In this section we present the second proof of Theorem 94 due to T.A. Springer. The proof is actually quite long as one needs to develop several setups. The key idea is to deduce purely inseparable descent through derivations and connections. This may be viewed as an extension of the Galois descent which handles only the case of separable extensions.
5.3.1 Derivations
Definition 102. Let R be a commutative ring and A a commutative R-algebra. Let M be an A-module. An R-derivation of A on M is an R-linear map D : A→ M such that
D(ab) = aD(b) + bD(a), ∀ a, b ∈ A. (5.5)
Let DerR(A, M ) denote the set of all R-derivations of A on M . It is equipped with a natural A-module structure by defining (bD)(x) := bD(x) for b∈ A and D ∈ DerR(A, M ).
Note that D(r· 1) = 0 for any r ∈ R.
If ϕ : A → B is an R-algebra homomorphism and N is a B-module, then one can regard N as an A-module. Thus, one obtains a map of A-modules
ϕ0 : DerR(B.N )→ DerR(A.N ), D7→ D ◦ ϕ. (5.6)
The map ϕ0 induces a short exact sequence of A-modules
0→ DerA(B.N )→ DerR(B.N )→ DerR(A.N ). (5.7)
5.3.2 Tangent spaces
We show how to use derivations to reformulate the tangent space of an algebraic variety at a point. Let X ⊂ An be a closed subvariety of the affine n-space over K, where K is an algebraically closed field, and let x = (a1, . . . , an) ∈ X be a K-point. We write the coordinate ring of X by K[X] = K[T ]/I, where T = (T1, . . . , Tn) and I = (f1, . . . , fs)⊂ K[T ] is the ideal of definition. Write the tangent space TxAn = Kn with the standard basis ∂/∂Ti for i = 1, . . . , n. Then the tangent space TxX ⊂ TxAn consists of vectors (v1, . . . , vn)∈ Kn satisfying the linear equations
∑n i=1
vi∂fj
∂Ti(x) = 0, j = 1, . . . , s. (5.8)
Let Mx ⊂ K[X] be the maximal ideal corresponding to x, and K(x) := K[X]/Mx = K[T ]/(T1− a1, . . . , Tn− an) the residue field at x, viewed as a K[Xmodule or a K[T ]-module. We can identify the tangent space TxAn with DerK(K[T ], K(x)) as
TxAn≃ DerK(K[T ], K(x)), v = (v1, . . . , vn)7→ Dv =
∑n i=1
viDi, (5.9)
where Di is the unique derivation such that Di(Tj) = δij for j = 1, . . . , n. One easily sees that the derivation Dv : K[T ] → K(x) factors through K[X] → K(x) if and only v satisfies the condition (5.8). Thus, the identification (5.9) induces a natural isomorphism
TxX ≃ DerK(K[X], K(x)). (5.10)
Note that Dv is uniquely determined its restriction on Mx, because K[X] = K ⊕ Mx. Thus, by (5.10), we obtain a natural isomorphism
TxX ≃ HomK(Mx/Mx2, K(x)) = (Mx/Mx2)∗. (5.11)
We call the K(x)-vector space Mx/Mx2 the cotangent space of X at x. The isomorphisms (5.10) and (5.11) provide two alternative definitions for the tangent space TxX of X at a point. These reformulations show that the tangent space TxX is an intrinsic property, in the sense that does not depend on the choice of an embedding of X into the affine space An. Furthermore, we can use these reformulations to define the tangent spaces of an arbitrary scheme.
Let f : X → Y be a morphism of algebraic varieties and let x ∈ X be a point. Then the differentiation of f gives a K-linear map of vector spaces
(df )x : TxX → TyY, y = f (x). (5.12)
If g : Y → Z is another morphism of algebraic varieties and put z = g(y). Then the differentiation of the composition g◦ f has the property
(d g◦f)x = (dg)y◦ (df)x (the chain rule). (5.13)
5.3.3 Kähler differentials
Definition 103. Let R and A be as in Definition 102. Let IA/R denote the kernel of the multiplication m : A⊗RA→ A. The Kähler differential Ω1A/R of A over R is defined by
Ω1A/R := IA/R/IA/R2 . (5.14)
This is an A-module by the isomorphism A⊗RA/IA/R ≃ A.
