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I find it is very inconvenient to handle group schemes by its defination(i.e. everything is defined by morphism). And I have noticed that for group varieties, one can treat them as actual groups(i.e. talking about elements). Moreover, in the book Geometric Invariant Theory by Mumford, though he gave the defination of group scheme by relations of morphism, what he really used was "point"! Not the closed point, but R-point(i.e. morphism Spec(R) $\to G$, where R is a ring ).

I don't know anything about thoery on R-point, and I guess this might be the way of handling group scheme by "point". Does anyone know the corresponding theory(or reference)? And how could them hook together?

Moreover, I am also curious about how does one think of group schemes. What could go wrong if one think of group scheme by element?

This may not be a well-asked question, because of my ignorance of group schemes, and the words above is just my own feeling while reading GIT. Any comments would be great appreciated!

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There is a definition of scheme as a functor from the cartegory of commutative $k$-algebras ($k$ and arbitrary commutative ring) to the category of sets. A group scheme is then a functor from the same category to the category of groups which, when composed with the forgetful functor to sets is a scheme. This way of viewing group schemes can have some advantages, as we then have the points over each algebra, and this is indeed a group. – Tobias Kildetoft Jul 4 '12 at 8:56
@Tobias:I think this point of view is the one adopted in GIT, could you point to me where I can find such definition? Thank you – Li Zhan Jul 5 '12 at 5:01
The definition used by Tobias is described in at least Jantzen's book on representations of algebraic groups. There the kernels of the Frobenius automorphism form an important family of group schemed that have just one point over any field, but do have a lot of structure (and also points over $k$-algebras with nilpotent elements). I don't know, whether his treatment would be endorsed by algebraic geometers. It is sufficient for representation theorists, though. – Jyrki Lahtonen Jul 5 '12 at 6:34
The definite reference for the functorial point of view is "Groupes algébriques" by Demazure-Gabriel. – Martin Brandenburg Jul 5 '12 at 6:47
up vote 5 down vote accepted

The main point is that a group scheme does not have enough points :-)

For example, given any number field $K$ (=finite-dimensional extension of $\mathbb Q$), Mazur-Rubin have recently proved that there exists an elliptic curve $E$ defined over $K$ such that $E(K)$ (=points with coordinates in $K$) is the trivial group with one element: this is absurdly insufficient for studying $E$.

More importantly the possibility to vary the field (or even scheme) and look at points of the group-scheme with coordinates in that field, i.e. the morphism point of view, is incredibly important in arithmetic geometry.

For example, if $E$ is an elliptic curve defined over $\mathbb F_p$ it has a zeta function $$Z(E)= \operatorname {exp} (\sum\operatorname {card}E(\mathbb F_{p^n})\cdot\frac{T^n}{n}) $$
which is, amazingly, a rational function in $T$ and has been the prototype of the arithmetic study of group-schemes.

That said, over an algebraically closed field $k$ , the classical study of a group-scheme just as a group $G(k)$ is quite interesting and fruitful: it is called the geometric study of $G$, as opposed to the arithmetic study evoked above.

Edit: points of a scheme
Given a scheme $X$ , a point with values in a another scheme $T$ is a morphism $T\to X$. This looks rather opaque (especially if you also consider that all schemes are over another scheme $S$ !) so let me try and show that this notion is not so unreasonable as it looks.

Consider the field (or even ring) $k$, the algebra $A=k[T_1,...,T_n]/I$ ( where $I\subset k[T_1,...,T_n]$ is an ideal) and the affine $k$-scheme $X=Spec(A)$.
Now, what is a $T$-point of $X$ if you take for $T$ the affine scheme $T=Spec(R)$ corresponding to some $k$-algebra $R$ ?
Answer: it just corresponds to $V(R)\subset R^n$, the set of solutions $(r_1,...,r_n)\in R^n$ of the system of equations $$P(r_1,...,r_n)=0 \quad \text{for all} \;P(T_1,...,T_n)\in I$$ which is the naïve interpretation of "points of $X$ with values in $R$".
And this is easy: a $k$-morphism $Spec(R)\to X=Spec(A)$ corresponds to a morphism of $k$-algebras $\phi: A=k[T_1,...,T_n]/I\to R$ to which you just associate the naïve $n$-tuple $(r_1,...,r_n)=(\phi(\bar T_1),...,\phi(\bar T_1))\in V(R)$

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Dear Georges Elencwajg: Thank you! By "point", or "points with coordinates in K", do you mean the morphism "Spec(K) $\to$ G"? But there do exist R-points (R is a ring), do there? Is this the way (as mentioned by Tobias above) to handle group scheme by points? – Li Zhan Jul 5 '12 at 5:07
Dear Li, I have written a little Edit in the hope of clarifying the notion of $R$-points you ask about. – Georges Elencwajg Jul 5 '12 at 6:28

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