DISCRIMINANTS AND THE MONOID OF QUADRATIC RINGS

arXiv:1504.05228v3 [math.AG] 13 Jan 2016

JOHN VOIGHT
Abstract. We consider the natural monoid structure on the set of quadratic rings over an
arbitrary base scheme and characterize this monoid in terms of discriminants.

Quadratic field extensions K of Q are characterized by their discriminants. Indeed, there
is a bijection
(
)
Separable quadratic
∼
algebras over Q
−
→ Q× /Q×2
up to isomorphism
√
Q[ d] = Q[x]/(x2 − d) 7→ dQ×2

where
√ a separable quadratic algebra over Q is either a quadratic field extension or the algebra
Q[ 1] ≃ Q × Q of discriminant 1. In particular, the set of isomorphism classes of separable
quadratic extensions of Q can be given the structure of an elementary abelian 2-group, with
identity element the class of Q × Q: we have simply
p
p
p
Q[ d1 ] ∗ Q[ d2 ] = Q[ d1 d2 ]
√
×
×2
up to isomorphism. If d1 , d2 , d√
1 d2 ∈
√ Q \ Q then Q( d1 d2 ) sits as the third quadratic
subfield of the compositum Q( d1 , d2 ):
√ √
Q( d1 , d2 )
√
Q( d1 )

◆◆◆
◆◆◆
◆◆◆
◆◆

♣♣♣
♣♣♣
♣
♣
♣♣♣

√
Q( d1 d2 )

❖❖❖
❖❖❖
❖❖❖
❖❖❖
❖

Q

√
Q( d2 )

♦♦
♦♦♦
♦
♦
♦♦♦
♦♦♦

√
Indeed, if√σ1 is√the nontrivial element
of
Gal(Q(
d1 )/Q), then there√
is a unique extension of
√
σ1 to Q( d1 , d2 ) leaving Q( d2 ) fixed, similarly with σ2 , and Q( d1 d2 ) is the fixed field
of the composition σ1 σ2 = σ2 σ1 .
This characterization of quadratic extensions works over any base field F with char F 6= 2
and is summarized concisely in the Kummer theory isomorphism
H 1 (Gal(F /F ), {±1}) = Hom(Gal(F /F ), {±1}) ≃ F × /F ×2 .

On the other hand, over a field F with char F = 2, all separable quadratic extensions have
trivial discriminant and instead they are classified by the (additive) Artin-Schreier group
F/℘(F ) where ℘(F ) = {r + r 2 : r ∈ F }
Date: January 14, 2016.
1

with the class of a ∈ F in correspondence with the isomorphism class of the extension
F [x]/(x2 − x + a). By similar considerations as above, we again find a natural structure
of an elementary abelian 2-group on the set of isomorphism classes of separable quadratic
extensions of F .
One can extend this correspondence between quadratic extensions and discriminants integrally, as follows. Let R be a commutative ring. A free quadratic R-algebra (also called
a free quadratic ring over R) is an R-algebra S (associative with 1) that is free of rank 2 as
an R-module. Let S be a free quadratic R-algebra. Then S/R ≃ ∧2 S ≃ R is projective,
so there is an R-basis 1, x for S; we find that x2 = tx − n for some t, n ∈ R and that S is
commutative. The map σ : S → S induced by x 7→ t − x is the unique standard involution
on S, an R-linear (anti-)automorphism such that yσ(y) ∈ R for all y ∈ S. The class of the
discriminant of S
d = d(S) = (x − σ(x))2 = t2 − 4n

in R/R×2 is independent of the choice of basis 1, x. A discriminant d satisfies the congruence
d ≡ t2 (mod 4R), so for example if R = Z then d ≡ 0, 1 (mod 4).
Now suppose that R is an integrally closed domain of characteristic not 2. Then there is
a bijection


Free quadratic rings over R ∼
−
→ {d ∈ R : d is a square in R/4R}/R×2
up to isomorphism
S 7→ d(S)
For example, over R = Z, the free quadratic ring S(d) over Z of discriminant d ∈ Z = Z/Z×2
with d ≡ 0, 1 (mod 4) is given by

2

Q[x]/(x2 ),
if d = 0;
Z[x]/(x ) ֒→
√
2
S(d) = Z[x]/(x − dx) ֒→ Q × Q, if d 6= 0 is a square;

Z[(d + √d)/2] ֒→ Q(√d),
otherwise.

The set of discriminants under multiplication has the structure of a commutative monoid , a
nonempty set equipped with a commutative, binary operation and identity element. Hence
so does the set of isomorphism classes of free quadratic R-algebras, an operation we denote
by ∗: the identity element is the class of R × R ≃ R[x]/(x2 − x). The class of the ring
R[x]/(x2 ) with discriminant 0 is called an absorbing element.
More generally, a quadratic R-algebra or quadratic ring over R is an R-algebra S which is
locally free of rank 2 as an R-module. By definition, a quadratic ring over R localized at any
prime (or maximal) ideal of R is a free quadratic R-algebra. Being true locally, a quadratic
R-algebra S is commutative and has a unique standard involution.
There is a natural description of quadratic R-algebras as a stack quotient, as follows. A free
quadratic R-algebra equipped with a basis 1, x has multiplication table uniquely determined
by t, n ∈ R and has no automorphisms, and so the functor which associates to a commutative
ring R the set of free quadratic R-algebras with basis (up to isomorphism) is represented by
two-dimensional affine space A2 (over Z). The change of basis for a free quadratic R-algebra
is of the form x 7→ u(x + r) with u ∈ R× and r ∈ R, mapping
(t, n) 7→ (u(t + 2r), u2(n + tr + r 2 )).
2

Therefore, we have a map from the set of free quadratic R-algebras with basis to the quotient
of A2 (R) by G(R) = (Gm ⋊ Ga )(R) with the above action. Working over Spec Z, the group
scheme G is naturally a subgroup scheme of GL2 , but it does not act linearly on A2 , and
the Artin stack [A2 /G] has dimension zero over Spec Z! Nevertheless, the set [A2 /G](R) is
in bijection with the set of quadratic R-algebras up to isomorphism.
Recall that a commutative R-algebra S is separable if S is (faithfully) projective as a
S ⊗R S-module via the map x ⊗ y 7→ xy. A free quadratic R-algebra S is separable if and
only if d(S) ∈ R× ; so, for example, the only separable (free) R-algebra over Z is the ring
Z × Z of discriminant 1, an impoverishment indeed! A separable quadratic R-algebra S is
étale over R, and R is equal to the fixed subring of the standard involution of S over R.
(In many contexts, one says then that S is Galois over R with Galois group Z/2Z, though
authors differ on precise terminology. See Lenstra [13] for one approach to Galois theory for
schemes.) If S, T are separable free quadratic R-algebras where R is a Dedekind domain of
characteristic not 2, having standard involutions σ, τ , respectively, then the monoid product
S ∗ T defined above (by transporting the monoid structure on the set of discriminants) is
the fixed subring of S ⊗R T by σ ⊗ τ , in analogy with the case of fields.
The characterization of free quadratic R-algebras by their discriminants is an example
of the parametrization of algebraic structures, corresponding to the Lie group A1 in the
language of Bhargava [3]. Results in this area direction go back to Gauss’s composition law
for binary quadratic forms and have been extended in recent years by Bhargava [5], Wood
[21], and others. Indeed, several authors have considered the case of quadratic R-algebras,
including Kanzaki [10] and Small [18]. In this article, we consider a very general instance
of this monoidal correspondence between quadratic R-algebras and discriminants over an
arbitrary base scheme.
Let X be a scheme. A quadratic OX -algebra is a coherent sheaf S of OX -algebras which
is locally free of rank 2 as a sheaf of OX -modules. Equivalently, a quadratic OX -algebra is
specified by a finite locally free morphism of schemes φ : Y → X of degree 2 (sometimes
called a double cover ): the sheaf φ∗ OY is a sheaf of OX -algebras that is locally free of rank
2. If f : X → Z is a morphism of schemes, and S is a quadratic OZ -algebra, then the pullback f ∗ S is a quadratic OX -algebra. Let Quad(X) denote the set of isomorphism classes
of quadratic OX -algebras, and for an invertible OX -module L let Quad(X; L ) ⊆ Quad(X)
V
be the subset of those algebras S such that there exists an isomorphism 2 S ≃ L of
OX -modules.
Our first result provides an axiomatic description of the monoid structure on the set
Quad(X) (Theorem 3.24).
Theorem A. There is a unique system of binary operations
∗X : Quad(X) × Quad(X) → Quad(X),
one for each scheme X, such that:
(i) Quad(X) is a commutative monoid under ∗X , with identity element the isomorphism
class of OX × OX ;
3

