1 Section 1 algebraic geometry

● 1.1 affine varieties

Review: Zariski topology, Hilbert’s Nullstellensatz, irreducibility, noetherian, reducibility (of algebras), maximal ideal spectrum.

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Equivalently, an $𝐹$-structure on $𝑘\left[𝑋\right]$ is an $𝐹$-algebra $𝐹\left[𝑋\right]$ such that $𝐹\left[𝑋\right]{\otimes }_{𝐹}𝑘\cong 𝑘\left[𝑥\right]$ as $𝑘$-algebras.

The same closed subset may admit two different $𝐹$-structures. For example, $ℂ\left[𝑥,𝑦\right]\text{/}\left({𝑥}^2+{𝑦}^2+1\right)$ admit two different $ℝ$-structures (with different $ℝ$-rational points): $ℝ\left[𝑥,𝑦\right]\text{/}\left({𝑥}^2+{𝑦}^2-1\right)$ and $ℝ\left[𝑥,𝑦\right]\text{/}\left({𝑥}^2+{𝑦}^2+1\right)$.

Review: regular functions, sheaves, stalks, affine varieties and morphisms, prevarieties.

Affine $𝐹$-varieties are affine varieties defined over $𝐹$. The category of affine varieties is anti-equivalent to the category of affine algebras (finitely generated reduced $𝑘$-algebras).

● 1.2 products and varieties

This product construction is the product in the categorical sense, and $𝑘\left[𝑋\right]\otimes 𝑘\left[𝑌\right]$ is the coproduct in the category of affine algebras. Products of prevarieties also exists in the categorical sense, defined chart by chart.

If $𝑋$ is a prevariety, there is a diagonal ${\mathrm{\Delta }}_{𝑋}=\left\{\left(𝑥,𝑥\right):𝑥\in 𝑋\right\}$ and a diagonal map ${𝑖}_{𝑋}:𝑋\to 𝑋\times 𝑋$ taking $𝑥$ to $\left(𝑥,𝑥\right)$. ${\mathrm{\Delta }}_{𝑋}$ has the induced topology from $𝑋\times 𝑋$. ${𝑖}_{𝑋}:𝑋\to {\mathrm{\Delta }}_{𝑋}$ is a homeomorphism for all prevariety $𝑋$.

Below is a useful condition for a prevariety to be a variety:

We can similarly define $𝐹$-varieties. The $𝐹$-rational points of $𝑋$ are $𝐹$-morphisms ${𝔸}^0\to 𝑋$.

● 1.3 projective varieties

We will denote everything projective by a ${}^{\ast }$.

An ideal is called homogeneous if it is generated by homogeneous polynomials. The closed sets of ${\mathbb{P}}^{𝑛}$ are the zero sets of homogeneous ideals.

Usual results for affine spaces carry over:

Products of ${\mathbb{P}}^{𝑛}$ can be defined as a closed subset of some projective space. Consider $𝜑:{\mathbb{P}}^{𝑚}\times {\mathbb{P}}^{𝑛}\to {\mathbb{P}}^{𝑚𝑛+𝑚+𝑛}$, ${\left({𝑥}_{𝑖}\right)}^{\ast },{\left({𝑦}_{𝑗}\right)}^{\ast }\mapsto {\left({𝑥}_{𝑖}{𝑦}_{𝑗}\right)}^{\ast }$. The image of this map defines an isomorphism ${\mathbb{P}}^{𝑚}\times {\mathbb{P}}^{𝑛}\to {𝑉}^{𝑚,𝑛}\subseteq {\mathbb{P}}^{𝑚𝑛+𝑚+𝑛}$, where ${𝑉}^{𝑚,𝑛}$ is a closed subset of ${\mathbb{P}}^{𝑚𝑛+𝑚+𝑛}$, defined by the zero set of ${𝑧}_{𝑖𝑗}{𝑧}_{𝑘𝑙}-{𝑧}_{𝑖𝑘}{𝑧}_{𝑗𝑙}$.

