Forschungsseminar Komplexe Analysis - Summer Semester 2014

A¹-homotopy theory of schemes

Wednesdays 4-6pm - Arnimallee 3, SR 005. ( 16th April 2014 - 16th July 2014 )

During this semester we will study the unstable A¹-homotopy category of schemes constructed by Morel and Voevodsky. The idea of A¹-homotopy theory is to provide a homotopy theory for varieties and schemes where the affine line plays the role of the unit interval. Our goal is to provide a homotopical classification of vector bundles on a smooth scheme.

A more detailed description of the talks can be found here and a list of useful references and links can be found here.


Schedule 2014

♠ 16.04.2014 // Introduction - Victoria Hoskins We provide some motivation coming from algebraic topology and state the classification of topological vector bundles up to homotopy. Then we give an overview of some of the ideas involved in the construction of a A¹-homotopy theory for schemes and outline the key ideas and techniques needed to construct the unstable A¹-homotopy category.
♠ 23.04.2014 // Classification of principal bundles in algebraic topology - Vincent Trageser In this talk, we give the homotopical classification of principal bundles in algebraic topology. For a topological group G, we show there is a classifying space BG such that homotopy classes of maps form a space X to BG are in bijection with isomorphism classes of principal G-bundles on X. In particular, we focus on the general linear group where we describe the classifying space as an infinite grassmannian.
♠ 30.04.2014 // An introduction to homotopical algebra - Elena Martinengo In this talk, we give the basic notions from homotopical algebra needed for the construction of the A¹-homotopy category. To invert a certain class of morphisms in a category it is often useful to introduce a model structure. We give a quick introduction to model categories, homotopy categories and state some important results from homotopical algebra. We focus on the key examples of topological spaces and simplicial sets.
♠ 07.05.2014 // Sheaves in the Nisnevich topology - Eva Martínez We define and compare the Zariski, étale and Nisnevich topologies. We explain how the Nisnevich topology incorporates useful features of the Zariski and étale topologies. We prove that, for the Nisnevich topology, it suffices to check the sheaf condition on 'distinguished squares'. Then we define fibres of sheaves in the Nisnevich topology.
♠ 14.05.2014 // The simplicial homotopy category of sheaves - Anna Wißdorf We consider the category of simplicial sheaves in the Nisnevich topology and give this a model category structure whose associated homotopy category is called the simplicial homotopy category of sheaves. We also briefly discuss Quillen functors and give some examples. Finally, we describe the B.G. property for a simplicial presheaf and note that fibrant simplicial sheaves in the Nisnevich topology have the B.G. property.
♠ 21.05.2014 // The A¹-homotopy category of smooth k-schemes - Nikolai Beck We start by recalling the naive notions of A¹-homotopy and then define A¹-weak equivalences in the category of simplicial sheaves. Morel and Voevodsky prove that there is a model structure on this category whose weak equivalences are A¹-weak equivalences and so we construct the A¹-homotopy category as the associated homotopy category. We also describe the A¹-localisation functor and discuss morphism groups in the A¹-homotopy category.
♠ 28.05.2014 // K₀ and K₁ for rings - Ángel Muñoz Castañeda We recall the definition of the Grothendieck group K₀ and describe the Grothendieck groups of schemes and rings. We define K₁ of a ring via a short exact sequence of groups and prove Whitehead's lemma. Finally, we define K₀ and K₁-regularity of a ring and prove that a regular ring is K₀ and K₁-regular.
♠ 04.06.2014 // Excision for Picard groups and Karoubi-Villamayor K-theory - Ángel Muñoz Castañeda We show that distinguished squares of schemes give rise to an exact sequence relating their Picard groups and prove that K₀-regularity implies Pic-regularity. We define the Karoubi-Villamayor K-theory groups of a ring and relate these to K₀ and K₁ of the ring. Then, for certain cartesian squares of rings, we obtain long exact sequences intertwining these groups.
♠ 11.06.2014 // Anodyne extensions I - Joana Cirici In this talk, we introduce an alternative construction of the A¹-homotopy category given by Morel that involves inverting certain 'anodyne extensions' in the category of presheaves on smooth affine schemes. This construction of the A^1-homotopy category will be used to prove the homotopic classification of vector bundles in the following talks.
♠ 18.06.2014 // Anodyne extensions II - Victoria Hoskins We complete the definition of elementary anodyne extensions by defining the class of fundamental geometric anodyne extensions and describe a key example: a section of a vector bundle over a k-space. We describe the cofibrant objects and prove that the naive A^1-homotopy equivalences are weak equivalences. Finally, we give a technical result for elementary anodyne extensions.
♠ 25.06.2014 // Homotopic properties of the canonical torsor on the Grassmannians - Arijit Dey We consider vector bundles and GL_n-torsors and focus on the example of the Grassmannian Gr_n of n-planes. By taking the colimit over n, we obtain the infinite Grassmannian Gr. We prove that the canonical torsor over the infinite Grassmannian Gr is a fibration and that Gr is fibrant and similarly that the infinite projective space is fibrant.
♠ 02.07.2014 // Homotopic classification of vector bundles on smooth k-schemes Finally, we reach the goal of our seminar and prove the homotopic classification of vector bundles! We start by proving the result for smooth affine schemes via a computation on naive homotopy classes of maps. Then we deduce the non-affine case from this by using the Jouanolou-Thomason theorem and strong A¹-homotopy invariance of K₀.
♠ In the final two weeks of the semester, we have two special talks.
♠ 09.07.2014 // Geometric invariant theory over the reals - Matthew Spong For the action of a real reductive group G on a real affine space W, the map from W to the spectrum of the ring of invariant polynomials is not always surjective. Therefore, we describe Luna's construction of a topological quotient of W by the action of G. Following work of Richardson--Slodowy and Schwarz, we equip this topological quotient with a real semi-algebraic structure. If time permits, we look at one or two examples of such quotients and determine their semi-algebraic structures.
♠ 09.07.2014 // Moduli spaces of decorated principal bundles over a curve (Phd Defense)- Nikolai Beck

