# Tagged Questions

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### 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: $\Omega^{d+1}_{A|k}\cong 0$?
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### 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 ...
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### 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 $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $R$, ...
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### 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 $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$ ? $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)=\{ (0),\ (\tilde{x}-a,\tilde{y}-b),\ b^2=a^3\}$.
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### 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. $$H=k+kt^{4\nu}(1+t)+kt^{6\nu}(1+t)+kt^{7\nu}(1+t)+k[[t]]t^{8\nu}$$ ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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, ...
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 $x$.