Intereting Posts

Prove $\int_0^1\frac{\ln2-\ln\left(1+x^2\right)}{1-x}\,dx=\frac{5\pi^2}{48}-\frac{\ln^22}{4}$
Linearly ordered sets “somewhat similar” to $\mathbb{Q}$
If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$.
Expected number of steps to transform a permutation to the identity
How do I prove this method of determining the sign for acute or obtuse angle bisector in the angle bisector formula works?
Looking for strictly increasing integer sequences whose gaps between consecutive elements are “pseudorandom”
A property of homogeneous of degree p functions:
Proving that $\sqrt{2}+\sqrt{3}$ is irrational
Show that the product of two consecutive natural numbers is never a square.
If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ then $\frac{1}{a^5}+\frac{1}{b^5}+\frac{1}{c^5}=\frac{1}{a^5+b^5+c^5}.$
How common are probability distributions with a finite variance?
w.r.t. which chain complex is $H^k_{sign}(M;R)$ computed?
Closed immersions are stable under base change
How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$?
Simple property of a valuation on a field

Martin Brandenburg pointed out elsewhere in the comments that he could give a one line proof, using the Yoneda lemma, of

$$\frac{\mathbf{C}[x_1,\ldots,x_{n+m}]}{I(X)^e+I(Y)^e} \cong \frac{\mathbf{C}[x_1,\ldots,x_n]}{I(X)} \otimes_\mathbf{C} \frac{\mathbf{C}[x_{n+1},\ldots,x_{n+m}]}{I(Y)}$$ (for $X,Y$ affine algebraic varieties), but apparently the proof was too long for his margin. How can this be done?

**Fundamental question aside**: why is there no abstract nonsense tag?

- Geometric meaning of primary decomposition
- Classic Circle and Adjacent Arrangement Problem
- What is the intuition behind the concept of Tate twists?
- If $X$ is affine reduced, show that $f\neq 0 \Rightarrow \overline {D(f)} = \operatorname {Supp} f$
- $f(x)\in\mathbb{Q}$ such that $f(\mathbb{Z})\subseteq \mathbb{Z}$ Then show that $f$ has the following form
- Why is the Artin-Rees lemma used here?

- Number of birational classes of dimension d, geometric genus 0 varieties?
- Compute the weights of a $(\mathbb C^*)^{m+1}$-action on $H^0(\mathbb P^m, \mathcal O_{\mathbb P^m}(1))$
- Special arrows for notation of morphisms
- Epic-monic factorisation in $\mathbf{Set}$.
- References on Inverse Problems, Approximation theory and Algebraic geometry
- What use is the Yoneda lemma?
- Is Category Theory geometric?
- Homogenous polynomials
- The projection formula for quasicoherent sheaves.
- Is it really possible to find primary decomposition of given ideal without using Macaulay2?

The claim, I suppose, is that for commutative $k$-algebras $R$ and $S$ and ideals $I \trianglelefteq R$ and $J \trianglelefteq S$, we have

$$(R \otimes_k S) / (I^e + J^e) \cong (R / I) \otimes_k (S / J)$$

where $I^e$ is the extension of $I$ along $R \to R \otimes_k S$ and similarly for $J^e$.

Well,

$$\mathbf{CAlg}_k ((R / I) \otimes_k (S / J), T) \cong \mathbf{CAlg}_k (R / I, T) \times \mathbf{CAlg}_k (S / J, T)$$

and

$$\mathbf{CAlg}_k (R / I, T) \cong \{ f \in \mathbf{CAlg}_k (R, T) : f (I) = 0 \}$$

$$\mathbf{CAlg}_k (S / J, T) \cong \{ g \in \mathbf{CAlg}_k (S, T) : g (J) = 0 \}$$

but

$$\mathbf{CAlg}_k (R, T) \times \mathbf{CAlg}_k (S, T) \cong \mathbf{CAlg}_k (R \otimes_k S, T)$$

so

$$\mathbf{CAlg}_k ((R / I) \otimes_k (S / J), T) \cong \{ h \in \mathbf{CAlg}_k (R \otimes_k S, T) : h (I S) = 0, h (R J) = 0 \}$$

but $I S = I^e$ and $R J = J^e$, and we have $h (I^e) = 0$ and $h (J^e) = 0$ if and only if $h (I^e + J^e) = 0$. Hence,

$$\mathbf{CAlg}_k ((R / I) \otimes_k (S / J), T) \cong \mathbf{CAlg}_k ((R \otimes_k S) / (I^e + J^e), T)$$

as required.

$$\hom(R/I \otimes S,-) = \hom(R/I,-) \times \hom(S,-) = \{f \in \hom(R,-):f(I)=0\} \times \hom(S,-)$$

$$ = \{h \in \hom(R \otimes S,-) : h(I \otimes 1)=0\}=\hom((R \otimes S)/I^e,-).$$

Hence, $R/I \otimes S = (R \otimes S)/I^e$. Then also $R/I \otimes S/J = (R \otimes S/J)/I^e = (R \otimes S)/(J^e,I^e)$.

- Difference between i and -i
- Name of corresponding objects in equivalent categories
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- Necessary condition for positive-semidefiniteness — is it sufficient?
- Which of the following sets are compact:
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- Definition of uniform structure
- Why is $^\mathbb{N}$ not countably compact with the uniform topology?
- Max distance between a line and a parabola
- Function which is not in $L^2(R^n)$
- sum of a series
- proving $\sum\limits_{k=1}^{n} \Bigl\lfloor{\frac{k}{a}\Bigr\rfloor} =\Bigl\lfloor{\frac{(2n+b)^{2}}{8a}\Bigr\rfloor} $
- Frechet derivative of square root on positive elements in some $C^*$-algebra
- For all $m\ge 2$, the last digit of $ 2^{2^m} $ is 6
- Model existence theorem in set theory