1
vote
1answer
35 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
63 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 ...
5
votes
1answer
76 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
15 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
38 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
38 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
99 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
98 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
53 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
43 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 ...
0
votes
1answer
46 views

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 ...
7
votes
2answers
186 views

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 ...
4
votes
1answer
143 views

$k$-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: ...
2
votes
1answer
33 views

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 ...
4
votes
1answer
81 views

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 ...
2
votes
1answer
46 views

Maximal homogeneous ideals of a graded $k$-algebra.

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Given any maximal ideal $\mathfrak{m}\subset A$, we can form the quotient to obtain a map $A\to ...
0
votes
2answers
61 views

Cohen-Macaulay and regularity

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
3
votes
2answers
39 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
0
votes
1answer
69 views

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 $C$ ...
1
vote
3answers
113 views

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
0
votes
1answer
27 views

Intuition behind the abstract definition of a node (singularity)

Look at the following definition of an ordinary double point (node). The source is the book:"Freitag, Kieh - Etale Cohomology and the Weil Conjectures:" I don't understand the geometry behind ...
0
votes
1answer
54 views

Castelnuovo-Mumford regularity and the maximal degree of generators

I am reading a few texts on Castelnuovo-Mumford regularity. If I understand correctly, almost all of them say: If $I$ is a homogeneous ideal in $k[X_0,...,X_n]$ where $k$ is algebraically closed ...
9
votes
2answers
133 views

$AB=z \mathrm{Id}_n$ implies $z^m BA = z^{m+1} \mathrm{Id}_n$ for what $m$?

This question builds on a series of questions looking for elementary proofs that $AB=\mathrm{Id}$ implies $BA=\mathrm{Id}$, for $A$ and $B$ both $n \times n$ matrices over a commutative ring. First ...
2
votes
0answers
33 views

G-equivariant invertible sheaves on affine curves

Let $A$ be a Noetherian integral domain, and $G$ a finite group of automorphisms acting on $A$. Let $B = A^G$, the ring of invariants. The inclusion $B \hookrightarrow A$ induces a surjective morphism ...
0
votes
1answer
75 views

How to prove this comment of Fulton

I'm trying to understand why this is true in Fulton's Algebraic Curves: Why we add this point $(0,\ldots, 0)$? Why this equality is true? I really need help. Thanks in advance.
2
votes
1answer
40 views

Can a rational map $X\leadsto Y$ be defined as a scheme morphism $Z\to Y$ for some $Z$?

Let $X=\operatorname{Spec}(R)$ be an integral scheme with generic point $\eta$ and let $Y$ be a separated scheme. A rational map $X\leadsto Y$ is a certain equivalence class and it is represented by ...
0
votes
1answer
44 views

Generating set of the algebra invariants of finite group.

Let a finite group $G$ acts on a complex vector space $V$ and let $\mathbb{C}[V]^G$ be corresponding algebra of polynomial invariants. Let $f_1,f_2,\ldots,f_m$ be a generating set of this algebra of ...
1
vote
1answer
86 views

How do we know that $f(x)\in Y$?

At page 19 in this book $f:X\to Y$ is defined to be $$f(a):=(\tilde\varphi(T_1')(a),\dots,\tilde\varphi(T_n')(a)).$$ To explain the notation above, $X\subseteq \mathbb{A}^m(k)$, $Y\subseteq ...
0
votes
0answers
29 views

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 ...
0
votes
1answer
43 views

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 ...
1
vote
1answer
38 views

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, ...
0
votes
2answers
45 views

Some questions on Hartshorne I.7: intersections in projective space

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 ...
4
votes
2answers
128 views

What are local homomorphisms, geometrically?

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 ...
8
votes
2answers
146 views

Is every affine scheme the complement of the closed point $x$ of the spectrum of a local ring $A$?

Let $R$ be a commutative ring with identity element and let $\operatorname{Spec}(R)$ be the associated affine scheme. Does for each affine scheme $\operatorname{Spec}(R)$ exist a local ring $A$ ...
4
votes
1answer
97 views

Question on calculating hypercohomology

I want to compute the algebraic de Rham cohomology of $ \mathbb{C}^* $, and I'm confused. I don't have much background in this, so I was hoping a very concrete example would clear up a lot of this ...
2
votes
1answer
110 views

Factorization of ideals in a coordinate ring (Dedekind domain)

Consider $f \in \mathbb{C}[X,Y]$ an irreducible curve non singular. Let $A = \mathbb{C}[X,Y] / (f)$ be the coordinate ring of $f$ and choose a curve $g \in \mathbb{C}[X,Y]$ with no component in common ...
2
votes
1answer
51 views

What kind of points are there in a finite type $k$-scheme?

