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We should be able to construct a counterexample as follows: Let $Y$ be an affine curve which is smooth away from a single node. We obtain $X$ from the normalization of $Y$ by removing one of the poin …

Let's fix a scheme $S$ and consider some Grothendieck topology $\tau\in\{Zariski,\acute{e}tale,fppf\}$ on the category of schemes over $S$. Define a $\tau$ fiber bundle with fiber type $F$ to be a map …

The affine group scheme $G$ you describe is not finite type. It is possible to describe $G$ explicitly. For $R$ a $k$-algebra, the $R$-points of $G$ are the tensor functorial automorphisms of $\mathr …

Another way to see this: if $\gamma$ is a small loop in $C_1$ winding once around $x$, then $f(\gamma)$ is a small loop in $C_2$ winding $e_x$ times around $f(x)$, so $$ \mathrm{Res}_x f^*\omega=\frac …

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 …

We should be able to construct such a real number $x$. Let $\epsilon\in(0,1/2)$ be fixed. If $S\subset \mathbb{R}_{>0}$ and $r>0$, we write $S^r$ for the set of positive $r$-th powers of elements of $ …

Write $X=\mathrm{Spec}\, k$, which is a $0$-dimensional variety. Motives of $0$-dimensional varieties are called Artin motives, and they are pure. The Betti realization is the Betti cohomology of $X$, …

Here is a counterexample. Fix a field $k$, and let $Y$ be built from two copies of the affine nodal curve $y^2=x^3+x^2$, glued together on the complement of the singular point. In other words $Y$ is a …

Let's say $k=\mathbb{C}$ (although something like this should work over any algebraically closed field). Let $V_1=\mathbb{P}^1-\{0,1,\infty,\pi\}$ and $V_2 = \mathbb{P}^1-\{0,1,\infty,e\}$. One can se …

For a variety $X$ defined over $\mathbb{Q}$, there's a (functorial) comparison isomorphism $$ H^i_{dR}(X)\otimes\mathbb{C}\to H^i_B(X,\mathbb{Q})\otimes\mathbb{C}. $$ If we pick $\mathbb{Q}$-bases for …

There is a notion of $p$-adic period coming from the Frobenius action on $p$-adic cohomology. Suppose $X$ is a smooth variety over $\mathbb{Q}$, and let $p$ be a prime. If there is a smooth model $\ti …

Take $$ R=\prod_{n=1}^\infty \mathbb{Z}/p^n\mathbb{Z}, $$ with the product topology. Every non-empty open set in $R$ contains an element that is $0$ in all but finitely many factors, and this element …

Suppose $X$ is a smooth (connected) variety over $\mathbb{C}$. Let $\mathscr{C}$ be the category of finite rank vector bundles on $X$ equipped with an integrable connection, and let $\mathscr{C}'$ be …

A counterexample is $X=\mathbb{A}^2\backslash\{y=0\}$, $A=\mathbb{C}[x,y,y^{-1}]$, and $\mathfrak{g}$ the span of the derivation $D(x)=1$, $D(y)=y$. Now $\mathfrak{g}$ is $1$-dimensional and $X$ is $2 …

Tate twists do indeed shift the Hodge filtration. Since the de Rham realization of the Tate motive $\mathbb{Q}(1)$ is concentrated in the degree $-1$ part of the Hodge filtration, we have $F^n M(m)_{d …