(ii) The association X 7→ (Quad(X), ∗X ) from schemes to commutative monoids is functorial in X: for each morphism f : X → Z of schemes, the diagram
Quad(Z) × Quad(Z)

∗Z

//

Quad(Z)
f∗



Quad(X) × Quad(X)

∗X

//



Quad(X)

is commutative; and
(iii) If X = Spec R and S, T are separable quadratic R-algebras with standard involutions
σ, τ , then S ∗Spec R T is the fixed subring of S ⊗R T under σ ⊗ τ .
The binary operation is defined locally (Construction 3.12): if X = Spec R, and S =
R ⊕ Rx and T = R ⊕ Ry are free quadratic R-algebras with x2 = tx − n and y 2 = sy − m
then we define the free quadratic R-algebra
where

S ∗ T = R ⊕ Rw

w 2 = (st)w − (mt2 + ns2 − 4nm).
This explicit description (in the free case over an affine base) is given by Hahn [9, Exercises
14–20, pp. 42–43].
A general investigation of the monoid structure on quadratic algebras goes back at least
to Loos [14]. Loos gives via a universal construction a tensor product on the larger category
of unital quadratic forms (quadratic forms representing 1); this category is equivalent to
the category of quadratic algebras for forms on a finitely generated, projective module of
rank 2 [14, Proposition 1.6] as long as one takes morphisms as isomorphisms in the category
[15, §1.4]. (See also Loos [15, §6.1] for further treatment.) The existence of the monoid
structure was also established in an unpublished letter of Deligne [7] by a different method:
he associates to every R-algebra its discriminant algebra (a quadratic algebra) and extends
the natural operation of addition of Z/2Z-torsors from the étale case to the general case by
geometric arguments. Our proof of Theorem A above carries the same feel as these results,
but it is accomplished in a more direct fashion and gives a characterization (in particular,
uniqueness).
Recently, there has been renewed interest in the construction of discriminant algebras
(sending an R-algebra A of rank n to a quadratic R-algebra) by Loos [15], Rost [17], and
more recently by Biesel and Gioia [4]. Indeed, Biesel and Gioia [4, Section 8] describe the
monoid operation in Theorem A over an affine base in the context of discriminant algebras.
We hope that our theorem will have some application in this context.
Our second result characterizes quadratic algebras in terms of discriminants. A discriminant (over X) is a pair (d, L ) such that L is an invertible OX -module and d ∈ (L ∨ )⊗2 is
a global section which is a square modulo 4: there exists a global section t ∈ L ∨ /2L ∨ =
L ∨ ⊗ OX /2OX such that t ⊗ t = d ∈ (L ∨ )⊗2 /4(L ∨ )⊗2 . We can of course also think of
d ∈ (L ∨ )⊗2 as an OX -module homomorphism d : L ⊗2 → OX ; but such a global section is
also equivalently given by a quadratic form D : L → OX (see section 2).
∼
An isomorphism of discriminants (d, L ), (d′ , L ′ ) is an isomorphism f : L −
→ L ′ such that
e
(f ∨ )⊗2 (d′ ) = d. For example, if X = Spec R for R a commutative ring and L = OX = R,
2
then as above a discriminant is specified by an element d ∈ R such that d ≡ t (mod 4R) for
4

some t ∈ R (noting that only t ∈ R/2R matters), and two discriminants d, d′ are isomorphic if
and only if there exists u ∈ R× such that u2d′ = d. Thinking of a discriminant as a quadratic
form D : L → OX , its image generates a locally principal ideal sheaf I ⊆ OX , and the
set of discriminants with given locally free image I ⊆ OX , if nonempty, is a principal
×
×2
homogeneous space for the group OX
/OX
. Let Disc(X) denote the set of isomorphism
classes of discriminants and Disc(X; L ) ⊆ Disc(X) the subset with underlying line bundle
L . Then the tensor product
(d, L ) ∗ (d′ , L ′ ) = (d ⊗ d′ , L ⊗ L ′ )
gives Disc(X) and Disc(X; OX ) the structure of a commutative monoid with identity element
the class of (1, OX ).
V
A quadratic OX -algebra S has a discriminant disc(S ) = (d(S ), 2 S ), defined by
V
d(S ) : ( 2 S )⊗2 → OX
(x ∧ y) ⊗ (z ∧ w) 7→ (xσ(y) − σ(x)y)(zσ(w) − σ(z)w)

where σ is the unique standard involution on S . Although a priori the codomain of d(S )
is S , in fact its image
lies in OX : indeed, if X = Spec R and S = Spec S and S is free with
V2 ⊗2
basis 1, x, then ( S) is free generated by (1 ∧ x) ⊗ (1 ∧ x) and
1 ∧ x ⊗ 1 ∧ x 7→ (x − σ(x))2 ∈ R.

We have a natural forgetful map Disc(X) → Pic(X) where (d, L ) maps to the isomorphism
class of L .
f
g
We say a sequence A −
→B −
→ C of commutative monoids is exact if f is injective, g is
surjective, and for all z, w ∈ B, we have
g(z) = g(w) if and only if there exists x, y ∈ A such that xz = yw;
equivalently, the sequence is exact if and only if f is injective and g induces an isomorphism
of monoids B/f (A) ≃ C. (For a review of monoids, see Section 1.) Similarly, a sequence
f

g

A −
→B −
→ C of sheaves of monoids is exact if the sheaf associated to the presheaf U 7→
gx
fx
→ Cx
B(U)/f (A (U)) is isomorphic to C , or equivalently if the induced sequence Ax −→ Bx −
of monoid stalks is exact for all x ∈ X.
We now describe the monoid Quad(X). We begin with the statement that the forgetful
map is compatible with the discriminant map, as follows.
Theorem B. Let X be a scheme. Then the diagram of commutative monoids
Quad(X; OX )

Quad(X)
//

disc



Disc(X; OX )

∧2

//

Pic(X)

//

disc



Disc(X)
//

Pic(X)

is functorial and commutative with exact rows and Zariski locally surjective columns.
By “Zariski locally surjective columns”, we mean that there is an (affine) open cover of X
where (under pullback) the columns are surjective. (Considering the corresponding sheaves
over X, we also obtain a surjective map of sheaves; see Theorem 3.25.)
5

We now turn to describe the morphism Quad(X; OX ) → Disc(X; OX ). For this purpose, we work locally and assume X = Spec R for a commutative ring R; we abbreviate
Quad(Spec R) = Quad(R) and Quad(Spec R; OSpec R ) = Quad(R; R) and similarly with discriminants.
disc
We would like to able to fit the surjective map Quad(R) −−→ Disc(R) of monoids into an
exact sequence by identifying its kernel, but unfortunately the fibers of this map vary over
the codomain. Instead, we will describe the action of a subgroup of Quad(R) on the fibers
of the map disc: this is a natural generalization, as the fibers of a group homomorphism
are principal homogeneous spaces for the kernel K and are noncanonically isomorphic as a
K-set to K with the regular representation.
Recalling the case of quadratic extensions of a field F with char F = 2, for a commutative
ring R we define the Artin-Schreier group AS(R) to be the additive quotient
AS(R) =

