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The first chapter of this paper (Gersten’s conjecture and the homology of schemes) defines a "Poincaré duality theory with supports" is a twisted cohomology theory satisfying certain properties on pag …

I was reading the proof of Milne étale cohomology on Brauer groups IV.2.1. Prove that if $A\otimes k(x)$ is a central simple algebra over $k(x)$ for all $x$. Then there exists (finite) etale covering …

I was reading the paper Swan conductors for characters of degree one in the imperfect residue field case by Kato. Is it easy to prove the property that the symbol {...} has the property if any two ele …

Let $E/K$ be an elliptic curve over a field (take it as a number field or a local field if necessary). If $E[2]$ is defined over $K$, then we can associate the multiplication by $2$ isogeny to a biqua …

Let's look at this paper on page 40. Let $K$ be a finite extension of $\mathbb Q_p$. One can define the derived de Rham algebra $L\Omega_{\mathcal{O}_{\overline{K}}/\mathcal{O}_K}^{\bullet}$. There is …

Let $\mathcal X$ be a scheme of finite type over $\mathbb Z$. We know that the number of geometrically irreducible components (of geometric fibers) is the constant at almost everywhere, by EGA. It is …

$\DeclareMathOperator\Br{Br}$Usually, we have $\Br_\text{Az}(X)\subseteq \Br_\text{Gr}(X)_\text{tor}\subseteq \Br_\text{Gr}(X):= H^2(X,\mathbb G_m)$. It is easy to find counter examples for the second …

Like explained in this passage that a period is a complex number whose real and imaginary parts are integrations of rational functions over $\mathbb{Q}$ on some $\mathbb{Q}$-semi-algebra set in $\math …