A Generalization of the Taylor Complex Construction

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Given multigraded free resolutions of two monomial ideals, we construct a multigraded free resolution of the sum of the two ideals.

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Regularity of Edge Ideals and Their Powers (2017, background): “estated in terms of edge ideals. Lemma 3.7. Let xbe a vertex in G:Then regG2freg(GnN G[x]) + 1;reg(Gnx)g: Kalai and Meshulam [39] proved the following result for squarefree monomial ideals and Herzog [31] generalized it to any monomial ideal. Theorem 3.8. Let I 1;:::;I s be squarefree monomial ideal”
Matchings, coverings, and Castelnuovo-Mumford regularity (2010, background): “Herzog [20] later generalized the result to monomial ideals that are not square-free.”
Resolutions of square-free monomial ideals via facet ideals: a survey (2006, background): “For more on the Taylor resolution, and a generalization of its construction, see Herzog [23].”
Edge ideals: algebraic and combinatorial properties (2010, background): “[73, 57] Let I1 and I2 be monomial ideals of R.”