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It seems that the page defines what is called locally of finite type. According to [1]https://stacks.math.columbia.edu/tag/01T1 a morphism of finite type should additionally be quasi-compact. This may or may not be included depending on the sizes of the index sets in which ${\displaystyle i,j}$ reside, however, such index sets are not specified.

Potentially there needs to be a split into two articles, one for locally of finite type and one for finite type. Note that the quasi-compact is mostly satisfied for morphisms between "reasonable" schemes which are of greatest interest. Since this articles makes a lot of concrete examples with affine schemes where only the condition for "locally of finite type" are checked, they could easily go to a "finite type" page with the additional remark that quasi-compactness is trivially satisfied. Tpreu (talk) 10:40, 15 August 2023 (UTC)[reply]