2
votes
1answer
38 views

Prove that $S$ is an integral domain and $T$ is not an integral domain.

Let $R = \mathbb{C}[x,y]$ $R^i \subset R$ be the abelian subgroup of $R$ generated by elements of $\mathbb{C}$ times monomials of degree at least $i$ $I = (x^3+x^2-y^2)$ $S = R/I$ $S^i$ be the group ...
5
votes
0answers
41 views

Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
0
votes
1answer
37 views

Showing local ring isomorphisms

This is a problem in K. Hulek's Elementary Algebraic Geometry. I figured out that $k[X]$ is the collection of polynomials of the form $f(x) + g(y)$ and also the local ring of an affine line at the ...
0
votes
2answers
55 views

Logic problem: Atiyah-Macdonald 1.11

Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following: "Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ ...
0
votes
1answer
17 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
2
votes
2answers
57 views

Hartshorne II Prop 6.8

My weaknesses with commutative alegbra are really slowing down my progress through Harthshorne. I hope someone can help me understand some statements in the proof of the proposition below. Prop ...
0
votes
2answers
20 views

If $\mathcal{I}(-)$ is the ideal map on subsets of affine space, why does $A\subseteq\overline{B}\iff\mathcal{I}(B)\subseteq\mathcal{I}(A)$?

I think this is a basic property of $\mathcal{I}(-)$, but I'm having trouble seeing it. I denote by $\mathbb{A}^n$ the affine $n$-space over an algebraically closed field $k$, where if ...
1
vote
2answers
53 views

Normalization of a variety

I'm currently in a number theory course and this question popped up. As I'm not super familiar with algebraic geometry, I was wondering if my reasoning is correct: Show that $\mathbb{C}[X,Y]/(Y^2 - ...
0
votes
0answers
57 views

When is $A = k[x_1,\ldots, x_n]/I$ integrally closed?

Suppose that it is not easy to determine that $A$ is a UFD (or that it is a local, noetherian dimension 1 domain with principal maximal ideal). Can someone suggest strategies for showing that a ...
2
votes
1answer
46 views

Dense open subsets of schemes

Let $X$ be a scheme. Let $U$ be an open subset of $X$. It is clear that if $U$ contains all the generic points of $X$ (by which I mean the generic points of irreducible components of $X$) then $U$ is ...
0
votes
0answers
21 views

Primality of homogeneous ideal

Let $R$ be the polynomial ring over the finite field $\mathbb{F}_p$ with $n$ variables. Let $I$ be an ideal of $R$ generated by homogeneous polynomials whose coefficients are 1 or -1. Are there any ...
0
votes
0answers
52 views

Hartshorne Chapter II exercise 5.7 on Invertible sheaves

I'm working on part c) which is to prove that for a Noetherian scheme $X$, a coherent sheaf $\mathscr{F}$ is invertible (locally free of rank 1) iff there exists a coherent sheaf $\mathscr{G}$ such ...
6
votes
1answer
80 views

What are the points of some schemes?

Let $X=\operatorname{Spec}\mathbb{C}[x,y,t]/(xy-t)$, $Y=\operatorname{Spec}K[x,y]/(xy-t)\rightarrow \operatorname{Spec}K$ and $Z=\operatorname{Spec}R[x,y]/(xy-t)\rightarrow \operatorname{Spec}R$, ...
1
vote
2answers
60 views

How to show rational function field of an affine subvariety with dim>0 is not algebraically closed?

