This book, titled "Residues and Duality," is a collection of lecture notes from a seminar on the work of A. Grothendieck, conducted at Harvard University in 1963-1964. The seminar, initiated by Robin Hartshorne, focused on Grothendieck's theory of duality for coherent sheaves, which had been previously hinted at but never systematically developed. Grothendieck provided an outline, and Hartshorne filled in the details, resulting in a series of six exposés circulated under the title "Séminaire Hartshorne." The book is divided into seven chapters, covering topics such as the derived category, applications to preschemes, duality for projective morphisms, local cohomology, dualizing complexes and local duality, residual complexes, and the duality theorem. It includes detailed proofs, examples, and discussions on various aspects of the theory, with contributions from other mathematicians like David Mumford, John Tate, Stephen Lichtenbaum, and John Fogarty. The preface acknowledges the support and encouragement of Grothendieck throughout the project and thanks all those who contributed to its preparation. The book is a comprehensive and revised version of the original seminar notes, providing a detailed exploration of the theory of residues and duality in algebraic geometry.This book, titled "Residues and Duality," is a collection of lecture notes from a seminar on the work of A. Grothendieck, conducted at Harvard University in 1963-1964. The seminar, initiated by Robin Hartshorne, focused on Grothendieck's theory of duality for coherent sheaves, which had been previously hinted at but never systematically developed. Grothendieck provided an outline, and Hartshorne filled in the details, resulting in a series of six exposés circulated under the title "Séminaire Hartshorne." The book is divided into seven chapters, covering topics such as the derived category, applications to preschemes, duality for projective morphisms, local cohomology, dualizing complexes and local duality, residual complexes, and the duality theorem. It includes detailed proofs, examples, and discussions on various aspects of the theory, with contributions from other mathematicians like David Mumford, John Tate, Stephen Lichtenbaum, and John Fogarty. The preface acknowledges the support and encouragement of Grothendieck throughout the project and thanks all those who contributed to its preparation. The book is a comprehensive and revised version of the original seminar notes, providing a detailed exploration of the theory of residues and duality in algebraic geometry.