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The structure of an abstract simplicial complex can be encoded into a quotient of a polynomial ring called the Stanley-Reisner ring. We can build a similar face ring for regular cell complexes, which can be thought of as simplicial posets, a generalization of simplicial complexes. We will introduce a colorful system of parameters for the Stanley-Reisner ring which exist when one has a proper vertex coloring of a simplicial complex. Introducing the colorful system of parameters allows us to also introduce a colorful generalization of the Hochster formula. We will also describe a universal system of parameters for the face ring of a regular cell complex, which we will show detects the depth of these face rings. Both of these parameters have the nice property of being fixed under symmetries. Moreover, when we resolve these rings over their parameter rings, we will show how we may conjecturally detect the shape of the equivariant resolution by the colorful Hochster formula.

Zoom: https://ucdavis.zoom.us/j/98164204505?pwd=cm50SmdL... pw: srrs2020

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