Algebraic Spaces and Stacks

Algebraic Spaces and Stacks presents core mathematical concepts at the intersection of algebraic geometry and category theory. The text builds from foundational material through to advanced topics in modern algebraic geometry. The book covers the theory of algebraic spaces, progressing from étale equivalence relations to more complex algebraic stacks. Technical proofs and detailed examples guide readers through key developments in the field's evolution since the 1960s. The material requires background knowledge in algebraic geometry, scheme theory, and category theory. Each chapter contains exercises that reinforce the theoretical framework. This work serves as both a reference text and a window into the methods that revolutionized algebraic geometry in the twentieth century. The progression from concrete to abstract concepts reflects broader patterns in how mathematical understanding advances through generalization.

There appear to be very few public reviews available for this specialized graduate mathematics text published in 2018. Readers note the book fills an important gap in the literature by providing a systematic introduction to algebraic spaces and stacks. Math Stack Exchange users reference it as a standard text for learning the foundations of the subject. Readers appreciate: - Clear explanations of technical concepts - Focus on geometric intuition - Detailed examples and exercises Critiques: - Requires significant background in algebraic geometry - Some sections could use more examples - Limited treatment of certain advanced topics No ratings or reviews found on Goodreads or Amazon. The text is primarily discussed in academic contexts and specialized math forums rather than consumer review sites. Several university course syllabi list it as a recommended reference text for graduate algebraic geometry courses.

The Geometry of Schemes by David Eisenbud and Joe Harris This text builds from basic algebraic geometry to develop scheme theory with the same rigorous treatment of foundations found in Artin's work on algebraic spaces.

Stacks Project by The Stacks Project Authors This comprehensive reference covers stacks, algebraic spaces, and schemes with detailed proofs and a systematic development that complements Artin's approach.

Algebraic Stacks and Moduli of Vector Bundles by Ulrich Görtz and Torsten Wedhorn The text provides a detailed treatment of moduli problems and their stack-theoretic solutions using methods parallel to those in Artin's book.

Foundational Methods of Algebraic Geometry by Brian Conrad The book develops the theory of algebraic spaces and stacks from first principles with an emphasis on the foundations and categorical methods.

Algebraic Geometry and Arithmetic Curves by Qing Liu This text presents the theory of schemes, algebraic spaces, and their arithmetic applications using similar categorical and functorial viewpoints as Artin's treatment.

🔷 Michael Artin was awarded the Wolf Prize in Mathematics in 2013 for his fundamental contributions to algebraic geometry, including the development of the theory of algebraic spaces and stacks. 🔷 Algebraic stacks were first introduced by Pierre Deligne and David Mumford in the 1960s, but Artin's work significantly expanded their theory and applications in modern algebraic geometry. 🔷 The book represents the culmination of over 50 years of research and development in the field, published in 2012 when Artin was 78 years old. 🔷 Algebraic spaces serve as a crucial intermediate step between schemes and stacks, providing a bridge that helps mathematicians understand more complex geometric objects. 🔷 The concepts covered in this book are essential tools in string theory and modern theoretical physics, helping physicists understand the geometry of spacetime in ways that classical geometry cannot.