The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert, and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi–Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.

http://www.gbv.de/dms/goettingen/51399310X.pdf

Central Simple Algebras and Galois Cohomology

This chapter and the next are adapted from a lecture published in the Proceedings of the Third Physics Workshop, held at the Institute of Theoretical and Experimental Physics, Moscow, in 1975. They focus on the uses of homotopy theory in physics, and especially on the calculation of homotopy groups. The next chapter summarizes briefly the ways in which homotopy theory can be applied to quantum field theory.

Elements of Homotopy Theory

1 Introduction.- 2 Unramified sheaves and strongly A1-invariant sheaves.- 3 Unramified Milnor-Witt K-theories.- 4 Geometric versus canonical transfers.- 5 The Rost-Schmid complex of a strongly A1-invariant sheaf.- 6 A1-homotopy sheaves and A1-homology sheaves.- 7 A1-coverings.- 8 A1-homotopy and algebraic vector bundles.- 9 The affine B.G. property for the linear groups and the Grassmanian

/pdf/a1-algebraic-topology-over-a-field-1acgpmg44e.pdf

A1-Algebraic Topology over a Field

The description for this book, Compositional Methods in Homotopy Groups of Spheres. (AM-49), will be forthcoming.

Composition methods in homotopy groups of spheres

The six operations in equivariant motivic homotopy theory

https://www.mathematik.uni-muenchen.de/~morel/Prepublications/hcobordismsRevPost7Apr2011.pdf

Smooth varieties up to A^1-homotopy and algebraic h-cobordisms

We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A^1-homotopy type. Arithmetic considerations involving the sheaf of A^1-connected components lead us to introduce several different notions of connectedness in A^1-homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of algebraic varieties (e.g., rational connectedness). 
We introduce the notion of an A^1-h-cobordism, an algebro-geometric analog of the topological notion of h-cobordism, and use it as a tool to produce non-trivial A^1-weak equivalences of smooth proper varieties. Also, we give explicit computations of refined A^1-homotopy invariants, such as the A^1-fundamental sheaf of groups, for some A^1-connected varieties. We observe that the A^1-fundamental sheaf of groups plays a central yet mysterious role in the structure of A^1-h-cobordisms. As a consequence of these observations, we completely solve the classification problem for rational smooth proper surfaces over an algebraically closed field: while there exist arbitrary dimensional moduli of such surfaces, there are only countably many A^1-homotopy types, each uniquely determined by the isomorphism class of its A^1-fundamental sheaf of groups.

We give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2. As a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first nonstable A1-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel’s A1-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in A1-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

A cohomological classification of vector bundles on smooth affine threefolds

We establish a relative version of the abstract “affine representability” theorem in Ahomotopy theory from Part I of this paper. We then prove some A-invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass-Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in A-homotopy theory.

https://msp.org/gt/2018/22-2/gt-v22-n2-p11-s.pdf

Affine representability results in 1–homotopy theory, II : Principal bundles and homogeneous spaces

We determine the first non-stable A1-homotopy sheaf of SL n. Using techniques of obstruction theory involving the A1-Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank at least d‐1 on split smooth affine quadrics of dimension 2d‐1. These computations allow us to answer a question posed by Nori, which gives a criterion for completability of certain unimodular rows. Furthermore, we study compatibility of our computations of A1-homotopy sheaves with real and complex realization.

Aravind Asok

Papers

Smooth varieties up to A^1-homotopy and algebraic h-cobordisms

Smooth varieties up to A^1-homotopy and algebraic h-cobordisms

A cohomological classification of vector bundles on smooth affine threefolds

Affine representability results in 1–homotopy theory, II : Principal bundles and homogeneous spaces

Algebraic vector bundles on spheres