Conventions and hypotheses carried by this pair

Choice conventions. Every item of this pair that needs the Axiom of Choice declares it (The Axiom of Choice) and passes the assumption on through the cited suppliers; an item that does not name AC uses none of the choice-dependent results. No incompatible-axiom branch is opened anywhere on the pair.

Jacobians have equation rows, and the presentation does not matter. The Jacobian of Equation rows and coordinate columns in an affine Jacobian is read with one row per defining equation and one column per coordinate, for the actual defining ideal of the scheme, not for the ideal of its reduction. The kernel statement The Jacobian kernel computes the tangent space is proved for every finite generating list of that ideal, so no result of this pair depends on the chosen presentation; a proper subset is also covered if it still generates the same ideal; if it does not, the theorem does not identify its kernel with the tangent space of the original scheme, and the scheme-theoretic tangent space is not computed from a reduced ideal.

Dual numbers are used only at rational points. The identification of the intrinsic tangent space with the fibre of the dual-number points Tangent vectors at rational points are dual-number points is asserted at $k$-rational points and at those points only. The intrinsic definition The intrinsic Zariski tangent space is the one used at a general scheme point, where no such identification is claimed; every statement of the pair names the kind of point it uses.

Tangent cones retain all initial forms. The tangent cone The scheme-theoretic tangent cone at a point is the spectrum of the full associated graded ring, without quotienting by nilpotents, and therefore remembers every initial form of the local equation. The scheme-theoretic linear span of the tangent cone shows that no proper linear closed subscheme contains this cone scheme-theoretically. The qualification is not cosmetic: the reduced cone can span strictly less than the tangent space. At the origin of the doubled line $\mathrm{Spec}k[x,y]\mathrm{/}(y^2)$ the reduced cone is the line $y=0$, of dimension one, while the tangent space is two-dimensional. The examples page of this pair records the corresponding cone computations for plane curves, and no computation there replaces a scheme-theoretic cone by its reduced support.

Regularity is absolute; smoothness is relative. Regularity is a property of the local ring of a scheme at a point (Regular points of locally Noetherian schemes), while smoothness is a property of a morphism, here of the structure morphism to $\mathrm{Spec}k$ (Smooth morphisms via local standard smooth presentations). Purely inseparable field algebras separate regularity from smoothness shows that the two notions diverge over imperfect fields: the spectrum of $L=k[t]\mathrm{/}(t^p-a)$, $a\mathrm{\notin }{k}^p$, is regular at its only point but not smooth over $k$, and no equivalence between regularity and smoothness may be quoted without the perfectness hypothesis that the pair's perfect-field items carry.

Target-open generic smoothness needs a smooth source. The theorem Generic smoothness over a dense target open assumes the source smooth over $k$; that hypothesis is not decoration. The examples page of this pair records a dominant morphism of irreducible classical varieties over a smooth target whose source has a singular point in every fibre, so that no nonempty target open has smooth restriction. In positive characteristic the Frobenius phenomenon defeats the arbitrary base-point-free form of Bertini; the counterexample recorded on the examples page is stated for general linear systems and deliberately makes no claim about the embedded hyperplane-section case, so no item of this pair quotes it as such a claim.

The Jacobian row and column convention, the tangent-cone construction from the associated graded ring, and the treatment of regularity as an absolute local condition follow Milne, Algebraic Geometry, Ch. 4 §§d–i. The positive- characteristic divergence between regularity and smoothness, and the Frobenius failure of Bertini for general linear systems, follow the source accounts read for the individual items; the exact locators are recorded on those items. This remark asserts no theorem of its own: it fixes which conventions and hypotheses the page's items actually use, and it points to the items that carry each claim.