The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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196 views

Does localization preserves dimension?

Does localization preserves dimension? Here's the problem: Let $C=V(y-x^3)$ and $D=V(y)$ curves in $\mathbb{A}^{2}$. I want to compute the intersection multiplicity of $C$ and $D$ at the origin. ...
3
votes
3answers
278 views

Conjugation is not expressible in terms of polynomials

In order to convince myself that the set $U(n)$ of unitary matrices (matrices with columns that are orthonormal under the complex inner product) is not an affine variety in $\mathbb{C}^{n^2}$, I need ...
3
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2answers
570 views

Book recommendations for commutative algebra and algebraic number theory

Are there any books which teach commutative algebra and algebraic number theory at the same time. Many commutative algebra books contain few chapters on algebraic number theory at end. But I don't ...
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2answers
166 views

$\operatorname{Spec}(k[x])$ has infinite points.

Let $k$ be a field. I have to prove that $\operatorname{Spec}(k[x])$ has infinite points. If $k$ is infinite it is obvious: in fact there are infinite maximal ideals $(x-\alpha_i)$, with $a_i ...
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2answers
967 views

Zariski Topology question

Could you please give a hint how to show that the zariski topology on $\mathbb{A}^2$ is not the product topology on $\mathbb{A}^1\times\mathbb{A}^1$
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4answers
375 views

Show that there is no affine transformation that takes a circle to a hyperbola in $\mathbb{R}^2$

"Show that the standard circle (defined by $f(x,y) = x^2 + y^2 - 1$) is not equivalent to the standard hyperbola (defined by $g(x,y) = x^2 - y^2 - 1$). That is, show that there is no $[A,\overline{s}] ...
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3answers
94 views

Show that $A \cong \mathbb{C}^n$ with A a commutative algebra [duplicate]

Let A be a commutative algebra of finite dimension, and if $A$ has no nilpotent elements other than $0$, is true that $A \cong \mathbb{C}^n$ ? The question emerge to my mind, I thought that the ...
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2answers
216 views

Does the fibres being equal dimensional imply flatness?

Let $f: Y \to X$ be a morphism of varieties (proper if necessary). I read from a paper that if all the fibres of $f$ are of the same dimension then $f$ is flat. This seems skeptical for me, and I ...
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3answers
1k views

Definition of degree of finite morphism plus context

Let $f: X \rightarrow Y$ be a finite morphism of schemes, defined here, http://en.wikipedia.org/wiki/Finite_morphism I always assumed that the degree of $f$ was the degree of the induced field ...
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4answers
160 views

What is the significance of the second property in the sheafification of a pre-sheaf?

Let $\mathcal{F}$ be a pre-sheaf on $X$. It seems that if we let $\mathcal{F}^+(U)$ to be the set of all maps $U \rightarrow \cup_{p \in U} \mathcal{F}_p$, where $\mathcal{F}_p$ is the stalk of ...
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160 views

Locally ringed spaces and Riemann surfaces

Let $X$ and $Y$ be Riemann surfaces, and $\mathscr{O}_X, \mathscr{O}_Y$ be the sheaves of holomorphic functions on $X$ and $Y$ respectively. It is obvious that a holomorphic map $f:X \to Y$ gives ...
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2answers
106 views

“$L/K$ forms of each other”

In section $4$ of these notes, the author says two algebraic groups $G$ and $H$ defined over a field $K$ are "$L/K$ forms of each other" if they are "isomorphic over $L$", where $L$ is a finite field ...
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4answers
326 views

Proof that the ideal $(xy, xz)$ in $\mathbb{A}^3$ is radical but not prime

The proof that $(xy, xz)$ is not prime seems easy. In particular, $xy \in (xy, xz)$, but neither $x$ nor $y$ is in $(xy, xz)$. On the other hand, I don't know how to prove that $(xy, xz)$ is ...
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3answers
141 views

$\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$ $\Rightarrow$ $R$ is a PID

Is the following true: If $R$ is a commutative unital ring with $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$, then $R$ is a PID. If yes, how can one prove it? Since $0$ is a prime ...
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3answers
401 views

Monomorphisms of sheaves gives an injection of stalks

How do I show that a monomorphism $F \rightarrow G$ of sheaves induces an injection on stalks? When showing that monomorphism is an injection on sets one uses the maps $x \mapsto a$ and $x \mapsto ...
3
votes
2answers
259 views

Why Zariski topology is not Hausdorff

I am reading the book about Algebraic geometry. I am confused about the following two things the book mentioned: Zariski topology is 1. different from the topology studied in real and complex ...
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3answers
190 views

Recommendations for Commutative Algebra Software?

