Perfectoid spaces : lectures from the 2017 Arizona Winter School

Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic $p$, and have been used to solve many important problems, including cases of the weight-monodromy conjecture and the association of Galois representations to torsion classes in cohomology. In recognition of the transformative impact perfectoid spaces have had on the field of arithmetic geometry, Scholze was awarded a Fields Medal in 2018. This book, originating from a series of lectures given at the 2017 Arizona Winter School on perfectoid spaces, provides a broad introduction to the subject. After an introduction with insight into the history and future of the subject by Peter Scholze, Jared Weinstein gives a user-friendly and utilitarian account of the theory of adic spaces. Kiran Kedlaya further develops the foundational material, studies vector bundles on Fargues–Fontaine curves, and introduces diamonds and shtukas over them with a view toward the local Langlands correspondence. Bhargav Bhatt explains the application of perfectoid spaces to comparison isomorphisms in $p$-adic Hodge theory. Finally, Ana Caraiani explains the application of perfectoid spaces to the construction of Galois representations associated to torsion classes in the cohomology of locally symmetric spaces for the general linear group. This book will be an invaluable asset for any graduate student or researcher interested in the theory of perfectoid spaces and their applications. "Introducidos por Peter Scholze en 2011, los espacios perfectoides son un puente entre la geometría en la característica 0 y la característica p, y se han utilizado para resolver muchos problemas importantes, incluidos los casos de la conjetura de peso-monodromía y la asociación de las representaciones de Galois a las clases de torsión en cohomología. En reconocimiento al impacto transformador que los espacios perfectoides han tenido en el campo de la geometría aritmética, Scholze fue galardonado con la Medalla Fields en 2018. Este libro, originado por una serie de conferencias impartidas en la Arizona Winter School de 2017 sobre espacios perfectoides, ofrece una amplia introducción al tema. Tras una introducción con una visión de la historia y el futuro del tema por parte de Peter Scholze, Jared Weinstein ofrece un relato fácil de usar y utilitario de la teoría de los espacios adic. Kiran Kedlaya desarrolla aún más el material fundacional, estudia los haces vectoriales sobre las curvas de Fargues-Fontaine e introduce los diamantes y las shtukas sobre ellos con vistas a la correspondencia local de Langlands. Bhargav Bhatt explica la aplicación de los espacios perfectoides a los isomorfismos de comparación en la teoría de Hodge p-ádica. Por último, Ana Caraiani explica la aplicación de los espacios de perfectoides a la construcción de representaciones de Galois asociadas a clases de torsión en la cohomología de espacios localmente simétricos para el grupo lineal general". Editor Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic p, and have been used to solve many important problems, including cases of the weight-monodromy conjecture and the association of Galois representations to torsion classes in cohomology. This book, originating from a series of lectures given at the 2017 Arizona Winter School on perfectoid spaces, provides a broad introduction to the subject. After an introduction with insight into the history and future of the subject by Peter Scholze, Jared Weinstein gives a user-friendly and utilitarian account of the theory of adic spaces. Kiran Kedlaya further develops the foundational material, studies vector bundles on Fargues–Fontaine curves, and introduces diamonds and shtukas over them with a view toward the local Langlands correspondence. Bhargav Bhatt explains the application of perfectoid spaces to comparison isomorphisms in p-adic Hodge theory. Finally, Ana Caraiani explains the application of perfectoid spaces to the construction of Galois representations associated to torsion classes in the cohomology of locally symmetric spaces for the general linear group. -- Publisher's description Originating from a series of lectures given at the 2017 Arizona Winter School on perfectoid spaces, this book provides a broad introduction to this subject. It includes an introduction with insight into the history and future of the subject by Peter Scholze.