R[4]
℘(R)[4]

where ℘(R)[4] = {n = r + r 2 ∈ R : r ∈ R} ∩ R[4]

and R[4] = {a ∈ R : 4a = 0}. We have a map i : AS(R) → Quad(R; R) ֒→ Quad(R) sending
the class of n ∈ AS(R) to the isomorphism class of the algebra S = R[x]/(x2 − x + n). The
group AS(R) is an elementary abelian 2-group since 2R[4] ⊆ ℘(R).
Our next main result is as follows (Theorem 4.3).
Theorem C. The fibers of the map disc : Quad(R) → Disc(R) have a unique action of the
group AS(R) compatible with the inclusion of monoids AS(R) ֒→ Quad(R). Moreover, the
kernel of this action on the fiber disc−1 (dR×2 ) contains annR (d)[4].
Roughly speaking, Theorems B and C together say that “a quadratic algebra is determined by its Steinitz class and its discriminant, locally up to an Artin-Schreier extension”.
These theorems could be rephrased in terms of the Grothendieck group; however, due to the
existence of an absorbing element, the group K0 (Quad(X)) is trivial for all schemes X.
The article is organized as follows. In section 1, we briefly review the relevant notions
from monoid theory. In section 2, we consider the monoid of discriminants; in section 3 we
define the monoid of quadratic R-algebras and prove Theorems A and B. In section 4 we
prove Theorem C.
The author would like to thank Asher Auel, Manjul Bhargava, James Borger, and Melanie
Wood for helpful suggestions. The author is also indebted to the anonymous referee and
Owen Biesel for very detailed and helpful comments and corrections. The author was supported by an NSF CAREER Award (DMS-1151047).
1. Monoids
To begin, we review standard terminology for monoids. A reference for the material in
this section is Bergman [2, Chapter 3]; more generally, see Burris–Sankappanavar [6] and
McKenzie–McNulty–Taylor [16].
A semigroup is a nonempty set A equipped with an associative binary operation
∗ : A × A → A.
A monoid is a semigroup with identity element 1 for ∗ (necessarily unique). Any semigroup
without 1 can be augmented to a monoid. Natural examples of monoids abound: the natural
6

numbers N = Z≥0 under addition, a ring R under its multiplication, and the set of endomorphisms of an algebraic object (such as a variety) under composition. A group is a monoid
equipped with an inverse map −1 : A → A.
Let A be a semigroup. We say A is commutative if xy = yx for all x, y ∈ A. An absorbing
element of A is an element 0 ∈ A such that 0x = x0 = 0 for all x ∈ A; a monoid has at
most one absorbing element. Multiplicative notation for A will be in general more natural
for us; however, we will occasionally write A additively with operation +, in which case the
identity element is denoted 0 and to avoid confusion A will have no absorbing element. An
element x ∈ A is (left) cancellative if xy = xz implies y = z for all y, z ∈ A.
A homomorphism of semigroups is a map f : A → B such that f (xy) = f (x)f (y) for all
x, y ∈ A, and a homomorphism of monoids is a homomorphism of semigroups such that
f (1A ) = 1B .
Let f : A → B be a homomorphism of monoids. Unlike groups, the kernel ker f = {x ∈
A : f (x) = 1} of a monoid homomorphism does not determine the structure of the image of
f ; instead, we define the kernel congruence of f by
Kf = {(x, y) : f (x) = f (y)} ⊆ A × A.

The set Kf defines a congruence on A, an equivalence relation compatible with the operation
on A, i.e., if (x, y), (z, w) ∈ Kf then (xz, yw) ∈ Kf . Conversely, given a congrence K on
a monoid A, the set operation [x] · [y] = [x · y] on equivalence classes [x], [y] ∈ A/K is
well-defined and the quotient map A → A/K via x 7→ [x] is a surjective homomorphism
of monoids with kernel K; any homomorphism f : A → B with Kf ⊇ K factors through
A → A/K.
The image f (A) = {f (x) : x ∈ A} is a submonoid of B, but if A, B are noncommutative,
then not every submonoid is eligible to be the kernel of a homomorphism (just as not
every subgroup is normal). As we will be interested only in commutative monoids, and this
assumption simplifies the presentation, suppose from now on that A, B are commutative.
Then the set
If = {(z, w) : f (x)z = f (y)w for some x, y ∈ A} ⊆ B × B,

is a congruence called the image congruence. (Without the hypothesis of commutativity, If
is a relation that is reflexive and symmetric, but not necessarily transitive nor a congruence;
if A, B are possibly nonabelian groups, then If is transitive and is a congruence if and only
if f (A) is a normal subgroup of B.) Note that if 0 ∈ f (A) then If = B × B. We write
B/f (A) = B/If .
A sequence
(1.1)

f

g

A−
→B−
→C

is exact if f is injective, g is surjective, and Kg = If , i.e.
g(z) = g(w) if and only if there exists x, y ∈ A such that xz = yw;

∼

→
equivalently, (1.1) is exact if f is injective and g induces an isomorphism B/f (A) = B/If −
C. A sequence of groups (1.1) is exact as a sequence of groups if and only if it is exact as a
sequence of monoids.
Remark 1.2. We will not make use of long exact sequences of monoids here nor write the
customary 0 or 1 at the ends of our short exact sequences. Indeed, the straightforward
7

extension of the notion from groups to monoids using the definitions above (kernel congruence
equals image congruence) has the following defect: the sequence
f

j

N−
→Z−
→0
of monoids under addition has If = Z × Z = Kj even though f is not surjective. (The map
f is, however, an epimorphism in the category of monoids.)
f

g

We will also have use of sheaves of monoids over a scheme X. A sequence A −
→B−
→C
of sheaves of monoids is exact if the sheaf associated to the presheaf U 7→ B(U)/f (A (U)) is
fx

gx

isomorphic to C , or equivalently if the induced sequence Ax −→ Bx −→ Cx of monoid stalks
is exact for all x ∈ X.
Like the formation of the integers from the natural numbers, one can construct the
Grothendieck group K0 (A) of a commutative monoid A, with the universal property that for
any monoid homomorphism A → G with G an abelian group, there exists a unique group
homomorphism K0 (A) → G such that the diagram
A
∃! ①



①

①

// G
①<<

①

K0 (A)
commutes. The group K0 (A) is constructed as A×A under the equivalence relation (x, x′ ) ∼
(y, y ′) if there exists z ∈ A such that xy ′ z = x′ yz. Note that if A has an absorbing element 0
then K0 (A) = {0}. The set of cancellative elements Acanc is the largest submonoid of A which
can be embedded in a group, and the smallest such containing group is the Grothendieck
group K0 (Acanc ). If A is a sheaf of monoids, we write K0 (A ) for the sheaf of abelian groups
associated to the presheaf U 7→ K0 (A (U)).
2. Discriminants
In this section, we define discriminants for quadratic rings over general schemes (Definition
2.10); for a discussion of discriminant modules overlapping the one presented here, see Knus
[12, §III.3] and Loos [14, §1.2]. We also relate semi-nondegenerate quadratic forms on line
bundles by their images (Lemma 2.7) and factor the monoid of discriminants over the Picard
group (Proposition 2.16).
Let X be a scheme. A quadratic form over X is a pair (M , Q) where M is a locally free
OX -module of finite rank and Q : M → OX is a quadratic map, i.e., for all open sets U ⊆ X,
we have
(i) Q(rx) = r 2 Q(x) for all r ∈ OX (U) and x ∈ M (U); and
(ii) The map T : M (U) × M (U) → OX (U) defined by
T (x, y) = Q(x + y) − Q(x) − Q(y)
is OX (U)-bilinear; we call T the associated bilinear form.
An isometry between quadratic forms Q : M → OX and Q′ : M ′ → OX is an OX -module
∼
isomorphism f : M −
→ M ′ such that Q′ ◦ f = Q. A similarity between quadratic forms Q, Q′
8

is a commutative square:
M

Q

OX
//

g ≀

≀ f



M′

Q′

//



OX

and so an isometry is just a similarity with g = id.
A quadratic form (M , Q) is also equivalently specified by M and an OX -module homomorphism Q : Sym2 M → OX , or a global section
Q ∈ Hom(Sym2 M , OX ) = (Sym2 M )∨ ≃ Sym2 (M ∨).

(Here, Sym2 M denotes the second symmetric power of M and Sym2 M the submodule of
symmetric second tensors of M .)
A quadratic form Q : M → OX with associated bilinear form T : L → L → OX induces
a homomorphism of OX -modules M → M ∨ defined by
M (U) → M ∨ (U)

y 7→ (x 7→ T (x, y))

Following Knus [12, (I.3.2)], we say Q : M → OX is nondegenerate if the associated map
M → M ∨ is injective and nonsingular (or regular ) if the associated map M → M ∨ is an
isomorphism.
Now let L be an invertible OX -module (i.e., locally free of rank 1). A quadratic form on L ,
the case of our primary concern, is a quadratic map Q : L → OX , but it is given equivalently
by a global section q ∈ Sym2 (L ∨ ) ≃ (L ∨ )⊗2 . Because will use this identification frequently,
we make it explicit. The identification is defined locally on X, so suppose X = Spec R
and L is a free module of rank 1 over R. Then Sym2 (L∨ ) ≃ (L∨ )⊗2 ≃ (L⊗2 )∨ . Suppose
q ∈ Sym2 (L∨ ), so q : L ⊗ L → R is an R-module homomorphism. Let L = Re for some
e ∈ L and define the quadratic map Q : L → R by Q(re) = r 2 q(e ⊗ e); this definition is
independent of the choice of e. Conversely, if Q : L → R is a quadratic map, then again
letting L = Re we define the R-module homomorphism q : L ⊗ L → R by q(e ⊗ e) = Q(e).
Remark 2.1. One must remember the domain L in this identification. Indeed, if i : T ⊗2 ≃
OX is an isomorphism of OX -modules, so that T ∈ Pic(X)[2], then i defines a quadratic
form I : T → OX , called a neutral form, giving rise to an isomorphism
∼

