Novikov ring

In mathematics, given an additive subgroup ${\displaystyle \Gamma \subset \mathbb {R} }$, the Novikov ring ${\displaystyle \operatorname {Nov} (\Gamma )}$ of ${\displaystyle \Gamma }$ is the subring of ${\displaystyle \mathbb {Z} [\![\Gamma ]\!]}$^[1] consisting of formal sums ${\displaystyle \sum n_{\gamma _{i}}t^{\gamma _{i}}}$ such that ${\displaystyle \gamma _{1}>\gamma _{2}>\cdots }$ and ${\displaystyle \gamma _{i}\to -\infty }$. The notion was introduced by Sergei Novikov in the papers that initiated the generalization of Morse theory using a closed one-form instead of a function. The notion is used in quantum cohomology, among the others.

The Novikov ring ${\displaystyle \operatorname {Nov} (\Gamma )}$ is a principal ideal domain. Let S be the subset of ${\displaystyle \mathbb {Z} [\Gamma ]}$ consisting of those with leading term 1. Since the elements of S are unit elements of ${\displaystyle \operatorname {Nov} (\Gamma )}$, the localization ${\displaystyle \operatorname {Nov} (\Gamma )[S^{-1}]}$ of ${\displaystyle \operatorname {Nov} (\Gamma )}$ with respect to S is a subring of ${\displaystyle \operatorname {Nov} (\Gamma )}$ called the "rational part" of ${\displaystyle \operatorname {Nov} (\Gamma )}$; it is also a principal ideal domain.