Schedule 2014

♠ 16.04.2014 // Introduction - Victoria Hoskins
We provide some motivation coming from algebraic topology and state the classification of topological vector bundles up to homotopy. Then we give an overview of some of the ideas involved in the construction of a A¹-homotopy theory for schemes and outline the key ideas and techniques needed to construct the unstable A¹-homotopy category.

♠ 23.04.2014 // Classification of principal bundles in algebraic topology - Vincent Trageser
In this talk, we give the homotopical classification of principal bundles in algebraic topology. For a topological group G, we show there is a classifying space BG such that homotopy classes of maps form a space X to BG are in bijection with isomorphism classes of principal G-bundles on X. In particular, we focus on the general linear group where we describe the classifying space as an infinite grassmannian.

♠ 30.04.2014 // An introduction to homotopical algebra - Elena Martinengo
In this talk, we give the basic notions from homotopical algebra needed for the construction of the A¹-homotopy category. To invert a certain class of morphisms in a category it is often useful to introduce a model structure. We give a quick introduction to model categories, homotopy categories and state some important results from homotopical algebra. We focus on the key examples of topological spaces and simplicial sets.

♠ 07.05.2014 // Sheaves in the Nisnevich topology - Eva Martínez
We define and compare the Zariski, étale and Nisnevich topologies. We explain how the Nisnevich topology incorporates useful features of the Zariski and étale topologies. We prove that, for the Nisnevich topology, it suffices to check the sheaf condition on 'distinguished squares'. Then we define fibres of sheaves in the Nisnevich topology.