(L ∨ )⊗2 −
→ ((L ⊗ T )∨ )⊗2 .
The notions of nondegenerate and nonsingular can be made quite explicit for quadratic
forms of rank 1. These conditions are local, so let Q : L → R be a quadratic form with L =
Re. Then Q is uniquely specified by the element Q(e) = a ∈ R, and the associated bilinear
form is specified by T (e, e) = 2a. A different choice of basis e′ = ue gives Q(e′ ) = u2 Q(e) =
au2 with u ∈ R× . We find that Q is nondegenerate if and only if 2a is a nonzerodivisor and
regular if 2a ∈ R× , and these are well-defined. We refine these notions (as with the halfdiscriminant [12, (IV.3.1.3)]) by saying that Q is semi-nondegenerate if a is a nonzerodivisor
and semi-nonsingular (or semi-regular ) if a is a unit; and we extend these notions globally to
a quadratic form Q : L → OX if they hold on an affine open cover.
9

Given two quadratic forms Q : L → OX and Q′ : L ′ → OX , corresponding to q ∈ (L ∨ )⊗2
and q ′ ∈ (L ′ ∨ )⊗2 , from the element
∨

q ⊗ q ′ ∈ (L ∨ )⊗2 ⊗ (L ′ )⊗2 ≃ ((L ⊗ L ′ )∨ )⊗2

we define the corresponding tensor product Q ⊗ Q′ : L ⊗ L ′ → OX : following the identification above, over X = Spec R where L = Re and L′ = Re′ , then L ⊗ L′ = R(e ⊗ e′ )
and (Q ⊗ Q′ )(e ⊗ e′ ) = Q(e)Q(e′ ). The tensor product gives the set of similarity classes of
quadratic forms of rank 1 over X the structure of commutative monoid.
Remark 2.2. The definition of the tensor product of quadratic forms is more subtle in general
for forms of arbitrary rank; here we find the correct notion because we can think of rank 1
quadratic forms as rank 1 symmetric bilinear forms.
Definition 2.3. Let Q : L → OX be a (rank 1) quadratic form. We say Q is cancellative if
it is cancellative in the monoidal sense, as q ∈ (L ∨ )⊗2 : if q ′ ∈ ((L ′ )∨ )⊗2 and q ′′ ((L ′′ )∨ )⊗2
have q ⊗ q ′ similar to q ⊗ q ′′ , then q ′ is similar to q ′′ .
We say Q is locally cancellative if for all x ∈ X there exists an affine open neighborhood
U ∋ x such that Q|U is cancellative.

Proposition 2.4. A (rank 1) quadratic form Q : L → OX is locally cancellative if and only
if Q is semi-nondegenerate.
Moreover, a locally cancellative rank 1 quadratic form Q over X is cancellative. If X is
affine then Q is cancellative if and only if it is locally cancellative.

Proof. Both properties are local, so it suffices to check this over a ring X = Spec R such
that the quadratic forms involved are free. To a quadratic form Q : L = Re → R, we have
Q(e) = a ∈ R. If Q′ , Q′′ are similarly other rank 1 quadratic forms with Q(e′ ) = a′ ∈ R and
Q(e′′ ) = a′′ ∈ R, then (Q ⊗ Q′ )(e ⊗ e′ ) = aa′ and (Q ⊗ Q′′ )(e ⊗ e′′ ) = aa′′ . We have Q′ ∼ Q′′
if and only if there exists u ∈ R× such that a′ = ua′′ , and similarly Q ⊗ Q′ ∼ Q ⊗ Q′′ .
Thus, if Q is semi-nondegenerate, then a is a nonzerodivisor, so Q ⊗ Q′ ∼ Q ⊗ Q′′ implies
aa′ = uaa′′ implies a′ = ua′′ implies Q′ ∼ Q′′ , so Q is locally cancellative. Conversely, if Q is
locally cancellative and a is a zero divisor, with aa′ = 0 and a′ 6= 0, then taking Q′ (e′ ) = a′
and Q′′ (e′′ ) = 0 we have Q ⊗ Q′ ∼ Q ⊗ Q′′ so Q′ ∼ Q′′ and thus there exists u ∈ R× such
that a′ = u(0) = 0, a contradiction.
Now for the second statement. Let Q : L → OX be locally cancellative. By the previous
paragraph, Q is semi-nondegenerate. Suppose that Q ⊗ Q′ ∼ Q ⊗ Q′′ for rank 1 quadratic
forms Q′ , Q′′ ; we will show that Q′ ∼ Q′′ . Cancelling in Pic(X), we may assume without
loss of generality that L ′ = L ′′ . Let U = Spec R be an affine open subset of X in which all
of L |U = L = Re is free and similarly L′ = Re′ = L′′ = Re′′ . Let a = Q(e) and similarly
a′ = Q′ (e′ ) and a′′ = Q′′ (e′′ ), so as in the previous paragraph we are given (a unique) u ∈ R×
such that aa′ = uaa′′ . Since Q is locally cancellative, we have a′ = ua′′ , and this defines a
similarity Q′ ∼ Q′′ . Repeating this on an open cover, the elements u glue to give an element
×
g ∈ OX
, and so together with the identity map on L ′ = L ′′ we therefore have a similarity
Q′ ∼ Q′′ .
The converse in the final statement follows immediately by taking U = X if X is affine. 
The following corollary is then immediate.
Corollary 2.5. The subset of locally cancellative quadratic forms over X is a submonoid of
the monoid of rank 1 quadratic forms over X.
10

Remark 2.6. The global notion of cancellative is not as robust as one may like. Kleiman [11]
gives an example of a scheme X and a global section t ∈ OX (X) that is a nonzerodivisor
such that it becomes a zerodivisor in an affine open t|U ∈ OX (U).
The similarity class of a locally cancellative quadratic form is determined by its image
(“effective Cartier divisors on a scheme are the same as invertible sheaves with a choice of
regular global section” [19, Tag 01X0]), as follows.
Lemma 2.7. There is a (functorial) isomorphism of commutative monoids




Similarity classes of rank 1


Locally free ideal




∼
locally cancellative quadratic forms
sheaves I ⊆ OX
−
→




Q : L → OX


such that [I ] ∈ 2 Pic(X)
modulo neutral forms
where the similarity class of a quadratic form Q : L → OX maps to the ideal I of OX
generated by the values Q(L ).
Ideal sheaves are a monoid under multiplication, so the “monoid” part of Lemma 2.7 says
that the tensor product Q ⊗ Q′ of two quadratic forms Q, Q′ maps to the product I I ′ of
their associated ideal sheaves I , I ′ .
Proof. First, we show the map is well defined, which we may do locally. Let Q : L → R be
a rank 1 locally cancellative quadratic form; then L is free so we may write L = Re and
then Q(L) = Q(e)R. By Proposition 2.4, we know that Q(e) is a nonzerodivisor, and this is
independent of the choice of e (up to a unit of R). If Q′ : L′ → R is similar to Q, then there
exist R-linear isomorphisms f : L → L′ and g : R → R such that Q′ (f (x)) = g(Q(x)) for
all x ∈ L. Letting e′ = f (e) we have L′ = Re′ . The map g must be of the form g(a) = ua
for some u ∈ R× , so Q′ (L′ ) = Q′ (f (L)) = uQ(L) = uQ(e)R = Q(e)R, and so the image is a
well-defined principal ideal.
Now let Q : L → OX be a locally cancellative quadratic form of rank 1, corresponding
to the global section q ∈ (L ⊗2 )∨ . We claim that q : L ⊗2 → OX is an isomorphism onto its
image I = Q(L ) = q(L ⊗2 ). If U = Spec R is an affine open set where L |U = Re, then
q(L⊗2 ) = q(e ⊗ e)R; if further U is such that q is cancellative over U, we have q(e ⊗ e) is a
nonzerodivisor, so q is injective on U (and q(L) is free). By hypothesis, such affine opens U
cover X, so we have [I ] = [L ⊗2 ] ∈ Pic(X).
Next, let q : L → OX and q ′ : L ′ → OX be locally cancellative quadratic forms such that
q(L ) = q ′ (L ′ ) = I . Since [I ] = [L ⊗2 ] = [(L ′ )⊗2 ], tensoring q ′ by a neutral form we may
∼
assume that f : L −
→ L ′ . Then on any open affine U = Spec R ⊆ X, where L |U = Re
and L ′ |U = Re′ , we have q(L) = q(e)R = q ′ (e′ )R. Therefore, there exists u ∈ R such that
q(e) = uq ′(e′ ) and u′ ∈ R such that q ′ (e′ ) = u′ q(e). Thus q(e)(1 − uu′) = 0. On an open
affine U where q is cancellative, we have uu′ = 1, so u ∈ R× . Moreover, the element u is
unique, since if q(e) = uq ′ (e′ ) = vq ′ (e′ ) then (u − v)q ′ (e′ ) = 0 so since q ′ is cancellative, we
have u = v. Repeating this argument on an open cover where both L and L ′ are free, there
∼
×
exists (a unique) u ∈ OX
→ OX such that q ′ f = uq, so
giving rise to an isomorphism OX −
that q, q ′ are similar.
Finally, the map is surjective. We are given that there exists an invertible bundle L
such that L ⊗2 ≃ I . The embedding L ⊗2 ≃ I ֒→ OX then defines a locally cancellative
quadratic form Q : L → OX with values Q(L ) = I , as can be readily checked locally. 
11

