Is $\text{Hom}_k(A, B)$ a group? What is its group operator? I want ask a question related to Hom operator.
I'm studying affine group schemes, and in the definition, it says that an affine group scheme is just a functor F from k-algebras to groups. If F is representable by a k-algebra A, then there is a natural correspondence between $F(R)$ and $\text{Hom}_k(A, R)$. Here A and R are k-algebras.
From this definition, I think it implies that $\text{Hom}_k(A, R)$ is a group, because the functor is from k-algebras to groups.
I'm quite confused here, because k-algebra homomorphisms need to map 0 and 1 in $A$ to 0 and 1 in $R$. So $\text{Hom}_k(A, R)$ is not a group in addition (because the zero map is not an algebra homomorphism). It is also not a group in multiplication, because there is no inverse element. So, how is this a group? And if so, what is the group operator? And can you please explain this situation in this definition? (I'm reading chapter 1 in the book "Introduction to Affine Group Schemes" of Waterhouse.)
 A: 
an affine group scheme is just a functor F from k-algebras to groups.

This is incorrect. An affine group scheme (over $k$) is a functor from (commutative) $k$-algebras to groups such that the underlying functor to sets is representable. (There are non-affine group schemes, such as elliptic curves, and there are also functors to groups whose underlying functor to sets is not representable by a scheme.)

So, how is this a group? And if so, what is the group operator?

$\text{Hom}_k(A, R)$ has no natural group structure, as you say. The group structure is part of the structure of being an affine group scheme, and is extra data not determined by $A$. The definition of representability only refers to a (natural) identification of sets $F(R) \cong \text{Hom}_k(A, R)$. Here are two simple examples:

*

*If $A = k[x]$, then $\text{Hom}_k(k[x], R) \cong R$ can be given a group structure via addition. This produces an affine group scheme structure on the affine line $\mathbb{A}^1 = \text{Spec } k[x]$ called the additive group scheme $\mathbb{G}_a$.

*If $A = k[x, x^{-1}]$, then $\text{Hom}_k(k[x, x^{-1}], R) \cong R^{\times}$ can be given a group structure via multiplication. This produces an affine group scheme structure on the punctured affine line $\mathbb{A}^1 \setminus \{ 0 \} = \text{Spec } k[x, x^{-1}]$ called the multiplicative group scheme $\mathbb{G}_m$.

More complicated examples are possible, of course. It's a nice exercise to verify that the functor $R \mapsto GL_n(R)$ is an affine group scheme, where the group operation on $GL_n(R)$ is matrix multiplication.
It's also a nice exercise in using the Yoneda lemma to verify that a (commutative) $k$-algebra $A$ represents an affine group scheme iff $A$ is equipped with the additional structure of a Hopf algebra, where the comultiplication $\Delta : A \to A \otimes_k A$ encodes the multiplication of the group scheme.
