Quasi-Perfect Maps

Quasi-perfect maps are scheme-maps f X Y characterized by any one of several nice properties preserved by tor-independent base change (see (4.7.3.1)). Among those properties are the following, the first two by (4.7.1), and the next two by (4.7.4) and (4.7.6)(d)  

It follows that quasi-perfection of / implies the following and in fact when Y is separated the converse is true, see (4.7.4)  

Further, though we won t prove it here, the main result Theorem 1.2 in [LN] is the equivalence of the following conditions  

We call a scheme-map / perfect if / is pseudo-coherent and of finite tor-dimension. (For pseudo-coherent/, being of finite tor-dimension is equivalent to boundedness of f, see [LN, Thm. 1.2]). 

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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Remark. The fact, mentioned above, that quasi-perfect maps are quasiproper results from (4.7.1)(ii) and [LN, Cor. 4.3.2], which says that / is quasiproper if and (clearly) only if R/ takes perfect complexes to pseudo-coherent complexes. 

Section (4.7) is concerned with quasi-perfect (= quasi-proper plus finite tor-dimension) maps of concentrated schemes. These maps have a number of especially nice properties with respect to /. ... 

Example 4.7.3 (a) Any quasi-proper scheme-map f of finite tor-dimension—so by (4.3.3.2), any proper perfect map, in particular, any flat finitely-presentahle proper map—is quasi-perfect. 

Remark. Using the analog of (4.7.3.l)(i) with quasi-proper in place of quasi-perfect [LN, Prop. 4.4], one shows similarly for locally embeddable / that / quasi-proper => f pseudo-coherent. The converse holds when / is also proper, see (4.3.3.2). Thus, e.g., a projective map is quasi-proper if and only if it is pseudo-coherent. 

Conversely, the following conditions on a scheme-map f X Y are equivalent and if Y is separated and f hounded above, they imply that f is quasi-perfect ... 

Exercises 4.7.6 (a). Let f. X Y be a quasi-perfect scheme-map. Assume that X is divisorial—i.e., X has an ample family of invertible -modules—so that by... 

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