λ-ring

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In algebra, a λ-ring or lambda ring, introduced by Grothendieck (1957, 1958, p.148), is a commutative ring together with some operations λn on it behaving like the exterior powers of vector spaces.

For more about λ-rings see Atiyah & Tall (1969), Knutson (1973), Hazewinkel (2009) and Yau (2010).

Definition[edit]

A λ-ring is a ring R together with operation λn for integer n behaving like exterior powers of vector spaces, in the sense that they have the same behavior on sums and products that exterior powers have on direct sums and tensor products of vector spaces, and behave in the same way as exterior powers under composition. In more detail, they have the following properties:

  • λn(x) is 0 if n < 0, and is 1 if n = 0, x if n = 1.
  • λn(1) = 0 if n ≥ 2
  • λn(x + y) = Σi+j=nλi(xj(y)
  • λn(xy) = Pn1(x), ..., λn(x), λ1(y), ..., λn(y))
  • λmn(x)) = Pm,n1(x), ..., λmn(x))

where Pn and Pm,n are universal polynomials with integer coefficients describing the behavior of exterior powers on tensor products and under composition, that can be described as follows.

Suppose a commutative ring has elements x = x1 + x2 + ...,y = y1 + y2 + ... and define λn(x) by

  • \displaystyle\sum \lambda^m(x)t^m=\prod_i (1+tx_i)

and similarly for y. Informally we think of x and y as vector bundles that are sums of line bundles xi, yj, and think of λn(x) as the nth exterior power of x. Then the polynomials Pn and Pm,n are the universal polynomials such that

  • \displaystyle\sum P_m(\lambda^1(x),\cdots,\lambda^m(x),\lambda^1(y),\cdots,\lambda^m(y))t^m=\prod_{i,j} (1+tx_iy_j)
  • \displaystyle\sum P_{m,n}(\lambda^1(x),\cdots,\lambda^{mn}(x))t^m=\prod_{i_1<i_2<\cdots<i_n} (1+tx_{i_1}x_{i_2}\cdots x_{i_n})

Some authors call these special λ-rings, and use λ-ring for a more general concept where the conditions on λn(1), λn(xy) and λmn(x)) are dropped.

Examples[edit]

  • The ring of integers, with the operations λn(x) = (x
    n
    ), is a λ-ring.
  • The K-theory K0(X) of a topological space X is a λ-ring, with the lambda operations induced by taking exterior powers of a vector bundle.
  • The ring of symmetric polynomials is a λ-ring. In fact it is the universal λ-ring generated by one element.
  • Any binomial ring is a λ-ring such that all Adams operations are the identity.

References[edit]

Further reading[edit]