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Published online by Cambridge University Press: 29 May 2025
This paper is a survey on infinite graded free resolutions. We discuss their numerical invariants: Betti numbers, regularity, slope (rate), and rationality of Poincaré series. We also cover resolutions over complete intersections, Golod rings and Koszul rings.
This paper is an expanded version of three talks given by I. Peeva during the Introductory Workshop in Commutative Algebra at MSRI in August 2013. It is a survey on infinite graded free resolutions, and includes many open problems and conjectures.
The idea of associating a free resolution to a finitely generated module was introduced in two famous papers by Hilbert [1890; 1893]. He proved Hilbert’s Syzygy Theorem (Theorem 4.9), which says that the minimal free resolution of every finitely generated graded module over a polynomial ring is finite. Since then, there has been a lot of progress on the structure and properties of finite free resolutions. Much less is known about the properties of infinite free resolutions. Such resolutions occur abundantly since most minimal free resolutions over a graded nonlinear quotient ring of a polynomial ring are infinite. The challenges in studying them come from:
The structure of infinite minimal free resolutions can be quite intricate.
The methods and techniques for studying finite free resolutions usually do not work for infinite free resolutions. As noted by Avramov [1992]: “there seems to be a need for a whole new arsenal of tools.”
Computing examples with computer algebra systems is usually useless since we can only compute the beginning of a resolution and this is nonindicative for the structure of the entire resolution.
Most importantly, there is a need for new insights and guiding conjectures. Coming up with reasonable conjectures is a major challenge on its own.
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