2
votes
1answer
51 views

The Zariski topology on $\mathrm{spec} \ A$ as an intial topology

Given any ring $A$ let $\mathrm{spec} \ A$ 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 maps from ...
0
votes
0answers
46 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
35 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
29 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
40 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 ...
2
votes
2answers
61 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
47 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 ...
4
votes
0answers
69 views

Counterexamples for lcm-gcd identity and modular law for rings

In Mile Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
5
votes
2answers
194 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 ...
3
votes
2answers
155 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 ...
0
votes
1answer
60 views

Resolution three noncollinear points [closed]

Let $p_1$, $p_2$ and $p_3$ three noncolinear points, and let $R$ be the homogeneous coordinate ring. Show that $R$ have a resolution $$0 \to S^{ \oplus 2}( - 3) \to S^{ \oplus 3}( - 2) \to S \to R ...
0
votes
1answer
70 views

Generalisation of a result on Kahler differentials

Let $B$ be a local ring which contains a field $k$ of characteristic zero, isomorphic to its residue field $B/\mathfrak{m}$. We know that the map $\delta:\mathfrak{m}/\mathfrak{m}^2 \to \Omega^1_{B/k} ...
1
vote
0answers
59 views

Local complete intersection scheme, conormal sheaves and differentials

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $Z \subset X$ be a local complete intersection subscheme in $X$. Denote by $I_Z$ the ideal sheaf of $Z$ in $X$ and $\Omega^1_X$ the sheaf ...
3
votes
0answers
53 views

Isomorphism between Ext groups in Huybrechts and Lehn's book Geometry of Moduli Spaces of Sheaves

On p.46 (or p. 43 in the 1st edition) of Huybrechts and Lehn book Geometry of Moduli Spaces of Sheaves, 2nd ed., they write: Since $K$ is $A$-flat and $I \otimes_k F_0$ is annilated by $m_A$, ...
6
votes
1answer
55 views

Hilbert Nullstellensatz and ring of continuous functions

Is there any relation between Hilbert's Nullstellensatz and the fact that the maximal ideals in $\mathcal C([0,1])$ correspond to a point in $[0,1]$ (which can be generalized to compact hausdorff ...
1
vote
1answer
43 views

What is $\overline{Y}$ in $\text{Spec}A$?

Consider a subset $Y$ of $\text{Spec}(A)$. (Here $A$ is a commutative ring.) What is the closure of $Y$ (or $\overline{Y}$)? I have been under the impression that $\overline{Y}$ is the set of ...
0
votes
1answer
33 views

an apparent contradiction regarding the local ring at a point

I have encountered an apparent contradiction: Let $Y$ be an affine variety of $\mathbb{A}^n$ and $P$ a point of $Y$. Then i have proved that $\mathcal{O}_P$ is an integral domain and it is also not an ...
3
votes
1answer
69 views

The family of schemes $\mathrm{spec} \ A[x]/(x^n)$

Consider the family $S_n:=\mathrm{spec} \ A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for $S_n$ ...
5
votes
1answer
55 views

From a vector bundle to a Koszul complex

Let $k = \mathbb C$. Given a commutative $k$-algebra $A$, an $A$-module $M$ and a homomorphism of $A$-modules $s:M \to A$, we can construct the Koszul dg algebra. $$K(A,M,s) = \wedge^{-\!*}_A(M)$$ ...
2
votes
1answer
37 views

A condition for a homogeneous ideal to be prime

The following is the problem 11 of Chaper 8 Section 4 of Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. A homogeneous ideal is said to be prime if it is prime as an ideal in ...
0
votes
1answer
43 views

Noether normalization and surjectivity (revisited)

Let $Y$ be an affine variety of dimension $d$ inside the affine space $\mathbb{A}^n$. Then $A(Y) = k[x_1,\dots,x_n]/I_Y=:k[\bar{x}_1,\dots,\bar{x}_n]$. By the Noether normalization theorem, there ...
0
votes
1answer
44 views

Grade of non principal Prime ideals in Noetherian UFDs

I want to prove that in any Noetherian UFD the grade of every non-principal prime ideal is at least $2$. I say in a UFD $R$ each nonzero prime ideal contains a prime element. Since the given ...
0
votes
0answers
27 views

Criterion of separability of function on a curve.

