Éléments de Géométrie Algébrique — English

§V.2.15 and §V.2.16. Jacobian and regularity supplements (formerly EGA IV §§17.15, 17.16)

These two short sections were originally drafted as §§17.15 and 17.16 of EGA IV, then re-allocated to Chapter V (§§2.15 and 2.16) without ever being published in either place. They sit logically as supplements to the Jacobian/regularity material of $(0_{IV},22)$ and (IV, 17): §V.2.15 introduces the notions of smooth form and of elementary singular point of multiplicity $n$, together with the local characterization that pins down complete intersections with an elementary singularity at a point. §V.2.16 is the technical appendix on smooth quadratic forms — the case $n=2$ of §V.2.15 — culminating in the equivalence between smoothness of $Q$, invertibility of the corrected discriminant $\tilde{d}(Q)$, and local isomorphism (fppf and, in good rank/characteristic, étale) of $Q$ to the standard quadratic form.

The §V↔§IV correspondence is given in the front matter; we lead with the V numbering and attach (formerly IV, M) parenthetically at the first occurrence of each cross-reference into the old numbering.

V.2.15.1. Smooth forms

Grothendieck note. I have just noticed that the terminology introduced in my formulation of §V.1 for supersingularity is unreasonable and conflicts in particular with the recent terminology. In any case, you must have noticed that in Definition (1.1)(c) of the text it should read "degenerate" in place of "non-degenerate". The canular¹ is given by varieties of even dimension in characteristic two defined, for example, by an equation $\varphi =0$ in an ambient variety of odd dimension: with the terminology of my notes such a variety cannot have an "ordinary singular point" (= ordinary quadratic singularity), i.e. $\varphi$ cannot have an "ordinary singular zero", because in characteristic two a quadratic form in an odd number of variables is always "degenerate". But the whole world has always considered that even in characteristic two the origin of the affine cone $X^2+YZ=0$ is an ordinary quadratic singularity. In the present notes I give the notion of a smooth quadratic form (or "ordinary") on a locally free module of finite type, in such a way that non-degenerate ⟹ smooth, the converse being true if $E$ is of even rank or all residual characteristics of the base $S$ are different from two. With this notion introduced, I propose to preserve the terminology "supersingular" (which does not conflict with any recent terminology) for the §V.1 notion, which corresponds to a non-degenerate quadratic form; we will also speak of a supersingular point of a locally Noetherian prescheme, or of a geometrically supersingular point for a prescheme locally of finite type over a field, in the same vein (by considering the homomorphism $Sym(\mathfrak{m}/{\mathfrak{m}}^2)\to >g{r}_{\mathfrak{m}}(A)$ and requiring that the kernel cannot be generated by a non-degenerate quadratic form). What we therefore wrongly called in §V.1 an "ordinary singular zero" is called "singular non-supersingular zero", or "singular zero with non-degenerate quadratic part", if one wishes a terminology close to the recent one in characteristic zero. In the same way we can speak of a "non-degenerate quadratic singularity" when one is locally of finite type over a field $K$.

By contrast the terminology "ordinary singular zero" — better "ordinary quadratic zero" — and "ordinary quadratic singularity", "geometrically ordinary quadratic singularity", can be extended (in conformity with usage) so as to correspond to the notion of a smooth quadratic form. In addition, it may be advisable to replace the word "ordinary" by "elementary" and to extend this terminology to singularities not necessarily quadratic but of any multiplicity. Tell me (Dieudonné) your impression on this. It seems on the other hand that the text of §V.1 is formally correct: in particular, in Proposition (1.7) the notion really introduced is that of a supersingular zero.²

This does not preclude that it would be proper, at least as a remark, to introduce the subscheme $V(\varphi {)}_{ultsing}$ of $V(\varphi {)}_{supsing}$ of ultra-singular zeros — those that are singular without being elementary quadratic; they will be described essentially by the same procedure as $V(\varphi {)}_{supsing}$, by taking the prescheme of zeros of the "corrected discriminant" introduced later in place of the ordinary discriminant. The only corrections to §V.1 thus seem to be of terminological order, and they should of course bear equally on the terminology of §V.1 page 10.

Grothendieck note. (IV, 23.9.2) is false as such except when the characteristic of $K$ is not two or the irreducible components of $X$ are of even dimension. In the general case one must suppress "$f^n$ satisfies the equivalent conditions of (IV, 8.8), in particular"; the rest of the proposition seems correct, and one should be able to prove it in an analogous manner by showing that for given $x\in X(k)$ there is a hypersurface $H$ of given degree $\ge 2$ tangent to $X$ at $x$ and such that $x$ is an ordinary quadratic point of $X\cap H$, of dimension one less.³ One must put one's foot in it, by remarking that if $k$ is of characteristic two and $X$ is connected non-empty of odd dimension then the conditions of (IV, 8.8) are not verified — i.e. $L/K$ is necessarily inseparable (I believe of degree two exactly, if $X={\mathbb{P}}^r$, but without guarantee).

