# Tagged Questions

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### Hilbert series and Cohen-Macaulay ring

Given a series, how can I find a ring which has exactly that Hilbert series? I know only a way, which in particular computes a lexicographic ideal. I need to solve this exercise: Find two rings ...
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### Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
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### 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 ...
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### How to prove the uniqueness

I'm trying to solve this question from Fulton's algebraic curves: I've already easily solved (a) and the existence part of (b). I'm having problems to prove the uniqueness of part (b). I need ...
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### Is there a local ring $O$ such that $\mathbb R\subsetneqq O$? and $\mathbb C\nsubseteq O$?

Is there a local ring $O$ such that $\mathbb R\subsetneqq O$? and $\mathbb C\nsubseteq O$? I need this to prove a another problem I'm proving. I hope there aren't such a ring. Thanks in advance
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### Why is this theorem important in fulton's book?

In Fulton algebraic curves book, we have the following proprieties which help us to find the intersection number of a pair of function at a given point. afterwards he states this theorem: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ${\Bbb C}^2$, ...
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### 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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### 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 ...
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### 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 ...
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### 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 $$J_1\cup J_2=V(I_1I_2).$$ ...