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On the search page you can search for examples of schemes and/or morphisms having (or not having) certain properties. Also it will show you additional properties that can be concluded using the available theorems (see the overview).

Example 1 (non-separated schemes)

Let's say we want to find examples of non-[prop:scheme.separated] schemes.
  1. Type a name for your scheme, e.g. $X$, and click .
  2. A list of scheme properties appears. You can click these properties to indicate that your scheme $X$ should (or should not) have one or more of these properties. In this case, click twice on 'separated' to indicate that your scheme should not be separated.
  3. Click and wait.
  4. Now you will see an example of a scheme $X$ which is not separated, e.g. [ex:AA_1_QQ_double_origin].
Suppose we are also interested in what other properties a non-separated scheme $X$ has.
  1. Make sure $X$ is selected, click and wait.
  2. Now you will see a table of scheme properties, indicating whether $X$ has (or does not have) certain other properties, together with a proof. For example, $X$ is also not [prop:scheme.affine] by [thm:sp_of_af].

Example 2 (is reduced an fppf-local property?)

Suppose we want to know if being reduced is an fppf-local property. That is, if $f : X \to Y$ is an [prop:morphism.fppf_cover] and $Y$ is [prop:scheme.reduced], does it follow that $X$ is reduced as well?
  1. Create a morphism $f : X \to Y$ using .
  2. Click on the properties 'fppf-cover' for $f$, and 'reduced' for $Y$ to indicate that $f$ and $Y$ should have these properties.
  3. Now select $X$ and click on to see if it follows that $X$ is also reduced.
  4. Looking through the conclusions, you will not find that $X$ must be reduced. Hence, instead, we will try to find a counterexample, that is, an example where $X$ is not reduced.
  5. Selecting $X$, click twice on 'reduced' to indicate that $X$ should not be reduced.
  6. Click and wait.
  7. We find (at least one) example of an fppf-cover with reduced target but non-reduced source, e.g. [ex:fat_point_to_Spec_QQ]. This shows that being reduced is not an fppf-local property.