We now turn to the smooth-form definition itself.

Let $S$ be a prescheme, $E$ a locally free module of finite type over $S$, and $\varphi$ a section of $Sy{m}^n(E)$, i.e. an "$n$-form on $E^{\vee }$", where $n\ge 0$. Giving $\varphi$ is equivalent to giving a section of ${\mathcal{O}}_P(n)$ over $P=\mathbb{P}(E)$ (cf. (III)), and therefore defines a subscheme $V(\varphi )$ of $P$. We say that the form $\varphi$ is smooth if $V(\varphi )$ is smooth and if in addition, for every $x\in S$, $V(\varphi {)}_s\ne P_s$ — i.e. $\varphi (s)\in Sy{m}_{k(s)}^n(E\otimes k(s))$ is not zero.

One sees immediately that $\varphi$ is smooth if and only if, for every $s\in S$, $\varphi (s)$ is smooth (which reduces matters to the spectrum of a field), and that the notion of a smooth form is invariant under change of base field (which reduces us to the case of an algebraically closed field). These two properties can be summarized by saying that if $S^{\prime }\to S$ is a surjective morphism then $\varphi$ is smooth if and only if its inverse image ${\varphi }^{\prime }$ on $S^{\prime }$ is such. Of course, if $E$ is a projective module of finite type over a ring $A$ and $\varphi \in Sy{m}_A^n(E)$, we say that $\varphi$ is smooth if the corresponding section on $\mathrm{Spec}(A)$ is smooth. Since for $E=A^{r+1}$ to give $\varphi$ is equivalent to giving a homogeneous polynomial of degree $n$ in the variables $X_0,X_1,\cdots ,X_r$, the Jacobian criterion implies immediately that $\varphi$ is smooth if and only if the ideal generated by $\varphi$ and the $\partial \varphi /\partial X_i$ contains a power of the augmentation ideal $(X_0,\cdots ,X_r)$ — equivalently, contains a power of each variable $X_i$. More precisely, the subscheme of ${\mathbb{P}}^r$ defined by the homogeneous ideal generated by $\varphi$ and the $\partial \varphi /\partial X_i$ is exactly the set of points of $V(\varphi )$ at which $V(\varphi )$ fails to be smooth over $\mathrm{Spec}(A)$ with relative dimension $r-1$.

V.2.15.2. Elementary augmentations

Definition (2.15.2).

Let $X$ be a prescheme and $\mathcal{J}$ a quasi-coherent ideal of ${\mathcal{O}}_X$. We say that the augmentation ${\mathcal{O}}_X\to {\mathcal{O}}_{X/\mathcal{J}}$ is an elementary augmentation of multiplicity $n$ if it satisfies the following conditions:

(a) $\mathcal{J}/{\mathcal{J}}^2={\mathcal{N}}_{X/Y}$ is locally free of finite type over $Y=V(\mathcal{J})$.

(b) The kernel of the canonical homomorphism $\Psi :Sy{m}_{{\mathcal{O}}_Y}(\mathcal{N})\to g{r}_{\mathcal{J}}({\mathcal{O}}_X)$ is generated by the kernel $\mathcal{K}$ of ${\Psi }_n$, which is an invertible submodule, locally a direct factor of $Sy{m}_{{\mathcal{O}}_Y}^n(\mathcal{N})$ — i.e. it is locally generated over $Y$ by a section $\varphi$ that is nowhere zero.

(c) The said $\varphi$ (which is defined locally only up to multiplication by a unit) is a smooth form.

Grothendieck note: one could have introduced in §V.2.15.1 the notion of an invertible smooth submodule of $Sy{m}^n(E)$. From the geometric standpoint this is more important than the notion of a smooth section, since $V(\varphi )$ is in fact defined by such a submodule. It would have the advantage of merging conditions (b) and (c) above into a single condition.

If $n=2$ we speak of an augmentation of elementary quadratic type. If $A$ is a ring with an augmentation $A\to A/\mathcal{J}$, we again agree to say that this augmentation is elementary of multiplicity $n$ if it is such on $\mathrm{Spec}(A)$.

If $A$ is a local ring we will say (by abuse of language) simply that $A$ is "elementary of multiplicity $n$" if the augmentation $A\to A/r(A)$ is elementary of multiplicity $n$; note that this implies $n\ge 2$, and if $A$ is Noetherian, that $A$ is necessarily not regular.