Remark 2.8. A (locally) cancellative rank 1 quadratic form might not pull back to a (locally)
cancellative form under an arbitrary morphism of schemes.
To work with discriminants, we will work modulo 2 and 4 as follows. The multiplication
by 4 map on OX gives a closed immersion
X[4] = X ×Spec Z Spec Z/4Z ֒→ X

and the pullback L[4] = L ⊗ OX /4OX is an invertible OX[4] -module, equipped with a map
[4] : L → L[4] . We can also further work modulo 2, obtaining L[2] .
Let R be a commutative ring. Then squaring gives a well-defined map of sets
sq : R/2R → R/4R
(2.9)
sq(t + 2R) = t2 + 4R
The map sq is functorial in R and canonically defined, so we can sheafify: if L is an invertible
OX -module, there is a unique map
sq : L[2] → (L[4] )⊗2

locally defined by (2.9). Explicitly, for an affine open U = Spec R of X where L (U) = L =
⊗2
Re, we may write L[2] (U) = (R/2R)e and L[4]
(U) = (R/4R)(e ⊗ e), and
sq((t + 2R)e) = sq(t + 2R)(e ⊗ e)

is well-defined (independent of the choice of e).

Definition 2.10. A discriminant over X is a pair (d, L ) where L is an invertible OX -module
∨
with
and d ∈ (L ∨ )⊗2 such that there exists t ∈ L[2]
∨ ⊗2
sq(t) = d[4] ∈ (L[4]
) .

(2.11)

If 2 is invertible on X, then X[4] is the empty scheme and the square condition (2.11) is
vacuously satisfied.
Definition 2.12. An isomorphism between discriminants (d, L ) and (d′ , L ′ ) is an isomor∼
phism f : L −
→ L ′ such that (f ∨)⊗2 : (L ′∨ )⊗2 → (L ∨ )⊗2 has (f ∨ )⊗2 (d′ ) = d.

Equivalently, an isomorphism between discriminants is an isometry (not a similarity!)
between the corresponding quadratic forms.
In what follows, we will often write abbreviate d for a discriminant (d, L ), and refer to
L as the underlying invertible sheaf.
Let Disc(X) denote the set of discriminants over X up to isomorphism. For an invertible
sheaf L on X, let Disc(X; L ) ⊆ Disc(X) denote the subset of isomorphism classes of
discriminants d whose underlying invertible sheaf is (isomorphic to) L . Define Disc(X) to
be the sheaf associated to the presheaf U 7→ Disc(U).

Lemma 2.13. There is a functorial bijection between
∼

Disc(X; OX ) −
→ {d ∈ OX : d is a square modulo 4OX (X)}/OX (X)×2 .

Proof. Suppose that f is an isomorphism between discriminants d, d′ . Let U = Spec R ⊆ X
be an affine open subset. Then d|U : R → R is a quadratic map with d(r) = r 2 d(1) for all
r ∈ R. The restriction f |U : OX (U) = R → R is an isomorphism and so is identified with a
unique element u ∈ R× , and so in R we have d′ |U (f |U (1)) = d′ |U (u) = u2 d′ |U (1) = d|U (1);
by gluing, there exists a (unique) global section u ∈ OX (X)× such that d = u2 d′ .

12

Example 2.14. If X = Spec R where R is a PID or local ring, then by Lemma 2.13, Disc(R) =
Disc(R; R) is canonically identified with
{d ∈ R : d is a square in R/4R}/R×2 .

So for R = Z, since Z×2 = {1} we recover the usual set of discriminants as those integers
d ≡ 0, 1 (mod 4).
Lemma 2.15. Disc(X) has the structure of commutative monoid under tensor product, with
identity element represented by the class of (1, OX ). Moreover, Disc(X; OX ) is a submonoid
of Disc(X) with absorbing element (0, OX ).
Proof. The binary operation of tensor product on quadratic forms restricts to a binary op∨
eration on discriminants: if (d, L ) and (d′ , L ′ ) are discriminants, with t ∈ L[2]
satisfying
∨ ⊗2
′
′
sq(t) = d[4] ∈ (L[4] ) and similarly for (d , L ), then
∨ ⊗2
′ ∨ ⊗2
sq(t ⊗ t′ ) = d[4] ⊗ d′[4] = (d ⊗ d′ )[4] ∈ (L[4]
) ⊗ ((L[4]
) ) ≃ ((L ⊗ L ′ )∨[4] )⊗2 .

This definition is independent of the choice of a representative discriminant in an isomorphism class, so we obtain a binary operation on Disc(X). This operation is associative and
commutative and (1, OX ) is an identity by definition of the tensor product. The subset
Disc(X; OX ) is closed under tensor product, and (0, OX ) is visibly an absorbing element. 
To a discriminant (d, L ), we can forget the quadratic map d and consider only the isomorphism class of the OX -module L : this gives a map
p : Disc(X) → Pic(X).
Proposition 2.16. The sequence
p

Disc(X; OX ) → Disc(X) −
→ Pic(X)

of commutative monoids is exact.

Proof. The map p : Disc(X) → Pic(X) is surjective because an invertible module L has the
zero quadratic form d = 0, which is a discriminant taking t = 0. (One can hardly do better
in general, since it may be the case that L (X) = {0} has no nonzero global sections.)
Let i : Disc(X; OX ) ֒→ Disc(X) be the inclusion map. We show that Ii = Kp . The
inclusion Ii ⊆ Kp is easy, so we show the reverse inclusion. Let d, d′ be discriminants and
suppose ([d], [d′]) ∈ Kp ; then the underlying invertible sheaves of d, d′ are isomorphic, so we
may assume without loss of generality that d, d′ ∈ (L ∨ )⊗2 . To show ([d], [d′ ]) ∈ Ii , we need
to show that there exist δ, δ ′ ∈ Disc(X; OX ) such that δ ⊗ d′ = δ ′ ⊗ d. For this purpose, we
may take δ = δ ′ = 0. More generally, we could take any δ ∈ (d : d′ ) = {δ ∈ OX (X) : δd′ ∈
hdi} ⊆ OX (X).

3. Quadratic algebras
In this section, we give a monoid structure on the set of isomorphism classes of quadratic
algebras. We begin by discussing the algebras over commutative rings, then work over a
general base scheme. For more on quadratic rings and standard involutions, see Knus [12,
Chapter I, §1.3] and Voight [20, §1–2].
Let R be a commutative ring. An R-algebra is an associative ring B with 1 equipped with
an embedding R ֒→ B of rings (mapping 1 ∈ R to 1 ∈ B) whose image lies in the center
13

of B; we identify R with this image in B. A homomorphism of R-algebras is required to
preserve 1.
Definition 3.1. A quadratic R-algebra (or quadratic ring over R) is an R-algebra S that is
locally free of rank 2 as an R-module.
Let S be a quadratic R-algebra. Then S is commutative, and there is a unique standard
involution on S, an R-linear homomorphism σ : S → S op = S such that σ(σ(x)) = x and
xσ(x) ∈ R for all x ∈ S. Consequently, we have a linear map trd : S → R defined by
trd(x) = x + σ(x) and a multiplicative map nrd : S → R by nrd(x) = xσ(x) = σ(x)x with
the property that x2 − trd(x)x + nrd(x) = 0 for all x ∈ S.
If S is free over R with basis 1, x, then the multiplication table is determined by the
multiplication x2 = tx − n: consequently, we have a bijection
(
)
Free quadratic R-algebras
∼
S over R equipped
−
→ R2
(3.2)
with a basis 1, x
S 7→ (trd(x), nrd(x)) = (x + x, xx) = (t, n).
A change of basis for a free quadratic R-algebra is of the form x 7→ u(x + r) with u ∈ R×
and a ∈ R, mapping
(t, n) 7→ (u(t + 2r), u2(n + tr + r 2 )).

