RESPONSABILIDAD CIVIL DERIVADA DEL USO Y CIRCULACION

In Section 3.1 we investigated the recently discovered formula Lδ∗π×f∗ for the twisted inverse image functorf!. We come to several conclusions:

1. If we redefine the twisted inverse image functorf!_{as Lδ}∗_π×_f∗_{, we can show that} it satisfies all the standard properties of the classically-definedf! as expected. In addition, all these are shown without falling back on the compactification. 2. Similar to the traditional treatment of the functor f!, the knowledge of base

change morphisms Φ and θ plays the most significant role in proving results about the functor Lδ∗π×f∗. This is due to the fact that most of the maps are defined in terms of Φ or at least implicitly contain Φ. Thus, proving statements about these maps boils down to reducing them to statements about Φ.

3. The noetherian hypothesis in all of the statements (as every morphism we are concerned with is in Se) is only there so we can use Proposition 2.5. In other

words, if we can drop the noetherian hypothesis from Proposition 2.5, then our results in 3.1 will generalize to the non-noetherian case.

In Section 3.2, we see how the formula can be implemented to computef!for morphisms of finite presentation between affine schemes. We also showed that this computation would be independent of the choice of factorization (at least up to canonical isomor- phism). This implies that given such morphism we can use our favorite factorization to compute the expression for f! using our formula.

We want to bring readers attention to Nayak’s result in [Nay05]. Note that this result is 45

46CHAPTER 4. CONCLUSION & SUGGESTIONS FOR FURTHER INVESTIGATIONS

not restricted to morphisms between noetherian schemes. The result (in our notation) is as follows:

Theorem 4.1 ([Nay05, Theorem 7.3.2]). 1. On the category Qnn of composites of

´

etale maps and proper flat pseudo-coherent maps of quasi-compact quasi-separated schemes, there exists a pseudofunctor (−)!, unique up to a unique isomorphism, taking values in D_qc+(X) for any scheme X, such that the following conditions hold.

(a) Over the subcategory of proper maps, (−)! _{gives the right adjoint to the}

derived direct-image pseudofunctor (−)∗.

(b) Over the subcategory of ´etale maps, (−)! is the inverse-image pseudofunctor

(−)∗.

(c) For the fiber-product diagram in Qnn

W X Z Y, g u (♦) f v

if u, vare ´etale andf, g are proper, then the associated base-change isomor- phism of Φ(♦) :u∗f×→g×w∗ equals, via the identifications in (a) and (b), the obvious pseudofunctorial isomorphism given by (−)!.

2. For the diagram♦, there is a flat base change isomorphism θ(♦) :u∗f! →g!w∗. Moreover θis horizontally and vertically transitive and is uniquely determined by the condition that whenf, gare proper,θ(♦)is the isomorphismΦ(♦)while iff, g

are ´etale, thenθ(♦)is the obvious isomorphism resulting from pseudofunctoriality of (−)∗.

As pseudo-coherent morphisms of quasi-compact quasi-separated schemes are always of finite presentation, we suggest that further investigation attempts to show that Lδ∗π×f∗ is well-defined and satisfies all the standard properties on some subclass of separated morphisms of finite presentation between quasi-compact quasi-separated (and not necessarily noetherian) schemes.

As discussed in point 3 of our concluding remark, we can drop the noetherian hypoth- esis from the whole theory in Section 3.1 if we can drop it from Proposition 2.5 and its subsequent usage in Section 3.1 (Lemma 3.7 and more generally Corollary 3.10). Throughout the rest of the chapter, let Sf be the class of separated morphisms of

finite-presentation between quasi-compact quasi-separated (not necessarily noetherian) schemes. We propose the following alternative to Proposition 2.5.

47 Conjecture 4.2. Let f :W → X and g :X → S be morphisms in Sf such that f is

proper and both g and g◦f are flat. If φis a morphism in Dqc(X), then

f×φ is an isomorphism ⇐⇒ Lf∗φis an isomorphism.

Remark 4.3. One difference between the conjecture and Proposition 2.5 is that we drop noetherian hypothesis in Conjecture 4.2. One possible approach that can be taken to rid the noetherian hypothesis is the absolute noetherian approximation ([TT88, Appendix C], [The16, Tag 01Z1]). Another difference is that we have an additional hypothesis in Conjecture 4.2. The additional hypothesis is the existence of a base scheme S for the morphism W → X such that both X and W are flat and of finite- type over this base scheme. This additional hypothesis should pose no problem as it is apparent in Chapter 3 that in most applications of Proposition 2.5 we had no shortage of flat maps.

Assuming Conjecture 4.2, the next conjecture can be proven by following the approach in the proof of Lemma 3.7. We claim that having both Conjecture 4.2 and Conjecture 4.4 is enough to generalize the formula Lδ∗π×f∗ toSf.

Conjecture 4.4. Let a:W →X, b:X →Y, c:Y →S be morphisms in Sf such that

b◦a is proper, and that b, c and c◦b◦a are flat, then if we restrict to the bounded below derived category D+_qc(Y)

1. La∗ψ(b) : La∗b×→La∗b! is an isomorphism. 2. a×ψ(b) :a×b×→a×b! is an isomorphism.