Intereting Posts

Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.
Calculating the Zeroes of the Riemann-Zeta function
Prove that if $\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$ then either $A \subseteq B$ or $B \subseteq A$.
Solving a Linear Diophantine Equation
$g^k$ is a primitive element modulo $m$ iff $\gcd (k,\varphi(m))=1$
Using CLT to calculate probability question
On the sum of digits of $n^k$
Continuous coloring of a Mandelbrot fractal
A binary quadratic form: $nx^2-y^2=2$
Solve Burgers' equation
Inverse Limits: Isomorphism between Gal$(\mathbb{Q}(\cup_{n \geq 1}\mu_n)/\mathbb{Q})$ and $\varprojlim (\mathbb{Z}/n\mathbb{Z})^\times$
Behaviour of ($\overline{a}_n)_{n=1}^{\infty}$ and ($\underline{a}_n)_{n=1}^{\infty}$
Prove that the number $19\cdot8^n+17$ is not prime, $n\in\mathbb{Z}^+$
Quadratic Extension of Finite field
Irrationals: A Group?

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?

- Two polynomial problem
- The fibers of a finite morphism of affine varieties are all finite
- Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$
- Exercise 5.5.F. on Ravi Vakil's Notes related to associated points
- Constructing the 11-gon by splitting an angle in five
- Isomorphism of Proj schemes of graded rings, Hartshorne 2.14

- How to show $\alpha_d : M_d \to \Gamma(X, \widetilde{M(d)})$ an isomorphism for sufficiently large $d$?
- Variety of Nilpotent Matrices
- Are monomorphisms of rings injective?
- Single $\text{GL}_n(\mathbb{C})$-conjugacy class, dimension as algebraic variety?
- $Z(I:J)$ is the Zariski closure of $Z(I)-Z(J)$
- Relation between blowing up at a point and at a variety
- Intersection of all maximal ideals containing a given ideal
- Quotient of a local ring at a point is a finite dimensional vector space
- Geometry or topology behind the “impossible staircase”
- (Certain) colimit and product in category of topological spaces

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

- Looking for an atlas with 1 chart
- Does $R \cong S$ imply $R \cong S$?
- Every two positive integers are related by a composition of these two functions?
- How to count the closed left-hand turn paths of planar bicubic graphs?
- Summation of series $\sum_{k=0}^\infty 2^k/\binom{2k+1}{k}$
- $p^3 + 2$ is prime if $p$ and $p^2 + 2$ are prime?
- Understanding dot product of continuous functions
- I would like to show that all reflections in a finite reflection group $W :=\langle t_1, \ldots , t_n\rangle$ are of the form $wt_iw^{-1}.$
- Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
- Prove that $\sqrt5 – \sqrt3$ is Irrational
- Show that the Area of image = Area of object $\cdot |\det(T)|$? Where $T$ is a linear transformation from $R^2 \rightarrow R^2$
- Prime Appearances in Fibonacci Number Factorizations
- Is a Cauchy sequence – preserving (continuous) function is (uniformly) continuous?
- Hermitian Matrices are Diagonalizable
- How $x^4$ is strictly convex function?