There is mention of the statements like the following which appears in every lecture notes or expository articles regarding algebraic group (or may be GIT). The first statement is more or less as follows.
For an (affine) algebraic group G over a field K , we know that for any K algebra R(sometimes they talk about finitely generated k algebra) there is a group called G(R).
Now some source mentions G(R):=Hom(R,G). My questions are as follows.
1)Since R is a ring and G is a scheme,so one is tempted to interpret it as Hom(SpecR,G),but Since morphisms of schemes does not necessarily form a group.So I interpret it as Hom(O(G),R). Does this interpretation make sense?
2)There is a fact as follows: Any group scheme G is a representable functor from the category of K algebras to the category of groups(some also takes this as the definition) and is represented by the K algebra K[G]. They call it the coordinate ring(algebra) of the scheme G. Now I know the concept of coordinate ring for an affine variety but not for a scheme.Also I observe that if I am correct in my first question then there is a chance that O(G)=K[G]. My question is what exactly K[G] is?Is it by definition O(G)?
3)Is it anyway true that Hom(O(G),R) is same as End($O(G) \otimes_{\mathbb{K}}R$)?
4)The definition of G equivariant morphism $f:X\to Y$ is that f(g.x)=g.f(x) for all g and x.Since for me action of an element g to an element x does not make sense(because $G \times X$ is fibered product and not a cartesian product of sets) I tried and figured out definitions that conveys the idea in terms of commutative diagram.But when I started reading more typical and advanced ideas of the topic namely orbits and stabilizers they are only talking in terms of elements and their image and not in terms of commutative diagram.I believe that everytime the definitions can be broken down to commutative diagrams to get fit in with the definitions of morphism of schemes and fiber product,BUT could anyone suggest me how to think about them without bothering about commutative diagram everytime ,I mean in terms of equation and elements?
5)I have come across the standard definition of action of an affine algebraic group G on a K algebra A(which is given by for every K algebra R action of G(R) on ($ A \otimes_{\mathbb{K}}R$) which is R module morphism and functorial in nature).
FACT:If G and X both are affine and if G acts on X then one obtains a homomorphism of rings from O(X) to $ O(G) \otimes_{\mathbb{K}}O(X)$.This fact is not very difficult to see but how from this fact one can prove G defines a unique action on O(X)(considering O(X) as a K algebra)?
It is given as g.f(x)=f(${x}^{-1}$.y),and failed to make sense of this in terms of commutative diagram.
Any help from anyone is welome.
An algebraic group is a specific type of group scheme.
A group scheme over $K$ is given by a scheme $G$ with an "identity section" $z: Spec(K) \to G$ of the structure map $G \to Spec(K)$, a "multiplication" map $\mu: G \times G \to G$, and an "inversion" map $i: G \to G$: these have to satisfy some natural axioms that are analogous to the axioms of a group (which you can find in any reference on group schemes: see the bottom for a link to one definition). So suppose you're given two $R$ points of $G$, i.e. two maps $x: Spec(R) \to G$ and $y: Spec(R) \to G$. Then you can define the product $x \cdot y$ to be the map $$Spec(R) \xrightarrow{(x,y)} G \times G \xrightarrow{\mu} G$$ Then $x^{-1}$ would be defined as $$Spec(R) \xrightarrow{x} G \xrightarrow{i} G$$ Then one has to prove that this endows $Hom_{Sch_K}(Spec(R),G)$ with the structure of a group, with identity $$Spec(R) \to Spec(K) \xrightarrow{z} G$$. If $G$ is affine, you can actually look at $O(G)$ the coordinate ring as a Hopf algebra, and make a dual argument about $Hom_{K-alg}(O(G),R))$.
Yes, $K[G]$ would be $O(G)$. In this case, not only is it a $K$-algebra, but also a Hopf algebra (this is basically the dual of a group object, in some sense).
For the rest, I'll just say that you can define a group action of a group scheme $G$ on a $K$-scheme $X$ as a map of schemes $a: G \times X \to X$, satisfying certain properties. I won't list all of these properties, but I'll give you one of them, to give you the idea. For an ordinary group action of a group $G$ on a set $X$, you would require that $1_G \cdot x = x$ for all $x \in X$. To express this scheme theoretically, you take the identity section: $z: Spec(K) \to G$ and then say that $$X \cong Spec(K) \times X \xrightarrow{z \;\times\;\text{id}_X} G \times X \xrightarrow{a} X$$ should be equal to the identity map $X \xrightarrow{\text{id}_X} X$. Can you figure out the rest of the axioms, and, for example, how to express $G$-equivariance of a morphism $f: X \to Y$ of $K$-schemes?
PS: a definition of a group scheme is a group object in the category of schemes, which you can find a reference for here: https://ncatlab.org/nlab/show/group+object, for example.
Honestly the best way to actually learn how this works is to study an example, like the multiplicative group scheme over a field: in this case $K[G] = K[x,y]/(xy-1)$, and multiplication is given by the map on schemes induced by $K[G] \to K[G] \otimes_K K[G] \cong K[x,y,x',y']/(xy-1,x'y'-1)$ sending $x$ to $xx'$ and $y$ to $yy'$ (can you figure out what the identity section and inverse should be?).