If $X$ is a prescheme and $n\ge 2$ is an integer, we say that $x\in X$ is an elementary singular point of multiplicity $n$ if its local ring ${\mathcal{O}}_{X,x}$ is elementary of multiplicity $n$. (Note that this terminology agrees with the general notion of multiplicity due to Samuel.) For $n=2$ we speak in particular of an elementary quadratic singularity (or ordinary, in the classical terminology). We also introduce, in conformity with general usage, the "geometric" variants for $X$ locally of finite type over a field: $x\in X$ is called a geometrically elementary singularity of multiplicity $n$ if for every (or, what is the same, for some) extension $K$ of $k$ and every (or some) point $z$ of X_K over $x$ rational over $K$, $z$ is an elementary singularity of multiplicity $n$.

V.2.15.3. Generalization of Proposition (1.6) to multiplicity $n$

Grothendieck note. "Generalize" the unwritten Proposition (1.6) (cf. the marginal aside in §V.1.5) to the case of multiplicity $n$.⁴

V.2.15.4. Generalization of Proposition (1.7) and Corollary to multiplicity $n$

Grothendieck note. "Generalize" Proposition (1.7) and its corollary to the case of multiplicity $n$. The idea is not so much to generalize as to give variants.

V.2.15.5. Elementary singular zeros of a section

Grothendieck note. Introduce the notion of an elementary singular zero of multiplicity $n$ (for $n=2$, a singular quadratic elementary zero) of a section $\varphi$ of ${\mathcal{O}}_X$, or more generally of a section of a locally free module on an $X$ locally Noetherian, and the corresponding geometric notion (over a base field $K$).

V.2.15.6. Local characterization of elementary singularities

Proposition (2.15.6).

Let $X$ be a subscheme of a locally Noetherian regular prescheme $X^{\prime }$, and let $x\in X$. For $x$ to be an elementary singularity of multiplicity $n$ of $X$ it is necessary and sufficient that there exist an open neighbourhood $U$ of $x$ and an ${\mathcal{O}}_U$-regular sequence ${\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_d$ such that $X{|}_U=V({\varphi }_1,\cdots ,{\varphi }_d)$ and such that $x$ is an elementary singular zero of multiplicity $n$ of $({\varphi }_1,\cdots ,{\varphi }_d)$. In particular, if $A$ is a Noetherian local ring that is elementary of multiplicity $n$, then $A$ is a complete-intersection ring. We see, using §V.2.15.4, that we can in addition find in a neighbourhood of $x$ a regular subprescheme $X^{\prime \prime }$ of $X^{\prime }$ containing $X$ such that $X$ is described in $X^{\prime \prime }$ by a single equation $\varphi =0$ admitting $x$ as an elementary singular zero of multiplicity $n$.

It follows in particular that if $A$ is a Noetherian local ring that is elementary of multiplicity $n$, then $A$ is Cohen–Macaulay. We prove in §V.5 (formerly EGA IV §20) (by an easy blow-up calculation) that the closed point of $\mathrm{Spec}A$ is the only singular point of $\mathrm{Spec}A$; from this it follows that $A$ is normal if and only if $\dim A\ge 2$, reduced if and only if $\dim A\ge 1$. If $\dim A=0$, then $A$ is elementary of multiplicity $n$ if and only if $\mathfrak{m}/{\mathfrak{m}}^2$ is of rank 1 and $n$ is the smallest integer such that ${\mathfrak{m}}^n=0$; such rings are also the quotients of discrete valuation rings by the $n$-th power of their maximal ideal.

V.2.15.7. Geometric variants of (2.15.6)

This is the place to state the "geometric" variants of (2.15.6). We find in particular that if $X$ is a prescheme locally of finite type over a field $k$, $x\in X$, and $n={\dim }_{x}X$, then $x$ is an elementary singularity of multiplicity $n$ of $X$ if and only if there exists an open neighbourhood $U$ of $x$ such that $U$ can be embedded as a subprescheme of a smooth prescheme $X^{\prime }$ over $k$, connected and of dimension $n+1$, defined by an equation $\varphi =0$, with $x$ a geometrically ordinary singular zero of multiplicity $n$ of $\varphi$. We will also say, as in §V.2.15.5, that this means that the value at $x$ of the principal part $d^n\varphi$ of $\varphi$ — which a priori is an element of ${\mathcal{P}}_{X^{\prime }/k}^n\otimes k(x)$, and more precisely lies in its augmentation ideal $I$ — is in fact an element of $I^n=Sy{m}_{{\mathcal{O}}_U}^n({\Omega }_{X^{\prime }/k}^1\otimes k(x))$, and as such is a smooth form.

Grothendieck note. Recall, from §V.2.15.1, that the set of points of smoothness of a form is open.

We remark also that such a point is isolated in the set of non-smooth points; it is geometrically normal (resp. geometrically reduced) if and only if $d\ge 2$ (resp. $d\ge 1$). If $d=0$ and $k(x)=k$ (i.e. $x$ is isolated and $k$-rational), then "$x$ is an elementary singular point of multiplicity $n$" means that ${\mathcal{O}}_{X,x}$ is $k$-isomorphic to $k[T]/(T^n)$.

