Algebraic K-theory of schemes

Inhaltsangabe / Literatur / empfohlene Vorkenntnisse Algebraic K-theory is an important invariant in algebraic geometry, encoding substantial arithmetic and geometric information. We will begin with a quick review of quasicoherent sheaves and the derived category of a scheme. We will then use the main results of the previous semester (namely, the additivity and localization theorems) to study the algebraic K-theory of schemes, with emphasis on the projective bundle theorem, Mayer-Vietoris, Bass' fundamental theorem, and Nisnevich descent, closely following the paper of Thomason-Trobaugh. We will then turn our attention to Adams' operations and Quillen's calculation of the K-groups of a finite field. Time permitting, we will treat additional topics, such as homotopy K-theory, K-theory with finite coefficients, Bott-inverted K-theory, etale descent, or the Grothendieck-Riemann-Roch theorem. Prerequisites are a basic knowledge of algebraic geometry, algebraic topology, and category theory; note that, while this is a continuation of a course on Waldhausen K-theory, the content will be largely orthogonal to that of the previous semester, where the general theory was developed (and is therefore not prerequisite for this course, provided one is willing to accept the basic results of Waldhausen K-theory on faith).

Course description Algebraic K-theory is an important invariant in algebraic geometry, encoding substantial arithmetic and geometric information. We will begin with a quick review of quasicoherent sheaves and the derived category of a scheme. We will then use the main results of the previous semester (namely, the additivity and localization theorems) to study the algebraic K-theory of schemes, with emphasis on the projective bundle theorem, Mayer-Vietoris, Bass' fundamental theorem, and Nisnevich descent, closely following the paper of Thomason-Trobaugh. We will then turn our attention to Adams' operations and Quillen's calculation of the K-groups of a finite field. Time permitting, we will treat additional topics, such as homotopy K-theory, K-theory with finite coefficients, Bott-inverted K-theory, etale descent, or the Grothendieck-Riemann-Roch theorem. Prerequisites are a basic knowledge of algebraic geometry, algebraic topology, and category theory; note that, while this is a continuation of a course on Waldhausen K-theory, the content will be largely orthogonal to that of the previous semester, where the general theory was developed (and is therefore not prerequisite for this course, provided one is willing to accept the basic results of Waldhausen K-theory on faith).