Depth of general fiber product rings R ×_T S

Determine the depth of the fiber product ring R ×_T S, defined as {(r,s) ∈ R × S | π_R(r) = π_S(s)} for surjective homomorphisms π_R: R → T and π_S: S → T between commutative Noetherian local rings R, S, and T sharing a common residue field, by characterizing it in terms of invariants of R, S, and T beyond the special cases already known (such as T = k).

Background

The paper studies fiber product rings R ×_T S of local rings R, S, and T with common residue field, with surjections to T. When T = k (the common residue field), the depth is known via Lescot’s formula depth(R ×_k S) = min{depth(R), depth(S), 1}, but for general T, especially 0-dimensional T ≠ k, analogous formulas need not hold.

Motivated by this gap, the authors develop tools using local cohomology and grade of certain extensions to provide exact depth results in specific cases (Theorem 4.3 and Corollary 4.4), but emphasize that a general description of depth of R ×_T S remains an open problem.