2
votes
1answer
42 views

Kahler forms of a smooth affine algebra vanish eventually?

If $k$ is a Noetherian ring, then do the Kahler forms of a smooth affine $k$-algebra of dimension $d$ vanish above $d$? I mean is: $\Omega^{d+1}_{A|k}\cong 0$?
2
votes
0answers
31 views

Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
3
votes
0answers
108 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 $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
3
votes
1answer
47 views

Kernels of power surjective maps

Suppose $k$ is an algebraically closed field, and $A$ and $B$ are finitely generated, commutative, graded $k$-algebras. Suppose $\varphi:A\to B$ is a map of $k$-algebras. Notice if $B$ is a domain, ...
5
votes
2answers
207 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
1answer
88 views

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

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for ...
0
votes
1answer
83 views

Studying $\operatorname{Spec}\mathbb{Z}[x]$, $\operatorname{Spec}\mathbb{R}[x]$, and $\operatorname{Spec}\mathbb{C}[x,y]$.

While there is a similar question here but that was marked as a duplicate to this question. The latter question, at the level that I am at doesn't give me much insight. I also thought that if I could ...
1
vote
1answer
56 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
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
61 views

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$?

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$ ? $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)=\{ (0),\ (\tilde{x}-a,\tilde{y}-b),\ b^2=a^3\}$.
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. $$H=k+kt^{4\nu}(1+t)+kt^{6\nu}(1+t)+kt^{7\nu}(1+t)+k[[t]]t^{8\nu}$$ ...
0
votes
1answer
117 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
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? ...
4
votes
2answers
95 views

Computing irreducible components of algebraic set

Consider the algebraic set $V(X^2-YZ,X-XZ)$. Find the irreducible components of this set and show that $I(V)=(X^2-YZ,X-XZ)$. I reasoned that $X-XZ=0$ iff $X=0$ or $Z=1$. If $X=0$, we get $Y=0$ or ...
1
vote
1answer
77 views

Find intersection multiplicities

Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$. If we let $f=x^2-3x+y^2$ and ...
3
votes
1answer
70 views

Kernel of a morphism of regular rings.

Let $k$ be a field and $f: A \rightarrow B$ be a surjective ring morphism between smooth Noetherian $k$-algebras. By smooth I mean that the module of Kahler Differentials $\Omega_{A|k}$ is a ...
0
votes
1answer
45 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
3
votes
1answer
103 views

Are $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ isomorphic?

I saw somewhere that $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ considered the same. Is it true? Why? I'm a beginner so please answer in details
1
vote
0answers
58 views

Prove this morphism is a derivation

Let $A$ a ring, $B$ a $A$-algebra, $I=\ker (B\otimes_A B \to B)$ given by multiplication, i wanna see why is $$d:B \to I/I^2$$ $$b \mapsto b\otimes 1 -1 \otimes b$$ well defined (in fact, why mod ...
2
votes
1answer
48 views

Noetherian local ring, detail in theorem 1.3.16 in Liu

I can't understand a detail in the proof of theorem 1.3.16 in Liu. The theorem is: let $(A,\mathfrak{m})$ a Noetherian local ring, $\hat{A}$ its $\mathfrak{m}$-adic completion, $(B,\mathfrak{n})$ an ...
3
votes
1answer
40 views

Complexifications of degree 3 subschemes in $\mathbb A^2_{\mathbb R}$

I am trying unsuccessfully to solve exercise II-20 (page 65) from the book "The geometry of schemes" by Eisenbud and Harris. In this exercise it is stated that there are two non-isomorphic subschemes ...
5
votes
1answer
54 views

Topological closure of ideal in $A[[T]]$ - Proposition 1.3.7 in Liu

In Proposition 1.3.7 of Liu's book, one proves that if a ring $A$ is noetherian then so is $A[[T]]$. We take an ideal $I$ of $A[[T]]$ and prove that there exist $F_1,\ldots,F_m\in I$ such that for all ...
1
vote
0answers
50 views

Are Ideals and Varieties Inclusion Reversing?

