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The Kock-Lawvere axiom for a topos $\mathcal{E}$ states that given a specified commutative ring object $R \in \mathcal{E}$, for all local Artinian $R$-algebra objects $A \in \mathcal{E}$, the morphism ...

Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$ Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0. Let $f$ be a polynomial which is zero ...

In this MO question I asked for some help with several definitions of formally etale maps. The definitions I'm asking about are 'isomorphism of tangent spaces', i.e the square below is a pullback $$\...

I'm writing some notes to myself on algebraic geometry and I'm trying to get the most conceptual definitions. Having arrived at formally étale morphisms, I am pretty desperate. Here is a list of ...