By Bernd Sturmfels, Ezra Miller
Contemporary advancements are lined comprises over a hundred figures and 250 workouts comprises entire proofs
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Extra resources for Combinatorial Commutative Algebra (Graduate Texts in Mathematics, Volume 227)
Borel-ﬁxed ideals enjoy the extra property that larger-indexed variables can be swapped for smaller ones without leaving the ideal. 3 The following are equivalent for a monomial ideal I. (i) I is Borel-ﬁxed. (ii) If m ∈ I is any monomial divisible by xj , then m xxji ∈ I for i < j. Proof. Suppose that I is a Borel-ﬁxed ideal. Let m ∈ I be any monomial divisible by xj and consider any index i < j. Let g be the elementary matrix in Bn (k) that sends xj to xj + xi and that ﬁxes all other variables.
Our inability to write down explicit canonical minimal (or at least “small”) resolutions leads us to examine intrinsic geometric properties of monomial ideals resulting from the inclusion of the lattice Zn into the vector space Rn . The coming chapters use convex geometric techniques, along with the combinatorial and algebraic topological methods surrounding them, to express data associated to arbitrary monomial ideals (and even some binomial ideals as well, in Chapter 9). The details of the multiple facets of this theory in higher dimensions are the subjects of later chapters in Part I.
This means that min(m ) ≥ max(mi ) and min(m ) ≥ max(mj ). 11. 1) constitute obner basis is reduced a Gr¨ obner basis for the submodule M ⊂ S r . 1). We have shown that the initial module in(M ) under the given term order is minimally generated by the monomials xu · ej for which u < uj . Hence this initial module decomposes as the direct sum r x1 , x2 , . . , xuj −1 · ej . 2). The minimal free resolution of the ideal x1 , x2 , . . , xuj −1 is a Koszul complex, which is itself a linear resolution.
Combinatorial Commutative Algebra (Graduate Texts in Mathematics, Volume 227) by Bernd Sturmfels, Ezra Miller