# Cohomology ring

A ring the additive group of which is the graded cohomology group

$$\oplus _ { n= } 0 ^ \infty H ^ {n} ( X , A ) ,$$

where $X$ is a chain complex, $A$ is a coefficient group and the multiplication is defined by the linear set of mappings

$$\nu _ {m,n} : H ^ {m} ( X , A ) \otimes H ^ {n} ( X , A ) \rightarrow H ^ {m+} n ( X , A ) ,$$

for all $m , n \geq 0$, which are the inner cohomology multiplications (cup products). The cohomology ring turns out to be equipped with the structure of a graded ring.

For the existence of the mappings $\nu _ {m,n}$ it is enough to have a set of mappings $\widehat \nu _ {m,n} : X _ {m+} n \rightarrow X _ {m} \otimes X _ {n}$ satisfying certain additional properties, and a mapping $A \otimes A \rightarrow A$, that is, a multiplication in the coefficient group $A$( see [2]). The $\nu _ {m,n}$ induce mappings

$$\mathop{\rm Hom} ( X _ {m} , A ) \otimes \mathop{\rm Hom} ( X _ {n} , A ) \rightarrow \mathop{\rm Hom} ( X _ {m+} n , A ) ,$$

which in their turn induce mappings $\nu _ {m,n}$ in cohomology.

In particular, a ring structure is defined on the graded group $H ( G , \mathbf Z ) = \oplus _ {n=} 0 ^ \infty H ^ {n} ( G , \mathbf Z )$, where $G$ is a group and $\mathbf Z$ is the ring of integers with a trivial $G$- action. The corresponding mappings $\nu _ {m,n}$ coincide with the $\cup$- product. This is an associative ring with identity, and for homogeneous elements $a , b \in H ( G , \mathbf Z )$ of degrees $p , q$ respectively, $a b = ( - 1 ) ^ {pq} b a$.

Analogously, the $\cup$- product defines a ring structure on the group $\oplus _ {n=} 0 ^ \infty H ^ {n} ( X , \mathbf Z )$, where $H ^ {n} ( X , \mathbf Z )$ is the $n$- dimensional singular cohomology group of a topological space $X$ with coefficients in $\mathbf Z$.

#### References

 [1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) [2] S. MacLane, "Homology" , Springer (1963)