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1. What the question asks (in plain language)

For a morphism of schemes

[ f : X\longrightarrow Y , ]

the pull‑back functor (f^{}) (left adjoint) and the direct‑image functor
(f_{}) (right adjoint) satisfy the adjunction

[ \operatorname{Hom}{\mathcal O_X}!\bigl(f^{*}\mathcal G,\mathcal F\bigr) \;\cong\; \operatorname{Hom}{\mathcal O_Y}!\bigl(\mathcal G, f_{*}\mathcal F\bigr). ]

Taking (\mathcal G = f_{*}\mathcal F) and the identity map on the right‑hand side gives the counit of the adjunction

[ \varepsilon_{\mathcal F}\;:\;f^{}f_{}\mathcal F\;\longrightarrow\;\mathcal F . ]

The problem is to describe when this canonical morphism is an isomorphism. In other words: for which morphisms (f) and for which (\mathcal O_X)-modules (\mathcal F) does the natural map

[ f^{}f_{}\mathcal F \xrightarrow{\;\varepsilon_{\mathcal F}\;} \mathcal F ]

have an inverse?

2. The answer – step by step

We treat the most useful and classical situations.
All statements are made for quasi‑coherent sheaves; the same formulas for arbitrary sheaves are false in general.

2.1 General description of the map

Write the adjunction counit in the concrete form

[ \varepsilon_{\mathcal F}\;:\; f^{-1}f_{*}\mathcal F\;\otimes_{f^{-1}\mathcal O_{Y}}\;\mathcal O_{X} \;\longrightarrow\;\mathcal F, \qquad a\otimes s\;\mapsto\;a\cdot s . ]

Surjectivity of (\varepsilon_{\mathcal F}) means that the sections of (\mathcal F) are generated by the sections that come from (f_{}\mathcal F); i.e. (\mathcal F) is relatively globally generated (or *(f)‑generated).

Injectivity is subtler; it asks that no non‑trivial relation among those generators appears after we tensor with (\mathcal O_{X}). This is automatically true when the structural sheaf (\mathcal O_{X}) is flat over (f^{-1}\mathcal O_{Y}) (equivalently, when (f) is flat).

2.2 The affine (hence finite, Stein, etc.) case

Theorem (affine morphism).
Let (f:X\to Y) be an affine morphism (in particular finite or Stein). For every quasi‑coherent (\mathcal O_{X})-module (\mathcal F) the counit [ \varepsilon_{\mathcal F}\colon f^{}f_{}\mathcal F\longrightarrow\mathcal F ] is an isomorphism.

Proof

Because (f) is affine we may work locally on (Y).
Take an affine open (V=\operatorname{Spec} B\subseteq Y) and set (U=f^{-1}(V)=\operatorname{Spec}A); the restriction of (f) corresponds to the ring homomorphism (B\to A).

A quasi‑coherent sheaf (\mathcal F) on (U) is the sheaf associated to an (A)-module (M); its direct image is the sheaf associated to the same (A)-module viewed as a (B)-module via restriction of scalars:

[ f_{*}\mathcal F \;\leftrightarrow\; {}_{B}M . ]

The pull‑back of a (B)-module (N) is the sheaf associated to the extension of scalars (A\otimes_{B}N). Therefore

[ f^{}f_{}\mathcal F \;\leftrightarrow\; A\otimes_{B}M . ]

The counit corresponds to the canonical (A)-linear map

[ \varepsilon_{M}\colon A\otimes_{B}M\longrightarrow M,\qquad a\otimes m\;\mapsto\;a\cdot m . ]

This map is an isomorphism for every (A)-module (M): it is surjective because any element (m\in M) equals (1\otimes m); it is injective because the tensor product (A\otimes_{B}M) is, by definition, the free (A)-module generated by symbols (a\otimes m) modulo the relations that make the map above well‑defined. Hence (\varepsilon_{M}) has inverse (m\mapsto 1\otimes m).

Since the statement holds on an affine open cover and the constructions are compatible with restriction, the sheaf morphism (\varepsilon_{\mathcal F}) is an isomorphism on all of (X). ∎

Consequences.

  • Finite morphisms are affine, so the same conclusion holds for them.
  • If (f) is affine and (\mathcal F) is coherent, the map is an isomorphism as well (coherence is not needed for the proof).

2.3 Flat morphisms – a criterion for injectivity

Assume now that (f) is flat.
Flatness guarantees that (\mathcal O_{X}) is a flat (f^{-1}\mathcal O_{Y})-module, which makes the functor (-\otimes_{f^{-1}\mathcal O_{Y}}\mathcal O_{X}) exact. Consequently the counit (\varepsilon_{\mathcal F}) is injective for every quasi‑coherent (\mathcal F). Hence we obtain:

Proposition.
If (f) is flat and (\mathcal F) is relatively globally generated, i.e. the counit is surjective, then (\varepsilon_{\mathcal F}) is an isomorphism.

Thus, for a flat morphism, the only extra condition needed is surjectivity (relative generation).

2.4 The projective case and relatively very ample line bundles

Let

  • (f:X\to Y) be a projective morphism,
  • (\mathcal L) a relatively very ample invertible sheaf on (X) (so (\mathcal L = \mathcal O_{X}(

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