Flat Base Change

Moreover is a formal J -scheme and the formation of commutes with flat base change. 

Additional basic properties of / are its compatibility with flat base change (Theorems (4.4.3), (4.8.3)), and the existence of canonical functorial maps, for Oy-complexes E and F having quasi-coherent homology ... 

On first acquaintance, [Del appears to offer a neat way to cut through the complexity—a direct abstract proof of the existence of / , with indications about how to derive the concrete special situations (which, after all, motivate and enliven the abstract formalism). Such an impression is bolstered by Verdier s paper [V j. Verdier gives a reasonably short proof of the flat base change theorem, sketches some corollaries (for example, the flnite tor-dimension case is treated in half a page [ibid., p. 396], as is the smooth case [ibid., pp. 397-398]), and states in conclusion that all the results of [H], except the theory of dualizing and residual complexes, are easy consequences of the existence theorem. In short, Verdier s concise summary of the main features, together with some background from [H] and a little patience, should suffice for most users of the duality machine. 

In (3.9) we consider the case when our ringed spaces are schemes. Under mild assumptions, we note that then R/ and Lf respect quasi-coherence (3.9.1), (3.9.2). We also show that some previously introduced functorial morphisms become isomorphisms (3.9.4) treats variants of the projection morphisms, while (3.9.5) signifies that R/ behaves well—even for unbounded complexes—with respect to flat base change. More generally, in (3.10) we see that such good behavior of Rf, characterizes tor-independent base changes. 

In this chapter we review and elaborate on—with proofs and/or references— some basic abstract features of Grothendieck Duality for schemes with Zariski topology, a theory initially developed by Grothendieck [Gr ], [H], [C], Dehgne [De ], and Verdier [V ]. The principal actor in this Chapter is the twisted inverse image pseudofunctor, described in the Introduction. The basic facts about this pseudofunctor—which may be seen as the main results in these Notes—are existence and flat base change, Theorems (4.8.1) and (4.8.3). 

There are other pasting techniques, due to Nayak [Nk], to establish the two basic theorems, (4.8.1) and (4.8.3). As mentioned in the Introduction, Nayak s methods avoid using Nagata s theorem, and so apply in contexts where Nagata s theorem may not hold. For example, the results in [Nk, 7.1] are generalizations of (4.8.1) and (4.8.3) to the case of noetherian formal schemes (except for thickening as in (4.8.11) below, which allows flat base-change isomorphisms for admissible squares (4.8.3.0) rather than just fiber squares, see Exercise (4.8.12)(d).)... 

To prove the theorem, it suffices to show that the map in question is an isomorphism after applying ( ) for any i G /. By Proposition 18.14, Lemma 19.2, and [26, (3.7.2), (iii)], the problem is reduced to the flat base change theorem (in fact open immersion base change theorem is enough) for schemes [41, Theorem 2, and we are done. ... 

By the flat base change theorem and Lemma 26.8, the question is local on Yj. Clearly, the question is local on Xj. Hence we may assume that Yj and Xj are affine. Set f = fj,Y = Yj, and X = Xj. Note that / is a closed immersion defined by an ideal of finite projective dimension, followed by an affine n-space. 

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