ALGEBRE SIMMETRICHE DI ALCUNE CLASSI DI IDEALI MONOMIALI

The purpose of this thesis is the study of the symmetric algebra 〖Sym〗_S (L) of an interesting class of monomial ideals: the ideals L ⊂S=K[x1,...,xn,y1,...,ym] of mixed products in two sets of variables. Recently, this class is been used in order to test some algebraic conjecture, including the conjecture of Eisenbud -Goto, on the symmetric algebra 〖Sym〗_S (L) .Since such conjecture involves fundamental invariants of Sym (L), such as the Krull dimension, the multiplicity and regularity of Castelnuovo-Mumford, it was necessary to calculate these invariants or their bounds. This problem is difficult, but if L is generated by a s-sequence, you can arrive at a concrete result. In the work Mixed Product Ideals generated by s-Sequences, Algebra Colloquium, 18.553 (2011), the authors M. La Barbiera, G.Restuccia identify subclasses of mixed products ideals, generated by a s-sequence, and proceed to the verification of the conjecture. For the remaining classes of mixed products ideals, not generated by a s - sequence, we calculate bounds for the invariants. A crucial role is played by the linear part of the ideal Jl ⊆in<(J) with respect to a order said admissible on the monomials in T1,...Tl, being T1,...,Tl the variables corresponding to generators of L of the relation J of the symmetric algebra 〖Sym〗_S (L) . Such an ideal Jl constructed by the ideals annihilators of L is such that in (J) = Jl + J 0, where J 0 is another ideal and J = 0 (0), if L is generated by a s-sequence. Therefore a key - part of the thesis concerned the calculation of the ideal Jl associated with some classes of ideals of mixed products in the ring S = K [x1, ..., xn, y1, ..., yn]. As the Veronese square free monomial ideal Ik ⊂ K [x1, ..., xn], k = 2, ..., n, is involved in the construction of an ideal of mixed products , we study initially Sym(Ik). Of Ik is known the importance in combinatorics (it is a polymatroid ideal). Subsequently we studied the algebra S[T1,...,Tl] / Jl and we prove that it is Cohen Macaulay. The following covered the calculation of bounds for the invariants of the symmetric algebra 〖Sym〗_S (L). A final part of the thesis concerned the monomial algebra, generated by a system of generators of Ik.They are Koszul, as for the sorted order, they admit a Groebner basis of degree 2. We studied in detail the Groebner basis of their toric ideal and compare with the one, best known ,lessicografic or reverse lessicografic, of which you do not know the degree.