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.


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 $L_{r-1} \to L_{r-2}$.

(I must have been sleeping.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.