Let $K$ be finitely generated extension of an algebraically closed field $k$ of transcendence degree $1$, $\operatorname{char }k =p$. Let $C(K)$ be set of discrete valuation rings $(\mathcal ...
1
vote
1answer
55 views

Ring extension and Jacobson rings

If $R\subseteq S$ are commutative rings, is it a fact that $R$ is a Jacobson ring if and only if $S$ is so? I guess the contraction of maximal and prime ideals of $S$ may be helpful in this ...
0
votes
1answer
43 views

Nullstellensatz non-valid for non-algebraically closed fields

I want an example (with details, please) showing that Nullstellensatz may be false over non-algebraically closed fields. Thanks in advance!
0
votes
1answer
40 views

Intersection of $max(R)$ with a closed subset in $Spec(R)$

Let $R$ be a commutative ring with unity and $E$ be a nonvoid closed subset of $Spec(R)$. If $U$ is an open subset of $Spec(R)$ with $E∩Max(R)⊆U$, where $Max(R)$ is the set of maximal ideals of $R$, ...
0
votes
0answers
33 views

Localization of Coordinate Ring isomorphic to ring of local regular functions

There is a very standard fact that I am having a hard time understanding. The claim is that if we have an affine variety $Y$ then the localization of the coordinate ring by the maximal ideal ...
0
votes
1answer
61 views

Krull dimension of $k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. [closed]

I need help to solve this exercise. If anyone can help, thanks in advance! Let $k$ a field and $R=k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. Find the Krull dimension of $R$.
2
votes
2answers
88 views

Two nonassociated functions defining the same hypersurface?

Let $X\subseteq\mathbb P^n$ be a complex, irreducible projective variety. Let $R$ be the projective coordinate ring of $X$, i.e. $R=\mathbb C[x_0,\ldots,x_n]/I$ for some homogeneous prime ideal $I$. ...
3
votes
0answers
54 views

Topological characterisations of freeness and separability?

According to wikipedia, a spectral space is defined to be homeomorphic to the spectrum of some commutative ring. They form a category $Spec$ where we take morphisms to be those whose preimage of open ...
-1
votes
1answer
49 views

Open set in the spectrum of a ring?

Consider $Spec(K[X])$ where $K$ is an algebraically closed field. Is $0$ open in the Zariski topology on spectrum? Does the spectrum have points which are neither open nor closed?
1
vote
0answers
28 views

Looking for an example for a very particular kind of cone

I am looking for a cone $X\subseteq\mathbb A^n_{\mathbb C}$ (defined by homogeneous equations) and an irreducible homogeneous polynomial $f$ in $n$ variables such that $U := D(f)\cap X = X_f = \{ x\in ...
1
vote
1answer
56 views

Prove the strong Nullstellensatz from these two conditions

It is an exercise. Let $R=k[x_1,\dots,x_n]$ where $k$ is an algebraically closed field. Assuming that (1) $R$ is Noetherian, and (2) the maximal ideas of $R$ are precisely the ideals of ...
3
votes
1answer
91 views

$\mathcal{O}_X(D)$ is invertible implies $D$ is locally principal

With words added for context, 14.2.G of Ravil Vakil's notes asks Suppose $X$ is an integral, normal and Noetherian scheme, and $D$ a Weil divisor. Let $\mathcal{O}_X(D)$ be the quasicoherent sheaf ...
0
votes
1answer
51 views

$\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$

In a proof (proof of theorem 4.3.36 in Liu's book) I need the equality $\mathcal{O}_{X_y,x}=\mathcal{O}_{X,x}/\mathfrak{m}_y\mathcal{O}_{X,x}$. The hypothesis of the theorem are the following: $Y$ ...
0
votes
0answers
82 views

Rings having the same characters but not isomorphic.

