Algebraic K-theory
This book provides a clear and comprehensive introduction to Algebraic K-theory with a strong pedagogical focus, making it accessible to advanced graduate students. It follows a carefully constructed path leading to Quillen’s homotopy-theoretic approach, presenting complete details and background to support a self-contained learning experience. The main chapters offer thorough expositions of Quillen’s foundational work, the relationship between classical and modern K-theory, and applications such as the K-theory of schemes and quadrics.
Beyond the core material, the book explores significant extensions of Quillen’s methods, including Waldhausen and Hermitian K-theory, negative K-theory, spectra, and triangulated categories. With foundational chapters covering category theory, homotopy, CW complexes, and simplicial sets, this volume serves as both an introduction and a comprehensive reference for students and researchers delving into Algebraic K-theory.
Projective Modules and Complete Intersections
This compact and self-contained volume (Lecture Notes in Mathematics, Vol. 1672) provides a clear and detailed treatment of projective modules, particularly over polynomial and commutative rings, with a focus on complete intersections. Mandal emphasizes Patching isotopic isomorphisms—a method rooted in Quillen's ideas and refined by Plumstead—alongside foundational results by Ferrand–Szpiro and Cowsik–Nori, and techniques developed by Lindel.
Structured into eight chapters, the book begins with preliminaries and patching theory, progresses through the theory of matrices and modules, and culminates in a rigorous treatment of complete intersections. Ideal for graduate students in commutative algebra and algebraic geometry, it delivers accessible proofs, rich examples, and modern insights—all within just over 120 pages .