THE INVARIANT e^{0}

Tatiana Howard

Abstract. The image of the group homomorphism e_{−}^{0} : W^{−}(ℙ^{1} ×ℙ^{1}) → E(ℙ^{1} ×ℙ^{1})

from the Witt group of skew symmetric bilinear forms over a product of two projective lines into the subquotient E(ℙ^{1} ×ℙ^{1}) of the Grothendieck group K_{0}(ℙ^{1} ×ℙ^{1}) is equal to ℤ_{2}H ⊠ H.

Introduction

Let X be a scheme with the trivial vector bundle O_{X}. If M is a vector bundle on X then its dual bundle is

M^{∧} = Hom_{O_{X}} (M, O_{X}).

Let K_{0}(X) denote the Grothendieck ring of X. The dualization functor^{∧} induces an involution ^{∧} : K_{0}(X) → K_{0}(X) given by the formula

[M] ⟼ [M^{∧}].

Consider now the following complex

⋯ ⟶^{1−^{∧}} K_{0}(X) ⟶^{1+^{∧}} K_{0}(X) ⟶^{1−^{∧}} K_{0}(X) ⟶^{1+^{∧}} K_{0}(X) ⟶^{1−^{∧}} ⋯

and define EX = ker(1−^{∧})/im(1+^{∧}). A pair (M, β) is said to be a symmetric (skew symmetric) bilinear space over

X if M is a vector bundle on X and β : M → M^{∧} is an isomorphism such that β = β^{∧} (β = −β^{∧}). Let W (X) (respectively W^{−}(X)) be a Witt ring of symmetric (resp.Witt group of skew symmetric) bilinear spaces over X.

M. Szyjewski introduced the invariant e_{+}^{0} : W (X) → EX

e_{+}^{0}(M, β) = [M]mod(im(1+^{∧}))

which is a natural generalization of the dimension index. We will focus our attention on the case when X = ℙ^{1} × ℙ^{1} is the product of two

projective lines over a fixed field k. Let p_{1} and p_{2} be the projections onto ℙ^{1}. For given two vector bundles F and G on ℙ^{1}, F ⊠ G = p_{1}^{∗}([F]) ⊗ p_{2}^{∗}([G]). We have

E(ℙ^{1} × ℙ^{1}) = ℤ_{2}[O_{ℙ^{1}×ℙ^{1}}] ⊕ ℤ_{2}H ⊠ H,

where H = [O_{ℙ^{1}}] − [O(−1)_{ℙ^{1}}] ∈ K_{0}(ℙ^{1}) and

im(e_{+}^{0}) = ℤ_{2}[O_{ℙ^{1}×ℙ^{1}}].

(see [Sz, 2.1.6, p. 458]) One can now consider e_{−}^{0} : W^{−}(ℙ^{1} × ℙ^{1}) → E(ℙ^{1} × ℙ^{1}). The main purpose of

this paper is to show that im(e_{−}^{0}) = ℤ_{2}H ⊠ H.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 11E81, 14C35.

The image of e_{−}^{0}.

Theorem. Let e_{−}^{0} : W^{−}(ℙ^{1} × ℙ^{1}) → E(ℙ^{1} × ℙ^{1}). Then im(e_{−}^{0}) = ℤ_{2}H ⊠ H.

Proof.

First we show that

im(e_{−}^{0}) ⊆ ℤ_{2}H ⊠ H.

Choose any (M, β) ∈ W^{−}(ℙ^{1} × ℙ^{1}). The elements 1, p_{1}^{∗}([O(−1)]), p_{2}^{∗}([O(−1)]), [O(−1)] form a basis of the free abelian group K_{0}(ℙ^{1} × ℙ^{1}). Thus

[M] = a + bp_{1}^{∗}([O(−1)]) + cp_{2}^{∗}([O(−1)]) + d[O(−1)],

for some a, b, c, d ∈ ℤ. We find the coefficients of [M^{∧}] with respect to the chosen basis:

[M^{∧}] = a + bp_{1}^{∗}([O(1)]) + cp_{2}^{∗}([O(1)]) + d[O(1)]
= a + b(2 − p_{1}^{∗}([O(−1)])) + c(2 − p_{2}^{∗}([O(−1)]))
+ d(4 − 2p_{1}^{∗}([O(−1)]) − 2p_{2}^{∗}([O(−1)]) + [O(−1)])
= a + 2b + 2c + 4d + (−2d − b)p_{1}^{∗}([O(−1)])
+ (−2d − c)p_{2}^{∗}([O(−1)]) + d[O(−1)]

Since [M] = [M^{∧}] we get the system of three linear equations

\{ \begin{matrix}a & = a + 2b + 2c + 4d \\ b & = −2d − b \\ c & = −2d − c \end{matrix} .

Hence c = b = −d and

[M] = a + bp_{1}^{∗}([O(−1)]) + bp_{2}^{∗}([O(−1)]) − b[O(−1)]
= a + b − b(1 − p_{1}^{∗}([O(−1)]) − p_{2}^{∗}([O(−1)]) + [O(−1)])
= a + b − bH ⊠ H
= dim(M) − bH ⊠ H.

