Clarificatons and Corrections for Grothendieck Duality and Base Change

In what follows, “the book” refers to Grothendieck duality and base change in its original form. The online version incorporates some of the minor changes below into the text (e.g., typos), but the more detailed discussions of certain points are not included in the on-line revision of the text (as this would have seriously disrupted the page numbering). All theorems, equations, page numbers, etc. which are mentioned without reference to [Be], [CE], [D], [G], or [RD] are to be understood to refer to the book. More than half of this write-up is devoted to explaining why the book is consistent with [D] and [Be]. It is important to confirm this consistency because there may be confusion caused by the fact that [D] and [Be] occasionally use definitions which do not coincide with the ones in the book (all such differences are recorded below). I am grateful to Ofer Gabber for pointing out many of the discrepancies below, particularly concerning Lemma 3.5.3 (which should have no sign), Theorem B.4.1 (which should have a sign), and the location of the main error in the proof of [Be], p. 532, Lemme 1.2.5. The errors in Lemma 3.5.3 and Theorem B.4.1 do not affect the truth or proof of any other theorems, lemmas, or corollaries in the book, and are caused by elementary explicit miscalculations (rather than subtle errors in homological algebra). The other errors and ambiguities are essentially all of expository nature and have no impact on the truth of any statements of results in the book, but a couple of proofs are affected in very very minor ways (as is made explicit below). In all cases of incorrectly formulated definitions (none of which affect the general foundations of duality theory or homological algebra used in the book), it is the “corrected” definitions which are actually used in the proofs. The only systematic homological mistake was an occasional failure to observe the sign implicit in [CE], Chapter V, Proposition 7.1 (nearly all comments below for pp. 228, 271–281 are due to this), even though on p. 113 this sign is properly noted. As is explained below, this mistake only affects the exposition and does not affect the statements of any results, because the essential steps in all affected proofs utilize the correct map (in accordance with [CE], Ch. V, Prop. 7.1).