A projective variety is a closed subset of some ${\mathbb{P}}^{𝑛}$, and a quasiprojective variety is an open subset of some projective variety.

● 1.4 dimension

If $𝑋$ is any irreducible affine variety, then $𝑘\left[𝑋\right]$ is an integral domain. Let $𝑘\left(𝑋\right)$ be its fraction field. For any principal open set $𝑈=𝐷\left(𝑓\right)$, $𝑘\left[𝑈\right]={𝑘\left[𝑋\right]}_{𝑓}$, so $𝑘\left(𝑈\right)=𝑘\left(𝑋\right)$.

For example, if $𝑋$ is irreducible affine, the dimension of $𝑋$ is the maximum number of algebraically independent elements in $𝑘\left[𝑋\right]$.

If $𝑌$ is a proper irreducible closed subvariety of $𝑋$, then $dim𝑌<dim𝑋$. If $𝑋$ and $𝑌$ are irreducible varieties, then $dim𝑋\times 𝑌=dim𝑋+dim𝑌$. $dim{𝔸}^{𝑛}=dim{\mathbb{P}}^{𝑛}=𝑛$, and $0$-dimensional varieties are all finite. For an irreducible polynomial $𝑓$, ${𝑍}_{{𝔸}^{𝑛}}\left(𝑓\right)$ always has codimension $1$.

● 1.5 a technical result

This implies that the image of any constructible set is constructible: a set is called constructible if it is the union of locally closed sets (the intersection of an open set and a closed set).

2 Section 2 linear algebraic groups, first properties

● 2.1 algebraic groups

If $𝐺$ is linear, let $𝐴=𝑘\left[𝐺\right]$. Then $𝜇:𝐺\times 𝐺\to 𝐺$ defines an algebra homomorphism, the comultiplication $\mathrm{\Delta }:𝐴\to 𝐴\otimes 𝐴$, and $𝑖:𝐺\to 𝐺$ defines the antipode $𝜄:𝐴\to 𝐴$. Roughly speaking, $\mathrm{\Delta }$ tells you how to split $𝑥$ and $𝑦$ apart when evaluating $𝑓\left(𝑥𝑦\right)$; it turns $𝑓$ into $\sum {𝑓}_{1𝑖}\otimes {𝑓}_{2𝑖}$, so that $𝑓\left(𝑥𝑦\right)=\sum {𝑓}_{1𝑖}\left(𝑥\right)\cdot {𝑓}_{2𝑖}\left(𝑦\right)$. For example, if group operation is addition and $𝑓\left(𝑥\right)={𝑥}^2$, then $𝑓\left(𝑥+𝑦\right)={𝑥}^2+2𝑥𝑦+{𝑦}^2$, so $\mathrm{\Delta }\left(𝑓\right)={𝑥}^2\otimes 1+2𝑥\otimes 𝑥+1\otimes {𝑥}^2$.

The identity $𝑒\in 𝐺$ is a map ${𝔸}^0\to 𝐺$, so it corresponds to a map $𝐴\to 𝑘$, taking $𝑓$ to $𝑓\left(𝑒\right)$. $𝑘$ can be embedded back to $𝐴$, taking $𝑐\in 𝑘$ to the constant function $𝑓\left(𝑥\right)=𝑐$. This way we get a map, the identity $𝜀:𝐴\to 𝐴$, which takes a function $𝑓$ and returns a constant function $𝑓\left(𝑒\right)$. This is the algebraic equivalent of the map $𝐺\to \left\{𝑒\right\}$.

