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 $i \leq j \leq k$.

## 111.2 Colimits

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

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

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

the colimit of its finitely generated submodules, and

in some way a colimit of finitely presented modules.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)