Novikov ring

In mathematics, given an additive subgroup formula_1, the Novikov ring formula_2 of formula_3 is the subring of formula_4 consisting of formal sums formula_5 such that formula_6 and formula_7. 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 formula_2 is a principal ideal domain. Let "S" be the subset of formula_9 consisting of those with leading term 1. Since the elements of "S" are unit elements of formula_2, the localization formula_11 of formula_2 with respect to "S" is a subring of formula_2 called the "rational part" of formula_2; it is also a principal ideal domain.
Novikov numbers.
Given a smooth function "f" on a smooth manifold formula_15 with nondegenerate critical points, the usual Morse theory constructs a free chain complex formula_16 such that the (integral) rank of formula_17 is the number of critical points of "f" of index "p" (called the Morse number). It computes the (integral) homology of formula_15 (cf. Morse homology): 
In an analogy with this, one can define "Novikov numbers". Let "X" be a connected polyhedron with a base point. Each cohomology class formula_20 may be viewed as a linear functional on the first homology group formula_21; when composed with the Hurewicz homomorphism, it can be viewed as a group homomorphism formula_22. By the universal property, this map in turns gives a ring homomorphism, 
making formula_24 a module over formula_25. Since "X" is a connected polyhedron, a local coefficient system over it corresponds one-to-one to a formula_25-module. Let formula_27 be a local coefficient system corresponding to formula_24 with module structure given by formula_29. The homology group formula_30 is a finitely generated module over formula_31 which is, by the structure theorem, the direct sum of its free part and its torsion part. The rank of the free part is called the Novikov Betti number and is denoted by formula_32. The number of cyclic modules in the torsion part is denoted by formula_33. If formula_34, formula_27 is trivial and formula_36 is the usual Betti number of "X".
The analog of Morse inequalities holds for Novikov numbers as well (cf. the reference for now.)