The diagonal map $𝑥\mapsto \left(𝑥,𝑥\right)$ corresponds to a map $𝐴\otimes 𝐴\to 𝐴$, taking $𝑓\left(𝑥\right)\otimes 𝑔\left(𝑦\right)$ to $\left(𝑓\cdot 𝑔\right)\left(𝑥\right)$. This is the multiplication map $𝑚:𝐴\otimes 𝐴\to 𝐴$, $𝑓\otimes 𝑔\mapsto 𝑓\cdot 𝑔$. Using $\mathrm{\Delta }$, $𝜄$, $𝜀$ and $𝑚$, we can rewrite the group axioms as follows:

The rules can be understood as follows. Like we said, $\mathrm{\Delta }$ does the following thing: give me a function $𝑓$ and two variables $𝑥$ and $𝑦$, I tell you how to evaluate $𝑓$ on $𝑥\cdot 𝑦$. So associativity is essentially saying that, give me $𝑓$ and three variables $𝑥$, $𝑦$ and $𝑧$, I have two equivalent ways to evaluate $𝑓\left(𝑥\cdot 𝑦\cdot 𝑧\right)$: first we get to know how to evaluate $𝑓\left(𝑥\cdot 𝑦\right)$, and then we get to know how to evaluate $𝑓\left(\left(𝑥\cdot 𝑦\right)\cdot 𝑧\right)$, or vice versa. Inverse is essentially saying that, given a function $𝑓:𝐺\to 𝑘$, first split $𝑥$ and $𝑦$ apart: $𝑓\left(𝑥𝑦\right)=\sum {𝑓}_{1𝑖}\left(𝑥\right){𝑓}_{2𝑖}\left(𝑦\right)$ and turn $𝑦$ into ${𝑦}^{-1}$, and multiply the two part together: $\sum {𝑓}_{1𝑖}\left(𝑥\right){𝑓}_{2𝑖}\left({𝑥}^{-1}\right)$. This is essentially $𝑓\left(𝑥{𝑥}^{-1}\right)$, and it asserts that this is equal to $𝑓\left(𝑒\right)=𝑓\left(𝜀\left(𝑥\right)\right)$. Finally, identity is just saying that $𝑓\left(𝑒\cdot 𝑥\right)$ is equal to $𝑓\left(𝑥\right)$.

For two examples, $\left(𝑘,+\right)$ is a linear algebraic group, called the additive group ${\text{G}}_{𝑎}$, and $\left({𝑘}^{\times },\cdot \right)$ is also a linear algebraic group, called the multiplicative group ${\text{G}}_{𝑚}$ or $GL\left(1\right)$. They are examples of connected groups.

Other linear algebraic groups include: $GL\left(𝑛,ℂ\right)$, $SL\left(𝑛,ℂ\right)$, ${\text{D}}_{𝑛}$ (diagonal), ${\text{T}}_{𝑛}$ (upper-triangular), ${\text{U}}_{𝑛}$ (unipotent upper-triangular), ${\text{O}}_{𝑛}$, ${\text{SO}}_{𝑛}$ and ${\text{Sp}}_{2𝑛}$. ${\text{O}}_{𝑛}$ is not connected when $char𝑘\ne 2$.

For an example of a non-linear algebraic group: the elliptic curve is a projective variety, with an abelian group structure.

● 2.2 some basic results

We first state some basic results we want to prove for algebraic groups, and then proceed to their proofs.

Therefore, irreducibility of algebraic groups is equivalent to connectedness. For this reason, we say that algebraic groups are connected rather than irreducible.

Recall that in the case of topological / Lie groups, any open neighbourhood of $𝑒$ generates the whole group. However in Zariski topology any open set is dense (at least in that irreducible component), so things are easier here:

We would like that operations on closed subgroups yield closed ones.

We now set out to prove them.

A non-example for this lemma would be $\left(\mathbb{Z},+\right)\subset \left(ℂ,+\right)$.

● 2.3 $𝐺$-spaces

For $𝑥\in 𝑋$, its isotropy group is ${𝐺}_{𝑥}=\left\{𝑔\in 𝐺:𝑔𝑥=𝑥\right\}$ and its orbit is $𝐺\cdot 𝑥=\left\{𝑔\cdot 𝑥:𝑔\in 𝐺\right\}$.

