The representation theory of the additive group scheme

In this post we’ll describe the representation theory of the additive group scheme Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ over a field Image may be NSFW.
Clik here to view.. The answer turns out to depend dramatically on whether or not Image may be NSFW.
Clik here to view. has characteristic zero.

Preliminaries over an arbitrary ring

(All rings and algebras are commutative unless otherwise stated.)

The additive group scheme Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ over a base ring Image may be NSFW.
Clik here to view. has functor of points given by the underlying abelian group functor

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{G}_a(-) : \text{CAlg}(k) \ni R \mapsto (R, +) \in \text{Ab}.$

This functor is represented by the free Image may be NSFW.
Clik here to view.-algebra Image may be NSFW.
Clik here to view. k[x] together with the Hopf algebra structure given by the comultiplication

Image may be NSFW.
Clik here to view. $\displaystyle \Delta(x) = x \otimes 1 + 1 \otimes x$

and antipode Image may be NSFW.
Clik here to view. S(x) = -x , and so it is an affine group scheme whose underlying scheme is the affine line Image may be NSFW.
Clik here to view. $\mathbb{A}^1$ over Image may be NSFW.
Clik here to view.. From a Hopf algebra perspective, this Hopf algebra is special because it is the free Hopf algebra on a primitive element.

A representation of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ can be described in a few equivalent ways. From the functor of points perspective, we first need to describe the functor of points perspective on a Image may be NSFW.
Clik here to view.-module: a Image may be NSFW.
Clik here to view.-module Image may be NSFW.
Clik here to view. has functor of points

Image may be NSFW.
Clik here to view. $\displaystyle V(-) : \text{CAlg}(k) \ni R \mapsto R \otimes_k V \in \text{Ab}$

making it an affine group scheme if Image may be NSFW.
Clik here to view. is finitely generated projective (in which case its ring of functions is the symmetric algebra Image may be NSFW.
Clik here to view. $S(V^{\ast})$ on the dual of Image may be NSFW.
Clik here to view.) but not in general, for example if Image may be NSFW.
Clik here to view. is a field and Image may be NSFW.
Clik here to view. is an infinite-dimensional vector space. Note that if Image may be NSFW.
Clik here to view. V = k we recover Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ , and more generally if Image may be NSFW.
Clik here to view. V = k^n we recover Image may be NSFW.
Clik here to view. $\mathbb{G}_a^n$ .

A representation of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ over Image may be NSFW.
Clik here to view. is, loosely speaking, a polynomial one-parameter group of automorphisms of a Image may be NSFW.
Clik here to view.-module. The simplest nontrivial example is

Image may be NSFW.
Clik here to view. $\mathbb{G}_a \ni x \mapsto \left[ \begin{array}{cc} 1 & x \\ 0 & 1 \end{array} \right]$

which defines a nontrivial action of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ on Image may be NSFW.
Clik here to view. k^2 .

Formally, we can define an action of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ on a Image may be NSFW.
Clik here to view.-module Image may be NSFW.
Clik here to view. as an action of the functor of points of the former on the functor of points of the latter which is linear in the appropriate sense. Explicitly, it is a natural transformation with components

Image may be NSFW.
Clik here to view. $\displaystyle \eta_R : \mathbb{G}_a(R) \times V(R) \to V(R)$

such that each Image may be NSFW.
Clik here to view. $\eta_R$ defines an Image may be NSFW.
Clik here to view.-linear action of the group Image may be NSFW.
Clik here to view. $\mathbb{G}_a(R) \cong (R, +)$ on the Image may be NSFW.
Clik here to view.-module Image may be NSFW.
Clik here to view. $V(R) \cong R \otimes_k V$ in a way which is natural in Image may be NSFW.
Clik here to view.. So we can think of the parameter Image may be NSFW.
Clik here to view. in the one-parameter group above as running over all elements of all Image may be NSFW.
Clik here to view.-algebras Image may be NSFW.
Clik here to view..

