Let $X$ be an affine variety of $\mathbb{C}^n$ with vanishing ideal $I_X$. Suppose that $I_X$ is a homogeneous ideal following the natural grading of the polynomial ring $\mathbb{C}[x_1,\cdots,x_n]$.

Question: is there a notion of "graded Zariski tangent space", where the variety defining the tangent space comes only from homogeneous elements of $I_X$ of equal degree? That is, for each $P \in X$ and each degree $k$, we have a Zariski tangent space $(T_PX)_k$, which is defined using only the elements of the homogeneous component $I_{X,k}$.

  • $\begingroup$ There is a Zariski tangent space defined in this eprint, at the end of section 2.2 (on the top of page 4) is extended to a Zariski tangent superspace for a particular example therein. $\endgroup$ – Alex Nelson Jul 20 '13 at 21:35
  • $\begingroup$ I think you are looking for something like $T_{X/C^n}$, which is $0$ for the usual tangent space. However, you can look at the "tangent complex", which is a complex of sheaves not just a sheaf. $\endgroup$ – Marci Jul 21 '13 at 20:34
  • $\begingroup$ @Marci: The tangent complex looks interesting... $\endgroup$ – Manos Jul 27 '13 at 12:43

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