---- > [!definition] Definition. ([[base scheme]]) > It is rare in algebraic geometry to work with [[scheme|schemes]] alone. Rather, one works over a **base scheme** $S$. Consider the [[slice category|slice]] [[category]] $\mathsf{Sch} / S$ whose objects are [[morphism of locally ringed spaces|morphisms of]] [[locally ringed space|locally ringed spaces]] $T \to S$ and whose morphisms are commutative diagrams > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRABUQBfU9TXfIRQAmclVqMWbdgHJuvEBmx4CRAIyk14+s1aIQAZW7iYUAObwioAGYAnCAFskZEDghI1PG-aeIXbpGEvEDtHD2oAxCCKLiA > \begin{tikzcd} > T \arrow[rr] \arrow[rd] & & T' \arrow[ld] \\ > & S & > \end{tikzcd} > \end{document} > ``` > > Given two objects $T \to S$ and $X \to S$ in $\mathsf{Sch}/S$, a **$T$-valued point** is an element of the homset $X(T):=\text{Hom}_{\mathsf{Sch} /S}(T \to S, X \to S)$, that is, a factorization > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRABUQBfU9TXfIRQAmclVqMWbABrdeIDNjwEiARlKrx9Zq0QgAyt3EwoAc3hFQAMwBOEALZIyIHBCTCe1u48TqXbxA95WwcnalckVS4KLiA > \begin{tikzcd} > T \arrow[rd] \arrow[rr] & & X \arrow[ld] \\ > & S & > \end{tikzcd} > \end{document} > ``` > and we write $X(T)$ for the set of $T$-valued points. The [[the Yoneda lemma|Yoneda philosophy]] is that $X(T)$ for all $T$ determines $X$. > > > [!basicexample] > A common choice is $\mathsf{Sch} / k=\mathsf{Sch} / \text{Spec }k$ : [[scheme over a field]] ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```