**Background:** Let $k$ be a field and denote by $P = k[x_1,\ldots,x_n]$ the polynomial ring in $n$ (commuting) variables over $k$. A *resolution* of an ideal $I \lhd P$ is an exact sequence of $P$-modules $$\ldots \to F_n \to F_{n-1} \to \ldots \to F_0 \to P/I \to 0.$$ This resolution if *minimal* if the rank of each $F_n$ is minimal.

Here's my question:

What are

minimalresolutions of ideals in polynomial rings good for?

**Why I'm asking:** I've been looking at some papers which apply a purely algebraic version of Robin Forman's discrete Morse theory to various algebraic contexts. One of the most fascinating applications out there involves the construction of minimal (cellular) resolutions of ideals, see for instance this paper. While I understand discrete Morse theory well and know the basic definition of resolutions etc., it is not clear why one would want to construct minimal ones.

More precisely, does the quest for a minimal resolution help in some practical way, such as with applying Buchberger's algorithm to a generating set of $I$, or is it a platonic search for the simplest algebraic object that resolves $I$?

Since I am not sure what areas of math are impinged upon by this question, I have only added a minimal number of tags, and I encourage/ask experts to please add other suitable ones.

The Geometry of Syzygies. This book discusses very heavily what geometric information can be gleaned from say minimal graded free resolutions of graded ideals. $\endgroup$ – Karl Schwede Jul 14 '12 at 3:34