The Stacks project

111.2 Colimits

Definition 111.2.1. A directed set is a nonempty set $I$ endowed with a preorder $\leq $ such that given any pair $i, j \in I$ there exists a $k \in I$ such that $i \leq k$ and $j \leq k$. A system of rings over $I$ is given by a ring $A_ i$ for each $i \in I$ and a map of rings $\varphi _{ij} : A_ i \to A_ j$ whenever $i \leq j$ such that the composition $A_ i \to A_ j \to A_ k$ is equal to $A_ i \to A_ k$ whenever $i \leq j \leq k$.

One similarly defines systems of groups, modules over a fixed ring, vector spaces over a field, etc.

Exercise 111.2.2. Let $I$ be a directed set and let $(A_ i, \varphi _{ij})$ be a system of rings over $I$. Show that there exists a ring $A$ and maps $\varphi _ i : A_ i \to A$ such that $\varphi _ j \circ \varphi _{ij} = \varphi _ i$ for all $i \leq j$ with the following universal property: Given any ring $B$ and maps $\psi _ i : A_ i \to B$ such that $\psi _ j \circ \varphi _{ij} = \psi _ i$ for all $i \leq j$, then there exists a unique ring map $\psi : A \to B$ such that $\psi _ i = \psi \circ \varphi _ i$.

Definition 111.2.3. The ring $A$ constructed in Exercise 111.2.2 is called the colimit of the system. Notation $\mathop{\mathrm{colim}}\nolimits A_ i$.

Exercise 111.2.4. Let $(I, \geq )$ be a directed set and let $(A_ i, \varphi _{ij})$ be a system of rings over $I$ with colimit $A$. Prove that there is a bijection

\[ \mathop{\mathrm{Spec}}(A) = \{ (\mathfrak p_ i)_{i \in I} \mid \mathfrak p_ i \subset A_ i \text{ and } \mathfrak p_ i = \varphi _{ij}^{-1}(\mathfrak p_ j)\ \forall i \leq j\} \subset \prod \nolimits _{i \in I} \mathop{\mathrm{Spec}}(A_ i) \]

The set on the right hand side of the equality is the limit of the sets $\mathop{\mathrm{Spec}}(A_ i)$. Notation $\mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Spec}}(A_ i)$.

Exercise 111.2.5. Let $(I, \geq )$ be a directed set and let $(A_ i, \varphi _{ij})$ be a system of rings over $I$ with colimit $A$. Suppose that $\mathop{\mathrm{Spec}}(A_ j) \to \mathop{\mathrm{Spec}}(A_ i)$ is surjective for all $i \leq j$. Show that $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A_ i)$ is surjective for all $i$. (Hint: You can try to use Tychonoff, but there is also a basically trivial direct algebraic proof based on Algebra, Lemma 10.17.9.)

Exercise 111.2.6. Let $A \subset B$ be an integral ring extension. Prove that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is surjective. Use the exercises above, the fact that this holds for a finite ring extension (proved in the lectures), and by proving that $B = \mathop{\mathrm{colim}}\nolimits B_ i$ is a directed colimit of finite extensions $A \subset B_ i$.

Exercise 111.2.7. Let $(I, \geq )$ be a directed set. Let $A$ be a ring and let $(N_ i, \varphi _{i, i'})$ be a directed system of $A$-modules indexed by $I$. Suppose that $M$ is another $A$-module. Prove that

\[ \mathop{\mathrm{colim}}\nolimits _{i\in I} M \otimes _ A N_ i\cong M \otimes _ A \Big( \mathop{\mathrm{colim}}\nolimits _{i\in I} N_ i\Big). \]

Definition 111.2.8. A module $M$ over $R$ is said to be of finite presentation over $R$ if it is isomorphic to the cokernel of a map of finite free modules $ R^{\oplus n} \to R^{\oplus m}$.

Exercise 111.2.9. Prove that any module over any ring is

  1. the colimit of its finitely generated submodules, and

  2. in some way a colimit of finitely presented modules.

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