Poset embeddings of Hilbert functions and Betti numbers (Q403111)

Let \(A=k[x_1,\ldots,x_n]\), where \(k\) is a field. Let \(\mathfrak a\) be a homogeneous \(A\)-ideal. We consider the following posets \({\mathcal I}_R=\{J : J \text{ is a homogeneous }R\)-ideal