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The answer to all three questions is yes and certainly is classical. One simple example is the following: Let $C_2$ act faithfully on the set $\{1,2,3,4\}$ in two ways. In the first the non-trivia …

In algebraic geometry it is well-known (see Hartshorne Exercise II.5.16 for example) that there is a 1-1 correspondence between rank $n$ (geometric) vector bundles $\pi\colon Y\to X$ on a scheme $X$ a …

Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional …

Suppose $p$ is a prime, that $F$ is a finite extension of the field $\mathbb{Q}_p$, $D$ is the division quaternion algebra over $F$ and $\mathcal{O}_D$ is the valuation ring of $D$. What is the abelia …

For motivation I would advise starting with Brauer's original paper. You'll need a JSTOR login though: https://www.jstor.org/stable/1968843?origin=crossref&seq=1#metadata_info_tab_contents

In case anyone else has the same question and discovers this page I have just found a more explicit reference for this result: Remark 3.4.4 of A Singular introduction to commutative algebra by Greuel …

The answer is now yes, I think http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.0896v2.pdf Edit: as requested: http://arxiv.org/abs/1011.0896

Suppose $G$ is a connected semisimple linear algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a homogeneous $G$-space with isotropy subgroup $H$ (associated Lie algebra $\mathfrak{h}$) that …

For the non-commutative case, whilst the result is not explicitly there, almost all the work is done in 'Modules over Iwasawa algebras' by Coates, Schneider and Sujatha (see https://ivv5hpp.uni-muenst …

You don't explicitly say your representation is complex but I think your example shows that this is the case you're interested in. If so, then $V_0$ has a $G$-invariant complement $V_1$ by Maschke's T …

The virtual DCC doesn't seem so different from the notion of Krull dimension $1$ that I explained in answer to this Different definitions of the dimension of an algebra question.