If $n=1$ and $x$ is rational over $k$, then the notion of elementary singular point of multiplicity $n$ corresponds in the classical terminology to a "point with $n$ distinct tangents".

Grothendieck note. We have already made explicit in §V.2.15.2 that if $x$ is a point of $X$ rational over $k$, then the notion of elementary singularity of multiplicity $n$ in the absolute or relative sense is the same; this remains true if $k(x)$ is a finite separable extension of $k$, and probably "finite" is not needed. This fact deserves to be inserted as a corollary or as a proposition.

Grothendieck note. I have included in §V.5 (formerly EGA IV §20) a section on blowing up a prescheme $X$ along a closed subprescheme $Y$ such that the augmentation ${\mathcal{O}}_X\to {\mathcal{O}}_Y$ is elementary of multiplicity $n$ (to which I allude in §V.2.15.6). What is involved there is a short section whose only ingredients are the general smoothness results of (IV, 17) and the definition at the start of §V.2.15.2 (and one could even avoid the latter). There would be no harm in incorporating those results from here on, for example immediately after Definition (2.15.1) above. Then the "geometric" range of §V.2.15.2 and the continuations §V.2.15.3 through §V.2.15.7 could be separated from those results by grouping them in a No. (2.16).

In any event, I noticed during the writing that the new foreseen No. (2.15) on smooth forms will expand into at least two quite distinct and independent sections: one containing the general "sorites" for "elementary of multiplicity $n$" stuff (any $n$), the other containing the characterization of smooth quadratic forms, which does not borrow from the preceding except in §V.2.15.1, i.e. essentially the definition of the smoothness of a form.

§V.2.16. Appendix on smooth quadratic forms (formerly EGA IV §17.16)

V.2.16.1. The discriminant of a quadratic form

Let $Q$ be a section of $Sy{m}^2(E)$ (with $E$ locally free of finite rank) such that $Q$ — which one can also interpret as a quadratic form on $E^{\vee }$ — defines a symmetric bilinear form $B$ on $E^{\vee }$, hence a homomorphism $E^{\vee }\to E$; by passing to determinant modules one finds $det{E}^{\vee }\to detE$, and finally a section $d(Q)\in \Gamma (det(E{)}^{\otimes 2})$, called (up to a minor abuse⁵) the discriminant of the quadratic form $Q$. If $E={\mathcal{O}}_S^n$ this is simply the section of ${\mathcal{O}}_S$ given by the determinant of the $n\times n$ matrix with coefficients in ${\mathcal{O}}_S$ expressing $Q$.

In all that follows, let us first note:

Proposition (2.16.2).

Suppose that for every $s\in S$ we have either $chark(s)\ne 2$, or that the rank of $E\otimes k(s)$ is even. Then in order for $Q$ to be smooth it is necessary and sufficient that $Q$ be "non-degenerate", i.e. that $d(Q)$ be invertible. (Without restriction on $S$ or $E$, the condition is sufficient: this last assertion is recorded only for convenient later reference, since it is trivially contained in the first, the hypothesis that $Q$ is non-degenerate forcing $E$ to be of even rank at every point where the characteristic is 2.)

This can be proved directly without consideration of characteristic, by placing oneself over an algebraically closed field and choosing a basis: the condition expresses that the $\partial Q/\partial X_i$ have no common non-trivial zeros, a fortiori they do not have a common non-trivial zero with $Q$. If the characteristic is not 2, the converse holds: by virtue of the formula

$$2Q=\sum X_i\cdot (\partial Q/\partial X_i),$$

we see that every common zero of the $\partial Q/\partial X_i$ is also a zero of $Q$.

Finally, if $chark=2$, the bilinear form $B$ associated with $Q$ (defined by the matrix of the $\partial Q/\partial X_i$) is alternating, so that its "kernel" $N\subset E^{\vee }$ is such that $E^{\vee }/N$ is of even rank; thus if $E$ is itself of even rank, the same is true of $N$. Consequently, if $N\ne 0$, the rank of $N$ is at least two, so there exists at least one non-trivial zero of $Q$ on $N$, i.e. $Q$ is not smooth.

Grothendieck note. This recourse to coordinates is decidedly offensive to us; we need it only to prove simply the following lemma.

Lemma (2.16.3).

Let $E$ be a vector bundle of finite rank over a field $k$, $Q\in Sy{m}^2(E)$, $x$ a non-zero element of $E^{\vee }$, and $x_0$ its image in $\mathbb{P}(E)$. Suppose $Q(x,x)=0$, i.e. $x_0\in V(Q)$. Then for $V(Q)$ to be smooth at $x_0$ and of dimension $(rankE)-1$ (i.e. $Q\ne 0$) it is necessary and sufficient that $x_0$ not belong to the kernel of the homomorphism $E^{\vee }\to E$ defined by the symmetric bilinear form associated with $Q$.