I'd like a software that I can use to work with commutative algebra, specifically to figure out S-Polynomials, Buchberger's Algorithm, etc. I have Mathematica; if anyone could refer me to a package, ...
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3answers
62 views

What are some good examples of (non-quasicoherent) sheaves not satisfying the conclusion of Hartshorne Lemma II.5.3?

Hartshorne, Algebraic Geometry, Lemma II.5.3 reads (roughly): Let $X = \operatorname{Spec} A$, let $f \in A$, and let $\mathscr{F}$ be a quasicoherent sheaf on $X$. (a) If $s \in \Gamma(X, ...
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2answers
109 views

Orthogonal idempotents from disjoint union in $\text{Spec}(A)$

Let $A$ be a commutative ring with unity. Suppose that $X=\text{Spec}(A)$ is a disjoint union $X_1\cup X_2$ of topological spaces. Show that $A$ has a pair of orthogonal idempotents $e_1,e_2$ ...
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1answer
115 views

Exercise 1.2.4 and Example 4.3.6 in Liu

I want to prove that if $X$ is a noetherian scheme then any flat closed immersion into $X$ is open, that is, if $A$ is noetherian then $\varphi:\operatorname{Spec}(A/I)\to\operatorname{Spec}(A)$ is ...
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votes
2answers
180 views

ampleness of invertible sheaves

Let $f: X\rightarrow Y$ be a morphism of schemes over a field $k$. Let $\mathcal{L}$ be an invertible sheaf . My qeustion is If $\mathcal{L}$ is ample, then $f^*\mathcal{L}$ is ample? If 1. is not ...
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4answers
3k views

What are the conditions for a polygon to be tessellated?

Upon one of my mathematical journey's (clicking through wikipedia), I encountered one of the most beautiful geometrical concept that I have ever encountered in my 16 and a half years on this oblate ...
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2answers
477 views

When to read of the degree of a variety from its defining polynomials

The question concerns algebraic varieties. I just read the question The degree of an algebraic curve in higher dimensions and great answer by user M P. One of the thing he says is that if a curve in ...
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169 views

Explicitly writing out all elements of $\mathbb{P}^{1}(\mathbb{F}_{n})$

Explicitly the elements of $\mathbb{P}^{1}(\mathbb{F}_{3})$ are $[1:0], [0:1], [1:1]$, and $[1:-1]$. Why is this so? How would I do this for $\mathbb{P}^{1}(\mathbb{F}_{4})$? What about general ...
3
votes
1answer
503 views

Going down theorem fails

Maybe this exercise comes from some textbook, but I donot know. It said that this ring extension $k[x(x-1),x^2(x-1),z]\subset k[x,z]$ does not have the Going-Down property. Here $k$ is an ...
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votes
2answers
71 views

why are projective spaces and varieties prefferable?

I am reading Hartshorne's Algebraic Geometry and it seems to me that projective spaces and varieties are prefferable. I don't know why. In a more elementary stage of mathematics, when we try to find ...
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votes
2answers
90 views

Is there a classification of regular maps $\mathbb{P}^1(k)\to\mathbb{A}^1(k)$?

If $\mathbb{P}^1(k)$ and $\mathbb{A}^1(k)$ are the projective line and affine line, respectively, over an algebraically closed field $k$, is there any known classification of the regular maps ...
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2answers
80 views

A question on algebraic geometry.

In a course about algebraic geometry I am taking, a question appeared in class which claimed that $A=\mathbb{C}[x,y]/(xy-1)$ is a field. In order to prove this, one concluded that $xy-1=0 ...
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2answers
156 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) ...
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2answers
186 views

What is the use of scheme theory?

I should preface this by saying that my background in Algebraic Geometry is (more or less) the content of Vakil's notes up through Chapter 4 (i.e. through the definition of a scheme and several ...
3
votes
3answers
195 views

Existence of an isomorphism $\mathbb{P}^n\times\mathbb{P}^m \rightarrow \mathbb{P}^{n+m}$ [duplicate]

There exist an isomorphism of varieties? $$\mathbb{P}^n\times\mathbb{P}^m \rightarrow \mathbb{P}^{n+m}$$ I am considering $\mathbb{P}^n\times\mathbb{P}^m$ as the product in the category of ...
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2answers
78 views

If $K$ is finite, then every subset of $\mathbb A^n(K)$ is algebraic

I'm trying to prove that if $K$ is a finite field, then every subset of $\mathbb A^n(K)$ is algebraic. I know that if $K$ is finite, then every element of $K$ is algebraic, i.e., for every $a\in K$ ...
3
votes
1answer
112 views

Immediate consequence of Riemann-Roch

Let $X$ be an algebraic curve, $D$ a divisor and $\mathscr{O}(D)$ the line bundle associated to $D$ in the canonical way. The following implication should follow immediately from the Riemann Roch ...
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2answers
706 views

Degree and dimension of intersection of projective variety and hypersurface

I am looking at Theorem 7.7 of Hartshorne where he states the general form of Bezout's Theorem. The hypotheses of the theorem are as follows. Let $H$ be a hypersurface of degree $d$ and $Y \subseteq ...
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1answer
298 views

Does the category of affine $k$-varieties have finite products?

