# Geometric interpretation of a (standard) commutative algebra fact

Which is your geometric interpretation (if any) of the following commutative algebra proposition?

Proposition. Let $M$ be a finitely generated $A$-module, $I\subseteq A$ an ideal, and $\phi\in \mathrm{End}_A (M)$ such that $\phi (M)\subseteq I\cdot M$. Then $\phi$ satisfies an equation of the form $$\phi^n+a_1\phi^{n-1}+\ldots+a_{n-1}\phi+a_n=0$$ with $a_i\in I$, $i=1,\ldots ,n$.

• It plays a role for integral ring extensions - for a geometric characterization of integral ring extensions, see Atiyah-MacDonald 1969, Ch 5. Exercise 35. Dec 11, 2013 at 10:59
• Inegral ring extensions... ok, I'll have a look to that place in Atiyah-MacDonald, thank you. Dec 11, 2013 at 11:10
• This amounts to finding a geometric interpretation of the Cayley-Hamilton theorem and I must say I already don't see one for $M_2(\mathbb R)$ acting on $\mathbb R^2$. Dec 11, 2013 at 13:05
• @Olivier: I agree Dec 11, 2013 at 13:08
• @Olivier this is not exactly true since Cayley Hamilton says the degree can be taken at most 2, and here there is no such claim. Dec 11, 2013 at 16:17

Well, you can think of $$M$$ as a module over $$A[T]$$ by letting $$T$$ act via $$\phi$$. Then $$M$$ corresponds to a quasi-coherent sheaf on $$\textrm{Spec}(A[T]) = A^1_{\textrm{Spec}(A)}$$. Since $$M$$ is finite there is a well defined scheme theoretic support $$Z$$. The geometric interpretation I would give is that $$Z$$ is supported in the $$n$$-th infinitimal neighbourhood of the zero section of $$A^1_A$$ over $$\textrm{Spec}(A)$$ union the inverse image of $$V(I) = \textrm{Spec}(A/I)$$ in $$A^1_A$$.