arXiv subjects
Publications and source records attributed to Samuel Alvite.
We study three classes of local homomorphisms and their behavior with respect to the ascent and descent of the \emph{complete intersection} property. Crucially, they fall in between the already studied classes of complete intersection and quasi-complete intersection homomorphisms, while also repairing some of the issues these presented.
Let $f \colon R \to B$ be a surjective homomorphism of rings with kernel $I$. Gulliksen (when $I$ is generated by a regular sequence) and later Mehta (in general) showed that for any $B$-modules $M$ and $N$, $\mathrm{Ext}_B^{\ast}(M,N)$ has a structure of graded $\mathrm{S}_B^{\ast}\left(\widehat{I/I^2}\right)$-module, where $\widehat{\quad}$ denotes dual and $\mathrm{S}$ denotes symmetric algebra. This construction is extended to the case where $f$ is not necessarily surjective in a way that allows one to regard these operators from a more natural perspective.
Valuation rings and perfectoid rings are examples of (usually non-noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck's definition of formal smoothness as in the noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the noetherian to the non-noetherian case, as well as generalizing some recent results in the non-noetherian case, such as the descent of regularity from perfectoid rings by B. Bhatt, S. Iyengar and L. Ma.
We show that a modification of the proof of a result of Gulliksen gives an elementary proof of the following important theorem by Avramov: if $(A,k) \to (B,l)$ is a homomorphism of noetherian local rings and $B$ is of finite flat dimension over $A$, then the homomorphism induced in Andr\'e-Quillen homology modules $H_2(A,l,l)\to H_2(B,l,l)$ is injective.