In the case of Lie group actions on smooth manifolds, the orbits can be bad in general; for example, the action of $ℝ$ on ${𝕋}^2$ by $\left(𝑥+𝑡,𝑥+\sqrt{2}\right)$ has orbits that are neither closed nor open. In the algebraic case, things are much simpler: it turns out that a closed orbit always exists, an essential statement in the theory of algebraic groups.

The proof also implies that an orbit is locally closed, so it can be made into a variety and hence a homogeneous $𝐺$-space.

For linear algebraic groups, an especially important kind of $𝐺$-actions is rational representations:

Intuitively, a representation is rational if it is defined using polynomials. So for example the representation of ${\text{G}}_{𝑎}$ by $𝑡\mapsto \left(\begin{array}{cc}1& 𝑡\\ & 1\end{array}\right)$ is rational, but $𝑡\mapsto \left(\begin{array}{cc}cos𝑡& sin𝑡\\ -sin𝑡& cos𝑡\end{array}\right)$ isn’t.

Every finite group has a faithful representation in some $GL\left(𝑉\right)$, as follows. Right translation permutes the group elements, and therefore can be viewed as actions on $𝑉=𝑘𝐺$; this action is a faithful representation. For linear algebraic groups, the case is similar; every linear algebraic group has a faithful representation in some finite-dimensional $GL\left(𝑉\right)$, via the action $𝑘\left[𝐺\right]$ by right translation. The gap here is that $𝑘\left[𝐺\right]$ is generally infinite dimensional. Therefore, we need the following lemma.

If $𝑋$ and $𝐺$ are affine, the group action $𝑎:𝐺\times 𝑋\to 𝑋$ defines a pullback ${𝑎}^{\ast }:𝑘\left[𝑋\right]\to 𝑘\left[𝐺\right]\otimes 𝑘\left[𝑋\right]$. Given $𝑓\in 𝑘\left[𝑋\right]$ and $𝑔\in 𝐺$, ${𝑎}^{\ast }𝑓$ tells us how to compute $𝑓$ on $𝑔𝑥$. This leads to a function $𝑠\left(𝑔\right)$ for every $𝑔\in 𝐺$: $𝑠\left(𝑔\right)$ is a function $𝑘\left[𝑋\right]\to 𝑘\left[𝑋\right]$ such that $𝑠\left(𝑔\right)𝑓\left(𝑥\right)=𝑓\left({𝑔}^{-1}𝑥\right)$. $𝑠\left(𝑔\right)$ reflects the action of $𝐺$ on $𝑘\left[𝑋\right]$.

Now, take $𝜌$ to be the right translation: $𝜌\left(𝑔\right)𝑓\left(𝑥\right)=𝑓\left(𝑥𝑔\right)$, then $𝜌$ defines a faithful rational representation of $𝐺$ in $GL\left(𝑘\left[𝐺\right]\right)$. Similarly, the left multiplication $𝜆\left(𝑔\right)𝑓\left(𝑥\right)=𝑓\left(𝑔𝑥\right)$ is a faithful rational representation of $𝐺$. We want to extract a finite dimensional faithful representation out of it. Recall that $𝑘\left[𝐺\right]$ is generated (as an algebra) by finitely many elements; these elements span a finite dimensional subspace of $𝑘\left[𝐺\right]$, so by the previous lemma $𝑘\left[𝐺\right]$ has a finite dimensional subspace $𝑉$ that is stable under $𝜌\left(𝐺\right)$ and generate (as an algebra) the whole $𝑘\left[𝐺\right]$. This is the representation we want:

