Lengths of tors determined by killing powers of ideals in a local ring. (Q5957508)

The author studies the functions \(f_k(a_1,\dots,a_n) = \text{length}\,(\text{Tor}_k^R\,(M_1/I_1^{a_1}M_1,\) \dots, \(M_n/I_n^{a_n}M_n))\) where \(R\) is a local ring, \(M_i\) is a finitely generated \(R\)-module, \(I_i\) is an ideal of \(R\) for each \(i = 1,\dots,n\), \(a_1,\dots,a_n\) are natural integers and \(I_1+\dots +I_n + \text{Ann}\,M_1 +\dots +\text{Ann}\, M_n\) is primary to the maximal ideal of \(R\). In the second item several properties of quasipolynomial functions are investigated and the next one is devoted to \(\mathbb N^n\)-graded algebras and modules. Paragraphs 4, 5 and 6 present some results on \(\text{Tor}_k(R/I_1^{a_1}, R/I_2^{a_2})\), the first one for \(k\geq 2\) and the following two for \(k=1\) and \(k=0\). The final part studies the monomial ideals in the polynomial ring \(R=K[[x_1,\dots,x_d]]\) over the field \(K\).