Let $S_1$, $S_2$ be sets or varieties (I don't think it matters, does it?). Then if $S_1 \subset S_2$, is it always the case that $I(S_2) \subset I(S_1)$ (where I is an ideal)? Also, is it always the ...
4
votes
1answer
78 views

Localization of Coordinate Rings: $\mathbb C[V_f] = \mathbb C[V]_f$.

Let $V\subseteq\mathbb C^n$ be an irreducible affine variety, then the coordinate ring $$\mathbb C[V] = \mathbb C[x_1,\dots,x_n]\big/\mathbf I(V)$$ is an integral domain. Let $f\in\mathbb ...
3
votes
0answers
43 views

Prime Ideals as Ring theoretic Ultrafilters

I am confused by the following statement in Awodey's Category Theory p. 35: Ring homomorphisms $A\to \mathbb Z$ into the initial ring $\mathbb Z$ play an analogous and equally important role [to ...
4
votes
0answers
51 views

The ring of homogeneous polynomials

I think I found an error in my textbook, but I am not completely sure. The book is Hulek, Elementary algebraic geometry, pag. 73. There is a theorem showing that $U_i$ and the affine space ...
0
votes
0answers
34 views

relation between quotient field of holomorphic functions and meromorphic functions

Let $U \in \mathbb{C}^n$ open connected. Is it true that the quotient field of $\mathcal{O}_{\mathbb{C}^n,z}$ for $z \in U$ is the stalk of the sheaf $\mathcal{K}$ where $\mathcal{K}(U)$ is the field ...
2
votes
1answer
71 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 ...
2
votes
1answer
65 views

Algebraic Geometry and Maximal ideals

I am solving the following problem but couldn't figure out a strategy to solve: Does $(x^3-17, y^2)$ generate maximal ideals in the quotient ring $R=\mathbb{C}[x,y]/I$ where $I$ is the principal ...
0
votes
0answers
47 views

Variety and algebraic curves

I am attempting the following problem from Artin: Every variety in $\mathbb{C^2}$ is the union of finitely many points and algebraic curves. I think the proof is trivial (unless I am missing ...
5
votes
1answer
84 views

Irreducible polynomials and algebraic geometry

I was reading Dummit and Foote and this was one of statements stated (without any proof), "An irreducible curve have finitely many singular points" I would like to know why is this true. Shouldn't it ...
3
votes
1answer
124 views

Number of common zeros of two quadratic polynomials in ${\Bbb C}[t,x]$

The following theorem is in Artin's Algebra(2nd edition): Theorem 11.9.10 Two nonzero polynomials $f(t,x)$ and $g(t,x)$ in two variables have only finitely many common zeros in ${\Bbb C}^2$, ...
3
votes
2answers
85 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 $e_1,e_2$ ...
1
vote
1answer
77 views

Is the completion of $(k[x,y]/f)_\mathfrak{m}$ isomorphic to $k[[x]][y]/f$?

Let $k$ be an algebraically closed field and let $f\in k[x,y]$ be an irreducible polynomial with no constant term that is not a polynomial in $x$ alone. Is it the case that the completion of the ...
4
votes
2answers
95 views

Integral morphism between varieties has finite fiber

I'm looking for a proof/counterexample of the following fact: Theorem Let $X \subseteq k^n$ and $Y \subseteq k^n$ be algebraic varieties over a field $k$ and let $\phi$ be a morphism from $X$ to ...
3
votes
1answer
99 views

What does $J_1\cap J_2=\emptyset$ mean algebraically for two varieties in $\Bbb{C}^n$?

Let $J_1, J_2$ be two varieties in ${\Bbb C}^n$. Then $$ J_i=V(I_i)\quad i=1,2. $$ for some $I_i\subset\Bbb{C}[x_1,\cdots, x_n]$ and $$ J_1\cap J_2=V(I_1\cup I_2) $$ and $$ J_1\cup J_2=V(I_1I_2). $$ ...
3
votes
2answers
71 views

What are the ideals in ${\Bbb C}[x,y]$ that contain $f_1,f_2\in{\Bbb C}[x,y]$?