♠ 14.05.2014 // The simplicial homotopy category of sheaves - Anna Wißdorf
We consider the category of simplicial sheaves in the Nisnevich topology and give this a model category structure whose associated homotopy category is called the simplicial homotopy category of sheaves. We also briefly discuss Quillen functors and give some examples. Finally, we describe the B.G. property for a simplicial presheaf and note that fibrant simplicial sheaves in the Nisnevich topology have the B.G. property.

♠ 21.05.2014 // The A¹-homotopy category of smooth k-schemes - Nikolai Beck
We start by recalling the naive notions of A¹-homotopy and then define A¹-weak equivalences in the category of simplicial sheaves. Morel and Voevodsky prove that there is a model structure on this category whose weak equivalences are A¹-weak equivalences and so we construct the A¹-homotopy category as the associated homotopy category. We also describe the A¹-localisation functor and discuss morphism groups in the A¹-homotopy category.

♠ 28.05.2014 // K₀ and K₁ for rings - Ángel Muñoz Castañeda
We recall the definition of the Grothendieck group K₀ and describe the Grothendieck groups of schemes and rings. We define K₁ of a ring via a short exact sequence of groups and prove Whitehead's lemma. Finally, we define K₀ and K₁-regularity of a ring and prove that a regular ring is K₀ and K₁-regular.

♠ 04.06.2014 // Excision for Picard groups and Karoubi-Villamayor K-theory - Ángel Muñoz Castañeda
We show that distinguished squares of schemes give rise to an exact sequence relating their Picard groups and prove that K₀-regularity implies Pic-regularity. We define the Karoubi-Villamayor K-theory groups of a ring and relate these to K₀ and K₁ of the ring. Then, for certain cartesian squares of rings, we obtain long exact sequences intertwining these groups.

♠ 11.06.2014 // Anodyne extensions I - Joana Cirici
In this talk, we introduce an alternative construction of the A¹-homotopy category given by Morel that involves inverting certain 'anodyne extensions' in the category of presheaves on smooth affine schemes. This construction of the A^1-homotopy category will be used to prove the homotopic classification of vector bundles in the following talks.

♠ 18.06.2014 // Anodyne extensions II - Victoria Hoskins
We complete the definition of elementary anodyne extensions by defining the class of fundamental geometric anodyne extensions and describe a key example: a section of a vector bundle over a k-space. We describe the cofibrant objects and prove that the naive A^1-homotopy equivalences are weak equivalences. Finally, we give a technical result for elementary anodyne extensions.

♠ 25.06.2014 // Homotopic properties of the canonical torsor on the Grassmannians - Arijit Dey
We consider vector bundles and GL_n-torsors and focus on the example of the Grassmannian Gr_n of n-planes. By taking the colimit over n, we obtain the infinite Grassmannian Gr. We prove that the canonical torsor over the infinite Grassmannian Gr is a fibration and that Gr is fibrant and similarly that the infinite projective space is fibrant.

♠ 02.07.2014 // Homotopic classification of vector bundles on smooth k-schemes
Finally, we reach the goal of our seminar and prove the homotopic classification of vector bundles! We start by proving the result for smooth affine schemes via a computation on naive homotopy classes of maps. Then we deduce the non-affine case from this by using the Jouanolou-Thomason theorem and strong A¹-homotopy invariance of K₀.

♠ In the final two weeks of the semester, we have two special talks.

♠ 09.07.2014 // Geometric invariant theory over the reals - Matthew Spong
For the action of a real reductive group G on a real affine space W, the map from W to the spectrum of the ring of invariant polynomials is not always surjective. Therefore, we describe Luna's construction of a topological quotient of W by the action of G. Following work of Richardson--Slodowy and Schwarz, we equip this topological quotient with a real semi-algebraic structure. If time permits, we look at one or two examples of such quotients and determine their semi-algebraic structures.

♠ 09.07.2014 // Moduli spaces of decorated principal bundles over a curve (Phd Defense)- Nikolai Beck

A1-homotopy theory of schemes

A¹-homotopy theory of schemes