I want to show that these two rings have the same characters but they are not isomorphic for $\nu>2$ Thank you for helping. $$H=k+kt^{4\nu}(1+t)+kt^{6\nu}(1+t)+kt^{7\nu}(1+t)+k[[t]]t^{8\nu}$$ ...
3
votes
0answers
58 views

Length of a composition series of a module

If $A=\mathbb{C}[x,y]_{(x,y)}$, then what is the length of $A$-module $$A/(x^3-x^2y^2+y^{100},x^3-y^{999})\ ?$$ Any suggestion ?
3
votes
2answers
194 views

Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
3
votes
1answer
67 views

Integral Domain with exactly two Prime Ideals

I am not looking for someone to give me an explicit example. I want to work this out myself if possible. Trying to learn schemes by reading The Geometry of Schemes by Eisenbud and Harris. Problem I-5 ...
0
votes
1answer
40 views

Points lying over a closed point in a separable extension of the base field are rationnal

At the end of the proof of Proposition 4.3.30 In Liu's book we have the following situation: $X$ is an algebraic variety over a field $k$, $x\in X$ is a regular closed point of $X$ with $k'=k(x)$ is a ...
0
votes
1answer
115 views

How to calculate $\operatorname{Spec} \mathbb{C}[x,y]/(y^2-x^3)$

Is there a general method for calculating things like $\operatorname{Spec} \mathbb{C}[x,y]/I$ ? Maximal ideals are $ \{(x-\tilde{a},y-\tilde{b}): b^2-a^3=0\}$ because of ...
0
votes
0answers
24 views

singular locus and jacobian matrix

Let $R=k[x_1, \cdots ,x_r] / I$ be an affine ring over a perfect field $k$ and suppose that $I$ has pure codimension $c$. Suppose that $I= (f_1, \cdots , f_s)$. If $J$ is the ideal of $R$ generated by ...
0
votes
0answers
42 views

Birational equivalence with surfaces

Let $X$ be an algebraic variety of dimension $n$. I'd like to prove that $X$ is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}$. I've already seen algebraic proofs using some ...
0
votes
1answer
50 views

When the ring of regular functions is a UFD?

Let $X$ be an irreducible affine variety over $\mathbb{k}$. There is the following theorem in algebraic geometry: the algebra $\mathbb{k}[X]$ of regular functions is a UFD if and only if each ...
2
votes
0answers
38 views

Parametric ideal decomposition

Let $x = \{x_{1},\dots, x_{n}\}$ be a set of variables and let $a = \{ a_{1}, \dots, a_{m}\}$ be a set of parameters. Let $\{f_{1}(a,x), \dots, f_{s}(a,x)\} \subset \mathbb{C}[a,x]$ be a set of ...
3
votes
1answer
109 views

A Question from Algebraic Geometry

For any two disjoint closed subsets $Y_1$ and $Y_2$ of $ \mathbb A ^n$ show that there exists $g \in\mathbb C [x_1, x_2, ..., x_n]$ such that $g(Y_1)=0$ and $g(Y_2)=1$.
0
votes
1answer
26 views

Ring of regular functions on an open set of spectrum of $R$ is a subring of the field of fractions of $R$.

Let $R$ be an integral domain, and let $X=\operatorname{Spec}(R)$. Show that all local rings $\mathcal{O}_X(U)$ - for nonempty open subsets $U\subseteq X$ - are subrings of the field of fractions ...
2
votes
0answers
40 views

Dimension of Kahler differentials of Laurent series over perfect field.

Let $k$ be perfect field, char $k=p>0$. How it can be shown that $\dim_{k((t))}\Omega^1(k((t)))=1$?
3
votes
2answers
57 views

Localization of a polynomial ring at a maximal ideal

Let $R$ be a regular local Noetherian ring, with maximal ideal $M$. Show that $N=R[x]M+(x)$ is a maximal ideal in the polynomial ring $R[x]$, and that the localization $R[x]_N$ is again regular local. ...
0
votes
2answers
36 views

intersection multiplicity at non-zero point

Compute the intersection multiplicity of $f=x+y-2$ and $g=x^2+y^2-2$ at $(1,1)$. Is this the same as the intersection multiplicity of $f(x+1)$ and $g(x+1)$ at $(0,0)$ which I have computed to be 2? ...