V.2.16.4. The standard quadratic form and the corrected discriminant

The study that follows is designed essentially to give a smoothness criterion for a quadratic form in the case not covered by Proposition (2.16.2), i.e. essentially the case of a vector bundle of odd rank over a field of characteristic two.

In this case every quadratic form $Q$ on $E^{\vee }$ is degenerate, but one sees easily (by taking, for instance, the "standard form") that it can still be smooth.

For every integer $n>0$ let us introduce the standard quadratic form $Q_n$ on ${\mathbb{Z}}^n$ as a form with integer coefficients in the $n$ variables $X_1,\cdots ,X_n$, and let us distinguish the two cases.

(a) $n=2m$:

  Q_{2m}(X_1, …, X_{2m}) = X_1 X_2 + X_3 X_4 + ⋯ + X_{2m−1} X_{2m}.

(b) $n=2m+1$:

  Q_{2m+1}(X_1, …, X_{2m+1}) = Q_{2m}(X_1, …, X_{2m}) + X_{2m+1}².

Lemma (2.16.5).

Let $n$ be an odd integer, and consider

  Q(X_1, …, X_n) = ∑_i a_i X_i² + ∑_{i < j} b_{ij} X_i X_j,

a quadratic form in $n$ variables with indeterminate coefficients $a_i$, $b_{ij}$. Then the discriminant $d(Q)$ is a polynomial with integer coefficients in the $a_i$, $b_{ij}$, and the content of this polynomial is equal to two — i.e. the greatest common divisor of its coefficients is 2.

Let us first prove that the content is a multiple of 2. This means that if we specialize the polynomial over the field $\mathbb{Z}/2\mathbb{Z}$ it is identically zero, which follows from the fact that a quadratic form of odd degree over a field of characteristic 2 is always degenerate — i.e. has discriminant zero. To prove that the content is exactly 2 it suffices to compute the discriminant of the standard form of degree $n$: on the one hand it must be a multiple of the content; on the other hand, the calculation gives 2 (since it is 1 for the even-degree summands and 2 for the single odd-degree summand).

It therefore makes sense to introduce the polynomial $\tilde{d}(Q)=(1/2)d(Q)$ in the coefficients of $Q$, which again has integer coefficients, so that it takes a well-defined value if we specialize the $a_i$, $b_{ij}$ to any ring $A$ — i.e. for any quadratic form of degree $n$ with coefficients in $A$.

We call $\tilde{d}(Q)$ the adjusted discriminant polynomial of the indeterminate quadratic form $Q$ in $n$ variables, and its value relative to the coefficients of a quadratic form $q$ with coefficients in any ring $A$ is called the adjusted (or corrected) discriminant of $q$.

More generally, one deduces in an essentially trivial fashion from Lemma (2.16.5) the following statement.

Proposition (2.16.6).

To every prescheme $S$, every locally free module of finite rank $E$ on $S$, and every quadratic form $Q\in \Gamma (Sy{m}^2(E))$ one can associate, in a unique way, a section $\tilde{d}(Q)$ of $det(E{)}^{\otimes 2}$ satisfying the following conditions:

(a) Compatibility with base change, and functoriality with respect to isomorphisms of $E$.

(b) If $E$ is everywhere of even rank, then $\tilde{d}(Q)=d(Q)$, where $d(Q)$ denotes the (ordinary) discriminant of $Q$.

(c) If $E$ is everywhere of odd rank, then $2\tilde{d}(Q)=d(Q)$.

Grothendieck note. It is not reasonable to announce the property of compatibility with base change without announcing at the same time, or even beforehand, functoriality with respect to isomorphisms — given that base change over $E$ itself cannot be defined except up to isomorphism. It would be otherwise if we restricted ourselves to the case $E={\mathcal{O}}_S^n$, which would not be convenient (suitable) for the references.

Definition (2.16.7).

The section $\tilde{d}(Q)$ of $det(E{)}^{\otimes 2}$ is called the corrected (or adjusted) discriminant of the quadratic form $Q$.

Corollary (2.16.8).

Let $Q$, $Q^{\prime }$ be two quadratic forms on $E$ and $E^{\prime }$, respectively, and suppose the parity of the rank of $E$ (resp. $E^{\prime }$) is constant over $S$. Then:

(a) If $E$ and $E^{\prime }$ are not both of odd rank, then $d(Q\oplus Q^{\prime })=d(Q)\cdot \tilde{d}(Q^{\prime })$.

(b) If $E$ and $E^{\prime }$ are both of odd rank, then $d(Q\oplus Q^{\prime })=2d(Q)\cdot \tilde{d}(Q^{\prime })$.⁶

The verification is trivial.