It is easy to see that the ideal IA/R is generated by elements a⊗1−1⊗a for all a ∈ A.
For a∈ A, write
da := [a⊗ 1 − 1 ⊗ a], a ∈ A (5.15)
the class in Ω1A/R. Then the map d : A→ Ω1A/R is an R-derivation of A on Ω1A/R. If M is an A-module and φ : Ω1A/R → M is a morphism A-module, then the composition
d◦ φ : A → Ω1A/R → M (5.16)
is an R-derivation of A on M . Conversely, any R-derivation D ∈ DerR(A, M ) arises in this way. That is, there is a unique A-homomorphism φD : Ω1A/R → M such that D = d◦ φD. This gives a canonical isomorphism of A-modules
HomA(Ω1A/R, M )−→ Der∼ R(A, M ) (5.17)
which is functorial for all A-modules M .
Consider a covariant functor from the category (A-mod) of A-modules to (A-mod) defined by
DerR(A,·) : A-mod → A-mod, M 7→ DerR(A, M ). (5.18)
Then by (5.17) the Kähler differential Ω1A/R represents this functor, and d is the universal family.
The isomorphism (5.17) linearizes the derivations. Thus, the computation of R-derivations DerR(A, M ) is reduced to calculating Ω1A/R, which can be done explicitly by linear relations as follows.
Suppose that A is essentially of finite type over R, say A = R[x1, . . . , xn]S = (R[T ]/I)S, where T = (T1, . . . , Tn), I = (f1, . . . , fs) and S is a multiplicatively closed subset which does not contain zero divisors. Then
Ω1A/R = A⟨ dx1, . . . , dxn ∑n
i=1
yjidxi = 0, j = 1, . . . , s⟩, (5.19)
where yji := (∂fj/∂Ti)(x1, . . . , xn)∈ A.
Let X = Spec A→ Spec K be an affine algebraic variety of dimension d. Then X is a non-singular algebraic variety if and only if the Kähler differential Ω1A/K is a locally free A-module of rank d. This follows from the Jacobian criterion for simple points and the computation of Ω1A/K (see (5.19)). More generally, if f : X → S is a morphism of locally Noetherian schemes locally of finite type, then Ω1X/S is a locally free OX-module of rank equal to the relative dimension dim(X/S) and f is flat if and only if f is smooth (see [31, I, Proposition 3.24] and see [31, I. Remark 3.22] for the definition of smooth morphisms).
Let ϕ : A → B be an R-algebra homomorphism. Then there is a unique morphism dϕ : Ω1A/R → Ω1B/R of A-modules making the following diagram commutative
A −−−→ϕ B
ydA ydB Ω1A/R −−−→ Ωdϕ 1B/R.
(5.20)
Note that dB◦ ϕ is an R-derivation of A on Ω1B/R. So the commutative diagram (5.20) follows from the universal property of (Ω1A/R, dA). One easily sees that dϕ(da) = dϕ(a) for all a∈ A. The map dϕ induces an exact sequence of B-modules
B⊗AΩ1A/R −−−−→ Ω1B⊗dϕ 1B/R −−−→ Ω1B/A −−−→ 0. (5.21)
Let N be a B-module and φ : Ω1B/R → N a B-linear homomorphism. The pull-back (dϕ)∗(φ) = φ◦ dϕ : Ω1A/R → N is a morphism of A-modules. We then have the commutative diagram
HomB(Ω1B/R, N ) −−−→ Der∼ R(B, N )
y(dϕ)∗ yϕ∗ HomA(Ω1A/R, N ) −−−→ Der∼ A(A, N ),
(5.22)
where the horizontal ones are canonical isomorphisms.
