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Depth of almost complete intersection rings

In general, depth can be zero. Let me give an example. Let $X\subset\mathbb{P}^n$ be a smooth variety of dimension $d$ which is subcanonical (this means $K_X=O_X(*)$ for some integer $*$) and NOT ...
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Analogue of Kock-Lawvere axiom for power series rings?

Yes, it is consistent, it even follows from the Kock-Lawvere axiom, as follows. We defined $\mathrm{Spf}(R[[\epsilon]]) := \mathrm{colim}_n \mathrm{Spec}(R[\epsilon]/(\epsilon^n))$, so we have $$R^{\...

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