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3 added 426 characters in body

The answer is given in the following - very interesting - paper:

Bachuki Mesablishvili, Descent Theory for Schemes, Applied Categorical Structures 12: 485–512, 2004.

The author generalizes Grothendieck's descent theory from faithfully flat morphisms to pure morphisms. These are morphisms of schemes $f$ such that every base change $f'$ of it is schematically dense in the sense that $f'$ does not factor through a proper subscheme. In the affine case, $\mathrm{Spec}(A) \to \mathrm{Spec}(R)$ is pure iff $R \to A$ is "stable injective" in the sense that for every $R$-algebra $B$ the map $B \to A \otimes_R B$ is injective.

One of the main results (Theorem 5.15) states that for a quasi-compact morphism $f : X \to Y$ the following are equivalent:

1) $f$ is pure

2) $f^* : \mathrm{Qcoh}(Y) \to \mathrm{Qcoh}(X)$ faithful

3) $f$ is a stable effective descent morphism for quasi-coherent sheaves

In the case that $X,Y$ are affine there are even more characterizations (Theorem 4.18), for example we can also add 4) $f^*$ is conservative, and 5) $f^*$ is comonadic.

For flat morphisms, it is well-known that $f^*$ is faithful iff $f$ is surjective iff $f$ is faithfully flat. In general, every pure morphism is surjective (since otherwise some base change $\emptyset \to \mathrm{Spec}(\text{field})$ is not schematically dense). But there are lots of surjective morphisms which are not pure, for example the zero section $s: \mathrm{Spec}(k) \to \mathrm{Spec}(k[\epsilon]/\epsilon^2)$.

2 added 174 characters in body

The answer is given in the following - very interesting - paper:

Bachuki Mesablishvili, Descent Theory for Schemes, Applied Categorical Structures 12: 485–512, 2004.

The author generalizes Grothendieck's descent theory from faithfully flat morphisms to pure morphisms. These are morphisms of schemes $f$ such that every base change $f'$ of it is schematically dense in the sense that $f'$ does not factor through a proper subscheme. In the affine case, $\mathrm{Spec}(A) \to \mathrm{Spec}(R)$ is pure iff $R \to A$ is "stable injective" in the sense that for every $R$-algebra $B$ the map $B \to A \otimes_R B$ is injective.

One of the main results (Theorem 5.15) states that for a quasi-compact morphism $f : X \to Y$ the following are equivalent:

1) $f$ is pure

2) $f^* : \mathrm{Qcoh}(Y) \to \mathrm{Qcoh}(X)$ faithful.

3) $f$ is a stable effective descent morphism for quasi-coherent sheaves

In the case that $X,Y$ are affine there are even more characterizations (Theorem 4.18), for example we can also add 4) $f^*$ is conservative, and 5) $f^*$ is comonadic.

1

The answer is given in the following - very interesting - paper:

Bachuki Mesablishvili, Descent Theory for Schemes, Applied Categorical Structures 12: 485–512, 2004.

The author generalizes Grothendieck's descent theory from faithfully flat morphisms to pure morphisms. These are morphisms of schemes $f$ such that every base change $f'$ of it is schematically dense in the sense that $f'$ does not factor through a proper subscheme. In the affine case, $\mathrm{Spec}(A) \to \mathrm{Spec}(R)$ is pure iff $R \to A$ is "stable injective" in the sense that for every $R$-algebra $B$ the map $B \to A \otimes_R B$ is injective.

One of the main results (Theorem 5.15) states that for a quasi-compact morphism $f : X \to Y$ the following are equivalent:

1) $f$ is pure

2) $f^* : \mathrm{Qcoh}(Y) \to \mathrm{Qcoh}(X)$ faithful.

3) $f$ is a stable effective descent morphism for quasi-coherent sheaves