**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:



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> What are **minimal** resolutions of ideals in polynomial rings good for?
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**Why I'm asking:** I've been looking at some papers which apply a purely algebraic version of Robin Forman's [discrete Morse theory](http://en.wikipedia.org/wiki/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](https://www.uni-marburg.de/fb12/diskrete_mathematik/pdf/publications/kres) 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. 



Why are minimal resolutions of polynomial ideals important?
As it has already been pointed out, the key concept is *uniqueness*.


We know that free (or, if you will, projective) resolutions exist in the case we are interested in. Yet, a module may possess many of them. However, each finitely generated module (over a polynomial algebra over a field) has a minimal free resolution, which is unique up to isomorphism. The degrees of the generators of its free modules not only yield the Hilbert function, but form a finer invariant, bringing us right next to, for example, the notion of *Castelnuovo--Mumford regularity* (discussed in Eisenbud's book).


For the sake of completeness: while the free (projective) dimension of a graded module in our case is defined to be the minimum length of a resolution by graded free modules (which coincides with the length of the minimal graded free resolution, thus minimality also servers as a notion of dimension), and it specifies in how many steps the module can be resolved, the CM regularity can be viewed as the width of the minimal resolution (in terms of the degrees of the generators at each free component), and it gives an indication of the complexity of the process.


Under similar circumstances, the opposite of a minimal free resolution is a *trivial complex*. Trivial complexes have no homology at all, thus taking the direct sum of a resolution with a trivial complex yields an other resolution, resulting in lack of uniqueness. It can be shown that every free resolution in our case is the direct sum of the minimal resolution and a trivial complex. Therefore, we may say that minimal resolutions carry most of the information we are looking for.


