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In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let σ and τ be simplicial complexes and σ*τ be their join. Let J σ be the exterior face ideal of σ and Δ(σ) the exterior algebraic shifted complex of σ. Assume that σ*τ is a simplicial complex on [n]={1,2,...,n}. For any d-subset S⊂[n], let  denote the number of d-subsets R∈σ which are equal to or smaller than S with respect to the reverse lexicographic order. We will prove that  for all S⊂[n]. To prove this fact, we also prove that  for all S⊂[n] and for all nonsingular matrices ϕ, where Δϕ(σ) is the simplicial complex defined by .

Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes