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15
comment twists of unipotent algebraic groups
As your base field $k$ is perfect and you consider only smooth connected group schemes (or do you?), not only is your group $U^\prime$ unipotent, it even has a composition series whose successive quotients are $\mathbb G_a$. See my answer below.
Dec
15
answered twists of unipotent algebraic groups
Dec
10
revised Is $Spec(R_p)$ to $Spec(R)$ is an open immersion?
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Dec
10
revised Is $Spec(R_p)$ to $Spec(R)$ is an open immersion?
added 69 characters in body
Dec
10
comment Is $Spec(R_p)$ to $Spec(R)$ is an open immersion?
You guys are right, I was thinking about a one-dimensional $R$. Let me fix that.
Dec
10
answered Is $Spec(R_p)$ to $Spec(R)$ is an open immersion?
Dec
9
comment Is the $n$-sphere $x_1^2+\cdots+x_n^2-1=0$ a rational variety in $\mathbb{A}^n$?
@GeorgesElencwajg Good point. My answer is a bit of overkill...
Dec
9
answered Is the $n$-sphere $x_1^2+\cdots+x_n^2-1=0$ a rational variety in $\mathbb{A}^n$?
Dec
8
comment GL_n unique group variety
Are you sure this is the statement you are looking for? Not every (one-dimensional) torus over $\mathbb Q$ is split. Consider for instance $x^2+2y^2=1$ over $\mathbb Q$; see page 106 in Poonen's www-math.mit.edu/~poonen/papers/Qpoints.pdf . This is a one-dimensional torus over $\mathbb Q$ which is isomorphic to $GL_1$ over $\overline{\mathbb Q}$, but not over $\mathbb Q$. Did you mean to ask whether $GL_n$ is isomorphic to $G$ over $\bar k$?
Dec
5
answered degree of extension of residue fields of cyclic covers
Dec
1
comment connected linear algebraic group over the algebraic closure of a field
The Galois group of $k$ permutes the connected components of $\bar G$ but leaves invariant the identity element.
Dec
1
revised What does the Tate module of an elliptic curve tell us?
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Nov
30
comment What is GAGA for dimension 1 ? (Historical Question)
@TeiHuang You probably meant to ask this question for compact Riemann surfaces, as the complex upper half-plane is not algebraic.
Nov
30
revised What does the Tate module of an elliptic curve tell us?
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Nov
30
comment Definition of a Elliptic curve
@AlexYoucis Thanks for the comment. I agree that the Lefschetz trace argument isn't necessarily easier. :)
Nov
29
comment Some basic questions about fibered surfaces
A morphism of fibered surfaces over $S$ (from $X$ to $Y$ say) is proper. In particular, a morphism of fibered surfaces is closed. As any $S$-birational morphism is dominant (ie, has dense image), this implies that the image is $Y$ (as it is dense and closed in $Y$). this answers your question in 2). Can you state a more precise question in 1)?
Nov
29
comment Definition of a Elliptic curve
@AlexYoucis You can also use the Lefschetz trace formula. The compactly supported Euler characteristic of an algebraic group over a field $k$ equals the trace of $t_a$ on $\ell$-adic cohomology (with $a \in E(\bar k)$ nonzero). This trace equals zero as there are no fixed points.
Nov
29
answered What does the Tate module of an elliptic curve tell us?
Nov
21
comment Intrepretation of $H^i(X, \mathcal{O}_X) = 0$
Vanishing of $H^1$ is equivalent to the vanishing of the Albanese variety. Equivalently, $H^1$ is zero iff any morphism to an abelian variety is constant.