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In this paper we study the regularity and the projective dimensionâ
âof the Stanley-Reisner ring of a $k$-decomposable simplicial complex and explain these invariants with a recursive formulaâ.
âTo this aimâ, âthe graded Betti numbers of decomposable monomial ideals which is the dual conceptâ
âfor $k$-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is givenâ.
âAs a corollaryâ, âfor a shellable simplicial complex $\Delta$â,
âa formula for the regularity of the Stanley-Reisner ring of $\Delta$ is presentedâ. âFinallyâ, âfor a chordal clutter $\mathcal{H}$â, âan upper bound for $\T{reg}(I(\mathcal{H}))$ is given in terms of the regularities of edge ideals of some chordal clutters which are minors of $\mathcal{H}$â.
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