Thanks.
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1 answers
2 votes
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Small confusion regarding singular homology
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I came to realise my mistake... the $0$-chain is not the zero simplex (which doesn't make any sense anyway, right?) but the neutral element in the group $S_n(X)$. I first deleted the question, but now ...

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2 answers
0 votes
136 views
Prove that $I$ is a finitely-generated ideal of $\mathbb{Z}[x]$ by finding a finite set of generators for $I$.
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Note that for a polynomial $f=a_m x^m+\cdots +a_0$ being in $I$ is equivalent to say that the constant term is even and all coefficients add up to an even number. Now I claim that with $J:=(x^2+x,x^2-...

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1 answers
2 votes
37 views
What does the vector space $\mathbb{Q}[z]/\mathbb{Q}[z]·P(z)$ of the polynomials modulo the multiples of $P(z)$ represent?
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1 votes

First of all note that $P(z)$ is irreducible in $\mathbb{Z}[z]$, because it is of degree $2$ and does not have a root in $\mathbb{Z}$. Then it is also irreducible in $\mathbb{Q}[z]$, so the ideal ...

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2 answers
2 votes
212 views
Tensor product of quotient of modules.
1 votes

Let‘s show that $R/(I+J)$ satisfies the universal property of the tensor product; then we‘re done by the usual universal property argument. First, we need a bilinear map $\phi: R/I \times R/J\...

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1 answers
0 votes
105 views
Non-Noetherian 0-dimensional ring
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1 votes

If one takes $R=K[x_1,x_2,\ldots]/(x_1^2,x_2^2,\ldots)$ to be the polynomial ring over a field in infinitely many variables modulo the ideal $(x_1^2,x_2^2,\ldots)$, then $R$ is not noetherian and $\...

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1 answers
0 votes
263 views
Show that every prime ideal of A (UFD) is generated by a set of prime elements
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1 votes

Consider an element f of the prime ideal and factor it as a product of prime elements in the UFD A. Since the ideal is prime, one factor must actually belong to the ideal and collecting these elements ...

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2 answers
2 votes
77 views
Intuition behind the definition of direct sum of linear subspaces
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So I'm far from an expert, hence this is just a try. But maybe we can think for example of a two dimensional subspace in euclidean space, thus a plane, and a one dimensional subspace that is not ...

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1 answers
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Trying to understand the connection of closed immersion/subscheme.
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I am trying to follow the hint from the comment. For a point $x\in X$, we can compute the stalk $$(i_*i^{-1}\mathcal{O}_X/\mathcal{I})_X)_x=0$$ if $x\notin Z$ since $Z$ is closed, and if $x\in Z$ then ...

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1 answers
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Trying to understand $\operatorname{Hom}(X,Y)=\operatorname{Hom}(A,\Gamma(Y,\mathcal{O}_Y)$ for $Y=\operatorname{Spec} A$ and $X$ a locally ringed.
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I think I came up with the answer myself, at least for my second question. I tried to write out the details: the map $\Gamma(D(s),\mathcal{O}_Y)=A_s\rightarrow \Gamma(X_{\varphi(s)},\mathcal{O}_X)$ is ...

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