Direct product of schemes

In this chapter we investigate various types of products arising naturally in scheme theory. We define direct products of closed subsets of S, direct products of schemes, quasi-direct products of schemes, and semidirect products of schemes. 

The relationship between quasi-direct products of schemes and direct products of closed subsets of S (as defined in the previous section) is described in the following theorem which generalizes [11 Proposition 3.13]. 

Various gasification schemes have been conceived for the direct production of H2 (and C02) instead of synthesis gas. Matsumura has reviewed the gasification of biomass with near- and super-critical water [42], The presence of liquid water suppressed the formation of char but not of tars. Full gasification proceeds in the presence of metal catalysts at 350-600 °C but also in absence of any catalysts at 500-750 °C. This subject is discussed elsewhere in this book [26],... 

The mechanism by which the hydroperoxide intermediate, (42) (Scheme 29) is converted into the products of Scheme 28 is not clear. The follow-up reaction of (42) may be diverted by reaction with an enone that undergoes epoxidation in 85 to 90% yield. Scheme 29, [121]. The epoxidation reaction does not take place directly from O2 and 02 but requires the formation of an intermediate of type (42) derived either from the enone or from an external carbon acid as in Scheme 29. Yields are considerably improved using an external carbon acid since the Michael addition between the enone and its anion otherwise competes with the epoxidation. For... 

In addition to the physical interactions described above, the tip may also be used to alter the local chemical conditions within the tunnel junction. For example, catalytic rehydrogenation of carbonaceous fragments on Pt(lll) by tip-directed production of atomized hydrogen in vacuum at the Pt-Ir tip has been described [524]. Similar modification schemes may also be envisioned based on limiting the transport of reactants and products into or away from the partly occluded tunnel junction. As noted earlier, such effects may be important in the study of electrodeposition and etching process [126-131]. Nonetheless, much remains to be understood about the detailed physics and chemistry of the immersed tunnel junction. 

Theorem. Let G be a finite group scheme over a perfect field. Then G is the semi-direct product of G° and n0 G. 

One technical point should be mentioned. Let G be the group scheme determined by N, and Gs and Gu the subgroups determined by N, and Nu. It would a priori be possible for G, and G to have nontrivial (finite connected) intersection even when N3 n Nu = e. By (8.3), however, that does not in fact happen here. Thus G is itself the direct product of G, and Gu. 

N x H into a group scheme which is the semi-direct product of N and H. 

Since our direct route to angularly fused triquinanes from cycloaddition of l-(4-pentynyl)cyclopen-tenes is limited to trisubstituted alkenes and simple terminal alkynes, bisnorisocomene, but not iso-comene itself, could be prepared (Scheme 22). However, this limitation is not a factor for most other compounds in this class of natural products, and the steric interactions described earlier worked to our advantage in a diastereocontrolled synthesis of pentalenene (see structure. Scheme 20). The natural product was obtained by subjecting the product of Scheme 14 to the sequence i, Li, NH3, MeOH ii, MeLi, Et20 iii, p-TsOH, benzene, reflux. ... 

---