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The answer is no. A counterexample: the quotient $\mathbb{F}_4[x]/(x(x-1))$ is isomorphic to $\mathbb{F}_4\times\mathbb{F}_4$. If $\mathbb{F}_2[x] / (f(x))$ were isomorphic to $\mathbb{F}_4\times\math …

Let $I'\subset B\otimes_A B$ be the ideal generated by the elements $b_i\otimes 1-1\otimes b_i$, and define $$ R=\{b\in B:b\otimes 1-1\otimes b\in I'\}. $$ It’s not hard to check that $R$ is an $A$-su …

A couple of years ago, David Speyer showed me the following counterexample. Let $E$ be an elliptic curve defined over a field, with points $P$, $Q$ that are linearly independent under the group law. …

For the first question, $\mathrm{Spec}(B)$ is quasi-compact for every ring $B$, so if $U\subset\mathrm{Spec}(A)$ is not quasi-compact, then there cannot exist $f:A\to B$ with $U=f^*(\mathrm{Spec}(B))$ …

Take $g(x)=x-x^2\in\mathbb{Z}[[x]]$. There is an $f(x)\in x\mathbb{Z}[[x]]$ which is an inverse to $g(x)$ under composition (this is true because $g(x)$ has constant term 0 and linear term a unit; we …

A counterexample is $$ A=\left[\begin{array}{cc}0&2\\8&0\end{array}\right],\hspace{5mm}B=\left[\begin{array}{cc}0&4\\4&0\end{array}\right]\in M_2(\mathbb{Z}_2). $$ The matrices are conjugate in $\math …

No, $\hat{R}_y$ need not be $\hat{x}$-adically complete. The polynomial ring $R=k[x,y]$ is a counterexample. The $x$-adic completion of $R$ is identified with $k[y][[x]]$, the ring of power series in …

Let $p$ be a prime. Take $k=\mathbb{F}_p$, $$ R=k[X,X^{1/p},X^{1/p^2},\ldots]. $$ For any $k$-algebra $A$, there is a natural bijection $$ \mathrm{Hom}_{k\text{-alg}}(R,A)\cong\big\{(a_0,a_1,\ldots):a …

The group algebra $\mathbb{Z}_p[[\mathbb{Z}_p\times \mathbb{Z}_p]]$ has the structure of a complete Hopf algebra, where $\mathbb{Z}_p\times \mathbb{Z}_p$ consists of precisely the group-like elements. …

The usual power series for $\log(1+x)$ determines an injection from $1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]$ into $\mathbb{Q}_p[[x_1,x_2]]$. A power series $f\in 1+(x_1,x_2)$ is in $\langle 1+x_1,1 …

Let $K$ be a field, and set $R=K[x]$ with the usual grading by degree. For each irreducible polynomial $p(x)\in R$, we get a valuation $v_p$ on $R$ given by $$ v_p(f)=2^{-\operatorname{ord}_p(f)}, $$ …

Choose a smooth projective $X/R$ of positive dimension, and pick a set-theoretic splitting $\varphi:X(\kappa(s))\to X(R)$ of the reduction map $\pi:X(R)\to X(\kappa(s))$. Take $a=\varphi\circ \pi$, an …

For (Q1): A finite dimensional $\mathbb{C}$-algebra $A$ is Artinian, so $A$ is a product of Artin local algebras. The units of a product of algebras is the product of the units, so we may assume $A$ i …

Since $K$ is generated by $p$ elements, $\mathbb{C}(x_0,\ldots,x_{q-1})$ cannot be an algebraic extension of $K$ if $q>p$. I claim that $\mathbb{C}(x_0,\ldots,x_{q-1})$ is a finite Galois extension of …