Vanishing direct limit of vector spaces

Question

Let $k$ be a field. Let $\{V_n\}_{n\in \mathbb{N}}$ be a direct system of (possibly infinite-dimensional) $k$-vector spaces. Is it true that if $\varinjlim_{n\in \mathbb{N}}V_n=0$, then $V_m=0$ for some $m\in \mathbb{N}$? If not, can you give an example?

Answer 1

Here's a counterexample along similar lines to the one Mars mentioned in the comments: For all $n \in \mathbb{N}$, let $V_n$ be the vector space of sequences of elements of $k$ that are eventually zero (i.e., sequences with finite support). For all $m \leq n$, let the morphism $f_{mn}\colon V_m \to V_n$ be the "left shift" map that sends a sequence $(v_0, v_1, v_2, \dots)$ to the sequence $(v_{n-m}, v_{n-m+1}, v_{n-m+2}, \dots)$. This forms a direct system of vector spaces over $k$.

All of the morphisms are nonzero, but any element is "eventually zero" in this direct system, i.e., for any $m \in \mathbb{N}$ and $v \in V_m$, there exists $n \geq m$ such that $\varphi_{mn}(v) = 0$. So $\varinjlim_{n \in \mathbb{N}} V_n = 0$. This shows how it's crucial to think about the morphisms, not just the objects, when working with (co)limits.