8 - The Merkurjev–Suslin theorem

Summary

This chapter is devoted to the central result of this book, the celebrated theorem of Merkurjev and Suslin on the bijectivity of the Galois symbol h²_{k, m} : K^M₂ (k)/m K^M₂ (k) → H²(k,μ^⊗2_m) for all fields k and all integers m invertible in k. Following a method of Merkurjev, we shall deduce the theorem by a specialization argument from the partial results obtained at the end of the last chapter, using a powerful tool which is interesting in its own right, the K₂- analogue of Hilbert's Theorem 90. Apart from the case when m is a power of 2, no elementary proof of this theorem is known. To establish it, we first develop the foundations of the theory of Gersten complexes in Milnor K-theory. This material requires some familiarity with the language of schemes. Next comes an even deeper input, a technical statement about the K-cohomology of Severi– Brauer varieties. Its proof involves techniques outside the scope of the present book, so at this point our discussion will not be self-contained. The rest of the argument is then much more elementary and requires only the tools developed earlier in this book, so some readers might wish to take the results of the first three sections on faith and begin with Section 8.4.

The theorem was first proven in Merkurjev [1] in the case when m is a power of 2, relying on a computation by Suslin of the Quillen K-theory of a conic. Later several elementary proofs of this case were found which use no algebraic K-theory at all, at the price of rather involved calculations.