● 2.4 Jordan decomposition

Recall that an endomorphism $𝑎$ of a vector space $𝑉$ is semi-simple if $𝑎$ is the direct sum of one-dimensional maps, i.e. $𝑎$ is diagonalizable, and nilpotent if ${𝑎}^{𝑠}=0$ for some $𝑠\in {\mathbb{Z}}_{+}$; $𝑎$ is called unipotent if $𝑎-1$ is nilpotent. Under a field of positive characteristic $𝑝$, $𝑎$ is unipotent if and only if ${𝑎}^{{𝑝}^{𝑠}}=1$ for some $𝑠\in {\mathbb{Z}}_{+}$. Also recall that a set of pairwise commuting matrices can be simultaneously upper-triangularized, and if they are also all semisimple they can be simultaneously diagonalized. The product, direct sum and tensor product of two semisimple (nilpotent, unipotent) matrices is semisimple (nilpotent, unipotent), and if $𝑎\in End\left(𝑉\right)$ and $𝑏\in End\left(𝑊\right)$ are semisimple (nilpotent) then so is $𝑎\otimes 1+1\otimes 𝑏$.

Similar to the additive Jordan decomposition $𝑎={𝑎}_{𝑠}+{𝑎}_{𝑛}$, we have a multiplicative version:

The importance of multiplicative Jordan decopmosition is that it can be carried over to linear algebraic groups. Recall that every linear algebraic group can be linearized as a closed subgroup of some ${GL}_{𝑛}$. We have a multiplicative Jordan decomposition there. Remarkably, this decomposition is intrinsic, i.e. not depending on the particular embedding:

The set of unipotent elements of $𝐺$, ${𝐺}_{𝑢}$, is a closed subset of $𝐺$. The same need not hold for semisimple elements. Also, if $𝑥\in 𝐺$ is an $𝐹$-point, then neither ${𝑥}_{𝑠}$ nor ${𝑥}_{𝑢}$ need to be an $𝐹$-point.

Hence, unipotent linear algebraic groups are nilpotent, and hence solvable. Another consequence is that whenever $𝐺\to GL\left(𝑉\right)$ is a rational representation of a unipotent linear algebraic group, there is a non-zero vector in $𝑉$ fixed by all of $𝐺$.

3 Section 3 commutative algebraic groups

● 3.1 structure of commutative algebraic groups

We recall that every element $𝑔\in 𝐺$ can be written as $𝑔={𝑔}_{𝑠}{𝑔}_{𝑢}$, and that ${𝐺}_{𝑢}$ is always a closed subset of $𝐺$. We now see that $𝐺$ splits cleanly into ${𝐺}_{𝑢}$ and ${𝐺}_{𝑠}$ in the case of commutative algebraic groups:

In the latter case, if $char𝑘=𝑝>0$, then the elements of $𝐺$ have order dividing $𝑝$.

● 3.2 diagonalizable groups and tori

The set of rational characters is denoted by ${𝑋}^{\ast }\left(𝐺\right)$. It has a natural abelian group structure, which we write additively. Characters are regular functions on $𝐺$ so lie in $𝑘\left[𝐺\right]$, and they are linearly independent in $𝑘\left[𝐺\right]$.

A homomorphism of algebraic groups $𝜆:{\text{G}}_{𝑚}\to 𝐺$ is called a cocharacter. The set of cocharacters is dentoed by ${𝑋}_{\ast }\left(𝐺\right)$, and it also have an abelian group structure, also written additively.

For example, if $𝐺={\text{D}}_{𝑛}$, then ${𝜒}_{𝑖}$ which takes $𝑥$ to the $𝑖$-th element off the diagonal of $𝑥$ is a character. Monomials in ${𝜒}_{𝑖}$ serves as a basis for $𝑘\left[{\text{D}}_{𝑛}\right]$, so they are all characters. As a result, ${𝑋}^{\ast }\left(𝐺\right)\cong {\mathbb{Z}}^{𝑛}$. A homomorphism ${\text{G}}_{𝑚}\to {\text{D}}_{𝑛}$ is always of the form $𝑥\mapsto \text{diag}\left({𝑥}^{{𝑎}_1},\cdots ,{𝑥}^{{𝑎}_{𝑛}}\right)$, so ${𝑋}_{\ast }\left({\text{D}}_{𝑛}\right)\cong {\mathbb{Z}}^{𝑛}$.

