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I am looking for a reference for (or a proof of) the following fact: Let $G$ be a profinite group. Let $X^\bullet$ be a complex of discrete $G$-modules. We assume that the cohomology $G$-modules of …

Proof (due to Joseph Bernstein). Assume that $H^i(X^\bullet)=0$ for $i>n$. We choose a $G$-morphism $A^n\to \ker[X^n\to X^{n+1}]$ such that the induced morphism $A^n\to H^n(X^\bullet)$ is surjective, …

I am asking for a reference for the following lemma (for which I know a proof). Lemma. Let $f\colon X\to Y$ be a surjective morphism of irreducible smooth complex algebraic varieties (separated, red …

In addition to Marty's reference, I would recommend to look into §6 "Le groupe fondamental algébrique des groupes algébriques linéaires connexes via les resolutions flasques" of Colliot-Thélène's pape …

I am looking for a reference for the assertion in the title. In more detail, let $\Gamma=\{1,\gamma\}$ be a group of order 2. Let $A$ be a $\Gamma$-module (an abelian group on which $\Gamma$ acts). T …

Find it here. (Edit: Link removed). I hope you can read Russian. Enjoy! Edit: Find here another (better?) scan, also in Russian.

(I write an answer rather than a comment in order to accommodate exact sequences.) Let $$0\to T\to E\to\Gamma\to 1\tag{$E_1$}$$ be your first group extension, where $T$ is the torus ${\Bbb R}^n/{\Bbb …

I can give a reference only for the second part of the question, namely, about central extensions. It was answered by Colliot-Thélène in 2008, not 30 years ago! Colliot-Thélène's paper Résolutions fl …

I give an elementary proof of the fact that a quasi-isomorphism of short complexes (complexes of length 2) of $\Gamma$-modules induces an isomorphism on hypercohomology. Actually, it is very close to …

Yes, the formula is correct, see Sansuc, J.-J. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327 (1981), 12–80, (10.1.2). In the ext …

In the book Lie groups and algebraic groups by Onishchik and Vinberg, Theorem 3 in Section 5.2.1 on page 240 says: Let $S$ be a real structure on a simply connected complex semisimple Lie group $G$. …

A reference: Lie Groups and Algebraic Groups (Springer Series in Soviet Mathematics) by Arkadij L. Onishchik, Ernest B. Vinberg and Dimitry A. Leites (new printing will appear on Amazon.com on Jul 31, …

It is written in Wikipedia http://en.wikipedia.org/wiki/Groupoid, that any connected groupoid $A\rightrightarrows X$ is isomorphic to an action groupoid $G\ltimes X$ coming from a transitive action o …

See Subsection 6.4.1 in my paper with Zeev Rudnick Hardy-Littlewood varieties and semisimple groups, Invent. Math. 119 (1995), 37-66, page 62 (or this PDF file, page 23). The equation is: $$ -9x^2+2x …

I am looking for a reference for the assertion in the title. This assertion is proved in a comment of user nfdc23 to this question. Has any proof of this assertion been published?