Verschiebungsterm

Verschiebungsterm, literally “shift term” in German, is a term used in the area of p-adic and Witt vector theory to denote the component that is introduced by the Verschiebung operator. The Verschiebung operator, denoted V, acts as a shift in indexing and, together with Frobenius F, encodes the structure of Witt vectors and related Dieudonné modules. The Verschiebungsterm is thus the part of an element that arises from applying V and reflects the displaced coordinates in the shifted object.

In the context of p-typical Witt vectors, elements are sequences (a0, a1, a2, …). The Verschiebung map V: W(R) → W(R) sends (a0, a1, a2, …) to (0, a0, a1, a2, …). The Verschiebungsterm corresponds to the new coordinates created by this shift, most notably the initial zero component and the subsequent placement of the original coordinates one position higher. This term interacts with Frobenius F, which satisfies F V = V F = p in the appropriate Witt-vector setting, encoding a fundamental compatibility between the two operators.

Beyond Witt vectors, Verschiebung also appears in Dieudonné theory, where V acts on Dieudonné modules associated with p-divisible groups. There, V is semilinear and together with F governs the p-adic structure, with the Verschiebungsterm referring to the contributions produced by the Verschiebung operator in the module or vector corresponding to a shifted level.

See also: Frobenius, Witt vectors, Dieudonné theory, p-divisible groups.