In this case, $𝑘\left[𝐺\right]$ is isomorphic to the group algebra of ${𝑋}^{\ast }\left(𝐺\right)$, and if $char𝑘=𝑝>0$, then ${𝑋}^{\ast }\left(𝐺\right)$ doesn’t have $𝑝$-torsion.

To recall, if $𝑀$ is a finitely generated abelian group, then its group algebra $𝑘\left[𝑀\right]$ is the $𝑘$-algebra with basis $𝑒\left(𝑚\right)$ ($𝑚\in 𝑀$), with multiplication $𝑒\left(𝑚\right)\cdot 𝑒\left(𝑛\right)=𝑒\left(𝑚+𝑛\right)$. The $𝑒(\cdot )$ is used solely to prevent confusing operations in $𝑀$ and in $𝑘\left[𝑀\right]$. We have $𝑘\left[{𝑀}_1\oplus {𝑀}_2\right]\cong 𝑘\left[{𝑀}_1\right]{\otimes }_{𝑘}𝑘\left[{𝑀}_2\right]$. Define homomorphisms $\mathrm{\Delta }:𝑒\left(𝑚\right)\mapsto 𝑒\left(𝑚\right)\otimes 𝑒\left(𝑚\right)$, $𝜄:𝑒\left(𝑚\right)\mapsto 𝑒\left(-𝑚\right)$ and $𝑒:𝑒\left(𝑚\right)\mapsto 1$. Assume $𝑀$ has no $𝑝$-torsion if $𝑝=char𝑘>0$. For example, if $𝑀=\mathbb{Z}\text{/}𝑑\mathbb{Z}$ where $𝑝\nmid 𝑑$, then $𝑘\left[𝑀\right]\cong 𝑘\left[𝑡\right]\text{/}\left({𝑡}^{𝑑}-1\right)$.

Since $\mathrm{𝒢︀}\left({\mathbb{Z}}^{𝑛}\right)={\text{D}}_{𝑛}$ and $\mathrm{𝒢︀}\left(\mathbb{Z}\text{/}𝑑\mathbb{Z}\right)$ is finite, we proved that

This proposition essentially says that homomorphisms between diagonalizable groups are rigid, or cannot be deformed; in other words, $Hom\left(𝐺,𝐻\right)$ behaves like a discrete set, think ${\mathbb{Z}}^{𝑛}$. As an application, recall the centralizer and the normalizer of a subgroup. They are both closed subgroups of $𝐺$, and ${𝑍}_{𝐺}\left(𝐻\right)⊴{𝑁}_{𝐺}\left(𝐻\right)$.

Now suppose $𝑇$ is a torus. For $𝜒\in {𝑋}^{\ast }\left(𝑇\right)$ and $𝜆\in {𝑋}_{\ast }\left(𝑇\right)$ the map $𝑎\mapsto 𝜒\left(𝜆\left(𝑎\right)\right)$ is a character ${\text{G}}_{𝑚}\to {\text{G}}_{𝑚}$, and such a map can only some monomial $𝑎\mapsto {𝑎}^{𝑑}$. We set $⟨𝜒,𝜆⟩=𝑑$. This defines a perfect pairing between ${𝑋}^{\ast }\left(𝑇\right)$ and ${𝑋}_{\ast }\left(𝑇\right)$; in particular ${𝑋}_{\ast }\left(𝑇\right)$ is free abelian. The map $𝑎\otimes 𝜆\mapsto 𝜆\left(𝑎\right)$ defines a canonical isomorphism between abelian groups ${𝑘}^{\times }\otimes {𝑋}_{\ast }\left(𝐺\right)\cong 𝑇$.

