The Stacks Project


Tag 01IJ

Chapter 25: Schemes > Section 25.9: Schemes

Definition 25.9.1. A scheme is a locally ringed space with the property that every point has an open neighbourhood which is an affine scheme. A morphism of schemes is a morphism of locally ringed spaces. The category of schemes will be denoted $\mathit{Sch}$.

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 1601–1616 (see updates for more information).

    \begin{definition}
    \label{definition-scheme}
    \begin{history}
    In \cite{EGA1} what we call a scheme was called a ``pre-sch\'ema'' and the
    name ``sch\'ema'' was reserved for what is a separated scheme in the
    Stacks project. In the second edition \cite{EGA1-second} the terminology
    was changed to the terminology that is now standard. However, one may
    occasionally encounter the terminology ``prescheme'', for example in
    \cite{Murre-lectures}.
    \end{history}
    A {\it scheme} is a locally ringed space with the property that
    every point has an open neighbourhood which is an affine scheme.
    A {\it morphism of schemes} is a morphism of locally
    ringed spaces. The category of schemes will be denoted
    $\Sch$.
    \end{definition}

    Historical remarks

    In [EGA1] what we call a scheme was called a ''pre-schéma'' and the name ''schéma'' was reserved for what is a separated scheme in the Stacks project. In the second edition [EGA1-second] the terminology was changed to the terminology that is now standard. However, one may occasionally encounter the terminology ''prescheme'', for example in [Murre-lectures].

    Comments (2)

    Comment #917 by Johan Commelin (site) on August 14, 2014 a 9:37 am UTC

    Maybe we want a historical remark here about the “prescheme/scheme” terminology, possibly with a short list of literature that is still influential but dates back to the “prescheme”-era.

    Comment #921 by Johan (site) on August 17, 2014 a 8:03 pm UTC

    OK, I made a short historical comment on this. See here.

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