5.3.4 Separable field extensions
Definition 104 (cf. [20, Chap. VI, Sect. 2], [37, Sect. 4.2, p. 63-64]). (1) A field extension
E/k is said to be separably generated if there is a subset {tα} ⊂ E such that (i) the extension k({tα})/k is purely transcendental, and
(ii) the extension E/k({tα}) is algebraic and separable.
The subset {tα} is called a separating transcendental base.
(2) We call E/k a separable extension if either
• char k = 0, or
• char k = p > 0 and for any k-linearly independent elements x1, . . . , xn in K, their pth powers xp1, . . . , xpn are also k-linearly independent.
Proposition 105. (1) If k is perfect, then any field extension K/k is separable.
(2) A field extension K/k of finite type is separably generated if and only if it is separable.
Proof. (1) This follows easily from the definition. Indeed, suppose that {xpi} is k-linearly dependent, say ∑
iaixpi = 0. Then one has ∑
ibixi = 0 and hence {xi} is k-linearly dependence. (2) See [37, Proposition 4.2.10 and Exercise 4.2.15 (5)]).
Note that Proposition105 (2) fails if K/k is not of finite type. Indeed, let k =Fp and K =∪n≥1Fp(t1/pn). Then K/k is not separably generated but K/k is separable.
Definition 106. Let X and Y be irreducible algebraic varieties over a field k, and
f : X → Y a morphism of finite type.
(1) We say that f is dominant if the image f (X) is Zariski dense. In this case, the function field k(Y ) can be viewed as a subfield of k(X) through the pull-back of functions.
(2) We say that f is separable if f is dominant and the field extension k(X)/k(Y ) is separable.
Proposition 107. Let f : X → Y be a separable morphism of irreducible normal alge-braic varieties over k such that f is a homeomorphism. Then f is an isomorphism.
Proof. The field extension k(X)/k(Y ) is separable of finite type. Since f is a home-omorphism, f is finite and dim X = dim Y = trdegkk(X) = trdegkk(Y ). It follows that k(X) = k(Y ) as k(X)/k(Y ) is both a separable and inseparable finite extension. Thus, f is a birational finite morphism of normal varieties and it is an isomorphism by the Zariski main theorem.
We give an example that f is birational and homeomorphic but not isomorphism.
Consider the normalization morphism f : eX → X, where X = Spec Q[x, y]/(y2 − x3).
Then f satisfies these properties.
Proposition 108. Let E/k be a field extension of finite type.
(1) dimEΩ1E/k ≥ trdegkE.
(2) The equality in (1) if and only if E/k separably generated.
Proof. See [37, Theorem 4.2.9].
Let E and E′ be field extensions of k of finite type with E′ ⊂ E. Applying the constructions (5.7) and (5.21), we get short exact sequences:
0 −−−→ DerE′(E, E) −−−→ Derk(E, E) −−−→ Derβ k(E′, E), (5.23)
E⊗E′ Ω1E′/k
−−−→ Ωα 1E/k −−−→ Ω1E/E′ −−−→ 0. (5.24)
Clearly, these two are dual with each other. Therefore, β is surjective and α is injective.
Corollary 109. Assume that k is perfect. The following statements are equivalent:
(a) E/E′ is separately generated.
(b) α is injective.
(c) β is surjective.
Proof. As is already shown, statements (b) and (c) are equivalent. By (5.24), α is injec-tive if and only if dimE′Ω1E′/k+ dimEΩ1E/E′ = dimEΩ1E/k. Since k is perfect, by Prop.105 and 108 dimBΩ1B/k = trdegkB for B = E or E′. On the other hand, we always have trdegkE = trdegE′E + trdegkE′. It follows that α is injective if dimEΩ1E/E′ = trdegE′E.
The latter is equivalent to that E/E′ is separably generated by Proposition 108.
5.3.5 Connections
Let A be a commutative k-algebra, where k is any field. LetD = DA:= Derk(A, A) denote the A-module of all k-derivations of A on itself. It admits a natural structure of Lie algebra over k. If D1, D2 ∈ D, then the Lie bracket is defined by [D1, D2] := D1D2− D2D1. The bracket is not A-bilinear. Let hor(A) :={a ∈ A|D(a) = 0, ∀ D ∈ D}. Elements of hor(A) are called horizontal elements. It is easy to check that hor(A) is a k-subalgebra of A, and that the bracket is hor(A)-bilinear.
