Definition 4.2.1. Let M and M1,· · · , Mr be R-module schemes over a base scheme S and let T be an S-scheme and r a positive natural number. Then we have a natural isomorphism M1T ×T · · ·×T MrT ∼= (M1×S· · ·×SMr)T.
(i) The two universal multilinear morphisms defining M1 ⊗R · · ·⊗RMr and M1T ⊗R· · ·⊗RMrT give rise to the following diagram
M1T ×T · · ·×T MrT
∼= ��
τ
��
(M1×S· · ·×SMr)T τT
��M1T ⊗R· · ·⊗RMrT (M1⊗R· · ·⊗RMr)T.
Now, the universal property of the tensor product M1T ⊗R · · · ⊗R MrT
completes the diagram by a unique morphism
τT /S :M1T ⊗R· · ·⊗RMrT →(M1⊗R· · ·⊗RMr)T
which we call the base change homomorphism of tensor product.
(ii) The two universal symmetric multilinear morphisms defining SRnM and SRn(MT) give rise to the following diagram
MTn ∼= ��
σ
��
(Mn)T σT
��SnR(MT) (SnRM)T.
Now, the universal property of the symmetric powerSRn(MT) completes the diagram by a unique morphism
σT /S :SRn(MT)→(SRnM)T
which we call the base change homomorphism of symmetric power.
4.2. BASE CHANGE AND WEIL RESTRICTION 37 (ii) The two universal alternating multilinear morphisms defining �
R
nM and
�
R
n(MT) give rise to the following diagram
MTn ∼= ��
�
(Mn)T λT
���
R
n(MT) (�
R
nM)T. Now, the universal property of the exterior power �
R
n(MT) completes the diagram by a unique morphism
λT /S :�
R
n(MT)→(�
R nM)T
which we call the base change homomorphism of exterior power.
Remark 4.2.2. The base change homomorphisms in the last definition need not be isomorphisms. However, as we will show (cf. Propositions 4.2.3 and 4.2.6), if S = SpecE and T = SpecL, where L/E is either a separable or a finite field extension, then the three base change homomorphisms are isomorphisms.
Proposition 4.2.3. Let E be a field and L/E a separable field extension. Then the three base change homomorphisms
M1,L⊗R· · ·⊗RMr,L→(M1⊗R· · ·⊗RMr)L,
�
R
r(ML)→(�
R rM)L
and
SR
r(ML)→(S
R rM)L
are isomorphisms.
Proof. We prove the statement just for the tensor product, and drop the similar proof of the exterior and symmetric powers. For simplicity, denote byT(ML) and T(M) the tensor productsM1,L⊗R· · ·⊗RMr,L andM1⊗R· · ·⊗RMr respectively.
By Theorem 4.1.4 we know that
T(ML)∼=Mult�R(M1,L×· · ·×Mr,L,Gm)∗ and
T(M)L ∼=�Mult�R(M1×· · ·×Mr,Gm)∗�
L.
By definition, we have
Mult�R(M1×· · ·×Mr,Gm)∗ = lim
←−H
H∗,
where H runs through all finite subgroup schemes of G := Mult�R(M1 ×· · ·× Mr,Gm). Consequently, we have
�Mult�R(M1×· · ·×Mr,Gm)∗�
L∼= (lim
←−H
H∗)L ∼= lim
←−H
(HL∗)
since the transition homomorphisms are epimorphisms and L/E is flat. Let H be a finite subgroup scheme of G. ThenHL is a finite subgroup scheme of
GL =Mult�R(M1×· · ·×Mr,Gm)L∼=Mult�R(M1,L×· · ·×Mr,L,Gm).
Therefore, there exists a natural homomorphism
π :Mult�R(M1,L×· · ·×Mr,L,Gm)∗ →Mult�R(M1×· · ·×Mr,Gm)∗L
which corresponds to the base change homomorphism b : T(ML) → T(M)L. Again, since the transition homomorphisms are epimorphisms, this homomorph-ism ( “projection to sublimit”) is an epimorphhomomorph-ism.
Let us assume at first that L is a separable closure of E and denote by Γ the absolute Galois group of E. We would like to show that π is an isomorphism.
It is sufficient to prove that the system of finite subgroups of GL, which are of the form HL for a finite subgroup H of G, is cofinal in the system of all finite subgroups ofGL. Let ¯H be a finite subgroup ofGL. So, elements of ¯H are closed points of GL, in other words, ¯H is a finite subgroup of the group GL(L) =G(L) of L-rational points of G. It follows that Γ·H¯ (the Galois conjugate of ¯H) is again a finite subset ofG(L) (note thatΓacts canonically onGLandG(L)). The subgroup of G(L) generated by Γ·H¯ is therefore finite, because G is Abelian.
