A note on the subadditivity of Syzygies

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Let R=S/I be a graded algebra with tᵢ and Tᵢ being the minimal and maximal shifts in the minimal S resolution of R at degree i. In this paper we prove that tₙ≤ t₁+Tₙ₋₁, for all n and as a consequence, we show that for Gorenstein algebras of codimension h, the subadditivity of maximal shifts Tᵢ in the minimal resolution holds for i ≥ h-1, i.e, we show that Tᵢ ≤ Tₐ+Tᵢ₋ₐ for i≥ h-1.