Poincaré series of fiber products and weak complete intersection ideals

The Golod property for powers of ideals and Koszul ideals

“…Complete intersection ideals provide another piece of evidence in support of a positive answer to Question 3.6, as the following proposition shows. This can also be concluded from [20,Remark 2.12]. But, we include a different proof here for the sake of completeness.…”

Generators of Koszul homology with coefficients in a J-closed module

“…Note that the minimal generators of I are contained in the ideal (g) since they are the entries in the first differential of the minimal free resolution F of Q/I over Q. And by (1), pd Q Q/I < ∞, thus I is a complete intersection ideal by [11,Remark 2.4]. The other direction is clear.…”

“…Proof. Since (g)/(g) 2 is a free Q/(g)-module and H (g; R) is a free Q/(g)-module for every i, it suffices to show that K(g; R) admits a trivial Massey operation by [11,Theorem 2.9].…”

“…Let Q be a commutative, Noetherian k-algebra where k is a field of characteristic zero and let g = g 1 , ..., g s be a regular sequence in Q. In this paper, we study a class of ideals which we call g-weak complete intersection ideals, inspired by the weak complete intersection ideals studied by Rahmati, Striuli, and Yang in their recent paper [11]. Weak complete intersection ideals in a local ring are the ideals I such that the differentials in the minimal free resolution F of the quotient Q/I land in IF .…”

Minimal Free Resolutions of Fiber Products

“…Both S and T come equipped with a natural surjection S πS − − → k πT ← − − T which are used to build the fiber product of S and T over k given by S × k T := {(s, t) ∈ S × T : π S (s) = π T (t)}. Much research has been conducted comparing and contrasting the homological properties of S and T with those of S × k T , e.g., the Cohen-Macaulay, Gorenstein, Golod, finite representation type, and Arf properties (see [1,2,4,5,[7][8][9][10][11][12][13]).…”