Lech's multiplicity conjecture

Theorem 1.1: for every flat local homomorphism $(R,\mathfrak m)\to(S,\mathfrak n)$ of nonzero Noetherian local rings, $e(R)\le e(S)$, in all dimensions and characteristics. The key input is a uniform lower bound on the zeroth homology of finite free complexes with entries in high powers of a parameter ideal, valid for generalized length functions, applied with Frobenius perfection in characteristic $p$ and perfectoid lengths in mixed characteristic. The paper does not claim the stronger Iyengar-Ma-Walker conjecture for all complete local rings, only its complete-domain characteristic-$p$ case.

Theorem 1.1: for every flat local homomorphism $(R,\mathfrak m)\to(S,\mathfrak n)$ of nonzero Noetherian local rings, $e(R)\le e(S)$, in all dimensions and characteristics. The key input is a uniform lower bound on the zeroth homology of finite free complexes with entries in high powers of a parameter ideal, valid for generalized length functions, applied with Frobenius perfection in characteristic $p$ and perfectoid lengths in mixed characteristic. The paper does not claim the stronger Iyengar-Ma-Walker conjecture for all complete local rings, only its complete-domain characteristic-$p$ case.