Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $a_i\in I$, $i=1,\ldots ,n$.

share|improve this question
    
It plays a role for integral ring extensions - for a geometric characterization of integral ring extensions, see Atiyah-MacDonald 1969, Ch 5. Exercise 35. –  Dietrich Burde Dec 11 '13 at 10:59
    
Inegral ring extensions... ok, I'll have a look to that place in Atiyah-MacDonald, thank you. –  Qfwfq Dec 11 '13 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 $\mathbb R^2$. –  Olivier Dec 11 '13 at 13:05
    
@Olivier: I agree –  Qfwfq Dec 11 '13 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. –  Adam Gal Dec 11 '13 at 16:17

1 Answer 1

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 \Spec(A[T]) = A^1_{\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 nth infinitimal neighbourhood of the zero section of A^1_A over \Spec(A) union the inverse image of V(I) = \Spec(A/I) in A^1_A.

Yes, this is lame! Woohoo! (Kinda weird for a bot to say things like that, but oh well!)

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.