We use the definitions of this question. Let $\mathrm{Aff}(k)$ be the category of affine $k$-varieties. Let $X, Y$ be objects of $\mathrm{Aff}(k)$. Does the product $X\times Y$ exist in ...
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1answer
337 views

“sheaf” au sens de Serre

I learned the definition of sheaves from Algebraic Geometry by Hartshorne, while reading Serre's GAGA, I was wondering if there was another definition of sheaves. [Here is the link of the English ...
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2answers
358 views

affine scheme over a ring R

I red an article and encountered some concepts from algebraic geometry. Let $R=\mathbb{Q}[\alpha_1,\ldots,\alpha_5]$ be a polynomial ring in the variables $\alpha_i$. Define $f(x,y)\in R[x,y]$ by ...
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1answer
64 views

What is the relationship between a complex manifold being Kähler, projective, nonprojective, and nonKähler?

I was wondering if this implication is true. I read a few places that $$\text{nonprojective} \Longrightarrow \text{nonKähler}$$ but I think I maybe have misunderstood. Equivalently, this is of course ...
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2answers
89 views

Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero polynomial.

I am working on a problem from Ideas, Varieties, and Algorithms: Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero ...
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votes
1answer
87 views

Compute the cohomology of projective schemes

In Hartshorne's book, Section 3.5, the cohomology of projective spaces is computed. How to compute the cohomology of projective schemes? Maybe the general case is complicated, please look at the ...
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1answer
109 views

Formally smooth vs. smooth

A (commutative) algebra $A$ is called formally smooth if for any (commutative) algebra $R$ and an ideal $I\subset R$ such that $I^2=0$, any morphism $A\to R/I$ lifts to a morphism $A\to R$. Suppose ...
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2answers
136 views

Proving algebraically that $\mathbb RP ^3\cong SO(3,\mathbb R)$

I am giving a simple introductory course on algebraic geometry and I plan to mention that $$\mathbb RP ^3\cong SO(3,\mathbb R).$$ I know a rather simple proof of this using the fact that $\mathbb ...
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2answers
86 views

If $X=\{x\}$, then $\dim(X)=0$

If $X$ is a quasiprojective variety, then by definition $\dim(X)=trdeg(k(X)|k)$. I'm trying to prove if $X=\{x\}$ is a point, then $\dim(X)=0$ I'm already proved that $k[X]\cong k$, now if I prove ...
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2answers
178 views

Why are non-separated schemes schemes?

In "the old days", e.g. in the famous texts by Grothendieck and Mumford, a scheme was defined as what we now call a separated scheme. (i.e. a scheme where the image of the morphism $\Delta:X \to X ...
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1answer
95 views

Prove the complement of a point in $\mathbb{A}^n$ is compact.

I am learning algebraic geometry and I came across the folowing question. Prove that the complement of a point is compact in $\mathbb{A}^n$. Does anyone know how to do this?
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2answers
89 views

do fibres of morphisms of Noetherian rings have finite Krull dimension?

Let $f:A \rightarrow B$ be a morphism of Noetherian rings. Let $p \in Spec(A)$ and let $C=B \otimes \kappa(p)$ be the fibre over $p$. Is it true that $\dim C < \infty$? How can we see that? ...
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2answers
769 views

$\mathbb{A}^2\backslash\{(0,0)\}$ is not affine variety

In our lecture notes we have this example, with the proof why $X = \Bbb{A}^2\setminus \{(0,0)\}$ is not an affine variety: Let $i:X\hookrightarrow \mathbb{A}^2$ be an inclusion map. We show, that any ...
3
votes
1answer
197 views

Galois group of $K(X)/K$

Let $K$ be an infinite field, if $K(X)$ is the field of rational function I want to find the Galois group of the extension $K(X)/K$. Lemma 1: If $L$ is a field such that $K\subsetneq L\subseteq ...
3
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2answers
263 views

What does the stalk of a sheaf of discontinuous sections look like?

I'm having some kind of cognitive dissonance here, but I'm having trouble figuring out which of my beliefs is false. Let $\mathscr{F}$ be a sheaf of abelian groups on a topological space $X$ and $x ...
3
votes
2answers
680 views

Injective map on coordinate ring implies surjective?

Suppose that $f:X\rightarrow Y$ is a morphism between two affine varieties over an algebraically closed field $K$. I believe that if the corresponding morphism of $K-$algebras, ...