Timeline for Projective Modules and their Determinants, Extended or not?

5 events

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Aug 1, 2012 at 21:04	comment	added	Piotr Achinger		@Mohan: Sorry for being sketchy and giving the wrong answer. Unfortunately, I don't understand your argument. Is my "vector bundle" intuition wrong? Are you saying that $L_a \oplus L_{-a}$ does not depend on $a$?
Jul 15, 2012 at 19:15	comment	added	Mohan		@Andrew Parker, Yes, $J(P,A)$ has height at least two.
Jul 15, 2012 at 16:09	comment	added	Andrew Parker		@Mohan In the above example, because the determinant is extended from A, the Quillen ideal, $J(P,A)$, is must have height $\geq 2$, correct? Which would mean that $J(P,A) = A$ because dim$(A) = 1$, implying $P$ must be extended?
Jul 14, 2012 at 13:32	comment	added	Mohan		I believe that such a $P$ is in fact extended, actually free. If $P$ is a projective module over $A[T]$, with $A$ Noetherian of dimension one and $P$ has determinant trivial, then $P$ is trivial. This is because, $A(T)$ ($A[T]$, localized with respect to the set of all monic polynomials) is one dimensional and so any projective module with trivial determinant is trivial. Thus, Quillen-Suslin imples $P$ is trivial.
Jul 13, 2012 at 23:33	history	answered	Piotr Achinger	CC BY-SA 3.0