V.2.16.9. The principal smoothness theorem

Theorem (2.16.9).

Let $E$ be a locally free sheaf of finite rank over a prescheme $S$, and $Q\in \Gamma (Sy{m}^2(E))$. The following conditions are equivalent:

(i) $Q$ is smooth.

(ii) The modified discriminant $\tilde{d}(Q)\in \Gamma (det(E{)}^{\otimes 2})$ is invertible.

(iii) If $E$ is of constant rank $n$, then there exists a surjective morphism $S^{\prime }\to S$ such that the form $Q_{S^{\prime }}$ deduced from $Q$ by base change is isomorphic to the standard quadratic form in $n$ variables.

(iii bis) As in (iii), but with $S^{\prime }\to S$ faithfully flat of finite presentation (fppf).

Corollary (2.16.10).

Let $k$ be a field, $E$ a vector space of finite dimension over $k$, $Q\in Sy{m}^2(E)$. The following are equivalent:

(i) $Q$ is smooth.

(ii) $\tilde{d}(Q)\ne 0$.

(iii) There exists an extension $k^{\prime }$ of $k$ such that $Q\otimes k^{\prime }$ is isomorphic to the standard form.

(iii bis) As in (iii), but with $k^{\prime }$ a finite extension of $k$.

This is a trivial consequence of (2.16.9). Note that if $k$ is algebraically closed, then (iii) and (iii bis) can be replaced by the more striking condition: $Q$ is isomorphic to the standard form.

V.2.16.11. Proof of Theorem (2.16.9) via principal homogeneous spaces

We may evidently suppose that $E$ has constant rank $n$. We obviously have (iii bis) ⟹ (iii) ⟹ (ii), taking into account that for the standard form the modified discriminant is 1. We now prove that (ii) ⟹ (iii bis), and also that (i) ⇔ (ii).

Let $E_0={\mathcal{O}}_S^n$, $Q_0$ a standard form on $E_0$, and consider the functor

  Isom((E₀, Q₀), (E, Q)) : (Sch/S)° → (Sets),

whose value at every $S^{\prime }\to S$ is the set of isomorphisms $E_{0,S^{\prime }}\stackrel{\sim }{\to }{E}_{S^{\prime }}$ compatible with the forms $Q_{0,S^{\prime }}$ and $Q_{S^{\prime }}$. It is immediate (without any condition on $E$ or $Q$) that this functor is representable by a prescheme $P$ affine and of finite presentation over $S$, which is a subprescheme of $W=\mathbb{V}({\mathrm{Hom}}_{{\mathcal{O}}_S}(E_0,E))$ and a closed subprescheme of the open subset $Isom(E_0,E)$ of $W$.

The implication (ii) ⟹ (iii bis) will be proved if we can show that, when $\tilde{d}(Q)$ is invertible, $P$ is faithfully flat over $S$: take $S^{\prime }=P$.

To simplify, write $\mathcal{Q}(E)=\mathbb{V}(Sy{m}^2(E))$ and similarly $\mathcal{Q}(E_0)$, related by the operation "transport of structure"; we thus have a morphism $u\mapsto Q(u)$:

  (∗)   Isom(E₀, E) → Isom(𝒬(E₀), 𝒬(E)).

Using the section $q_0$ of $\mathcal{Q}(E_0)$ corresponding to $Q_0$, we get a morphism $u\mapsto Q(u)(q_0)$:

$$(\ast )Isom(E_0,E)\to \mathcal{Q}(E).$$

On the other hand, the form $Q$ corresponds to a section $q$ of $\mathcal{Q}(E)$ over $S$, and $P$ is nothing else than the inverse image of $q(S)$ by $(\ast )$, as follows trivially from the definitions. (This, incidentally, establishes in passing the announced representability of $P$ as an affine prescheme of finite presentation over $S$.)

Now introduce the open subset

$$\mathcal{Q}(E)\ast =\mathcal{Q}(E{)}_{\tilde{d}}$$

of $\mathcal{Q}(E)$ corresponding to quadratic forms with invertible corrected discriminant, representing the functor

  S′ ↦ {sections of Sym²(E_{S′}) with invertible corrected discriminant}.

Indeed, by transport of structure we have $Isom(E_0,E)\to Isom(\mathcal{Q}(E_0)\ast ,\mathcal{Q}(E)\ast )$, and since $q_0$ is a section of $\mathcal{Q}(E)\ast$ (since the corrected discriminant of the standard form has value 1), the morphism $(\ast )$ factors through $\mathcal{Q}(E)\ast$. There is thus a morphism

$$(\ast \ast )Isom(E_0,E)\to \mathcal{Q}(E)\ast ,$$

which is evidently of finite presentation, since the two terms are of finite presentation over $S$.

It now suffices to prove:

Proposition (2.16.11).