If $𝑉$ is a $𝑇$-space, then we have a (locally finite) representation $𝑠$ of $𝑇$ in $𝑘\left[𝑉\right]$. For a regular character $𝜒$, let ${𝑘\left[𝑉\right]}_{𝜒}$ denote its eigenspace:

The subspaces ${𝑘\left[𝑉\right]}_{𝜒}$ form an ${𝑋}^{\ast }\left(𝐺\right)$-grading of $𝑘\left[𝑉\right]$:

If $𝜑$ is a morphism ${\text{G}}_{𝑚}\to 𝑍$ we write ${lim}_{𝑎\to 0}𝜑\left(𝑎\right)=𝑧$ if $𝜑$ extends to a morphism $\widetilde{𝜑}:{𝔸}^1\to 𝑍$ with $\widetilde{𝜑}\left(0\right)=𝑧$. If ${𝜑}^{\prime }\left(𝑎\right)=𝜑\left({𝑎}^{-1}\right)$ then we write ${lim}_{𝑎\to \infty }𝜑\left(𝑎\right)={lim}_{𝑎\to 0}{𝜑}^{\prime }\left(𝑎\right)$. If $𝑉$ is a $𝑇$-space and $𝜆$ is a cocharacter, let $𝑉\left(𝜆\right)$ be the set of $𝑣\in 𝑉$ such that ${lim}_{𝑎\to 0}𝜆\left(𝑎\right)\cdot 𝑣$ exists. Then $𝑉\left(-𝜆\right)$ is the set of $𝑣$ such that ${lim}_{𝑎\to \infty }𝜆\left(𝑎\right)\cdot 𝑣$ exists.

Intuitively, $𝜆\left(𝑎\right)\cdot 𝑉$ defines a one-parameter family of morphisms $𝑉\to 𝑉$, and $𝑉\left(𝜆\right)$ is those points which “converges” as $𝑎\to 0$. For example, if $𝜆:𝑎\mapsto \left(\begin{array}{cc}{𝑎}^2& \\ & {𝑎}^{-1}\end{array}\right)$, then $𝜆\left(𝑎\right)\cdot \left(\begin{array}{c}𝑥\\ 𝑦\end{array}\right)=\left(\begin{array}{c}{𝑎}^2𝑥\\ {𝑎}^{-1}𝑦\end{array}\right)$, so $𝑉\left(𝜆\right)$ is the $𝑥$-axis and $𝑉\left(-𝜆\right)$ is the $𝑦$-axis. Intuitively $𝑉\left(𝜆\right)$ are the components without negative exponents, and $𝑉\left(-𝜆\right)$ are those without positive exponents.

● 3.3 additive functions

Assume that $char𝑘=𝑝>0$ and $𝐹$ is perfect, i.e. ${𝐹}^{𝑝}=𝐹$. Define a ring $𝑅$ as a “twisted” polynomial ring: the underlyinngn abelian group is $𝐹\left[𝑡\right]$, while the multiplication is

where $𝜑:𝑥\mapsto {𝑥}^{𝑝}$ is the Frobenius isomorphism.

Left and right ideals in $𝑅$ are all principal and $𝑅$ is left and right noetherian. Therefore, every finitely generated left- or right-$𝑅$-module is a direct sum of cyclic modules, either free or torsion. $\mathrm{𝒜︀}\left(𝐹\right)$ can then be seen as a left $𝑅$-module, by

If $𝑝=0$ then $𝑅=𝐹$ and $\mathrm{𝒜︀}\left(𝐹\right)$ is an $𝑅$-module.

As an example, the additive $𝐹$-functions on ${\text{G}}_{𝑎}^{𝑛}$ are the additive polynomials in $𝐹\left[{𝑡}_1,\cdots ,{𝑡}_{𝑛}\right]$. The only such polynomials are linear combinations of monomials of the form ${𝑡}_{𝑖}^{{𝑝}^{𝑗}}$, which form a free $𝑅$-module with basis ${𝑡}_{𝑖}$.

