Let \(M\) be a finitely generated graded module over the graded algebra \(A\) and \(S\) be a Gorenstein graded algebra mapping onto \(A\), with maximal graded ideal \(m\). Let \(\dim S=r\) and \(\dim M=d\). The author defines the homological degree of \(M\) by the recursive formula \[ \text{hdeg} (M):= \text{deg} (M)+ \sum^r_{i=r-d+1} {d-1 \choose i-r+d-1} \cdot \text{hdeg} (\text{Ext}^i_S (M,S) \bigr) \] where \(\deg(M)\) is the multiplicity of \(M\). \(\text{hdeg} (M)\) is independent of \(S\). The following conjecture is stated:
Conjecture: Let \(M\) be a graded module and let \(h\) be a regular hyperplane section. Then \(\text{hdeg} (M)\geq \text{hdeg} (M/hM)\).
The validity of it is established in the following case: \(S\) is a Gorenstein graded algebra, \(M\) a finitely generated graded module of depth at least 1, and \(h\in S\) is a special generic hyperplane section on \(M\) (a special element is a superficial element for all finite iterations of the Ext-modules of non-decreasing degrees of \(M\) with coefficients in \(S)\). – The behavior of the homological degree in certain short exact sequences is discussed. – In the last section the homological degree is defined for modules over local rings and that degree is compared with other invariants of the module and the ring.