conte.html

<p>An important feature of schemes is that they are well-suited to be viewed &quot;in families&quot;.  This means that, in the back of our minds, there should always be a morphism of schemes 
\(X \to Y\) more or less explicitly mentioned.  The scheme \(Y\) is the <em>base scheme</em>.  This point of view is helpful even in the case in which we are not thinking of a morphism.  In 
such cases, likely the base is a single point, e.g. the spectrum of a field.</p>

<p>A <em>base ring</em> is simply the case above when there is a ring \(R\) such that \(Y = \spec R\); equivalently, the base scheme \(Y\) is affine.  If \(R\) is a field, then \(R\) is also 
called a <em>ground field</em>.</p>

<p><strong>Exercise.</strong> Prove or disprove: <em>A scheme with only one point is the spectrum of a field</em>.</p>

<p>A notational convention is to write <em>families</em> and <em>structure morphisms to the ground field/ring</em> vertically:</p>

<p>
\(
  \begin{array}{c}
  X \cr
  \downarrow \cr
  Y
  \end{array}
\)
<img src="pictures/2020_11_03_small_lavagna.png" alt="ima" width="400px;" height="auto;">
</p>


<p>The morphism \(X \to Y\) need not satisfy any further property.  However, it is common that <em>families</em> satisfy extra conditions, implicitly or explicitly.  Common assumptions are: 
smooth, proper, projective, flat, equi-dimensional, finite type, locally of finite type...</p>

<h3>Running example</h3>

<p>As usual, our running example is an elaboration of
\( \oh \in \Q\).</p>

<ol>
<li><p><strong>The family \(\Z \subset \Z[\oh]\).</strong><br>
The base is \(\spec \Z\), the total space of the family is \(\spec \Z [\oh]\).  These two schemes coincide away from the prime \((2)\): the family

<table><thead>
<tr>
<th>\(\begin{array}{c} \spec \zoh \cr \downarrow \cr \spec \Z \end{array}\)</th>
<th><img src="pictures/2020_11_03_inclusion.jpeg" alt="ima" width="400px;" height="auto;"></th>
</tr>
</thead><tbody>
</tbody></table>
is the inclusion of the complement of the closed point \((2) \in \spec \Z\).<br>
This family <em>is</em> open, smooth, flat, of relative dimension 0...<br>
It <em>is not</em> projective, proper, surjective, closed, universally closed...  </p></li>
<li><p><strong>The family \(\Z[\oh] \subset \Q\).</strong><br>
The base is \(\spec \Z[\oh]\), the total space of the family is \(\spec \Q\).  The total space of the family is a single point: the family
\[
\begin{array}{c}
\spec \Q \cr
\downarrow \cr
\spec \Z [\oh]
\end{array}
\]
is the inclusion of the generic point \( (0) \in \spec \Z[\oh]\) of the base.<br>
This family <em>is not</em> locally of finite type, open, smooth, flat, projective, proper, surjective, closed, universally closed...  In fact, it is a bit of a struggle to find <em>any</em> 
property that this morphism has!</p></li>
</ol>