Cotangent space appears in both differential geometry (DG) and algebraic geometry (AG).

In DG, given a smooth manifold $M$ and $x\in M$ one has an isomorphism $I_x/I_x^2 \cong T^*_xM$, where $I_x$ is an ideal of functions vanishing at $x$ in ring $C^\infty(M)$ (or $C_x^\infty(M)$). One can prove this isomorphism algebraically, or using Hadamard's lemma like argument.

In AG, given an affine variety $V$ and $x\in V$ one defines Zariski cotangent space as $\mathfrak{m}_x/\mathfrak{m}_x^2$, where $\mathfrak{m}_x$ is defined as an ideal of functions vanishing at $x$ in ring $O_{V,x}$.

Clearly AG introduced its version of tangent and cotangent spaces based on DG. However, It seems to me that in DG the description of cotangent space as $I_x/I_x^2$ is rarely used.

Question. Was the description of the cotangent space $T^*_xM$ as $I_x/I_x^2$ been known before cotangent Zariski space was introduced?


1 Answer 1


Zariski formulated the criterion for smoothness at a point in terms of the dimension of $I_x/I_x^2$ (to use your notation) as Theorem 3.2 in the paper https://www.jstor.org/stable/pdf/2371499.pdf from 1939 (see p. 260). See the paragraph and theorem preceding that also. He singles out Theorem 3.2 in the first paragraph of the paper, with no reference to earlier work by anyone else. In his later paper http://www.ams.org/journals/tran/1947-062-01/S0002-9947-1947-0021694-1/S0002-9947-1947-0021694-1.pdf from 1947, the discussion on pages 2 and 3 compares two ways of describing simple points: what Zariski calls a "novel and intrinsic" definition A from his earlier work (which involves $I_x/I_x^2$) and a "time-honored and classical" definition B (which involves a Jacobian rank). He writes on page 3 "There is ample evidence in the present paper, as well as in previous papers of ours, in support of the thesis that it is the more general concept of a simple point, as defined in A, that constitutes the natural generalization of the classical concept of simple point."

In summary, Zariski is the person who realized the importance of $I_x/I_x^2$ in geometry, so it first arose in algebraic geometry rather than differential geometry.


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