(3.3)

We have identified R ⊆ S as a subring; the quotient S/R is locally free of rank 1. Therefore,
we have a canonical identification
∼ V
S/R −
→ 2S
(3.4)
x + R 7→ 1 ∧ x.
Lemma 3.5. Let S be a quadratic R-algebra. Then the map
V
d : ( 2 S)⊗2 → R
(3.6)
(x ∧ y) ⊗ (z ∧ w) 7→ (xσ(y) − σ(x)y)(zσ(w) − σ(z)w).
is a discriminant.
We have
(3.7)

d((1 ∧ x)⊗2 ) = (x − σ(x))2 = (2x − trd(x))2

= 4x2 − 4x trd(x) + trd(x)2 = trd(x)2 − 4 nrd(x)

in the lemma, as one might expect. We accordingly call the quadratic map d = d(S) in
Lemma 3.5 the discriminant of S.
Proof. We define the map
t:

V2

(S/2S) → R/2R

t(1 ∧ x) = trd(x)

via the identification (3.4). The map t is well-defined since trd(x + r) = trd(x) + 2r ≡ trd(x)
(mod 2R). We then verify that
sq(t)((1 ∧ x)⊗2 ) = t(1 ∧ x)2 = trd(x)2 ≡ trd(x)2 − 4 nrd(x) = d((1 ∧ x)⊗2 )

by (3.7).

14

(mod 4R)


Recall that a commutative R-algebra B is separable if B is projective as a B ⊗R B-module
via the map x ⊗ y 7→ xy. If B ≃ R[x]/(f (x)) with f (x) ∈ R[x], then B is separable if and
only if the ideal generated by f (x) and its derivative f ′ (x) is the unit ideal.
Lemma 3.8. A quadratic R-algebra S is separable if and only if its discriminant d is an
isomorphism.
V
Proof. The map d : ( 2 S)⊗2 → R is an isomorphism if and only if it is locally an isomorphism, so we reduce to the case where S = R[x]/(x2 − tx + n) = R[x]/(f (x)). Then by (3.7),
d is an isomorphism if and only if trd(x)2 − 4 nrd(x) ∈ R× if and only if S is separable [12,
Chapter I, (7.3.4)].

V
Corollary 3.9. If S is a separable quadratic R-algebra over R then 2 S ∈ Pic(R)[2].
Now let X be a scheme.

Definition 3.10. A quadratic OX -algebra is a sheaf S of OX -algebras that is locally free of
rank 2 as a sheaf of OX -modules: there is a basis of open sets U of X such that S (U) is
free of rank 2 as an OX (U)-module.
Equivalently, a quadratic OX -algebra is given by a double cover φ : Y → X, a finite locally
free morphism of schemes of degree 2: the sheaf φ∗ OY is a sheaf of OX -algebras that is locally
free of rank 2. By uniqueness of the standard involution, we obtain a standard involution on
S , a standard involution on S (U) for all open sets U (covering each by affine opens where
S is free), and in particular maps trd and nrd on S .
Analogous to Lemma 3.5, we have the following result.
Proposition
3.11. Let S be a quadratic OX -algebra. Then there is a unique discriminant
V
d : 2 S → OX that coincides locally with the one defined by (3.6).
V
∨
∨ ⊗2
Proof. Let L = 2 S . We must exhibit t ∈ L[2]
such that sq(t) = d[4] ∈ (L[4]
) , where [4]
denotes working modulo 4, as in the previous section. We adapt the argument in Lemma
3.5 to a global setting. Working
V2 first locally, let U = Spec R ⊆ X be an open set where
S (U) = S and L (U) = L =
S. Since trd(x + r) = trd(x) + 2r for r ∈ R and x ∈ S, the
map
V
t : 2 (S/2S) ≃ L/2L → R/2R
x ∧ y 7→ trd(x + y)

is well-defined (since t(x ∧ x) = trd(2x) = 0) and R-linear. This map does not depend on
∨
any choices, so repeating this on an open cover, we obtain an element t ∈ L[2]
.
∨ ⊗2
Now we verify that sq(t) = d[4] ∈ (L[4] ) . We may do so on an open cover, where S is
V
free, so let S = R ⊕ Rx, and 2 S = L = R(1 ∧ x)⊗2 . Then
d((1 ∧ x)⊗2 ) = trd(x)2 − 4 nrd(x) ≡ trd(x)2

(mod 4R).

On the other hand, by definition we have
sq(t)((1 ∧ x)⊗2 ) ≡ t(1 ∧ x)2 = trd(1 + x)2 = (2 + trd(x))2 ≡ trd(x)2
The result follows.

(mod 4R).


15

A quadratic
X -algebra S is separable if and only if d induces an isomorphism of OX V2 O⊗2
∼
modules ( S ) −
→ OX , as this is true on any affine open set.
Let Quad(X) denote the set of isomorphism classes of quadratic OX -algebras, and for an
invertible OX -module L let Quad(X;VL ) ⊆ Quad(X) be the subset of those algebras S
such that there exists an isomorphism 2 S ≃ L of OX -modules. Define similarly Quad(X)
to be the sheaf associated to the presheaf U 7→ Quad(U).
We now give Quad(X) the structure of a commutative monoid.
Construction 3.12. Let X = Spec R and let S = R ⊕ Rx and T = R ⊕ Ry be free quadratic
R-algebras with choice of basis. Let x2 = tx − n and y 2 = sy − m so
t = trd(x), n = nrd(x), s = trd(y), m = nrd(y)

Then we define the free quadratic R-algebra
where
(3.13)

with t, n, s, m ∈ R.

S ∗ T = R ⊕ Rw
w 2 = (st)w − (mt2 + ns2 − 4nm).

Construction 3.12 has been known for some time, e.g., it is given by Hahn [9, Exercises
14–20, pp. 42–43]. (See the introduction for further context and references.)
Lemma 3.14. Construction 3.12 is functorial with respect to the base ring R. The operation
∗ gives the set of free quadratic R-algebras with basis the structure of commutative monoid
with identity element R × R = R[x]/(x2 − x) and absorbing element R[x]/(x2 ).
Proof. Functoriality is clear, and S ∗ T = T ∗ S for all free quadratic R-algebras S, T by the
symmetry of the construction. It is routine to check associativity. To check that E = R × R
is the identity element for ∗ we simply substitute s = 1, m = 0 to obtain S ∗ E = S; and a
similar check works for the absorbing element.

Remark 3.15. Construction 3.12 generalizes the Kummer map, presented in the introduction.
Indeed, suppose that R is a PID or local ring and 2 ∈ R× . Then √
by completing the square,
2
any quadratic
√ R-algebra S√is of the form S = R[x]/(x − n) = R[ n]
√ where n = d(S)/4. So
if S = R[ n] and T = R[ m], then S ∗ T = R[x]/(x2 − 4nm) ≃ R[ nm].
At the same time, Construction 3.12 generalizes Artin-Schreier extensions of fields. Suppose that R = k is a field of characteristic 2. Then every separable extension of k can be
written in the form k[x]/(x2 − x + n), and
k[x]/(x2 − x + n) ∗ k[x]/(x2 − x + m) = k[x]/(x2 − x + (m + n)).