● 3.4 elementary unipotent groups

With all the notions introduced, Proposition 3.2 implies that every connected linear algebraic group must be toral or elementary unipotent. Therefore,

4 Section 4 derivations, differentials, Lie algebras

● 4.1 derivations and tangent spaces

If $𝜑:𝐴\to 𝐵$ is a homomorphism of $𝑅$-algebras and $𝑁$ is a $𝐵$-module, pullback along $𝜑$ gives ${𝜑}_0:{\text{Der}}_{𝑅}\left(𝐵,𝑁\right)\to {\text{Der}}_{𝑅}\left(𝐴,𝑁\right)$ with kernel ${\text{Der}}_{𝐴}\left(𝐵,𝑁\right)$.

If $𝜑:𝑋\to 𝑌$ is a morphism of affine algebraic varieties, then ${𝜑}^{\ast }:𝑘\left[𝑌\right]\to 𝑘\left[𝑋\right]$ pulls back to a map ${𝜑}_0^{\ast }:{𝑇}_{𝑥}𝑋\to {𝑇}_{𝜑𝑥}𝑌$, denoted $d{𝜑}_{𝑥}$ and called the differential of $𝜑$ at $𝑥\in 𝑋$.

The third description tells us that if $𝜑:𝑋\to 𝑌$ is an isomorphism of $𝑋$ onto an affine open subvariety of $𝑌$, then $d{𝜑}_{𝑥}$ is an isomorphism.

● 4.2 differentials, separability

Suppose $𝑅$ is a commutative ring and $𝐴$ a commutative $𝑅$-algebra. Let $𝑚:𝐴{\otimes }_{𝑅}𝐴\to 𝐴$ be the algebra multiplication map, and $𝐼=ker𝑚$. $𝐼$ is generated by elements of the form $𝑎\otimes 1-1\otimes 𝑎$, and $𝐴\otimes 𝐴\text{/}𝐼\cong 𝐴$.

If $𝐴$ is the ring of functions over a space $𝑋$, then $𝐼$ represent all functions of the form $𝑓\left(𝑥\right)-𝑓\left(𝑦\right)$, and $𝐼\text{/}{𝐼}^2$ decodes the first-order behaviour of such functions. Denote by $d𝑎$ or $d_{𝐴\text{/}𝑅}𝑎$ the image of $𝑎\otimes 1-1\otimes 𝑎$ in ${\mathrm{\Omega }}_{𝐴\text{/}𝑅}$, then $d$ is a derivation in ${\text{Der}}_{𝑅}\left(𝐴,{\mathrm{\Omega }}_{𝐴\text{/}𝑅}\right)$.

${\mathrm{\Omega }}_{𝐴\text{/}𝑅}$ is the universal module of $𝑅$-derivations of $𝐴$, in the sense that it is the unique (up to isomorphism) $𝐴$-module with a derivation $d$ such that for every $𝐴$-module $𝑀$ the map $\mathrm{\Phi }:{Hom}_{𝐴}\left({\mathrm{\Omega }}_{𝐴\text{/}𝑅},𝑀\right)\to {\text{Der}}_{𝑅}\left(𝐴,𝑀\right)$, $𝜑\mapsto 𝜑\circ d$ is an isomorphism of $𝐴$-modules.

If $𝜑:𝐴\to 𝐵$ is an $𝑅$-algebra homomorphism, there is a unique homomorphism of $𝐴$-modules ${𝜑}^0:{\mathrm{\Omega }}_{𝐴\text{/}𝑅}\to {\mathrm{\Omega }}_{𝐵\text{/}𝑅}$ with ${𝜑}^0\circ d_{𝐴\text{/}𝑅}=d_{𝐵\text{/}𝑅}\circ 𝜑$. In this sense, the above mentioned map $\mathrm{\Phi }$ is natural.