If char k = p > 0, then Dp := D◦ D ◦ · · · ◦ D (p times) is again a k-derivation of A.
This gives a p-Lie (or restricted Lie) algebra structure onD. More precisely, the operator D7→ Dp satisfies the following three conditions (cf. [37, Sect. 4.4]:
(a) (aD)p = apDp for a∈ k and D ∈ D.
(b) ad(Dp) = (adD)p for all D∈ D.
(c) We have (Jacobson’s formula)
(D + D′)p = Dp+ D′p+
p−1
∑
i=1
i−1si(D, D′), (5.25)
where si(D, D′) is the coefficient of ai in ad(aD + D′)p−1(D′) for 1 ≤ i ≤ p − 1, i.e. ∑p−1
i=1si(D, D′)ai = ad(aD + D′)p−1(D′).
We now introduce connections. We consider a field extension E/k of finite type, and let D = DE := Derk(E, E). The subfield hor(E) of horizontal elements consists of all algebraic and separable elements of E over k. Recall that E/k is said to be primary if k is the algebraic separable closure of k in E. Then E/k is primary if and only if k = hor(E).
We shall give two definitions of connections and then show that they represent the same notion through a natural transformation.
Definition 110. Let E/k and DE be as above, and let A be an E-vector space.
(1) A connection on A for the extension E/k is a map
∇ : A → Ω1E/k⊗E A (5.26)
such that ∇(ax) = da ⊗ x + a∇(x) for all x ∈ A and a ∈ E.
(2) A connection of DE on A is an E-linear map
c :DE → Endk(A) (5.27)
such that c(D)(ax) = D(a)x + a(c(D)(x)) for all D ∈ DE, a∈ E and x ∈ A.
Suppose ∇ : A → Ω1E/k ⊗E A is a connection. For any D ∈ DE, let φD : Ω1E/k → E the corresponding E-linear map so that D = φD· d. The contraction ∇D at D is defined to be the composition
∇D : A −−−→ Ω∇ 1E/k⊗ A −−−→ A.φD⊗1 (5.28)
Lemma 111.
(1) One has ∇D(ax) = D(a)x + a∇D(x) and ∇aD = a∇D for all a∈ E and x ∈ A.
(2) Conversely, let c : DE → Endk(A) be a connection then there exists a unique ∇ such that c(D) =∇D for all D ∈ DE.
Proof. (1) By definition, ∇D(ax) = φD(da⊗ x + a∇(x)) = D(a)x + a∇D(x). The equality ∇aD = a∇D follows from φaD = aφD and ∇D = φD· ∇.
(2) Choose an E-basis{dfi} for Ω1E/k. Let{φi} be the dual basis for HomE(Ω1E/k, E).
Recall that Ω1E/k is an finite dimensional E-vector space. Put Di := φi · d. Then the map c is uniquely determined by c(Di) ∈ Endk(A). Put xi := c(Di)(x) ∈ A. Define
∇(x) =∑
idfi⊗ xi. The condition of E/k being of fintie type is used here. It is easy to see∇Di(x) = xi = c(Di)(x) for all i. Thus, ∇D(x) = c(D)(x) for all D ∈ DE.
Given a connection ∇, one extends morphisms
∇i : ΩiE/k⊗ A → Ωi+1E/k⊗ A, ΩiE/k :=∧iΩ1E/k, ∇0 =∇ (i ≥ 1) (5.29)
by
∇i(ω⊗ x) = dω ⊗ x + (−1)iω∧ ∇i−1(x). (5.30)
Definition 112.
(1) The curvature of ∇ is K(∇) := ∇1· ∇ : A → Ω2E/k⊗ A. K(∇) is a 2-form with values in Endk(A).
