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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.

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  • $\begingroup$ 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. $\endgroup$
    – Thomas Kahle
    Commented Jul 10, 2012 at 7:45
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    $\begingroup$ 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 $0$. $\endgroup$
    – Adam Boocher
    Commented Jul 12, 2012 at 3:22
  • $\begingroup$ 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. $\endgroup$
    – Karl Schwede
    Commented Jul 14, 2012 at 3:34

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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.

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  • $\begingroup$ Dear gruff, thank you for this nice answer. I have been going through Eisenbud in light of comments to my original question, but it is not yet clear what one "does" with this uniqueness of minimal resolutions. What is it good for? It is my understanding that any free resolution yields the hilbert function, and so minimality is completely unnecessary in this context. Does the minimality provide, for instance, any algorithmic benefits (as in computing Grobner bases via Buchberger)? $\endgroup$
    – Vidit Nanda
    Commented Aug 7, 2012 at 22:43
  • $\begingroup$ Well, actually, minimality (and hence uniqueness) is of no practical importance. It is just the opposite: we do not even have a general method, an algorithm to compute such a resolution, even though we know it exists. Of course, for special cases (like generic ideals, where we have the minimality of the algebraic Scarf complex), we do. The power of uniqueness is some sort of invariance, so that we can attach other measures (like CM regularity) to the resolution, which we cannot do in general. We need to start with something certain and unique. $\endgroup$
    – user25585
    Commented Aug 7, 2012 at 23:10

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