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Minimal free resolutions of lattice ideals of digraphs. (English) Zbl 1445.13015
Let \(I(\mathcal L)=\langle x^{m^+}-x^{m^-}| m \in \mathcal L\rangle \subset K[x]\), \(x=(x_1,\ldots,x_n)\), be the lattice ideal associated to the lattice \(\mathcal L\) spanned by the columns of the Laplacian matrix \(L\) of a finite, strongly connected, weighted, directed graph \(G\). A free resolution of \(I(\mathcal L)\) is computed. It is proved that this resolution is minimal if and only it \(G\) is strongly complete.
MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05E40 Combinatorial aspects of commutative algebra
05C20 Directed graphs (digraphs), tournaments
05C57 Games on graphs (graph-theoretic aspects)
Software:
SINGULAR
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