Additive group of ring of Witt vectors inherits algebraic group structure

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For ring of Witt vectors

Suppose K is a field and p is a prime number. Suppose W denotes the ring of Witt vectors over K for the prime p. Then, the additive group of W naturally inherits the structure of an algebraic group over K, with the algebraic group structure arising from the fact that the operations are defined by formulas with coordinates in K, and that these formulas work to give a group in any field extension of K.

For ring of Witt vectors truncated to finite length

Suppose K is a field, p is a prime number, and d is a natural number. Consider the ring of truncated Witt vectors of length d, obtained by projecting the ring of Witt vectors to the first d coordinates. The additive group of this ring is an algebraic group over K.

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