With the preceding notation (to be recalled), the morphism $(\ast \ast )$ is faithfully flat.

It will follow that if $Q$ satisfies (ii) — i.e. if $q$ is a section of $\mathcal{Q}(E)\ast$ — then $P$, deduced from $(\ast \ast )$ by the base change $q$, is again faithfully flat over $S$. This proves (ii) ⟹ (iii bis), and therefore Theorem (2.16.9).

Since the two terms in $(\ast \ast )$ are smooth over $S$, it suffices to prove flatness fibre by fibre, which brings us to the case where $S$ is the spectrum of an algebraically closed field $k$. We may evidently suppose that $E_0=E$, so that $(\ast \ast )$ takes the form $GL(E)\to \mathcal{Q}(E)\ast$; this morphism is deduced from the natural action of the group scheme $GL(E)$ on $\mathcal{Q}(E)\ast$ by $u\mapsto u(q_0)$, where $q_0$ is the section of $\mathcal{Q}(E)\ast$ corresponding to the standard form $Q_0$. By the generic flatness theorem, since $\mathcal{Q}(E)\ast$ is smooth over $k$ — therefore reduced — there exists an open dense subset $U$ of $\mathcal{Q}(E)\ast$ over which the preceding morphism is flat. For every $u\in GL(E)(k)$, i.e. every automorphism $u$ of $E$, $u(U)$ satisfies the same condition, and it suffices, in order to establish the flatness of $(\ast \ast )$, that the translates $u(U)$ cover $\mathcal{Q}(E)\ast$. For this it suffices to prove that

$$GL(E)(k)=\mathrm{Aut}(E)$$

acts transitively on $\mathcal{Q}(E)\ast (k)$, the set of quadratic forms on $E^{\vee }$ with non-zero corrected discriminant — i.e. that $(\ast \ast )$ is surjective; this also proves (2.16.11).

We are therefore reduced to proving the following lemma.

Lemma (2.16.11.1).

Let $Q$ be a quadratic form in $n$ variables over an algebraically closed field $k$ such that $\tilde{d}(Q)\ne 0$. Then $Q$ is isomorphic to the standard form (i.e. it can be put in standard form by a suitable choice of basis).

If $E$ is of even rank, or if $k$ is of characteristic $\ne 2$, then the hypothesis means that $Q$ is non-degenerate and the conclusion is found in Bourbaki.

In the opposite case ($chark=2$ and rank E odd) we see that $Q$ is degenerate; let E_1 be a line in $E$ lying in the kernel of the associated form, and E_2 a complement, so that $Q$ decomposes as a direct sum $Q_1\oplus Q_2$, with Q_1 of rank one and Q_2 of even rank. By Corollary (2.16.8), Q_1 and Q_2 are of corrected discriminant $\ne 0$; by the previous case, Q_2 is isomorphic to the standard form; on the other hand, Q_1 is non-zero, so it is isomorphic to the standard form of degree one. Thus $Q$ is isomorphic to the standard form, which proves (2.16.11.1).

It remains to prove the equivalence of (i) and (ii) in (2.16.9). We may evidently suppose $S=\mathrm{Spec}(k)$, $k$ algebraically closed, and we are reduced to proving:

Lemma (2.16.11.2).

$Q$ smooth ⟹ $Q$ standard.

By Proposition (2.16.2) we may assume $chark=2$ and $E$ of odd rank. If $Q$ is isomorphic to the standard form, we verify immediately, by Lemma (2.16.3), that it is smooth: the kernel of $E^{\vee }\to E$ is of dimension one and $Q$ is not identically zero on its kernel, so it has only the trivial zero. Conversely, still by (2.16.3), $Q$ smooth implies that the restriction of $Q$ to the kernel $N$ of $Q$ has only the trivial zero — which evidently means that $N$ is of dimension one and that $Q{|}_N\ne 0$. Taking a complement of $N$, we see immediately that $Q$ is isomorphic to the standard form. This proves (2.16.11.2) and completes the proof of (2.16.9).

Grothendieck note. We could have given (2.16.11.2) at the beginning as a corollary to (2.16.2) or (2.16.3) (which could be interchanged); then the part of this section that is independent of the "corrected discriminant" would be amalgamated at the start of the section.

Also, you should know better than I do to what extent the notions of corrected discriminant and (2.16.9) are known, so as to give the correct credit. Perhaps there is some other recent terminology?

I also remark that it is really the corrected (adjusted) discriminant that deserves the name "discriminant": it is this one that can be generalized to the discriminant of any form (see the following section).⁷ The discriminant of a quadratic form in the terminology here adopted — which, if I am not mistaken, is the recent terminology, though I don't have Bourbaki at hand to check — ought to be called the determinant of a quadratic form, not the discriminant. I would love to know your opinion on this. If you agree, we use this occasion to correct at this point the recent terminology, which induces an error (since until the last few days I had myself confused the discriminant and the determinant).

