I'm reading Knutson's book on algebraic spaces, and I stumbled over the quasi-separatedness axiom in his definition of algebraic spaces (Definition 1.1, Chapter II). He defines an algebraic space A as a sheaf on the site of schemes with the étale topology satisfying:

I) Local representability. There exists an a representable étale covering $U -> A\rightarrow A$, U $U$ a scheme.
II) Quasi-separatedness. The map U x_A $U -> \times_A U x \rightarrow U \times U$ is quasi-compact.

In a technical remark (1.9) at the end of the section, he argues that the quasi-separatedness assumption is needed for the existence of fibre products in the categeory of algebraic spaces. As I see it, this is wrong. For instance, the proof of existence of fibre products for schemes given i Hartshorne carries over to algebraic spaces without problems, even if we just assume local representability.

This makes me wonder, would it be more natural to take only the local representability as requirement for algebraic spaces, or do we run into problems later on?

Sometimes one sees informal definitions of algebraic spaces as the closure of schemes under étale equivalence relations in the category of étale sheaves. In what sence is this true? Here it seems to me that we really do need the quasi-separatedness axiom (or something similar) since we need an étale equivalence relations $R -> UxU \rightarrow U \times U$ to satisfy effective descent in order to get local representability for its quotient.

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# Quasi-separatedness for Algebraic Spaces

I'm reading Knutson's book on algebraic spaces and I stumbled over the quasi-separatedness axiom in his definition of algebraic spaces (Definition 1.1, Chapter II). He defines an algebraic space A as a sheaf on the site of schemes with the étale topology satisfying:

I) Local representability. There exists an étale covering U -> A, U a scheme.
II) Quasi-separatedness. The map U x_A U -> U x U is quasi-compact.

In a technical remark (1.9) at the end of the section, he argues that the quasi-separatedness assumption is needed for the existence of fibre products in the categeory of algebraic spaces. As I see it, this is wrong. For instance, the proof of existence of fibre products for schemes given i Hartshorne carries over to algebraic spaces without problems, even if we just assume local representability.

This makes me wonder, would it be more natural to take only the local representability as requirement for algebraic spaces, or do we run into problems later on?

Sometimes one sees informal definitions of algebraic spaces as the closure of schemes under étale equivalence relations in the category of étale sheaves. In what sence is this true? Here it seems to me that we really do need the quasi-separatedness axiom (or something similar) since we need an étale equivalence relations R -> UxU to satisfy effective descent in order to get local representability for its quotient.