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4. Vector bundles. ) and the gij (x) respect this structure, then each fiber also carries this structure. In particular, if N is a K-module and the transition maps respect this structure, then F is called a vector bundle. In this case, the bundle atlas is of the form A = (p−1 (Ui ), ϕi )i∈I with ϕi : p−1 (Ui ) → V × W, [i, x, w] → (x, w). Change of charts is given by the transition functions gij (x, w) = (ϕij (x), gij (x)w). Homomorphisms are required to respect the extra structure on the fibers.

1. The ring of dual numbers over K. The ring of dual numbers over K is the truncated polynomial ring K[x]/(x2 ) ; it can also be defined in a similar way as complex numbers, replacing the condition i2 = −1 by ε2 = 0 : K[ε] := K ⊕ εK, (a + εb)(a + εb ) = aa + ε(ab + ba ). 1) In analogy with the terminology for complex numbers, let us call z = x + εy a dual number, x its spacial part, y its infinitesimal part and ε the infinitesimal or dual number unit. A dual number is invertible iff its spacial part is invertible, and inversion is given by x − εy .

14) where a “hat” means “omit this coefficient”. 7) is, for any choice 1 ≤ j1 < . . < j ≤ k , 0 → i=1 εji R → R pji −→ i=1 K[ε1 , . . , εji , . . , εk ] → 0. 15) In particular, for the “maximal choice” ji = i , i = 1, 2, . . , k , we get 0 → ε1 . . εk R → R → k i=1 K[ε1 , . . , εj , . . , εk ] → 0. 16) corresponds to the “most vertical” bundle ε1 . . εk T M → T kM → k i=1 T k−1 M. There are also various projections T k K → T K for all = 0, . . , k −1 ; their kernels are certain sums of the ideals considered so far.