Since 4 = 0, the discriminant of every such algebra has class 1 in R/R×2 .
In the above construction, if S, T are separable over R, so that they are (étale) Galois [13]
extensions of R (with the standard involutions σ, τ respectively as the nontrivial R-algebra
automorphism), then the algebra S ∗ T is the subalgebra of S ⊗R T fixed by the product of
the involutions σ ⊗ τ acting on S ⊗R T [18, Proposition 1].
In all cases, a direct calculation shows that Equation 3.13 is satisfied by the element
w = x ⊗ y + σ(x) ⊗ τ (y) ∈ S ⊗R T ;

this will figure in the proof of Theorem 3.24. However, there is no reason why the R-algebra
generated by w need be free of rank 2 over R; for example, if R has characteristic 2 and
16

σ(x) = x, τ (y) = y, then w = 0. Thus, Construction 3.12 can be thought of as a formal way
to create a fixed subalgebra of S ⊗R T under the involution given by the product of standard
involutions.
Lemma 3.16. Construction 3.12 is functorial with respect to isomorphisms: if
∼

φ : S = R ⊕ Rx −
→ S ′ = R ⊕ Rx′
∼

ψ : T = R ⊕ Ry −
→ T ′ = R ⊕ Ry ′
are R-algebra isomorphisms of quadratic R-algebras, then there is a canonical isomorphism
∼

φ∗ψ :S∗T −
→ S ′ ∗ T ′.
Proof. There exist unique u, v ∈ R× and r, q ∈ R such that φ(x) = ux′ +r and ψ(y) = vy ′ +q.
Because φ is an R-algebra homomorphism, both φ(x) and x satisfy the same unique monic
quadratic polynomial, and from
(ux′ + r)2 = t(ux′ + r) − n
we conclude that
(x′ )2 = u−1 (t − 2r)x′ − u−2 (n − tr − r 2 ) = t′ x − n′

so t = ut′ +2r and n = u2 n′ +tr+r 2 . Similarly, we obtain s = vs′ +2q and m = v 2 m′ +sq+q 2 .
We claim then that the map
∼

φ∗ψ :S∗T −
→ S′ ∗ T ′

(φ ∗ ψ)(w) = (uv)w ′ + (qt + rs − 2qr)

is an isomorphism; for this we simply verify that
((uv)w ′ + (qt + rs − 2qr))2 = st((uv)w ′ + (qt + rs − 2qr)) − (mt2 + ns2 − 4nm)
and the result follows.



Lemma 3.17. Let S , T be quadratic OX -algebras. Then there is a unique quadratic OX algebra S ∗ T up to OX -algebra isomorphism with the property that on any affine open set
U ⊆ X such that S = S (U) and T = T (U) are free, we have
(S ∗ T )(U) ≃ S ∗ T
as in Construction 3.12.
Proof. This proposition is a standard application of gluing; we give the argument for completeness. Let {Ui = Spec Ri } be an open affine cover of X on which
S (Ui ) = Si = Ri ⊕ Ri xi and T (Ui ) = Ti = Ri ⊕ Ri yi
are free. We define (S ∗ T )(Ui ) = Si ∗ Ti = Ri ⊕ Ri wi according to Construction 3.12.
We glue these according to the isomorphisms
on S and T using Lemma 3.16, as follows.
S
We have Ui ∩ Uj = Uj ∩ Ui = k Uijk covered by open sets Uijk = Spec Rik ≃ Spec Rjk
distinguished in Ui and Uj . Because S is a sheaf, we have compatible isomorphisms
φijk : Rik ⊕ Rik xi = S (Spec Rik ) ≃ S (Spec Rjk ) = Rjk ⊕ Rjk xj
17

for each such open set. Similarly, we obtain compatible isomorphisms ψijk for T over the
same open cover. By Lemma 3.16, we obtain compatible isomorphisms
φijk ∗ ψijk : (S ∗ T )(Spec Rik ) ≃ (S ∗ T )(Spec Rjk )

and can thereby glue on X to obtain a quadratic OX -algebra, unique up to OX -algebra
isomorphism.

Corollary 3.18. Construction 3.12 gives Quad(X) the structure of a commutative monoid,
functorial in X, with identity element the isomorphism class of OX × OX .
Proof. Lemma 3.17 shows that Construction 3.12 extends to X and is well defined on the
set of isomorphism classes Quad(X) of quadratic OX -algebras. To check that we obtain a
functorial commutative monoid, it is enough to show this when X is affine, and this follows
from Lemmas 3.14 and 3.16.

Lemma 3.19. If S is a separable quadratic OX -algebra, then S ∗ S ≃ OX × OX .

Proof. By gluing, it is enough to show this on an affine cover. Suppose S = R[x]/(x2 −tx+n)
has discriminant d = t2 − 4n. Then by definition, we have
S ∗ S = R[w]/(w 2 − t2 w + 2n(t2 − 2n));

with the substitution w ← w − 2n, we find that S ∗ S ≃ R[w]/(w 2 − dw). Since d ∈ R× , the
replacement w ← wd−1 yields an isomorphism S ∗ S ≃ R × R.

Lemma 3.20. If S , T ∈ Quad(X) then
and

d(S ∗ T ) = d(S )d(T ) ∈ Disc(X)
V2

(S ∗ T ) ≃

V2

S ⊗

V2

T.

Proof. If S = S (U) and T = T (U) are as in Construction 3.12, then
(3.21)

d(S ∗ T )((1 ∧ (x ⊗ y))⊗2 ) = (st)2 − 4(mt2 + ns2 − 4nm) = (t2 − 4n)(s2 − 4m)
= d(S)((1 ∧ x)⊗2 )d(T )((1 ∧ y)⊗2)

The first statement then follows. For the second, again on affine opens we have the isomorphism
V2
V
V
S ⊗R 2 T → 2 (S ∗ T )
(3.22)
(1 ∧ x) ⊗ (1 ∧ y) 7→ 1 ∧ w,
which glues to give the desired isomorphism globally.



Lemma 3.23. The discriminant maps
disc : Quad(X) → Disc(X) and disc : Quad(X; OX ) → Disc(X; OX )
are surjective homomorphisms of sheaves of commutative monoids.
Proof. The fact that these maps are homomorphisms of monoids follows locally from Lemma
3.20. We show these maps are surjective locally, and for that we may assume X = Spec R
and L = Re. We refer to Lemma 2.13 and Example 2.14: given any d ∈ R such that
d = t2 − 4n with t, n ∈ R, we have the quadratic ring R[x]/(x2 − tx + n) of discriminant
d.

18

We are now ready to prove Theorem A.
Theorem 3.24. Construction 3.12 is the unique system of binary operations
∗ = ∗X : Quad(X) × Quad(X) → Quad(X),

one for each scheme X, such that:
(i) Quad(X) is a commutative monoid under ∗, with identity element the class of OX × OX ;
(ii) For each morphism f : X → Y of schemes, the diagram
Quad(Y ) × Quad(Y )

∗Y

//

Quad(Y )
f∗



Quad(X) × Quad(X)

∗X

//



Quad(X)

is commutative; and
(iii) If X = Spec R and S, T are separable quadratic R-algebras with standard involutions
σ, τ , then S ∗ T is the fixed subring of S ⊗R T under σ ⊗ τ .
Proof. By (3.2), the universal free quadratic algebra with basis is the algebra
S univ = Runiv [x]/(x2 − tx + n)

where Runiv = Z[t, n] is the polynomial ring in two variables over Z: in other words, for
any commutative ring R and free quadratic R-algebra S with basis, there is a unique map
f : Runiv → R such that S = f ∗ S univ = S univ ⊗f,Runiv R. By (ii), then, following this argument
on an open affine cover, we see that the monoid structure on Quad(Spec Runiv ) determines
the monoid structure for all schemes X.
Dispensing with subscripts, consider S = R[x]/(x2 − tx + n) and T = R[y]/(y 2 − sy + m)
where R = Z[t, n, s, m]; we show there is a unique way to define S ∗ T .
To begin, we claim that S ∗ T is free over R. As R-modules, we can write S ∗ T = R ⊕ Iz
where I ⊆ F = Frac(R) is a projective R-submodule of F and the class [I] ∈ Pic(R)
well-defined. But Pic(Z[t, n, s, m]) ≃ Pic(Z) = {0} (Z is seminormal [8]), so I ≃ R.
Now let
D = (t2 − 4n)(s2 − 4m).
Then S[1/D] and T [1/D] are separable over R[1/D], with involutions σ(x) = t − x and
τ (y) = s − y. By (iii), the product S[1/D] ∗ T [1/D] is the subring of S[1/D] ⊗R[1/D] T [1/D]
generated by
Then

z = x ⊗ y + σ(x) ⊗ τ (y) = 2(x ⊗ y) − s(x ⊗ 1) − t(1 ⊗ y).
z 2 = x2 ⊗ y 2 + 2nm + σ(x)2 ⊗ τ (y)2

= (tx − n) ⊗ (sy − m) + 2nm + (tσ(x) − n) ⊗ (sτ (y) − m)

= ts(x ⊗ y + σ(x) ⊗ τ (y)) − mt((x + σ(x)) ⊗ 1) − ns(1 ⊗ (y + τ (y))) + 4nm

= (st)z − (mt2 + ns2 − 4nm).