This subgroup, denoted by HΓ, is a finite subgroup scheme of GL, which is in-variant under the action of Γ. By Galois descent, there exists a finite subgroup H of G such that HL = HΓ. As ¯H ⊆ HΓ, we have that ¯H ⊆ HL, proving the claim.
Now, assume that L is a separable extension of E and thus subextension of a separable closure Es, ofE. The base change homomorphisms yield the following commutative triangle
T(MEs)
bEs/E
���
��
��
��
��
bEs/L �
������������
T(ML)Es bEs ��T(M)Es.
4.2. BASE CHANGE AND WEIL RESTRICTION 39 By the above discussion, bEs/E is an isomorphism and bEs/L is an epimorphism.
It follows that bEs/L is also a monomorphism and thus an isomorphism. This implies that bEs is an isomorphism. This homomorphism is the extension to Es of the base change homomorphism b:T(ML)→T(M)L. As this extension is an isomorphism, we conclude thatb is an isomorphism as well.
Recall that ifE is a field andLis a finite extension ofE, then the Weil restriction ResL/E is the right adjoint of the base change functor from the category of affine schemes over E to the category of affine schemes over L, i.e., for every affine E-scheme X and every affine L-scheme Y we have a bijection of sets:
MorL(XL, Y)∼= MorE(X,ResL/EY).
In fact, the Weil restriction is the restriction, to the full subcategory of affine schemes, of the pushforward functor from the category of fppf sheaves overSpecL to the category of fppf sheaves overSpecE. Explicitly, this functor sends an fppf sheafF overSpecL to the sheaf
T �→ResL/EF(T) :=F(TL).
Recall also that the Weil restriction preserves the group objects, i.e., ResL/EH is an affine group scheme over E if H is an affine group scheme over L and the above bijection restricts to a group isomorphism
HomL(GL, H)∼= HomE(G,ResL/EH)
for every affine group scheme G overE. It also follows from the adjunction that if H is an affine R-module scheme over L, then ResL/EH is an affine R-module scheme over E and the above isomorphism induces an R-module isomorphisms, when restricted to the subgroup ofR-linear homomorphisms. Finally, recall that the Weil restriction commutes with base change, that is, if T is an E-scheme, then there exists a canonical isomorphism
(ResL/EY)T ∼= ResL×ET /T(Y ×E T).
Lemma 4.2.4. Let E be a field and L/E a finite field extension. Let M be an affine R-module scheme over E and N an fppf sheaf of R-modules over SpecL.
Then there exists a canonical sheaf isomorphism
ResL/EHomL(ML, N)∼= HomE(M,ResL/EN)
which is the “sheafified” version of the Weil restriction, i.e., the global sections of this isomorphism deliver the usual Weil restriction.
Proof. Let T be a scheme over E. We have the following isomorphisms HomE(M,ResL/EN)(T) = HomT(MT,(ResL/EN)T)∼= HomT(MT,ResTL/T(N ×E T)∼= HomT(MT,ResTL/T(N ×LTL))∼=
HomTL((MT)TL, N×LTL))∼= HomTL((ML)TL, N ×LTL)) = HomL(ML, N)(TL)∼= ResL/E(HomL(ML, N))(T),
where the first isomorphism (not equality) follows from the fact that the Weil restriction commutes with base change. The second isomorphisms is induced by the canonical isomorphism N ×E T ∼= N ×LTL. The third isomorphism follows from the adjunction property of the Weil restriction. The fourth isomorphism follows from the canonical isomorphism (MT)TL ∼= (ML)TL. Finally, the last isomorphism is given by the “definition” of the Weil restriction.
Proposition 4.2.5. Let E be a field and L/E a finite field extension. Let M1, M2, . . . , Mr and M be affine R-module schemes overE and N an fppf sheaf of Abelian groups over L.
a) The bijection
MorL(M1,L×· · ·×Mr,L, N)∼= MorE(M1×· · ·×Mr,ResL/EN) restricts to an isomorphism
Mult�RL(M1,L×· · ·×Mr,L, N)∼=Mult�RE(M1×· · ·×Mr,ResL/EN).
b) The isomorphismMult�RL(MLr, N)∼=Mult�RE(M,ResL/EN)of part a) restricts to an isomorphism
Alt�RL(MLr, N)∼=Alt�RE(M,ResL/EN).
c) The isomorphismMult�RL(MLr, N)∼=Mult�RE(M,ResL/EN)of part a) restricts to an isomorphism
Sym�RL(MLr, N)∼=Sym�RE(M,ResL/EN).
