By Wolfgang Franz

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Moreover, ∆m is equal to ∆LCM{mi |i∈I} for some subset ∆ of the vertex set of ∆. A full subcomplex of ∆ is a subcomplex of all the faces of ∆ that involve a particular set of vertices. Note that all the subcomplexes ∆ m are full. 2 (Bayer, Peeva, and Sturmfels). Let ∆ be a simplicial complex labeled by monomials m1 , . . , mt ∈ S, and let I = (m1 , . . , mt ) ⊂ S be the ideal in S generated by the vertex labels. The complex C (∆) = C (∆; S) is a free resolution of S/I if and only if the reduced simplicial homology Hi (∆m ; K) vanishes for every monomial m and every i ≥ 0.

3. Consider the ideal I = (x0 , x1 ) ∩ (x2 , x3 ) of two skew lines in P 3 : Prove that I = (x0 x2 , x0 x3 , x1 x2 , x1 x3 ), and compute the minimal free resolution of S/I. In particular, show that S/I has projective dimension 3 even 28 2. First Examples of Free Resolutions though its associated primes are precisely (x0 , x1 ) and (x2 , x3 ), which have height only 2. 2 can’t be extended to give the projective dimension in general. 4. Show that the ideal J = (x0 x2 − x1 x3 , x0 x3 , x1 x2 ) defines the union of two (reduced) lines in P 3 , but is not equal to the saturated ideal of the two lines.

2A Monomial Ideals and Simplicial Complexes 17 We set the degree of the basis element corresponding to the face A equal to the exponent vector of the monomial that is the label of A. With respect to this grading, the differential δ has degree 0, and C (∆) is a Z r+1 -graded free complex. For example we might take S = K and label all the vertices of ∆ with 1 ∈ K; then C (∆; K) is, up to a shift in homological degree, the usual reduced chain complex of ∆ with coefficients in S. Its homology is written Hi (∆; K) and is called the reduced homology of ∆ with coefficients in S.

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