On Some Permanence Properties of (Derived) Splinters

Gröbner deformations and F‐singularities

“…Then, $$R_{\mathfrak {m}}/xR_\mathfrak {m}$$ is F ‐full (strongly F ‐injective) by definition. Hence, $$R_{\mathfrak {m}}$$ is F ‐full (strongly F ‐injective) by [25, Theorem 4.2, Corollary 5.16], thus R is F ‐full (strongly F ‐injective) by Proposition 4.8 and [13, Theorem 5.12]. By [12, Lemma 3.4] and [13, Theorem 3.3], $$R_x$$ is F ‐full (strongly F ‐injective), so condition (1) of G ‐deforming property definition is satisfied.…”

“…Hence, $$R_{\mathfrak {m}}$$ is F ‐full (strongly F ‐injective) by [25, Theorem 4.2, Corollary 5.16], thus R is F ‐full (strongly F ‐injective) by Proposition 4.8 and [13, Theorem 5.12]. By [12, Lemma 3.4] and [13, Theorem 3.3], $$R_x$$ is F ‐full (strongly F ‐injective), so condition (1) of G ‐deforming property definition is satisfied. Now condition (2) follows from Lemma 4.7 and [13, Theorem 3.9].$$\Box$$…”

“…By [12, Lemma 3.4] and [13, Theorem 3.3], $$R_x$$ is F ‐full (strongly F ‐injective), so condition (1) of G ‐deforming property definition is satisfied. Now condition (2) follows from Lemma 4.7 and [13, Theorem 3.9].$$\Box$$…”

Untitled

“…The study of splinters has been a celebrated endeavor with a rich history. The direct summand conjecture, now a theorem, says precisely that regular rings are splinters (refer to [DT20] for a survey, basic properties, and additional references). However we only need to use the following well known and basic facts [Ho73, Lemma 2]: Lemma 2.6.…”

On the canonical, fpqc, and finite topologies on affine schemes. The state of the art

“…Remark 6.3. Given a finite extension R → S of complete local domains, with R regular, it is not known whether there exists an R-free finitely generated S-module, but there is a countably generated such module M [28] 18 . It would be interesting to know whether there is a countably generated S-algebra which is R-free.…”