nLab
Novikov field

Novikov fields

Novikov fields

Idea

A Novikov field (or Novikov ring) is a field of generalized formal power series that allows non-integer exponents. In contrast to a Hahn series field, in a Novikov field the series are only ω\omega-long rather than transfinitely long. This requires a “left-finiteness” condition on the exponents to ensure closure under multiplication.

Definition

There are many variant definitions of Novikov fields in the literature, see e.g. this question. Here we give a reasonable-seeming general and abstract definition.

Let kk be a commutative ring and GG a linearly ordered abelian group.

Definition

The Novikov ring of kk with value group GG is the ring of functions f:Gkf:G\to k such that for any yGy\in G the set {xGx<yf(x)0}\{ x\in G \mid x \lt y \wedge f(x)\neq 0\} is finite. Such functions are added pointwise, and multiplied by the formula

(fg)(z)= x+y=zf(x)g(y). (f\cdot g)(z) = \sum_{x+y=z} f(x) \cdot g(y).

Left-finiteness of the support of ff and gg implies that the above sum is finite. Specifically, if g0g\neq 0 then since GG is totally ordered, there is a least y 0y_0 such that g(y 0)0g(y_0)\neq 0. Then the set {xxzy 0f(x)0}\{ x\mid x \le z-y_0 \wedge f(x)\neq 0\} is finite, and hence so is its subset {xy.x+y=zf(x)0g(y)0}\{ x \mid \exists y. x+y = z \wedge f(x)\neq 0 \wedge g(y)\neq 0 \}. Note that this depends on the fact that GG is totally ordered and a group; a partially ordered monoid would not suffice.

Notationally, we write such a function as xGf(x)t x\sum_{x\in G} f(x)\, t^x for tt a formal variable. If kk is a field, then so is the Novikov ring.

Examples

  • When G=G=\mathbb{R}, the Novikov field is sometimes called the “universal Novikov field”.

  • When G=G=\mathbb{Q} and k=k =\mathbb{R}, the Novikov field is known as the Levi-Civita field.

Characterizations

The Novikov field embeds into the Hahn series field k[[t G]]k[[t^G]]. It can (probably) be characterized therein as

  • The set of Hahn series with order type ω\omega that converge to themselves in the valuation topology.

  • The closure of the field k(t G)k(t^G) of generalized rational functions? inside the Hahn series field.

It can also (probably) be characterized abstractly as

  • The Cauchy completion of k(t G)k(t^G) in its valuation uniformity.

  • The completion of k(t G)k(t^G) as a valued field, i.e. the unique (up to isomorphism) dense valued field extension without proper dense valued field extension.

Applications

  • The universal Novikov field of \mathbb{R} is a natural context in which to relate magnitude homology? of finite metric spaces to their magnitude?. (Hahn series also suffice, but all the action actually takes place in the Novikov field.)

  • In the Fukaya category, the chain complexes defining Hom’s between objects are defined over a Novikov ring.

Other rings of generalized power series include:

Hahn series are a special kind of Ribenboim power series, but Puiseux and Novikov series are not. However, they are all instances of the linearization of a finiteness space.

References

Last revised on July 22, 2019 at 23:01:24. See the history of this page for a list of all contributions to it.