Image may be NSFW.
Clik here to view. $\eta$ can equivalently, by currying, be thought of as a natural transformation of group-valued functors from Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ to the functor

Image may be NSFW.
Clik here to view. $\displaystyle \text{Aut}(V)(-) : \text{CAlg}(k) \ni R \mapsto \text{Aut}_R(V \otimes_k R)$

even though the latter is generally not representable by a scheme, again unless Image may be NSFW.
Clik here to view. is finitely generated projective; note that if Image may be NSFW.
Clik here to view. V = k^n then Image may be NSFW.
Clik here to view. $\text{Aut}(V)$ is an affine group scheme, namely the general linear group Image may be NSFW.
Clik here to view. GL_n .

The advantage of doing this is that we can appeal to the Yoneda lemma: because Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ is representable, such natural transformations correspond to elements of the group

Image may be NSFW.
Clik here to view. $\text{Aut}_{k[x]}(V \otimes_k k[x]) \subseteq \text{End}_{k[x]}(V \otimes_k k[x]) \cong \text{Hom}_k(V, V \otimes_k k[x])$

at which point we’ve finally been freed of the burden of having to consider arbitrary Image may be NSFW.
Clik here to view.. Writing down the conditions required for a map Image may be NSFW.
Clik here to view. $V \to V \otimes_k k[x]$ to correspond to an element of Image may be NSFW.
Clik here to view. $\text{Aut}_{k[x]}(V(k[x]))$ giving a homomorphism of group-valued functors Image may be NSFW.
Clik here to view. $\mathbb{G}_a \to \text{Aut}(V)$ , we are led to the following.

Definition-Theorem: A representation of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ over Image may be NSFW.
Clik here to view. is a comodule over the Hopf algebra of functions Image may be NSFW.
Clik here to view. $\mathcal{O}_{\mathbb{G}_a} \cong k[x]$ .

This is true more generally for representations of any affine group scheme.

Let’s get somewhat more explicit. A comodule over Image may be NSFW.
Clik here to view. k[x] is in particular an action map Image may be NSFW.
Clik here to view. $\alpha : V \to V \otimes_k k[x]$ . Isolating the coefficient of Image may be NSFW.
Clik here to view. x^n on the RHS, this map breaks up into homogeneous components

Image may be NSFW.
Clik here to view. $\displaystyle \alpha_n : V \to V \otimes_k x^n$

which each correspond to an element of Image may be NSFW.
Clik here to view. $\text{End}_k(V)$ , which we’ll call Image may be NSFW.
Clik here to view. M_n . The entire action map can therefore be thought of as a power series

Image may be NSFW.
Clik here to view. $\displaystyle M(x) = M_0 + M_1 x + M_2 x^2 + \dots$

and the condition that Image may be NSFW.
Clik here to view. $\alpha$ is part of a comodule structure turns out to be precisely the condition that 1) Image may be NSFW.
Clik here to view. M(0) = M_0 is the identity, 2) that we have

Image may be NSFW.
Clik here to view. $\displaystyle M(x + y) = M(x) M(y)$

as an identity of formal power series, and 3) that for any Image may be NSFW.
Clik here to view. $v \in V$ , Image may be NSFW.
Clik here to view. M_i v = 0 for all but finitely many Image may be NSFW.
Clik here to view.. Which of these is nonzero can be read off from the value of Image may be NSFW.
Clik here to view. $\alpha(v)$ , which takes the form

Image may be NSFW.
Clik here to view. $\displaystyle \alpha(v) = M(x)(v) = \sum_{i \ge 0} M_i v \otimes x^i$ .

