Chapter 3 - The Woods Hole trace formula

Summary

The Woods Hole trace formula

The Grothendieck group of crystals

Let X be a noetherian scheme over F_q. Denote by K₀(X) the Grothendieck group of the abelian category CrysX. This is the abelian group generated by isomorphism classes [ℱ] of crystals, modulo relations [ℱ₂] = [ℱ₁] + [ℱ₃] for every short exact sequence 0→ℱ₁→ℱ₂→ ℱ₃→ 0 in CrysX.

Proof. For every i the exact sequence 0 → ℱⁱ⁺¹ → ℱⁱ → ℱⁱ/ℱⁱ⁺¹ → 0 gives [ℱⁱ] − [ℱⁱ⁺¹] = [ℱⁱ/ℱⁱ⁺¹] in K₀(X) which summing over all i ∈ {0, …, n − 1} gives the claimed identity.

Lemma 3.2. Let ℱ• be a bounded complex of crystals on X. Then

in K₀(X).

It follows that if 0 →ℱ₁→ … → ℱ_n → 0 is exact in CrysX then

Proof of Lemma 3.2. On the one hand, the complex ℱ• splits into short exact sequences

while on the other hand the cohomology crystals sit in short exact sequences

Taking the alternating sum over i of the resulting identities in K₀(X) and comparing the terms yields the desired identity

Since the functor − ⊗ − on crystals is exact in both arguments (Corollary 2.8), it induces a bi-additive map

which gives K₀(X) the structure of a commutative ring.

Similarly, for a map ℱ : X → Y of noetherian schemes over F_q the functor ℱ : CrysY →CrysX is exact (Corollary 2.2), so it induces a map

This map is a ring homomorphism.

Finally, if ℱ : X → Y is a separated map of finite type between noetherian schemes over F_q then the long exact sequence of Theorem 2.33 combined with Lemma 3.2 shows that the map

is well-defined. It is additive, but in general not a ring homomorphism. It has the following important transitivity property.

Proposition 3.3. Let ℱ : X → Y and g : Y → Z be separated morphisms of finite type between noetherian schemes overF_q. Then R(gℱ)! = Rg! ◦ Rℱ! as maps from K₀(X) to K₀(Z).