(2) A connection ∇ is said to be integrable or flat if K(∇) = 0.
Lemma 113. Let c : DE → Endk(A) be the connection of DE on A associated to a connection ∇ on A. Then ∇ is integrable if and only if c is a Lie algebra homomorphism.
Proof. This follows from a straightforward computation which we omit.
5.3.6 Inseparable descent
Suppose that A = A0⊗kE is endowed with a k-structure defined by a k-subspace A0. We define a flat connection cA by cA(D)(∑
ai⊗ xi) =∑
D(ai)⊗ xi for ai ∈ E and xi ∈ A0.
cAis the unique flat connection such that cA(D)(A0) = 0 for all D∈ DE, and it is called the canonical flat connection on A (of course, with respect to the k-structure A0). We now formulate inseparable descent in terms of connections. This is the key ingredient used in Springer’s proof of Theorem 94
Proposition 114 (Inseparable descent). Let E/k be a field extension of finite type and DE the k-derivations on E to itself. Assume that hor(E) := {a ∈ E|D(a) = 0, ∀ D ∈ DE} = k.
(1) Let A0 be a k-vector space and A := A0 ⊗k E. One has {x ∈ A|cA(D)(x) = 0 ∀ D ∈ DE} = A0.
(2) If W ⊂ A be an E-subspace, then W is defined over k if and only if cA(D)(W )⊂ W for all D ∈ DE.
(3) Let B0 be another k-vector space and B := B0⊗kE. Let f : A→ B be an E-linear map. Then f is defined over k if and only if the diagram
A −−−→ Bf
cA(D)
y ycB(D) A −−−→ Bf
(5.31)
commutes for all D ∈ DE.
Proof. See [37, Proposition 11.1.4 and Corollary 11.1.5].
5.3.7 Proof of Theorem 94
We may assume that char k = p > 0. It suffices to show that the map Homk(T,Gm) → HomE(T,Gm) is surjective for any finite purely inseparable extension E/k. That is, any character χ defined over E is defined over k.
We may also assume that Ep ⊂ k. Indeed, suppose that Epn ⊂ k for some n. Then we have a filtration En = kEpn ⊂ kEpn−1 ⊂ · · · ⊂ E1 = kEp ⊂ E0 = E and Eip ⊂ Ei+1.
By induction, it suffices to show the case Ep ⊂ k.
Let A0 = k[T ] and A = A0⊗kE = E[T ], and cAthe flat connection ofDE on A. Denote by ∆ : A→ A ⊗E A the co-multiplication; this is an E-algebra homomorphism. Since T is defined over k, one has ∆ = ∆0⊗ E with the co-multiplication ∆0 : A0 → A0 ⊗kA0. Thus, by Proposition 114 one has a commutative diagram
A −−−→ A ⊗∆ E A
ycA(D) ycA⊗A(D) A −−−→ A ⊗∆ E A.
(5.32)
As χ∈ X(T ), one has ∆(χ) = χ ⊗ χ. Put f := χ−1· cA(D)(χ). One computes
∆(f ) = ∆(χ−1)· ∆(cA(D)(χ))
= χ−1⊗ χ−1· cA⊗A(D)(χ⊗ χ)
= χ−1⊗ χ−1· [1 ⊗ χ · (cA(D)(χ)⊗ 1) + χ ⊗ 1 · (1 ⊗ cA(D)(χ))]
= f ⊗ 1 + 1 ⊗ f.
(5.33)
Write f =∑
ψ∈X(T )cψψ with characters cψ in ¯k[T ]. It follows from
∆(f ) = f ⊗ 1 + 1 ⊗ f = ∑
cψ(ψ⊗ 1 + 1 ⊗ ψ) =∑
cψ(ψ⊗ ψ)
that cψ = 0 if ψ ̸= 1. For ψ = 1, it follows from c1(1⊗ 1 + 1 ⊗ 1) = c1(1⊗ 1) that c1 = 0. Thus f = 0 and cA(D)χ = 0 for all D ∈ DE. This proves that χ ∈ A0, by Proposition114.