In particular, S[1/D] ∗ T [1/D] ≃ R[1/D] ⊕ R[1/D]z.
By (ii), (S ∗ T )[1/D] ≃ S[1/D] ∗ T [1/D], and we have S ∗ T ⊆ (S ∗ T )[1/D]. Since R
is a UFD and S ∗ T is free over R, it is generated as an R-algebra by an element of the
19

form (az + b)/D k for some a, b ∈ R and k ∈ Z≥0 . But by (3.21), d((S ∗ T )[1/D]) = D, so
d(S ∗ T ) = (a/D k )2 D ∈ R, thus a/D k ∈ R. Since trd((az + b)/D k ) = (ast + 2b)/D k ∈ R,
we conclude that 2b/D k ∈ R; since D is not divisible by 2, by Gauss’s lemma we have
b/D k ∈ R, so without loss of generality we may take b = 0 and S ∗ T is generated by az for
some a ∈ R. Since R× = {±1} and (S ∗ T )[1/D] is generated by z, we must have a = D k
for some k ∈ Z≥0 . Finally, we consider
(x2

Z[x]
Z[y]
Z[z]
∗ 2
= 2
k
− tx + n) (y − y)
(z − D tz + D 2k n)

over Z[t, n]. The algebra on the right has discriminant D 2k (t2 − 4n), but by (i), it must be
isomorphic to the algebra on the left of discriminant (t2 − 4n), so we must have D 2k = 1, so
k = 0. Therefore S ∗ T = R ⊕ Rz.

Having given the monoid structure, we conclude this section by proving Theorem B.
Theorem 3.25. Let X be a scheme. Then the following diagram of sheaves of commutative
monoids is functorial and commutative with exact rows and surjective columns:
Quad(X; OX )

Quad(X)
//

disc



∧2

//

Pic(X)
≀

disc

Disc(X; OX )
//



Disc(X)
//



Pic(X)

Proof. The exactness of the bottom row follows from Proposition 2.16. The exactness of
the top row and commutativity of the diagram follows by the same (trivial) argument.
Surjectivity follows from Lemma 3.23.

4. Proof of Theorem C
In this section, we prove Theorem C and conclude with some final discussion.
Let R be a commutative ring and let R[4] = {a ∈ R : 4a = 0}. Let
℘(R) = {r + r 2 : r ∈ R}.

and let ℘(R)[4] = ℘(R) ∩ R[4]. Note that 4(r + r 2 ) = 0 if and only if (1 + 2r)2 = 1, so we
have equivalently
℘(R)[4] = {r + r 2 : r ∈ R and (1 + 2r)2 = 1}.
Lemma 4.1. ℘(R)[4] is a subgroup of R[4] under addition.
Proof. We have 0 = 0 + 02 ∈ ℘(R)[4]. If n = r + r 2 ∈ ℘(R)[4] and m = s + s2 ∈ ℘(R)[4] then
(r + s + 2rs) + (r + s + 2rs)2 = (r + r 2 ) + (s + s2 ) + 4(r + r 2 )(s + s2 ) = n + m

and 4(n + m) = 0, so n + m ∈ ℘(R)[4]. Finally, if n ∈ ℘(R)[4] then −n = 3n ∈ ℘(R)[4] by
the preceding sentence.

We define the Artin-Schreier group AS(R) to be the quotient
AS(R) =

R[4]
.
℘(R)[4]

Since 2R[4] ⊆ ℘(R)[4], the group AS(R) is an elementary abelian 2-group.
20

We define a map i : AS(R) → Quad(R; R) sending the class of n ∈ AS(R) to the isomorphism class of the algebra S = R[x]/(x2 − x + n).
Proposition 4.2. The map i : AS(R) → Quad(R, R) is a (well-defined) injective map of
commutative monoids.
Proof. Let S = i(n) = R[x]/(x2 −x+n) and T = i(m) = R[y]/(y 2 −y+m) with n, m ∈ AS(R).
Then S ≃ T if and only if y = u(x + r) for some u ∈ R× and r ∈ R, which by (3.3) holds if
and only if u(1 + 2r) = 1 and u2 (n + r + r 2) = m; these are further equivalent to 1 + 2r ∈ R×
and
n + r + r 2 = m(1 + 2r)2 = (1 + 4r + 4r 2 )m.
But 4m = 0 so n + r + r 2 = m and since 4n = 0 we have 4(r + r 2 ) = 0. Thus S ≃ T if and
only if (1 + 2r)2 = 1 and n + r + r 2 = m, as desired. It follows from Construction 3.12 that
S ∗ T = R[w]/(w 2 − w + (n + m)), since 4nm = 0, and i(0) = R[w]/(w 2 − w) is the identity,
so i is a homomorphism of monoids.

We now prove Theorem C, and recall Quad(R; R) is the set of isomorphism classes of free
quadratic R-algebras.
Theorem 4.3. Let R be a commutative ring and let d ∈ R be a discriminant. Then the fiber
disc−1 (d) of the map
disc : Quad(R; R) → Disc(R; R)
above d has a unique action of the group AS(R)/ annR (d)[4] compatible with the inclusion of
monoids AS(R) ֒→ Quad(R; R).

Proof. A (free) quadratic R-algebra with basis 1, x such that (x − σ(x))2 = d is of the form
S = R[x]/(x2 − tx + n) with t2 − 4n = d. Let m ∈ R[4]. Then by Construction 3.12 we have
S ∗ (R[y]/(y 2 − y + m)) = (R[y]/(y 2 − y + m)) ∗ S = R[w]/(w 2 − tw + dm + n)

since 4m = 0 so dm = (t2 − 4n)m = t2 m. Thus we have an action of R[4] on the set of
these quadratic R-algebras with basis and a free action of R[4]/ annR (d)[4]. Two quadratic
R-algebras S and S ′ are in the same orbit if and only if t′ = t and n′ = dm + n for
some m ∈ R[4] if and only if t′ = t and n′ − n ∈ dR[4]; therefore, the orbits are indexed
noncanonically by the set
{t ∈ R : t2 ≡ d (mod 4R)} × R[4]/dR[4].
We now descend to isomorphism classes: by Proposition (4.2), the monoid multiplication ∗
is well-defined on isomorphism classes, giving the unique action of AS(R)/ annR (d)[4] on the
fiber over d.

Under favorable hypotheses, the action of AS(R) is free, and so we make the following
definition.
Definition 4.4. An element t ∈ R is sec (an abbreviation for square even cancellative) if the
following two conditions hold:
(i) t is a nonzerodivisor, and
(ii) r 2 , 2r ∈ tR implies r ∈ tR for all r ∈ R.
21

A quadratic R-algebra S is sec if disc(S) is a nonzerodivisor and there a basis 1, x for S
such that trd(x) is sec. A quadratic OX -algebra S is sec if S is sec on an open affine cover
of X.
Proposition 4.5. Let R be a commutative ring. Then the action of AS(R) on the set of sec
quadratic R-algebras of discriminant d is free.
Proof. We continue as in the proof of Theorem 4.3. Let m ∈ R[4] and let S be a sec quadratic
R-algebras with discriminant d. Let 1, x be a basis for S = R[x]/(x2 − tx + n) such that
t = trd(x) is sec. To show the action is free, suppose that
(4.6)

S ∗ (R[y]/(y 2 − y + m)) = R[w]/(w 2 − tw + dm + n) ≃ S;

we show that m ∈ ℘(R). Equation (4.6) holds if and only if there exist u ∈ R× and r ∈ R
such that t = u(t + 2r) and dm + n = u2(n + tr + r 2 ). Consequently, u2 d = d. Since S is
sec, d is a nonzerodivisor, so u2 − 1 = 0, and thus t2 m = dm = tr + r 2 so r 2 = (tm − r)t.
We also have 2r = (1 − u)t, so since t is sec and r 2 , 2r ∈ tR, we conclude that r ∈ tR. Let
r = at with a ∈ R. Then substituting, we have t2 (a2 + a − m) = r 2 + tr − t2 m = 0. Since t
is a nonzerodivisor, we conclude that a2 + a − m = 0, so m ∈ ℘(R) as claimed.

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[19] The Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2015.
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Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave,
Burlington, VT 05401, USA; Department of Mathematics, Dartmouth College, 6188 Kemeny
Hall, Hanover, NH 03755, USA
E-mail address: jvoight@gmail.com

23