Equating the coefficients of Image may be NSFW.
Clik here to view. x^i y^j and of Image may be NSFW.
Clik here to view. x^j y^i on both sides of the identity Image may be NSFW.
Clik here to view. M(x + y) = M(x) M(y) (the two coefficients are equal on the LHS, hence must be equal on the RHS) gives

Image may be NSFW.
Clik here to view. $\displaystyle {i + j \choose i} M_{i+j} = M_i M_j = M_j M_i$

from which it follows in particular that the Image may be NSFW.
Clik here to view. M_i commute. This is necessary and sufficient for us to have Image may be NSFW.
Clik here to view. M(x + y) = M(x) M(y) , so we can say the following over an arbitrary Image may be NSFW.
Clik here to view.:

Theorem: Over a ring Image may be NSFW.
Clik here to view., representations of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ can be identified with modules Image may be NSFW.
Clik here to view. over the divided power algebra

Image may be NSFW.
Clik here to view. $\displaystyle k[M_1, M_2, \dots] / \left( {i + j \choose i} M_{i + j} = M_i M_j \right)$

which are locally finite in the sense that for any Image may be NSFW.
Clik here to view. $v \in V$ we have Image may be NSFW.
Clik here to view. M_i v = 0 for all but finitely many Image may be NSFW.
Clik here to view.. Given the action of each Image may be NSFW.
Clik here to view. M_i , the corresponding action of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ is

Image may be NSFW.
Clik here to view. $\displaystyle \sum_i M_i x^i : V \mapsto V \otimes_k k[x]$ .

If Image may be NSFW.
Clik here to view. is torsion-free as an abelian group, for example if Image may be NSFW.
Clik here to view. $k = \mathbb{Z}$ , then the divided power algebra can be thought of as the subalgebra of Image may be NSFW.
Clik here to view. $(k \otimes \mathbb{Q})[x]$ spanned by Image may be NSFW.
Clik here to view. $\frac{x^i}{i!}$ , where the isomorphism sends Image may be NSFW.
Clik here to view. M_i to Image may be NSFW.
Clik here to view. $\frac{x^i}{i!}$ . This is because the key defining relation above can be rewritten Image may be NSFW.
Clik here to view. $(i + j)! M_{i + j} = (i! M_i) (j! M_j)$ if there is no torsion.

These representations should really be thought of as continuous representations of the profinite Hopf algebra given as the cofiltered limit over the algebras spanned by Image may be NSFW.
Clik here to view. $M_1, M_2, \dots M_n$ for each Image may be NSFW.
Clik here to view., with the quotient maps given by setting all Image may be NSFW.
Clik here to view. M_i past a certain point to zero. This profinite Hopf algebra is dual to the Hopf algebra Image may be NSFW.
Clik here to view. k[x] of functions on Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ , and we can more generally relate comodules over a coalgebra Image may be NSFW.
Clik here to view. to modules over a profinite algebra dual to it in this way under suitable hypotheses, for example if Image may be NSFW.
Clik here to view. is a field.

In characteristic zero

If Image may be NSFW.
Clik here to view. is a field of characteristic zero, or more generally a Image may be NSFW.
Clik here to view. $\mathbb{Q}$ -algebra, then the divided power algebra simplifies drastically, because we can now divide by all the factorials running around. Induction on the relation Image may be NSFW.
Clik here to view. ${i + j \choose i} M_{i + j} = M_i M_j$ readily gives

Image may be NSFW.
Clik here to view. $\displaystyle M_i = \frac{M_1^i}{i!}$

and we conclude the following.

Theorem: Over a Image may be NSFW.
Clik here to view. $\mathbb{Q}$ -algebra Image may be NSFW.
Clik here to view., representations of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ can be identified with endomorphisms Image may be NSFW.
Clik here to view. $M_1 : V \to V$ of a Image may be NSFW.
Clik here to view.-module which are locally nilpotent in the sense that, for any Image may be NSFW.
Clik here to view. $v \in V$ , we have Image may be NSFW.
Clik here to view. M_1^i v = 0 for all but finitely many Image may be NSFW.
Clik here to view.. Given such an endomorphism, the corresponding action of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ is

Image may be NSFW.
Clik here to view. $\displaystyle \exp (M_1 x) : V \to V \otimes_k k[x]$ .

The corresponding profinite Hopf algebra is the formal power series algebra Image may be NSFW.
Clik here to view. $k[[\partial]]$ in one variable, with comultiplication Image may be NSFW.
Clik here to view. $\Delta(\partial) = \partial \otimes 1 + 1 \otimes \partial$ . If Image may be NSFW.
Clik here to view. is a field of characteristic zero, this means we can at least classify finite-dimensional representations of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ using nilpotent Jordan blocks.

