Topic Algebraic K-Theory and K2
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Algebraic K-Theory and K2

Algebraic \(K\)-theory provides the cohomological tools needed to construct and classify central extensions of algebraic groups. The relevant pieces for the Langlands roadmap are \(K_0\), \(K_1\), and \(K_2\) (Milnor's "low-dimensional" K-theory), which are exactly the ingredients of the Brylinski–Deligne covers ([[node:covers.brylinski_deligne_k2_extensions|BD \(K_2\)-extensions]]).

K_0, K_1, K_2

\(K_0(R)\)

The Grothendieck group of finitely generated projective \(R\)-modules. For a field, \(K_0(F) = \mathbb{Z}\); for a Dedekind domain, \(K_0(R) = \mathbb{Z} \oplus \operatorname{Pic}(R)\). Records "rank" and "determinant" information.

\(K_1(R)\)

\(K_1(R) = \operatorname{GL}(R)^{\mathrm{ab}} = \operatorname{GL}(R) / [\operatorname{GL}(R), \operatorname{GL}(R)]\). For a commutative ring (with mild hypotheses), \(K_1(R) = R^\times \times \operatorname{SK}_1(R)\) where \(\operatorname{SK}_1(R)\) is the kernel of the determinant; for a field, \(K_1(F) = F^\times\).

\(K_2(R)\) (Milnor)

The Steinberg group \(\operatorname{St}(R)\) is presented by generators \(x_{ij}(r)\) (for \(i \ne j\) and \(r \in R\)) subject to the Steinberg relations. There is a surjection \(\operatorname{St}(R) \twoheadrightarrow E(R)\) (elementary matrices) whose kernel is \(K_2(R)\):

\[ 1 \to K_2(R) \to \operatorname{St}(R) \to E(R) \to 1. \]

This is the universal central extension of \(E(R)\). For a field \(F\), Matsumoto's theorem gives a presentation of \(K_2(F)\) by symbols \(\{a, b\}\) for \(a, b \in F^\times\) modulo bilinearity and the Steinberg relation \(\{a, 1 - a\} = 1\).

Milnor \(K_*^M\)

Higher Milnor \(K\)-groups \(K_n^M(F) = (F^\times)^{\otimes n} / \text{Steinberg}\) extend the \(K_2\) symbol picture. They have a coherent structure parallel to Galois cohomology (the Bloch–Kato conjecture, now theorem).

Tame symbols and reciprocity

For a discrete valuation \(v : F^\times \to \mathbb{Z}\) with residue field \(k\), the tame symbol is the bilinear map

\[ \{-,-\}_v \colon K_2(F) \to k^\times, \qquad \{a, b\}_v = (-1)^{v(a) v(b)} \cdot \overline{a^{v(b)} b^{-v(a)}}. \]

These symbols satisfy a reciprocity law for curves over a field: the sum over all closed points of the tame symbols of a pair of rational functions is trivial.

Connection to algebraic groups

  • \(K_1(R) = \operatorname{GL}(R)^{\mathrm{ab}}\) is essentially the group of units \(R^\times\) (when commutative + nice).
  • \(K_2(R)\) is the universal central extension of \(\operatorname{SL}(R)\) (equivalently, \(E(R)\)).
  • For \(G\) a split semisimple simply connected algebraic group over a ring \(R\), \(G(R)\) has a universal central extension by \(K_2(R)\).
  • The Brylinski–Deligne theory constructs central extensions of reductive groups by \(K_2\), parametrising metaplectic covers in a uniform way over local and global fields.

Mathlib formalisation

Mathlib has limited K-theory at present. The relevant pieces:

  • Mathlib.Algebra.Module.LinearMap.Basic and friends — modules over a ring, projective modules.
  • Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs — the abstract group \(\operatorname{GL}_n(R)\).
  • No native \(K_n(R)\) typeclass — to be developed.

References

  • Milnor, Introduction to Algebraic K-Theory (Annals of Mathematics Studies 72, 1971).
  • Weibel, The K-Book: An Introduction to Algebraic K-Theory (AMS GSM 145, 2013).
  • Bass, Algebraic K-Theory (Benjamin, 1968).
  • Brylinski–Deligne, Central extensions of reductive groups by \(K_2\) (Publ. IHÉS 94, 2001).

Prerequisite Topics

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