Martin's Blog

Functors of points and base ring

Posted by martin on Thursday, 08 October 2009 at 09:58

So far in my series on functors of points, I have considered functors $k\textbf{-Alg} \to \textbf{Set}$ for some fixed field $k$. We begin by observing that we may allow $k$ to be any ring. Then I consider whether it is possible to relate functors with base ring $k$ to functors with base ring $\mathbb{Z}$, with only partial success.

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Morphisms and functors of points

Posted by martin on Thursday, 01 October 2009 at 15:45

This post will discuss the fact that $A$-points of an affine $k$-scheme $X$ (and more general objects) are the same as morphisms $\mathop{\mathrm{Spec}_k} A \to X$. James already brought this up in his comment last time. As well as proving this in the affine $k$-scheme case, I shall attempt to give an intuitive explanation of this fact, although I don’t find this entirely satisfying.

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