In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λn on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results (Lascoux (2003)). λ-rings were introduced by Grothendieck (1957, 1958, p.148). For more about λ-rings see Atiyah & Tall (1969), Knutson (1973), Hazewinkel (2009) and Yau (2010).
Motivation If V and W are finite-dimensional vector spaces over a field k, then we can form the direct sum V ⊕W, the tensor product V ⊗W, and the n-th exterior power of V, Λn(V). All of these are again finite-dimensional vector spaces over k. The same three operations of direct sum, tensor product and exterior power are also available when working with k-linear representations of a finite group, when working with vector bundles over some topological space, and in more general situations. λ-rings are designed to abstract the common algebraic properties of these three operations, where we also allow for formal inverses with respect to the direct sum operation. (These formal inverses also appear in Grothendieck groups, which is why the underlying additive groups of most λ-rings are Grothendieck groups.) The addition in the ring corresponds to the direct sum, the multiplication in the ring corresponds to the tensor product, and the λ-operations to the exterior powers. For example, the isomorphism
Λ 2 ( V ⊕ W ) ≅ Λ 2 ( V ) ⊕ ( Λ 1 ( V ) ⊗ Λ 1 ( W ) ) ⊕ Λ 2 ( W ) {\displaystyle \Lambda ^{2}(V\oplus W)\cong \Lambda ^{2}(V)\oplus \left(\Lambda ^{1}(V)\otimes \Lambda ^{1}(W)\right)\oplus \Lambda ^{2}(W)}
corresponds to the formula
λ 2 ( x + y ) = λ 2 ( x ) + λ 1 ( x ) λ 1 ( y ) + λ 2 ( y ) {\displaystyle \lambda ^{2}(x+y)=\lambda ^{2}(x)+\lambda ^{1}(x)\lambda ^{1}(y)+\lambda ^{2}(y)}
valid in all λ-rings, and the isomorphism
Λ 1 ( V ⊗ W ) ≅ Λ 1 ( V ) ⊗ Λ 1 ( W ) {\displaystyle \Lambda ^{1}(V\otimes W)\cong \Lambda ^{1}(V)\otimes \Lambda ^{1}(W)}
corresponds to the formula
λ 1 ( x y ) = λ 1 ( x ) λ 1 ( y ) {\displaystyle \lambda ^{1}(xy)=\lambda ^{1}(x)\lambda ^{1}(y)}
valid in all λ-rings. Analogous but (much) more complicated formulas govern the higher order λ-operators.
Motivation with Vector Bundles If we have a short exact sequence of vector bundles over a smooth scheme X {\displaystyle X}
0 → E ″ → E → E ′ → 0 , {\displaystyle 0\to {\mathcal {E}}''\to {\mathcal {E}}\to {\mathcal {E}}'\to 0,} then locally, for a small enough open neighborhood U {\displaystyle U} we have the isomorphism
⋀ n E | U ≅ ⨁ i + j = n ⋀ i E ′ | U ⊗ ⋀ j E ″ | U {\displaystyle \bigwedge ^{n}{\mathcal {E}}|_{U}\cong \bigoplus _{i+j=n}\bigwedge ^{i}{\mathcal {E}}'|_{U}\otimes \bigwedge ^{j}{\mathcal {E}}''|_{U}}
Now, in the Grothendieck group K ( X ) {\displaystyle K(X)} of X {\displaystyle X} (which is actually a ring), we get this local equation globally for free, from the defining equivalence relations. So
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