# Why are minimal resolutions of polynomial ideals important?

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 minimal resolutions 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.

• In a polynomial ring over a field the resolution has length at most $n$, that is $F_{i}=0$ for $i>n$. If $I$ is homogeneous, then there exists a unique minimal free resolution that simultaneously minimizes various measures of minimality. It turns out that this minimal resolution is a direct summand of any other resolution. – Thomas Kahle Jul 10 '12 at 7:45
• I second Thomas' point about the uniqueness. The minimal free resolution of a module is an important invariant of a module and is a finer invariant than say the Hilbert function. Although you can retrieve most numerical information (say codimension or degree) from a nonminimal resolution, it's often easier to compute these from a minimal resolution. For example, the rank of $F_i$ in a minimal free resolution is equal to the dimension of the $k$ vector space $Tor_i(P/I,k)$. Equivalently, if you tensor a minimal free resolution with $k$ all the maps become $0$. – Adam Boocher Jul 12 '12 at 3:22
• Just to add to what Adam says, you should check out David Eisenbud's book. The Geometry of Syzygies. This book discusses very heavily what geometric information can be gleaned from say minimal graded free resolutions of graded ideals. – Karl Schwede Jul 14 '12 at 3:34