Spherical subcategories in algebraic geometry

Introductory Geometry: Algebraic Geometry

Smashing Subcategories and the Telescope Conjecture – an Algebraic Approach

We prove a modified version of Ravenel’s telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presen...

Algebraic Geometry

Physical objects and constraints may be modeled by polynomial equations and inequalities. For this reason algebraic geometry, the study of solutions to systems of polynomial equations, is a tool for scientists and engineers. Moreover, relations between concepts arising in science and engineering are often described by polynomials. Whatever their source, once polynomials enter the picture, notio...

Algebraic Geometry

Algebraic Geometry

Examples • If S = ∅, then Z(S) = An. • If S = {1}, then Z(S) = ∅. • If S = {x1 − a1, . . . , xn − an}, then Z(S) = {(a1, . . . , an)}. Remark. All rings in this course are commutative and have 1. Remark. If A is a ring, then any subset S ⊆ A generates a minimal ideal 〈S〉 ⊆ A. In fact, we have 〈S〉 = {∑ aj xj : aj ∈ A, xj ∈ S}. Lemma. Z(S) = Z(〈S〉) for all S ⊆ k[x1 , . . . , xn]. Proof. Since 〈S〉...