Grothendieck duality theory assigns to essentially finite-type maps $f$ of noetherian schemes a pseudofunctor $f^{\times }$ right-adjoint to $\mathsf{R}f_{\ast }$, and a pseudofunctor $f^{!}$ agreeing with $f^{\times }$ when $f$ is proper, but equal to the usual inverse image $f^{\ast }$ when $f$ is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by ‘compactly supported’ versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes.

We show that a certain category 𝓖 whose objects are pairs G ⊃ H of groups subject to simple axioms is equivalent to the category of ≥ 2-dimensional vector spaces and injective semi-linear maps; and deduce via the "Fundamental Theorem of Projective Geometry" that the category of ≥ 2-dimensional projective spaces is equivalent to the quotient of a suitable subcategory of 𝓖 by the least equivalence relation which identifies conjugation by any element of H with the identity automorphism of G.

The effects of four thermal parameters on the temperature distribution during Zone-Melting-Recrystalllzation processing of Slllcon-On-Insulator devices with graphite strip as the heat source were investigated numerically. Results indicate that the convective heat losses, the variability of the thermal conductivity of silicon as a function of temperature, and the multilayer nature of the structure do not affect the temperature distribution significantly. However, the velocity of the graphite strip can significantly affect the temperature distribution. The temperature profile remains the same for graphite strip velocities below 0.03 cm/sec. The relationship between required graphite strip temperature and velocity of the strip for film melting to occur is presented in graphical form.