Let $k$ be an arbitrary field and $X$ a $k$-scheme of finite type (i.e. a scheme with a finite cover of spectra of finitely generated $k$-algebras). How can I think of the points $x\in X$? What ...
3
votes
2answers
61 views

Kähler differentials of the cuspidal cubic

I want to compute $\Omega^1_{A,\mathbb{C}}$ for $A = \mathbb{C}[X,Y]/(Y^2 - X^3)$, or more precisely, I want to show that the module of Kähler differentials is free of rank 2 at the origin, and free ...
1
vote
1answer
63 views

The geometric interpretation for extension of ideals?

Suppose $f\colon B\to A$ is a ring homomorphism, and $I\subseteq B$ is an ideal. What's the geometric interpretation for the extension $f(I)A$ of the ideal $I$? Especially, I'm interested in the case ...
3
votes
1answer
51 views

Is the unique morphism from the empty scheme $\operatorname{Spec}((0))$ to some other scheme $X$ smooth?

This is a very pedantic question, but Is the unique morphism from the empty scheme $\emptyset = \operatorname{Spec}((0))$ to some other scheme $X$ smooth?
2
votes
1answer
101 views

The Zariski topology on $\operatorname{Spec} A$ as an intial topology

Given any commutative ring $A$ let $\operatorname{Spec} A$ be the space of prime ideals of $A$. Can we interpret the Zariski topology as an initial (or final) topology with respect to some canonical ...
0
votes
0answers
78 views

Atiyah & Macdonald's Introduction to Commutative Algebra, Exercise 8.5

The exercise asks the reader to prove that $X$ is a finite covering (i.e., the number of points of $X$ lying over a given point of $L$ is finite and bounded) of $L$, where the affine varieties $X$ and ...
1
vote
1answer
46 views

Support of a quasicoherent sheaf

When $M$ is a finitely generated module over a commutative ring $R$, it is easy to see that the support of $\tilde{M}$ on $\mathrm{Spec}\,R$ is given by $V(\mathrm{ann}_R(M))$. This is not true for ...
0
votes
0answers
35 views

Integral dependence of coordinate ring

In Hartshorne P18-P19, the proof of Thm. 3.4 shows that the ring $S(Y)_{(x_{i})}$ is contained in the integral closure of the coordinate ring $S(Y)$ (all regarded as subrings of the quotient field of ...
0
votes
1answer
62 views

Irreducible components in the spectrum of a ring

I have a question concerning page 43 of this book. In Corollary 2.7 it says that the map $\mathfrak{p}\mapsto \overline{\{\mathfrak{p}\}}$ is a bijection from Spec($A$) onto the sets of closed ...
1
vote
2answers
71 views

Help with $\sqrt{I}$, where $I=(y^2,x+yz)$ in $\mathbb{C}[x,y,z]$

$a)$ $\sqrt{I}$ where $I=(y^2,x+yz)$ in $\mathbb{C}[x,y,z]$. first it's clear $y \in \sqrt{I}$ then $x=(x+yz)-yz \in \sqrt{I}$ because $yz \in \sqrt{I}$ is it $\sqrt{I}=(x,y)$ ? $b)$ ...
2
votes
1answer
57 views

In $\Bbb Z[x,y]$ is $(x^2+1,y^2+1,-xy+1)$ prime?

This is a reality check for the following computations that I did: Consider the map $(\operatorname{id}, \iota): \Bbb A_\Bbb Z^1 \rightarrow \Bbb A_\Bbb Z^1\times \Bbb A_\Bbb Z^1$ from the definition ...
3
votes
0answers
110 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
6
votes
2answers
214 views

What is an example of two k-algebras that are isomorphic as rings, but not as k-algebras?

Let $k$ be a field. Let $A$ and $B$ be two $k$-algebras, ie. two rings that are also $k$-vector spaces and their multiplication is $k$-bilinear. Any isomorphism of $k$-algebras is also a ring ...
2
votes
2answers
195 views

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 ...