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My question arises from a brief remark about the affine tangent cone in Chapter 5 of Commutative Algebra: with a View Toward Algebraic Geometry by Eisenbud. To paraphrase, if $X$ is an affine variety, set $A(X) = A/I$ to be its coordinate ring and $I$ its defining ideal. Recall that the tangent cone at $p$, denoted by $TC_p(X)$ is an algebraic set that is given by the zero set of the initial ideal, $Z(\text{in}(I))$, where $\text{in}(I) = ( \text{in}(f) \, | \, f \in I)$, and $\text{in}(f)$ is the first non-zero homogeneous component of $f$. Geometrically, this is the cone composed of limiting positions of secants to $X$ passing through $p$. Now if $m_p$ is the maximal ideal in $A(X)$ corresponding to the point $p$, Eisenbud's claim is that $$\text{gr}_{m_p} A(X) = \bigoplus_{i=0}^{\infty} m_p^i / m_p^{i+1}$$ is isomorphic to the coordinate ring of the tangent cone at $p$.

It is clear to me that $\text{gr}_{m_p} A(X)$ is a finitely generated $k$-algebra, and if $a_i$ are the generators of $m_p$, then $\xi_i = a_i \text{ mod } m_p^2$ generate $\text{gr}_{m_p} A(X)$. However I'm struggling to show that the kernel of this (homogeneous of degree zero) homomorphism, which sends $a_i$ to $\xi_i$ is given by $\text{in}(I)$.

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