Since (M, β) is nonsingular its dimension has to be an even number, so

e_{−}^{0}(M, β) = b(mod2)H ⊠ H.

Now we prove that im(e_{−}^{0}) ⊇ ℤ_{2}H ⊠ H. It is enough to show that there exists a bilinear skew symmetric form (P, β) such that e_{−}^{0}( P, β) = H ⊠ H. Let ℙ^{1} × ℙ^{1} be given in ℙ^{3} = Proj k[x, y, z, t] by the equation xy = zt. Consider the open cover of ℙ^{1} × ℙ^{1} consisting of the sets D(x), D(y), D(z), D(t) (for any homogenous polynomial f ∈ k[x, y, z, t], D(f) = {p ∈ Proj k[x, y, z, t] : f ∉ p}). Denote M = p_{2}^{∗}(O(1)) ⊕ p_{1}^{∗}(O(1)) ⊕ O. Then M is given by the following transition maps

χ_{y,x}^{#} = [ \begin{matrix}\frac{x}{z} & 0 & 0 \\ 0 & \frac{x}{t} & 0 \\ 0 & 0 & 1 \end{matrix} ] ,  χ_{z,x}^{#} = [ \begin{matrix}\frac{x}{z} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] ,  χ_{t,x}^{#} = [ \begin{matrix}1 & 0 & 0 \\ 0 & \frac{x}{t} & 0 \\ 0 & 0 & 1 \end{matrix} ] .

One may notice that

[ \begin{matrix}1 & 0 & 0 \\ −1 & 1 & 0 \\ −\frac{y}{x} & 0 & 1 \end{matrix} ] [ \begin{matrix}\frac{x}{z} & 0 & 0 \\ 0 & \frac{x}{t} & 0 \\ 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}\frac{z}{y} & 0 & 1 \\ \frac{t}{y} & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ] = [ \begin{matrix}\frac{x}{y} & 0 & \frac{x}{z} \\ 0 & \frac{x}{t} & −\frac{x}{z} \\ 0 & 0 & −\frac{y}{z} \end{matrix} ]

[ \begin{matrix}1 & 0 & 0 \\ −1 & 1 & 0 \\ −\frac{y}{x} & 0 & 1 \end{matrix} ] [ \begin{matrix}\frac{x}{z} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 0 & 0 \\ \frac{x}{z} & 1 & 0 \\ \frac{y}{z} & 0 & 1 \end{matrix} ] = [ \begin{matrix}\frac{x}{z} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ]

[ \begin{matrix}1 & 0 & 0 \\ −1 & 1 & 0 \\ −\frac{y}{x} & 0 & 1 \end{matrix} ] [ \begin{matrix}1 & 0 & 0 \\ 0 & \frac{x}{t} & 0 \\ 0 & 0 & 1 \end{matrix} ] [ \begin{matrix}\frac{x}{t} & 1 & 0 \\ 1 & 0 & 0 \\ \frac{y}{t} & 0 & 1 \end{matrix} ] = [ \begin{matrix}\frac{x}{t} & 1 & 0 \\ 0 & −1 & 0 \\ 0 & −\frac{y}{x} & 1 \end{matrix} ] ,

so there is an exact sequence

0 → O(1) → M → P → 0

where P be is the subbundle given by transitions:

π_{y,x}^{#} = [ \begin{matrix}\frac{x}{t} & −\frac{x}{z} \\ 0 & −\frac{y}{z} \end{matrix} ] ,  π_{z,x}^{#} = [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] ,  π_{t,x}^{#} = [ \begin{matrix}−1 & 0 \\ −\frac{y}{x} & 1 \end{matrix} ] .

Since Λ^{2}P ≅ O (notice that det π_{y,x}^{#} = det π_{t,x}^{#} = −1 and det π_{z,x}^{#} = 1), there exists a skew symmetric nonsingular bilinear form

P × P → Λ^{2}P ≅ O.

Moreover

[p_{2}^{∗}(O(1))] + [p_{1}^{∗}(O(1))] + [O] = [O(1)] + [P],

thus the class [P] in the group E(ℙ^{1} × ℙ^{1}) is

[P] = [p_{2}^{∗}(O(1))] + [p_{1}^{∗}(O(1))] + [O]−[O(1)]
= 2[O] − ([O]−[p_{1}^{∗}(O(1))] − [p_{2}^{∗}(O(1))] + [O(1)])
= H ⊠ H.

References

[Sz] M. Szyjewski, Quadratic Forms over Schemes, Documenta Mathematica 1 (1996), 449-478.

[Sz2] M. Szyjewski, On the Witt ring of a relative projective line, Colloquium Mathematicum 75 (1998), 53-78.

Mathematics Department, University of Maryland College Park, College Park, MD 20742-4015 USA

E-mail address: thoward@math.umd.edu