# Why is this Hilbert's Syzygy theorem?

In Lang's Algebra, chapter XXI, §4, on p. 861 he describes the standard construction of a graded (in principle infinite) free resolution of a finite graded module $M$ over the polynomial ring $A = k[x_1,\ldots,x_r]$.

Then it is announced:

We want this resolution to stop, and the possibility of its stopping is given by the next theorem.

Namely,

Theorem 4.15 (Hilbert Syzygy Theorem). Let $k$ be a field and $$A = k[x_1,\ldots,x_r]$$ the polynomial ring in $r$ variables. Let $M$ be a graded module over $A$, and let $$0 \to K \to L_{r-1} \to \cdots \to L_0 \to M \to 0$$ be an exact sequence of graded homomorphisms of graded modules, such that $L_0, \ldots, L_{r-1}$ are free. Then $K$ is free. If $M$ is in addition finite over $A$ and $L_0, \ldots, L_{r-1}$ are finite free, then $K$ is finite free.

My question: What does this have to do with the infinite resolution stopping?

The $L_{r-1} \to \ldots \to L_0 \to M \to 0$ may come from the possibly infinite resolution, but where does the $0 \to K \to L_{r-1}$ come from?

Let $K$ be the kernel of $L_{r-1} \to L_{r-2}$.