Pairing

In mathematics, a pairing is an R-bilinear map of modules, where R is the underlying ring. In mathematics, a pairing is an R-bilinear map of modules, where R is the underlying ring. Let R be a commutative ring with unity, and let M, N and L be three R-modules. A pairing is any R-bilinear map e : M × N → L {displaystyle e:M imes N o L} . That is, it satisfies for any r ∈ R {displaystyle rin R} and any m , m 1 , m 2 ∈ M {displaystyle m,m_{1},m_{2}in M} and any n , n 1 , n 2 ∈ N {displaystyle n,n_{1},n_{2}in N} . Equivalently, a pairing is an R-linear map where M ⊗ R N {displaystyle Motimes _{R}N} denotes the tensor product of M and N. A pairing can also be considered as an R-linear map Φ : M → Hom R ⁡ ( N , L ) {displaystyle Phi :M o operatorname {Hom} _{R}(N,L)} , which matches the first definition by setting Φ ( m ) ( n ) := e ( m , n ) {displaystyle Phi (m)(n):=e(m,n)} . A pairing is called perfect if the above map Φ {displaystyle Phi } is an isomorphism of R-modules. A pairing is called non-degenerate on the right if for the above map we have that e ( m , n ) = 0 {displaystyle e(m,n)=0} for all m {displaystyle m} implies n = 0 {displaystyle n=0} ; similarly, $e$ is called non-degenerate on the left if e ( m , n ) = 0 {displaystyle e(m,n)=0} for all n {displaystyle n} implies m = 0 {displaystyle m=0} . A pairing is called alternating if N = M {displaystyle N=M} and for the above map we have e ( m , m ) = 0 {displaystyle e(m,m)=0} for all m.