LetXbe a scheme with a topology andGbe a sheaf of groups onX. For the following general theorem, see Giraud [23, Th´eor`eme 1.4.5, Exemple 2.1.2, Corollaire 2.2.6].
Theorem A.5. LetXbe a scheme with a topology andGbe a sheaf of group onX. a) For any objectE ofTors(G), the presheaf of groupsAutG(E)defined by
AutG(E)(U) = AutTors(G)(E|U)
overU →X, is a sheaf of groups onX.
b) For any objectsE andE0ofTors(G), the presheaf of setsIsomG(E,E0)defined by
IsomG(E,E0)(U) = IsomTors(G)(E|U,E0|U)
overU → X, is a sheaf onX and has a canonical structure of (right) sheaf torsor for
AutG(E)overX.
c) There’s a canonical isomorphism,
G→AutG(G),
of sheaves of groups onX, where on the right,Gdenotes the trivial sheaf torsor for G
More generally, for eachU →X,Tors(G|U)is a category overU. We denote byTORS(G)
the associated fibered category over the topology onX. Then TORS(G)is astack, see Giraud [23, II] for a precise definition, but colloquially we say thatTORS(G) satisfies descent. Con- versely, we fix a stack T over the topology onX (e.g. the stack associated to the category of schemes overX, sheaves of groups on X, vector bundles onX, or bilinear forms on X, etc). LetE be an object of T|X. We call an objectE0 ofT|X a (twisted) formofE if these objects are locally isomorphic, i.e. if there exists a coverU ofXso thatE andE0are isomorphic when restricted toU. Denote byForms(E)the category of forms ofE andFORMS(E)the associated sub-stack ofT of twisted forms of E, i.e.FORMS(E)|U = Forms(E|U) for U → X. For an objectE inT|X denote byForms(E)be the set of isomorphism classes of forms ofE overX,
which is a pointed set with distinguished element the class ofE.
A.2.1 Twisted forms vs. torsors
In parallel with Theorem A.5, we can compare twisted forms and torsors for the sheaf of auto- morphism group, see Giraud [23, Th´eor`eme 2.5.1].
Theorem A.6. LetTbe a stack over the topology of a schemeXand letE be an object ofT|X.
LetAut(E)be the sheaf onXof automorphism groups ofE inT. Then there’s an equivalence of stacksFORMS(E)→TORS(Aut(E)), given by the equivalences of categories
Forms(E|U) → Tors(Aut(E|U))
E0 7→ Isom
Aut(E|U)(E|U,E
0)
A.2.2 The case of symmetric bilinear forms
We consider the case of symmetric bilinear forms. The case of general bilinear forms can be treated similarly.
Theorem A.7. LetXbe a scheme with12 ∈OX considered in the ´etale topology. Let(H, h,L)
be a fixedL-valued symmetric bilinear space of ranknonX.
a) The category ofO(E, b,L)-torsors is equivalent to the category of whose objects areL- valued symmetric bilinear spaces of ranknand whose morphisms are isometries.
b) The category ofSO(E, b,L)-torsors is equivalent to the category whose objects are pairs
((E0, b0,L), ψ0) consisting of an L-valued symmetric bilinear space of rankn together with an isometryψ0 : disc(E, b,L) → disc(E0, b0,L)of discriminant forms, and whose morphisms between objects ((E0, b0,L), ψ0) and ((E00, b00,L), ψ00) are isometries ϕ : (E0, b0,L)→(E00, b00,L)such thatψ00= disc(ϕ)◦ψ0.
c) The category ofGO(E, b,L)-torsors is equivalent to the category whose objects are all symmetric bilinear spaces of ranknwith values in a line bundle and whose morphisms are similarity transformations.
d) Letnbe even. The category ofGSO(E, b,L)-torsors is equivalent to the category whose objects are pairs((E0, b0,L0), ψ0)consisting of anL0-valued symmetric bilinear space of rankn(for some line bundleL0 onX) together with an isometryψ0 : disc(E, b,L) → disc(E0, b0,L0) of discriminant forms, and whose morphisms between any two objects
((E0, b0,L0), ψ0)and((E00, b00,L00), ψ00)are similarity transformationsϕ: (E0, b0,L0)→ (E00, b00,L00)such thatψ00= disc(ϕ)◦ψ0.
Moreover, the above sheaves of groups are smooth, locally of finite-type, and affine overX, hence all respective sheaf torsors are representable as schemes.
Proof. In each case, we identify the stated category of sheaf torsors with a corresponding category of twisted forms (inside the sub-stack ofOX-modules that locally have the structure of a bilinear
form) of a base object, then we’ll appeal to Theorem A.6. There are two steps. First, we show that the sheaf of automorphism groups of the base object is isomorphic to the stated sheaf of groups. Second, we identify all twisted forms of the corresponding base object. To this end, we prove that every object of the stated form is a twisted form of the base object by finding a suitable covering, we then prove that every twisted form is of the stated type by using properties of the contracted product.
For a)andc), first note that O(H, h,L) (resp.GO(H, h,L)) is defined as the sheaf of isometry (resp. similitude) groups of(H, h,L). Second, let(E, b,M)be a symmetric bilin- ear space of rank nonX and letU = {Ui → X}i∈I be an ´etale cover ofX trivializing both
L andM viali : OUi −→∼ L|Ui andmi : OUi −→∼ M|Ui. Since everyOX-valued symmet-
ric bilinear form is locally isomorphic for the ´etale topology, see for example Demazure/Gabriel [14, III,§5.2], we can refine the cover U, finding isometries ϕi : (H|Ui, l
−1
i ◦h|Ui,OUi) −→∼ (E|Ui, m
−1
i ◦ b|Ui,OUi) for each i ∈ I. Now note that these isometries induce similarities (ϕi, mi ◦l−i 1) : (H|Ui, h|Ui,L|Ui) −→∼ (E|Ui, b|Ui,M|Ui), which are themselves isometries
of the corresponding line bundle-valued forms if and only ifM|Ui =L|Ui andmi◦li−1 is the
identity map for alli∈I, i.e.M =L. Thus(H, h,L)and(E, b,M)are locally similar in the ´etale topology, and are locally isometric if and only ifM =L. Now we prove that every twisted formE (i.e.OX-module with the structure of a bilinear form on some ´etale cover) of(H, h,L)
this end, let GbeO(H, h,L) (resp.GO(H, h,L)) and consider the corresponding (sheaf)
G-torsorP =IsomG(E,(H, h,L))of isometries (resp. similitudes). Then there’s a canonical isomorphism ofOX-modulesP
G
∧H −→∼ E. The mapP×(H ⊗H)−−−→id×h P ×L induces a symmetricOX-bilinear morphism,
P G∧(H ⊗H)→P G∧L,
whereL has the structure of left G-sheaf via the multiplier coefficient. In particular, ifG = O(H, h,L), then by definitionGacts trivially onL and there’s a canonical isomorphismP G∧
L −→∼ L, thus there’s a symmetricOX-bilinear morphismE⊗E →L. IfG=GO(H, h,L), then there are canonicalOX-module morphisms,
P G∧L −→∼ (P G∧Gm)G∧mL =µP G∧mL,
where µP is the canonical (sheaf) Gm-torsor induced from P by extension of structure group GO(H, h,L) −→µ Gm. In particular,P
G
∧ L is some line bundleM, and thus there’s a sym- metricOX-bilinear morphismE ⊗E →M.