### When is $k(X)$ algebraic over $k(Y)$ for a dominant morphism $f:X\rightarrow Y$ between varieties.

Let $f:X\rightarrow Y$ be a dominant morphism between irreducible varieties over an algebraically closed field $k$. When is $k(X)$ algebraic over $k(Y)$? Is there an if and only if criterion? What if ...
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### Ring of rational power series

Let $A$ be any commutative ring with 1. A power series $f\in A[[t]]$ is called rational if we can find a $g\in A[t]$ such that $fg\in A[t]$. It is clear that the set of rational power series forms a ...
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### Help in this notation in Fulton's Algebraic Curves book

I'm reading Fulton's Algebraic Curves book, I'm stuck in the following proposition (page 105): In fact, what I didn't understand is the following notation in the proof of this proposition: Why ...
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### Proof that presheaf is a sheaf for Spec

Atiyah Macdonald define presheaf (chapter 3, exercise 23) on the base of $Spec(A)$, where $A$ is commutative ring with $1$, as follows $$\mathfrak{F}(X_f) = A_f,$$ where $X_f$ is a basic open set ...
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### Direct image of the exceptional divisor along a blow-up

Let $X=\mathrm{Spec}(k[x_1,\ldots,x_n])$ for $n\geq 2$, and let $\mathcal{I}=\widetilde{I}\subseteq\mathcal{O}_X$ for an ideal $I\subseteq k[x_1,\ldots,x_n]$. Let ...
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### Completion of quotient of polynomial ring

Hartshorne's Algebraic Geometry uses the following facts on page 35 without proof: The completion of $(k[x,y]/(y^2-x^2-x^3))_{(x,y)}$ is $k[[x,y]]/(y^2-x^2-x^3)$ and that of ...
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### Showing that $x^3+y^3+z^3=0$ is not rational

Is there a short proof that $F:x^3+y^3+z^3=0$ in $\mathbf{P}^2$ is not rational, apart from using the genus? Perhaps this is an elliptic curve, so every morphism $\mathbf{P}^n\rightarrow F$ is ...
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### $k$-algebra homomorphism of the polynomial ring $k[x_1,\dots,x_n]$

Let $\phi:k[x_1,\dots,x_n]\mapsto k[x_1,\dots,x_n]$ be a $k$-algebra homomorphism with $\phi(x_i)=f_i$, where $k$ is algebraically closed and has characteristic zero. I have the following questions: ...
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### Example of a ring which is not CM at all its prime ideals

A commutative ring $A$ is said to be CM at a maximal ideal $\mathfrak{m}$ if and only if $Depth(A_{\mathfrak{m}})=Krull(A_{\mathfrak{m}})$. What is an example of a connected commutative ring $A$ which ...
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### Flatness after dividing out a minimal prime ideal

Let $A \hookrightarrow B$ be an extension of finitely generated, reduced $k$-algebras, where $k$ is a field of characteristic zero such that $B$ is a free $A$-module of finite rank. Let $A$ be an ...
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### Support of the pullback module

Let $X$ be an algebraic variety, let $\Delta : \mathrm X \to \mathrm X^2$ be the diagonal embedding and let $\mathrm M$ be a quasi-coherent sheaf of modules on $\mathrm X^2$. Make the supposition ...
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### Weak nullstellansatz in Atiyah-Macdonald 5.17

$\newcommand{\fm}{\mathfrak{m}}$ Problem 17 in the exercises after the 5th chapter of Atiyah-Macdonald is the following (with some references and hints omitted): Let $X$ be an affine algebraic ...
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### A question regarding Hilbert's Nullstellensatz.

Let $k$ be an algebraically closed field, and $a$ an ideal of the polynomial ring $k[x_1,x_2,\dots,x_n]$. The strong form of Hilbert's Nullstellensatz says that $I(Z(a))=\sqrt{a}$. Note:- Initially, ...
I am reading I.7 of Hartshorne, and here are some questions I don't understand. 1) Prop. 7.4. Let $M$ be a finitely generated graded module over a noetherian graded ring $S$. Then there exists a ...
For want of a better name, let us say that a ring homomorphism $f : A \to B$ is local if it (preserves and) reflects invertibility, i.e. $f (a)$ is invertible in $B$ (if and) only if $a$ is invertible ...