Perfect Maps of Noetherian Schemes

In this section all schemes are assumed noetherian and all scheme-maps finite-type and separated. The abbreviations introduced at the beginning of 4.4 will be used throughout. 

The main result, Theorem (4.9.4), inspired by [V, p. 396, Lemma 1 and Corollary 2], gives several criteria for / to be perfect (i.e., since / is pseudo-coherent, to have finite tor-dimension). Included there is the implication / perfect Xe f isomorphism. 

In [Nk, Theorem 5.9] Nayak extends these results to separated maps that are only essentially of finite type. 

In this connection, recall that by Nagata s compactification theorem, any (finite-type separated) scheme-map / factors as / = fu. 

Straightforward—if a bit tedious—considerations, nsing the definitions of the maps involved (see, e.g., (4.8.4)), translate Lemma (4.9.2) into commutativity of the natural diagram 

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Perfect maps of noetherian schemes will be treated in 4.9. 

Thus, there is a canonical dualizing pair (/, / R/ / — 1) when / is smooth and there are explicit descriptions of its basic properties in terms of differential forms. But it is not at all clear that there is a canonical such pair for all /, let alone one which restricts to the preceding one on smooth maps. At the (homology) level of dualizing sheaves the case of varieties over a fixed perfect field is dealt with in [Lp, 10], and this treatment is generalized in [HS, 4] to generically smooth equidimensional maps of noetherian schemes without embedded components. 

Analogously, section (4.9) deals with perfect (= finite tor-dimension) finite-type separated maps of noetherian schemes. These maps behave nicely with respect to the twisted inverse image. For example, if / X —> T is a finite-type separated map of noetherian schemes, and / is the associated twisted inverse image functor, perfectness of / is characterized by boundedness of fOy plus the existence of a functorial isomorphism... 

For example, since finite-type maps of noetherian schemes are always pseudo-coherent, the foregoing and (4.3.9) show that a separated such map is quasi-perfect if and only if it is proper and perfect. 

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