Proof. a) We will proceed by induction on r. If r = 1, then the statement follows from the adjunction of the Weil restriction or more generally of the pushforward, discussed before Lemma 4.2.4. So, assume that r > 1 and that we know the result for smaller integers. We have
Mult�RL(M1,L×· · ·×Mr,L, N)∼=
4.2. BASE CHANGE AND WEIL RESTRICTION 41 Mult�RL(M1,L×· · ·×Mr−1,L,HomL(Mr,L, N))ind’n∼=
Mult�RE(M1×· · ·×Mr−1,ResL/EHomL(Mr,L, N))4.2.4∼= Mult�RE(M1×· · ·×Mr−1,HomE(Mr,ResL/EN))∼=
Mult�RE(M1×· · ·×Mr,ResL/EN).
b) By the adjunction property, we know that there exists a unique R-linear homomorphism σN : (ResL/EN)L →N with the following universal prop-erty: for every E-scheme X the map
MorE(X,ResL/EN)→MorL(XL, N)
induced by base change to Land composing with σN is an isomorphism. It follows that if ϕ :Mr →ResL/EN is alternating, the morphism σN ◦ϕL is alternating too and therefore, the R-linear isomorphism of part a) restricts to an R-linear monomorphism
Alt�RE(Mr,ResL/EN)�→Alt�RL(MLr, N).
We have to show that this is surjective too. Take an alternating morphism ϕ : MLr → N. Because of part a), we know that there exists a pseudo-R-multilinear morphism ψ :Mr →ResL/EN such that ϕ=σN ◦ψL. We want to show that ψ is alternating. For any 1≤i < j ≤r, let
∆ri,j :Mr−1 →Mr, (m1, . . . , mr−1)�→(m1, . . . , mj−1, mi, mj, . . . , mr) denote the generalized diagonal embedding equating the ith and jth com-ponents. By functoriality of the Weil restriction, the following diagram commutes:
Mult�RE(Mr,ResL/EN) ∼= ��
∆∗i,j
��
Mult�RL(MLr, N)
∆∗i,j
��
Mult�RE(Mr−1,ResL/EN) ∼= ��Mult�RL(MLr−1, N).
Since ϕ is mapped to zero under the homomorphism Mult�RL(MLr, N)→Mult�RL(MLr−1, N)
(because it is alternating), the morphism ψ lies in the kernel of the homo-morphism
Mult�RE(Mr,ResL/EN)→Mult�RE(Mr−1,ResL/EN), and this holds for every pair i < j. Hence ψ is alternating.
c) Similar arguments prove the desired isomorphism.
Proposition 4.2.6. Let E be a field and L/E a finite field extension. Let M1, M2, . . . , Mr and M be finite R-module schemes over E. Then the three base change homomorphisms
M1,L⊗R· · ·⊗RMr,L→(M1⊗R· · ·⊗RMr)L,
�
R
r(ML)→(�
R rM)L and
SR
r(ML)→(S
R rM)L
are isomorphisms.
Proof. We prove the statement for the tensor product and drop the proofs for the exterior and symmetric power, as they can be similarly proved.
LetN be an affine group scheme over L. By functoriality of the Weil restriction, we have the following commutative diagram:
HomE(M1⊗R· · ·⊗RMr,ResL/EN)
∼=
��
∼= ��HomL((M1⊗R· · ·⊗RMr)L, N)
��
Mult�RE(M1×R· · ·×RMr,ResL/EN) ∼= ��Mult�RL(M1,L×R· · ·×RMr,L, N) where the vertical homomorphisms are induced by the universal R-multilinear morphisms. The left vertical morphism is an isomorphism by the definition of M1 ⊗R· · ·⊗RMr. The horizontal morphisms are isomorphism by the Weil re-striction. Thus the right vertical homomorphism is an isomorphism as well. The base change morphism induces a commutative triangle
HomL((M1⊗R· · ·⊗RMr)L, N) ��
��
HomL(M1,L⊗R· · ·⊗RMr,L, N)
�����������������������
Mult�RL(M1,L×R· · ·×RMr,L, N)
where the vertical and oblique morphisms are isomorphisms. In fact the vertical arrow is an isomorphism because of the last diagram and the oblique arrow is an isomorphism by the definition of M1,L ⊗R · · · ⊗R Mr,L. Consequently, the homomorphism
HomL((M1 ⊗R· · ·⊗RMr)L, N)→HomL(M1,L⊗R· · ·⊗RMr,L, N)
is an isomorphism, and this holds for every affine group scheme N over L. It follows that the base change homomorphism is an isomorphism.