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We note that, given a scheme X = Spec (R) and a coherent O X -algebra F such that each affine restriction F | U i is associated with some faithful finitely generated projective R i -algebra A i , if σ i is an anti-automorphism of A i such that x σ i (x) is in R i for all x ∈ A i , then F admits one standard involution σ ~ , which commutes with all automorphisms and anti-automorphisms of F . Next, given a locally finitely presented O X -module E on an affine scheme X, and an involution of the first kind σ on the sheaf of endomorphisms E n d O X (E) , there exist an invertible O X -module L and isomorphisms φ : E ⊗ O X L → ∼ E ∗ and Φ : E n d O X (E) → ∼ E n d O X (E ∗) such that, locally, σ ⊗ id = Φ ∘ m , where m is the natural isomorphism E n d O X (E ⊗ L) ≃ E n d O X (E) on any open U in X. And finally, under the same conditions that (X , O X) is a locally ringed space, E a locally finitely presented O X -module, and σ an involution of the first kind on E n d O X (E) , for any x ∈ X , there is u ∈ L x such that σ x (f) = u - 1 ∘ f ∗ ∘ u , for any f ∈ E n d O X , x (E x). Moreover, for any local gauge V of L at x, there is a unit ε ∈ O X (V) such that ε x u (q) (p) = u (p) (q) , for all p, q ∈ E x . [ABSTRACT FROM AUTHOR]