Marked bases over quasi-stable modules

Let $K$ be a field of any characteristic, $A$ be a Noetherian $K$-algebra and consider the polynomial ring $A[x_0,\dots,x_n]$. The present paper deals with the definition of marked bases for free $A[x_0,\dots,x_n]$-modules over a quasi-stable monomial module and the investigation of their properties. The proofs of our results are constructive and we can obtain upper bounds for the main invariants of an ideal of $A[x_0,\dots,x_n]$ generated by a marked basis, such as Betti numbers, regularity and projective dimension.