Exactness of an additive left Kan extension
Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily...
View ArticleDo forms of $\mathbb{G}_a$ split over perfect rings?
I'll try to pursue my understanding of forms of $\mathbb{G}_a^d$ over general bases.Let $R$ be a ring of positive characteristic $p$ and let $G$ be a smooth affine group scheme over $R$ that is a form...
View ArticleFlatness of a simple ring extension
Assume $A \subseteq B=A[b]$ are integral domains, $b \in B$ is algebraic over $A$ (but not necessarily integral over $A$), and $A$ and $B$ have the same field of fractions.(Notice that $b=u/v$ for some...
View ArticleWhen is every $s$-pure exact sequence pure, and every $s$-flat module flat?
Let $R$ be a ring. Consider a short exact sequence of left $R$-modules:$$\eta: \quad 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 .$$We say that $\eta$ is $s$-pure if it remains exact...
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