Example. The representation Image may be NSFW.
Clik here to view. $\left[ \begin{array}{cc} 1 & x \\ 0 & 1 \end{array} \right]$ we wrote down earlier corresponds to exponentiating the Jordan block Image may be NSFW.
Clik here to view. $\left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]$ .

Example. Any coalgebra has a “regular representation,” namely the comodule given by its own comultiplication. The regular representation of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ is the translation map

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{G}_a \ni x \mapsto (f(y) \mapsto f(y+x)) \in \text{Hom}_k(k[y], k[y] \otimes_k k[x])$ .

and the corresponding locally nilpotent endomorphism is the derivative Image may be NSFW.
Clik here to view. $f \mapsto \partial f$ (hence the use of Image may be NSFW.
Clik here to view. $\partial$ above). More generally, locally nilpotent derivations on a Image may be NSFW.
Clik here to view.-algebra Image may be NSFW.
Clik here to view. correspond to actions of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ on the affine Image may be NSFW.
Clik here to view.-scheme Image may be NSFW.
Clik here to view. $\text{Spec } R$ .

The regular representation has subrepresentations given by restricting attention to the polynomials of degree at most Image may be NSFW.
Clik here to view. for each Image may be NSFW.
Clik here to view.. These subrepresentations are finite-dimensional, and are classified by the Image may be NSFW.
Clik here to view. $(n+1) \times (n+1)$ nilpotent Jordan block, which follows from the fact that they have a one-dimensional invariant subspace given by the constant polynomials.

In positive characteristic

In positive characteristic the binomial coefficient Image may be NSFW.
Clik here to view. ${i + j \choose i}$ in the relation Image may be NSFW.
Clik here to view. ${i + j \choose i} M_{i + j} = M_i M_j$ is sometimes zero, so things get more complicated. For starters, the same induction as before shows that if Image may be NSFW.
Clik here to view. is a field of characteristic Image may be NSFW.
Clik here to view., or more generally an Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ -algebra, we have

Image may be NSFW.
Clik here to view. $\displaystyle M_i = \frac{M_1^i}{i!}, i < p$ .

However, when we try to analyze Image may be NSFW.
Clik here to view. M_p , we find that we can put no constraint on it in terms of smaller Image may be NSFW.
Clik here to view. M_i but instead that Image may be NSFW.
Clik here to view. M_1^p = 0 . In fact we have Image may be NSFW.
Clik here to view. M_i^p = 0 for all Image may be NSFW.
Clik here to view., because the identity

Image may be NSFW.
Clik here to view. $\displaystyle M(x + y) = M(x) M(y)$

gives by induction

Image may be NSFW.
Clik here to view. $\displaystyle M(px) = M(x)^p = \sum_{i \ge 0} M_i^p x^{ip} = 1$

(here we use the fact that we know the Image may be NSFW.
Clik here to view. M_i commute). So things are not determined just by Image may be NSFW.
Clik here to view. M_1 ; we at least also need to know Image may be NSFW.
Clik here to view. M_p . Note that Image may be NSFW.
Clik here to view. M_1^p = 0 guarantees that the exponential Image may be NSFW.
Clik here to view. $\exp (M_1 x)$ still makes sense in characteristic Image may be NSFW.
Clik here to view., because it’s no longer necessary to divide by factorials divisible by Image may be NSFW.
Clik here to view..

