Definition 2.1.1. Let M be a commutative group scheme over S. If there is a ring homomorphism αM : R → EndS(M), then M together with αM is called anR-modules scheme over S, and αM is called the module structure ofM. Equivalently, anR-module structure onM is a factorization of the representable functor hM = HomS( , M) : Sch/S → Ab through the forgetful functor R-Mod→ Ab from the category of R-modules to the category of Abelian groups.
By abuse of terminology, we callM anR-module scheme (overS). For simplicity, we writer·:M →M or simply r:M →M for α(r).
Definition 2.1.2. Let M and N be R-module schemes over S. An R-linear homomorphism over S or an R-module homomorphism ϕ over S from M to N is a group scheme homomorphism ϕ : M → N over S, such that αN(r)◦ϕ = ϕ◦αM(r) for every r∈R. We denote by HomR(M, N) the group of all R-linear homomorphisms from M to N, which is in fact anR-module using the action of R on M or N. If the ring R is understood from the context and there is little risk of confusion with the group of all homomorphisms (not necessarilyR-linear) fromM toN, we denote this module by Hom(M, N).
Remark 2.1.3.
1) A sequence
0→M� −→M −→M��→0
of R-module schemes is exact, if it is exact as a sequence of commutative group schemes.
2) If T is an S-scheme, then the R-module structure ofM gives an R-module structure of the base extension MT.
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3) IfM1, . . . , MrareR-module schemes overS, then the productM1×SM2×S
· · ·×SMr is again an R-module scheme over S.
4) A commutative group scheme over S is a Z-module scheme over S. So, we can think of the theory of R-module schemes as a generalization of the theory of commutative group schemes.
5) LetM be anR-module scheme over S andGa group scheme over S. Then the group HomS(M, G) has a natural structure of R-module through the action of R on M.
6) LetM be a finite flatR-module scheme overS. The Cartier dual ofM, i.e., the group scheme Hom(M,Gm,S) has a naturalR-module scheme structure given by the action of R onM.
7) Let M be a finite flat R-module scheme over Spec(A), where A is a Henselian local ring. We have the connected-´etale sequence of M as a group scheme overSpec(A)
0→M0 →M →M´et→0.
The functoriality of this sequence implies that the action ofRonM induces actions on connected and ´etale factors, i.e., for every r ∈ R, we have the following commutative diagram:
0 ��M0
r·
�� ��M
r·
�� ��M´et
r·
�� ��0
0 ��M0 ��M ��M´et ��0.
Therefore, the connected and ´etale factors of M have natural structures of R-module schemes and the connected-´etale sequence of M is an exact sequence of R-module schemes over Spec(A).
Definition 2.1.4. Let M and N be R-module schemes over S. Define a con-travariant functor HomR(M, N) from the category of schemes over S to the cat-egory of R-modules as follows:
T �→HomR(M, N)(T) := HomRT(MT, NT).
If this functor is representable by a group scheme over S, that group scheme, which is in fact an R-module scheme is also denoted by HomR(M, N) and is called the inner Hom from M to N.
Remark 2.1.5. Note that the condition of a homomorphism to be R-linear, is a closed condition, therefore, if Hom(M, N) exists as a group scheme, then
2.1. R-MODULE SCHEMES 11 HomR(M, N) exists and is a closed subscheme of Hom(M, N). So, we can apply the existence results that we have for group schemes, to the new setting of R-module schemes. For instance, if M is finite flat over S, and N is affine, then HomR(M, N) is representable by an affine R-module scheme (cf. theorem 1.3.5 in [Pink] 1). If in addition, N is of finite type over S, then HomR(M, N) is of finite type too.
It is known that exact sequences of group schemes are stable under base change, and therefore, the same holds for exact sequences ofR-module schemes. However, we give a proof of this fact in the following special case:
Lemma 2.1.6. Suppose that0→K −−→i N −−→p Q→0is a short exact sequence of affine R-module schemes over a field k and let T = SpecC be a k-scheme.
Then the sequence
0→KT iT
−−−→NT pT
−−−→QT →0 obtained by base change is exact.
Proof. Denote by A, B the Hopf algebras representing N, Q and by IB the aug-mentation ideal ofB. Then the Hopf algebra representingK isA/(IB·A). Since Cis flat overk, we have an injectionB⊗kC �→A⊗kCand thereforeNT pT
−−−→QT
is a quotient morphism. We also have (IB·A)⊗kC = (IB⊗kC)·(A⊗kC) and so by flatness we have
(A/(IB·A))⊗kC ∼=A⊗kC/((IB·A)⊗kC) =A⊗kC/(IB⊗kC)(A⊗kC).
It implies thatKT is the kernel ofNT πT
−−−→QT. Consequently the short sequence 0→KT iT
−−→NT pT
−−−→QT →0 is exact.
Proposition 2.1.7. Let M be an affine R-module scheme over a field k. Then the functorsHomR(−, M)andHomR(M,−)from the category of affineR-module schemes over k to the category of presheaves of R-modules are left exact.
Proof. Let 0→ K −−→i N −−→p Q →0 be a short exact sequence of R-module schemes overk. We have to show that the sequence
0→HomR(Q, M)−−→p∗ HomR(N, M)−−→i∗ HomR(K, M) is exact. It is equivalent to the exactness of the sequence
0→HomR(Q, M)(C)−−→p∗ HomR(N, M)(C)−−→i∗ HomR(K, M)(C) for every k-algebra C, i.e., the exactness of the sequence
0→HomRC(QC, MC)−−→p∗ HomRC(NC, MC)−−→i∗ HomRC(KC, MC).
1Note that since the paper [Pink] is not yet published and is in preparation, its numbering is subject to change. In this treatise, we will use the numbering of the last version available so far.
By previous lemma, the R-morphism NC pC
−−−→ QC is the cokernel of the in-jection KC �→ NC in the category of affine R-module schemes. So, for any R-homomorphism ϕ : NC → MC such that ϕ◦iC = 0, there exists a unique
The exactness now follows; indeed, pick an R-morphism f : QC → MC with f ◦pC = 0, then putting ϕ := 0 the zero morphism, there are twoR-morphisms QC →MC, namely f and the zero morphism, whose composition withpC are ϕ and from the above observation they should be equal. This shows the injectivity of
HomRC(QC, MC)−−−→p∗C HomRC(NC, MC).
Clearly we have Imp∗C ⊂Keri∗C. Let g :NC →MC be an element of Keri∗C, i.e., g◦i∗C = 0, then according to what we said above, there is a ψ :QC →MC with pC ◦ψ =g, or in other words g =p∗C(ψ) and thus Keri∗C ⊂Imp∗C.
Similarly, the fact that KC iC
−−−→ NC is the kernel of the quotient morphism NC −−−→pC QC implies that given any R-homomorphism ϕ : MC → NC with trivial composition pC ◦ϕ there is a unique R-homomorphism ψ : MC → KC
such that the following diagram is commutative 0 ��KC
And this implies, as above, the exactness of the following short sequence 0→HomRC(MC, KC)−−−→i∗C HomRC(MC, NC)−−−→p∗C HomRC(MC, QC)
for every k-algebra C, and consequently the following sequence of R-module schemes is exact
0→HomR(M, K)−−→i∗ HomR(M, N)−−→p∗ HomR(M, Q).