Novikov ring

In mathematics, given an additive subgroup

, the Novikov ring

[1] consisting of formal sums

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

is a principal ideal domain.

consisting of those with leading term 1.

with respect to S is a subring of

called the "rational part" of

; it is also a principal ideal domain.

with nondegenerate critical points, the usual Morse theory constructs a free chain complex

such that the (integral) rank of

is the number of critical points of f of index p (called the Morse number).

It computes the (integral) homology of

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

may be viewed as a linear functional on the first homology group

; when composed with the Hurewicz homomorphism, it can be viewed as a group homomorphism

ξ : π =

By the universal property, this map in turns gives a ring homomorphism, making

Since X is a connected polyhedron, a local coefficient system over it corresponds one-to-one to a

be a local coefficient system corresponding to

is a finitely generated module over

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

The number of cyclic modules in the torsion part is denoted by

is the usual Betti number of X.

The analog of Morse inequalities holds for Novikov numbers as well (cf.