V.2.16.12. The orthogonal-group fibration

Lemma (2.16.12).

If we let the absolute group scheme $GL(n)=\mathrm{Aut}({\mathbb{Z}}^n)$ act on $\mathcal{Q}({\mathbb{Z}}^n)$, the stabilizer of the standard quadratic form is denoted by $O(n)$, called the absolute orthogonal group. (More generally, for every quadratic form $Q\in \Gamma (Sy{m}^2(E))$ we introduce the group subscheme $O(Q)\subset GL(E)$ formed of the $GL(E)$-stabilizer of $Q$, called the orthogonal group scheme relative to $Q$; if $Q$ is isomorphic to the inverse image of the standard form in $n$ variables, $O(Q)$ is isomorphic to $O(n{)}_S$.) Granted this, it is immediate that two points $Q,Q^{\prime }$ of the first term in $(\ast \ast )$ with values in $S^{\prime }$ have the same image in the second term ⇔ we have $u^{\prime }=uv$ with $v\in O(n)(S^{\prime })$. Taking into account (2.16.11) and the terminology to be developed in EGA V and VI⁸ (and awaiting it, in SGAD IV), we see that the natural right-action of $O(n{)}_S$ on the first term of $(\ast \ast )$, together with the projection $(\ast \ast )$, makes $Isom(E_0,E)$ a principal homogeneous fibration over $\mathcal{Q}(E)\ast$ with group $O(n{)}_{\mathcal{Q}(E)\ast }$.

More specifically, $GL(n)$ is a principal homogeneous fibration over $\mathcal{Q}(n)\ast =\mathcal{Q}({\mathbb{Z}}^n)\ast$ with structure group $O(n{)}_{\mathcal{Q}(n)\ast }$. It follows (with the notation of the proof of (2.16.9)) that $P$ is in fact a principal homogeneous fibration with group $O(n{)}_S$, canonically associated to the form $Q$ (functorially in isomorphisms of forms, and compatibly with base change). From purely formal arguments and from the "theory of flat descent" of EGA V,⁹ one then proves that the functor $(E,Q)\mapsto P$ gives an equivalence between the fibred category of smooth quadratic forms on locally free modules of rank $n$ over an arbitrary prescheme $S$, and the fibred category of principal homogeneous fibrations under $O(n{)}_S$ over an arbitrary prescheme $S$.

V.2.16.13. Étale local triviality in good rank/characteristic

Let $Q$ be a smooth quadratic form on $E$ locally free of finite rank. One verifies easily that for $s\in S$, $O(Q)$ is smooth over an open neighbourhood of $s$, except exactly in the case where $k(s)$ is of characteristic two and the rank of $E$ at $s$ is odd. Suppose that such a circumstance does not arise for any $s\in S$. Then (and only then) the $P$ anticipated above is smooth over $S$, being a principal homogeneous fibration under $O(n{)}_S$, which is smooth. Using "Hensel's lemma", it follows that under the conditions of (2.16.9) one can adjoin the equivalent condition:

(iii ter) As in (iii), but with $S^{\prime }\to S$ étale and surjective.

Indeed, it is better — preserving always the previous hypothesis — that it follows from the general theory of reductive group schemes and principal homogeneous fibre bundles over them (cf. SGAD XXIV) that if $Q$ is smooth, then every point $s$ has an open neighbourhood $U$ and a finite, étale, surjective morphism $S^{\prime }\to U$ such that $Q_{S^{\prime }}$ has the standard form. If, for example, $S$ is local, we can in (iii ter) assume in addition that $S^{\prime }\to S$ is finite.

V.2.16.14. Failure outside the good rank/characteristic range

These results — those of (2.16.13), starting with condition (iii ter) for a smooth $Q$ — break down if we abandon the additional hypothesis on the rank and on the characteristic. For example, if $S=\mathrm{Spec}(k)$, $k$ an imperfect field of characteristic two, and $E$ of rank 1, the quadratic form $a{T}^2$ for $a\in k-k^2$ obviously cannot be put in standard form after a separable extension of $k$ (note that $k(a^{1/2})$ is not separable over $k$).

However, in the general case we can find a finite locally free surjective morphism $S^{\prime }\to S$ — indeed, a principal homogeneous fibration under the group scheme ${\mu }_{2,S}\cong O(n{)}_S/SO(n{)}_S$ of square roots of unity over $S$; the base change $S^{\prime }\to S$ has the effect of reducing the structure group of $O(n)$ to $SO(n)$, which is smooth — so that for $Q_{S^{\prime }}$ the local-isotriviality result mentioned earlier holds. In particular, if $S$ is local, we can in (iii bis) assume in addition that $S^{\prime }\to S$ is finite.