nLab Z-functor

This page is about the concept in functorial algebraic geometry. For functors between Z-categories see there.

Definition

A $\mathbb{Z}$-functor or integers functor is a functor from the category of commutative rings CRing to the category of sets Set. The category of $\mathbb{Z}$-functors is thus the functor category $\mathrm{Set}^\mathrm{CRing}$.

Examples

• The forgetful functor $\mathbb{A}^1:\mathrm{CRing} \to \mathrm{Set}$ is a $\mathbb{Z}$-functor.

• More generally, any affine scheme is a $\mathbb{Z}$-functor.

Properties

Definition

Let $X$ be a $\mathbb{Z}$-functor. Then the ring of fractions $\mathcal{O}(X)$ of $X$ is the set $X \Rightarrow \mathbb{A}^1$ of natural transformations from $X$ to the forgetful functor $\mathbb{A}^1$.

The ring structure on $X \Rightarrow \mathbb{A}^1$ is defined pointwise by the following, for every commutative ring $R$ and element $x \in X(R)$:

$$0_R(x) \coloneqq 0 \qquad 1_R(x) \coloneqq 1$$

$$(a_R + b_R)(x) \coloneqq a_R(x) + b_R(x)$$

$$(- a_R)(x) \coloneqq - a_R(x)$$

$$(a_R \cdot b_R)(x) \coloneqq a_R(x) \cdot b_R(x)$$

Related concepts

• affine scheme

References

The terminology “$\mathbb{Z}$-functor” is due to

• Michel Demazure, Pierre Gabriel, Groupes algebriques, tome 1, avec un appendice Corps de classes local par Hazewinkel M, Mason and Cie, Paris (1970)

  (later volumes never appeared)

popularized through the English translation of the first couple of chapters:

• Michel Demazure, Peter Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland mathematics studies 39 (1980) [ISBN:9780080871509]

Formulation in univalent homotopy type theory:

• Max Zeuner: Univalent Foundations of Constructive Algebraic Geometry [arXiv:2407.17362]