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Let A be a connected graded ring over the base field K, with augmentation ideal m. In this chapter we study derived m-torsion, both for complexes of graded A-modules and for complexes of graded bimodules. The graded bimodule A* := HomK(A,K) is graded-injective and m-torsion on both sides. One of the main results is on the representability of the right derived m-torsion functor RΓm. Under quite general conditions the functor RΓm is isomorphic to the left derived tensor functor P⊗LA(-), where P := RΓm(A). We also prove the NC MGM Equivalence in the connected graded context and a theorem on symmetric derived m-torsion. The χ condition of Artin and Zhang is introduced in Section 16.5. We study how this condition interacts with symmetric derived m-torsion.
It was exciting, no doubt, to watch silent films to the accompaniment of musical excerpts played by cinema orchestras, but the 1930s gave audiences the chance to see stars sing and act. That decade consequently offers valuable historical insight into vocal practice and performance technique. This chapter begins with an overview of freshly created screen operettas and of films adapting stage operettas. It briefly examines Die Drei von der Tankstelle (1930) and other German films, before moving to British and American films. The demands of film are contrasted with techniques required in the theatre. The chapter then looks at the practice of adaptation in Hollywood, and it ends with a discussion of the operetta Heimat film in Germany.
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