Continuing from here we find that

Image may be NSFW.
Clik here to view. $\displaystyle M_{i_0 + i_1 p} = \frac{M_1^{i_0}}{i_0!} \frac{M_p^{i_1}}{i_1!}, i_0, i_1 < p$

using the fact that Image may be NSFW.
Clik here to view. ${ip \choose p} M_{ip} = M_p^i$ and Image may be NSFW.
Clik here to view. ${ip \choose p}$ is not divisible by Image may be NSFW.
Clik here to view. for Image may be NSFW.
Clik here to view. i < p by Kummer’s theorem, as well as Lucas’s theorem which gives more precisely Image may be NSFW.
Clik here to view. ${ip \choose p} \cong i \bmod p$ . This gets us up to Image may be NSFW.
Clik here to view. p^2 - 1 and then when we try to analyze Image may be NSFW.
Clik here to view. $M_{p^2}$ we find that we cannot relate it to any of the smaller Image may be NSFW.
Clik here to view. M_i , because Image may be NSFW.
Clik here to view. ${p^2 \choose i}$ is always divisible by Image may be NSFW.
Clik here to view. (using either Kummer’s or Lucas’s theorems), and that Image may be NSFW.
Clik here to view. M_p^p = 0 , which we already knew.

The general situation is as follows. We can write Image may be NSFW.
Clik here to view. M_n as a product of Image may be NSFW.
Clik here to view. M_i, i < n precisely when we can find Image may be NSFW.
Clik here to view. $1 \le i \le n-1$ such that Image may be NSFW.
Clik here to view. ${n \choose i}$ is nonzero Image may be NSFW.
Clik here to view. $\bmod p$ . Kummer’s theorem implies that this is possible precisely when Image may be NSFW.
Clik here to view. is not a power of Image may be NSFW.
Clik here to view., from which it follows that each Image may be NSFW.
Clik here to view. M_n is a product of Image may be NSFW.
Clik here to view. $M_{p^i}$ . We can be more precise as follows. Write Image may be NSFW.
Clik here to view. in base Image may be NSFW.
Clik here to view. as

Image may be NSFW.
Clik here to view. $\displaystyle n = i_0 + i_1 p + \dots + i_k p^k.$

Then induction gives

Image may be NSFW.
Clik here to view. $\displaystyle {i_0 + i_1 p + \dots \choose i_0, i_1 p, \dots } M_n = M_{i_0} M_{i_1 p} \dots M_{i_k p^k}$

where the LHS is congruent to Image may be NSFW.
Clik here to view. $1 \bmod p$ by a mild generalization of Lucas’s theorem and the RHS can be rewritten as above, giving

Image may be NSFW.
Clik here to view. $\displaystyle M_n = \frac{M_1^{i_0}}{i_0!} \frac{M_p^{i_1}}{i_1!} \dots \frac{M_{p^k}^{i_k}}{i_k!}$ .

This is enough for the Image may be NSFW.
Clik here to view. M_n to satisfy every relation defining the divided power algebra, and so we conclude the following.

Lemma: Over an Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ -algebra Image may be NSFW.
Clik here to view., the divided power algebra can be rewritten

Image may be NSFW.
Clik here to view. $\displaystyle k[M_1, M_p, \dots]/(M_1^p, M_p^p, \dots)$ .

Corollary: Over an Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ -algebra Image may be NSFW.
Clik here to view., representations of the additive group scheme Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ correspond to modules Image may be NSFW.
Clik here to view. over the algebra Image may be NSFW.
Clik here to view. $k[M_1, M_p, \dots]/(M_1^p, M_p^p, \dots)$ which are locally finite in the sense that for all Image may be NSFW.
Clik here to view. $v \in V$ , we have Image may be NSFW.
Clik here to view. $M_{p^i} v = 0$ for all but finitely many Image may be NSFW.
Clik here to view.. Given the action of each Image may be NSFW.
Clik here to view. $M_{p^i}$ , the corresponding action of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ is

Image may be NSFW.
Clik here to view. $\displaystyle \exp (M_1 x + M_p x^p + \dots) : V \mapsto V \otimes_k k[x]$ .

As before, we can equivalently talk about continuous modules over a suitable profinite Hopf algebra.