I do not know how to show the following statement. If $X\subset A^n$ is an irreducible subvariety, $\dim X>0$, then the rational function field of $X$, $K(X)$ is not algebraic closed. What ...
2
votes
1answer
56 views

$I(Y) = \{ p(x,y,z) \in k[x,y,z] \mid p (t,t^2,t^3) = 0, \forall t \in k \}$ is prime

I've been working on the following problem from Hartshorne: Let $Y\subseteq \mathbb{ A }^3 $ be the set $Y = \{(t,t^2 , t^3) \mid t \in k \}$. Show that $Y$ is an affine variety of dimension $1$. To ...
2
votes
1answer
51 views

Algebraic Curves: Valuation at a point

I would like to understand the notion of valuation on the local ring of a curve at a point. In the Book The Arithmetic of Elliptic Curves in chapter 2, Example 1.3 $$V:\ Y^{2}=X^{3}+X$$ I don't ...
0
votes
1answer
66 views

Prove that a subset is a finitely generated subring

Consider $\mathbb{A}^2$ with $\rho : (x, y) \mapsto (-x, -y)$. Can anyone help me prove that $S = \{f \in \mathbb{C}[x, y] : f \circ \rho = f\}$ is a finitely generated subring? Also, can $S$ be ...
3
votes
1answer
76 views

Atiyah-Macdonald 5.2

Exercise 5.2 in Atiyah-Macdonald asks to show the following: "Let $A$ be a subring of a ring $B$ such that $B$ is integral over $A$, and let $f: A \to \Omega$ be a homomorphism of $A$ into an ...
2
votes
2answers
77 views

Going-up and going-down theorems: motivation

I am reading about the going-up and going-down theorems in Atiyah & Macdonald's commutative algebra book. I'm wondering if anyone could give me some basic facts/examples to help me understand why ...
1
vote
0answers
40 views

A query on Veronese mapping

The Veronese mapping defined as usual on some $P^n$. Then it is certainly regular. I want to prove that the inverse map to this map is also regular. I have an idea to use projections with ...
1
vote
2answers
28 views

minimal prime ideals over the union of two prime ideals

When two subvarieties intersect properly ($X_1\cap X_2$), it should end up with a new subvariety($X_3$=$X_1\cap X_2$). I do not know how to keep track of the intersection operation from the algebraic ...
1
vote
0answers
30 views

K[x,y,z,w]/(xw-yz) not UFD [duplicate]

I am trying to prove its not UFD. I started by assuming x=ab in K[v] , where v=v(xw-yz) then x-ab=(xw-yz)f, for some f in K[x,y,w,z] I tried to say that deg of a, and b is less or equal 1, and ...
4
votes
3answers
84 views

stalk of projective variety in terms of the coordinate ring

Let $X \subseteq \mathbb{P}^n$ be an embedded projective variety over some field $k$ with its corresponding homogeneous coordinate ring $R = k[X_0,\dots,X_n]/I(X)$. Let further $X = \bigcup_{i=0}^n ...
2
votes
0answers
38 views

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 ...
1
vote
0answers
61 views

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 ...
-1
votes
0answers
30 views

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, ...
1
vote
1answer
69 views

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 ...
2
votes
0answers
79 views

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 $a_{ij}\in k[x,y,z]$ ...
0
votes
1answer
57 views

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 ...
3
votes
1answer
71 views

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 ...
1
vote
0answers
35 views

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 ...
0
votes
0answers
32 views

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 ...
1
vote
1answer
56 views

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 ...
0
votes
0answers
29 views

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 ...
1
vote
0answers
57 views

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 ...
0
votes
0answers
104 views

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 $K[x,y]$ ? ...
1
vote
0answers
49 views

question about gradation of a ring

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 ...
0
votes
0answers
48 views

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. ...
7
votes
1answer
200 views

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 ...
5
votes
1answer
80 views

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 ...
1
vote
1answer
45 views

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 ...
1
vote
1answer
80 views

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 ...
6
votes
1answer
145 views

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 ...
0
votes
0answers
23 views

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 ...
3
votes
0answers
40 views

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) ...
1
vote
1answer
47 views

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 ...
3
votes
0answers
102 views

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 ...
2
votes
1answer
100 views

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 ...
0
votes
1answer
66 views

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 ...
1
vote
0answers
55 views

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 ...