Semi-fiber products of algebras and lifting of complexes

Let k k be a field. In this paper, we introduce the notion of semi-fiber products of commutative k k -algebras and show that the class of such rings contains several classes of commutative rings, including that of the fiber products of local k k -algebras over their common residue field k k . For a noetherian local k k -algebra R R and an ideal I I of R R , under certain conditions, we characterize the liftability of k k along the natural surjection R ↠ R / I R\twoheadrightarrow R/I in terms of retractions, sections, and the existence of semi-fiber product decompositions of R R .