Example. We can classify all nontrivial Image may be NSFW.
Clik here to view.-dimensional representations over a field Image may be NSFW.
Clik here to view. of characteristic Image may be NSFW.
Clik here to view. as follows. There is some smallest Image may be NSFW.
Clik here to view. such that the action of Image may be NSFW.
Clik here to view. $M_{p^i}$ is nonzero. In a suitable basis, Image may be NSFW.
Clik here to view. $M_{p^i}$ acts by a nilpotent Jordan block Image may be NSFW.
Clik here to view. $J = \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]$ , and all the other Image may be NSFW.
Clik here to view. $M_{p^j}$ commute with it. A calculation shows that this implies that each Image may be NSFW.
Clik here to view. $M_{p^j}$ must also have the form Image may be NSFW.
Clik here to view. $\left[ \begin{array}{cc} 0 & c_j \\ 0 & 0 \end{array} \right]$ for some scalars Image may be NSFW.
Clik here to view. c_j , and hence our representation has action map

Image may be NSFW.
Clik here to view. $\displaystyle \exp \left( J (x^{p^i} + c_{i+1} x^{p^{i+1}} + \dots) \right) = \left[ \begin{array}{cc} 1 & x^{p^i} + c_{i+1} x^{p^{i+1}} + \dots \\ 0 & 1 \end{array} \right]$ .

Contrast to the case of characteristic zero, where there is only one isomorphism class of nontrivial Image may be NSFW.
Clik here to view.-dimensional representation (given by Image may be NSFW.
Clik here to view. M_1 = J ).

Example. Consider again the regular representation given by the translation action of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ on itself:

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{G}_a \ni x \mapsto (f(y) \mapsto f(y+x)) \in \text{Hom}_k(k[y], k[y] \otimes_k k[x])$ .

We find as before that Image may be NSFW.
Clik here to view. $M_1 = \partial$ is the derivative. Now, in characteristic Image may be NSFW.
Clik here to view. the derivative of a polynomial is well-known to have the curious property that Image may be NSFW.
Clik here to view. $\partial^p = 0$ ; among other things this means that translation is no longer given by just exponentiating the derivative. However, since over Image may be NSFW.
Clik here to view. $\mathbb{Z}$ we have

Image may be NSFW.
Clik here to view. $\displaystyle \partial^p y^n = n(n-1) \dots (n-(p-1)) y^{n-p}$

we have, over Image may be NSFW.
Clik here to view. $\mathbb{Z}$ and hence over any Image may be NSFW.
Clik here to view., a well-defined differential operator

Image may be NSFW.
Clik here to view. $\displaystyle \frac{\partial^p}{p!} y^n = {n \choose p} y^{n-p}$

(for Image may be NSFW.
Clik here to view. $n \ge p$ ; otherwise zero) and this differential operator must be Image may be NSFW.
Clik here to view. M_p ; similarly for higher powers of Image may be NSFW.
Clik here to view. we have

Image may be NSFW.
Clik here to view. $\displaystyle M_{p^k} y^n = \frac{\partial^{p^k}}{(p^k)!} y^n = {n \choose p^k} y^{n-p^k}$

(for Image may be NSFW.
Clik here to view. $n \ge p^k$ ; otherwise zero, as above).

Endomorphisms

Actually we should have expected an answer like this all along, or at least we should have known that we would also need to write down representations involving powers of Frobenius in addition to the obvious representations of the form Image may be NSFW.
Clik here to view. $\exp (M_1 x)$ . This is because in characteristic Image may be NSFW.
Clik here to view., the Frobenius map gives a nontrivial endomorphism Image may be NSFW.
Clik here to view. $\mathbb{G}_a \to \mathbb{G}_a$ , and the pullback of any representation along an endomorphism is another representation. Pulling back along the Frobenius map sends Image may be NSFW.
Clik here to view. $\exp (Mx)$ to Image may be NSFW.
Clik here to view. $\exp (Mx^p)$ . More generally, any sum of powers of the Frobenius map gives a nontrivial endomorphism of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ as well.

The fact that this sort of thing doesn’t happen in characteristic zero means that Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ can’t have any non-obvious endomorphisms like the Frobenius map there. In fact we can say the following.

Proposition: Over a ring Image may be NSFW.
Clik here to view., endomorphisms Image may be NSFW.
Clik here to view. $\mathbb{G}_a \to \mathbb{G}_a$ are in natural bijection with additive polynomials, namely polynomials Image may be NSFW.
Clik here to view. $f(x) \in k[x]$ such that Image may be NSFW.
Clik here to view. f(0) = 0 and Image may be NSFW.
Clik here to view. f(x + y) = f(x) + f(y) .