This question is based on an exercise in Artin's Algebra: Which ideals in the polynomial ring $R:={\Bbb C}[x,y]$ contain $f_1=x^2+y^2-5$ and $f_2=xy-2$? Using Hilbert's (weak) nullstellensatz, ...
2
votes
1answer
81 views

Are these two theorems about algebraic varieties the same?

In Artin's Algebra, there is a theorem (1) stated as the following: Let $J\subset\Bbb{C}[x]$ be an ideal such that $J=(f_1,\cdots,f_r)$ where $f_1,\cdots,f_r\in\Bbb{C}[x_1,\cdots,x_n]$. Let ...
4
votes
4answers
117 views

Why are roots of polynomials called geometric objects?

I read the following from the Wikipedia article about algebraic varieties: Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by ...
1
vote
1answer
80 views

Can this quick way of showing that $K[X,Y]/(Y-X^2)\cong K[X]$ be turned into a valid argument?

I've been trying to show that $$ K[X,Y]/(Y-X^2)\cong K[X] $$ where $K$ is a field, $K[X]$ and $K[X,Y]$ are the obvious polynomial rings over the indeterminates $X$ and $Y$ and $(Y-X^2)$ is the ...
3
votes
1answer
61 views

Radical ideal of leading terms and Grobner

Let $k$ be a field, let $A$ be an ideal of $k[x_1,\ldots,x_n]$, and let $>$ be a monomial order. I'm asked to show that $A$ is radical if $\langle LT(A)\rangle$ is radical. So, suppose $\langle ...
1
vote
1answer
79 views

regarding finite integral ring extension

My question regards understanding (and possibly a source for proof) of the following, cited in the book Complex Geometry by Huybrechts (Theorem 1.1.30.) (Also, it is there stated that this is a ...
2
votes
1answer
68 views

localization in algebraic geometry

It is often asserted in commutative algebra texts that localization is important in algebraic geometry. I would appreciate some precise examples which show the utility of the concept in this context. ...
14
votes
7answers
470 views

Introduction to ring theory?

I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear ...
2
votes
1answer
57 views

All maximal ideals in the ring of polynomials of are of the kind $N_p=\langle x_i-p_i:i=\overline {1,n}\rangle$ for some point p in the affine space

I am reading a proof on the coincidence of the functional field of a variety (defined by equivalence classes of regular functions) and the field of quotients of its coordinate ring. It turns out I ...
2
votes
1answer
38 views

Regular functions, their zeros and irreducible components

By $X$ we denote some affine variety embedded in $\mathbb{A}^n$. Suppose $\phi\in \mathbb{k}[X]$ divides zero, i.e. there is $\psi\in\mathbb{k}[X]$ such that $\phi\psi=0$. Furthermore, it means that ...
3
votes
1answer
52 views

Why does $\operatorname{Spec}(\prod_1^\infty \Bbb F_2)$ have connected components that are not open?

To be honest, I don't even know how to describe all prime ideals in $\prod_1^\infty \Bbb F_2$. I know we get one for each $n \in \Bbb N$ corresponding to the set of elements that are zero in the ...
4
votes
0answers
94 views

The importance of being Cohen-Macaulay

I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book. In that book there are many examples and sentences of the type "If something satisfies this properties, then it is ...
0
votes
0answers
64 views

Dimension of local ring as vector space over $\mathbb C$

I want to know what the dimension of each of the local ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$ over $\mathbb C$-vector space. I know the dimension of it in the origin point, ...
0
votes
1answer
101 views

What is the Krull dimension of this local ring

I want to know what is the dimension of this ring $\mathbb C[x,y]_{(0,0)}/(y^2-x^7,y^5-x^3)$. I don't know how to do that. If I suppose $y^2=x^7$ I will get a higher degree of $x$.