Proof. This is mostly a matter of unwinding definitions. By the Yoneda lemma, maps Image may be NSFW.
Clik here to view. $\mathbb{G}_a \to \mathbb{G}_a$ (on underlying affine schemes, maps Image may be NSFW.
Clik here to view. $\mathbb{A}^1 \to \mathbb{A}^1$ ) correspond to elements of Image may be NSFW.
Clik here to view. $\mathbb{G}_a(k[x]) \cong k[x]$ , hence to polynomials Image may be NSFW.
Clik here to view. f(x) , and the condition that such a map corresponds to a group homomorphism is precisely the condition that Image may be NSFW.
Clik here to view. f(0) = 0 and Image may be NSFW.
Clik here to view. f(x + y) = f(x) + f(y) . Image may be NSFW.
Clik here to view. $\Box$

Now let’s try to classify all such polynomials. Writing Image may be NSFW.
Clik here to view. $f(x) = \sum f_i x^i$ , equating the coefficient of Image may be NSFW.
Clik here to view. x^i y^j on both sides as before gives

Image may be NSFW.
Clik here to view. $\displaystyle {i + j \choose i} f_{i + j} = \begin{cases} 0 & \mbox{if } i, j \ge 1 \\ f_i & \mbox{if } j = 0 \\ f_j & \mbox{if } i = 0 \end{cases}$

If Image may be NSFW.
Clik here to view. is torsion-free, and in particular if Image may be NSFW.
Clik here to view. is a Image may be NSFW.
Clik here to view. $\mathbb{Q}$ -algebra, then this implies that Image may be NSFW.
Clik here to view. f_n = 0 for Image may be NSFW.
Clik here to view. $n \ge 2$ (by setting Image may be NSFW.
Clik here to view. n = i + j where Image may be NSFW.
Clik here to view. $i, j \ge 1$ ), and since Image may be NSFW.
Clik here to view. f(0) = f_0 = 0 the only possible nonzero coefficient is Image may be NSFW.
Clik here to view. f_1 . So Image may be NSFW.
Clik here to view. is scalar multiplication by the scalar Image may be NSFW.
Clik here to view. f_1 .

On the other hand, if Image may be NSFW.
Clik here to view. is an Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ -algebra, then it can again happen as above that Image may be NSFW.
Clik here to view. f_n doesn’t vanish if Image may be NSFW.
Clik here to view. ${n \choose i}$ is always divisible by Image may be NSFW.
Clik here to view., which as before happens iff Image may be NSFW.
Clik here to view. is a power of Image may be NSFW.
Clik here to view.. In this case Image may be NSFW.
Clik here to view. is a linear combination

Image may be NSFW.
Clik here to view. $\displaystyle f(x) = \sum_i f_{p^i} x^{p^i}$

of powers of the Frobenius map, which is clearly an endomorphism. In summary, we conclude the following.

Proposition: If Image may be NSFW.
Clik here to view. is torsion-free, the endomorphism ring of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ is Image may be NSFW.
Clik here to view. acting by scalar multiplication. If Image may be NSFW.
Clik here to view. is an Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ -algebra, the endomorphism ring of Image may be NSFW.
Clik here to view. $\mathbb{G}_a$ is Image may be NSFW.
Clik here to view. k[F] , with Image may be NSFW.
Clik here to view. acting by Frobenius.

The endomorphisms generated by Frobenius can be used to write down examples of affine group schemes which only exist in positive characteristic. For example, the kernel of Frobenius is the affine group scheme Image may be NSFW.
Clik here to view. $\alpha_p \cong \text{Spec } k[x]/x^p$ with functor of points

Image may be NSFW.
Clik here to view. $\displaystyle \alpha_p : \text{CAlg}(k) \ni R \mapsto \{ r \in R : r^p = 0 \} \in \text{Ab}$ .

The representation theory of the additive group scheme

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