Differences

This shows you the differences between two versions of the page.

--- blog:2023-12-06 [2023/12/06 23:59] – created pzhou
+++ blog:2023-12-06 [2023/12/07 15:13] (current) – [Mittag-Leffler] pzhou
@@ Line 12: / Line 12: @@
 I need to say, non-characteristic deformation lemma. what is 2.7.2?
+===== Mittag-Leffler =====
+what is Mittag-Leffler condition? if I give you a projective system of abelian group, (no, not assuming surjectivity at each level), we can ask for projective limit.
+It is reasonable to assume ML condition. For example, if each level is a finite dimensional vector space, then it is automatically ML.
+It is also easy to get a system that is not ML, for example,
+$$ k[x] \gets k[x] \gets k[x] \gets \cdots $$
+where the map is $x$, well, the limit is zero. I really wonder what is not a ML system.
+Well, consider the following examples
+$$ 0 \to (x^n) \to k[x] \to k[x] / (x^n) \to 0 $$
+if we take projective limit on these things, we get
+$$ 0 \to 0 \to k[x] \to k[[x ] ] $$
+note, the right side is not exact, and it should'nt. Is there some derived projective limit? hmm? It should be $k[[x ] ] / k[x]$. This should be derived inverse limit is zero.
+So, ML condition implies the acyclic for derived inverse limit functor. This is the poit of 1.12.3.
+Then we consider a projective system of chain complex, or equivalently, a chain complex of projective system (since the category of projective system in an abelian category also forms an abelian category). It is natural to compare two things: one is, apply limit functor, two is compute the cohomology.
+$$ \phi^k H^k \varprojlim \to \varprojlim H^k $$
+Why? Because we have
+$$ H^k (X_\infty) = \frac{Z_\infty^k}{B_\infty^k} \to \frac{Z_n^k}{B_n^k} = H^k_n $$
+Now, is this map $H^k (\varprojlim_n X_n) \to \varprojlim H^k(X_n)$ surjective?
+We consider taking two exact seq
+$$ 0 \to B^k(X_\infty) \to Z^k(X_\infty) \to H^k(X_\infty) \to 0 $$
+$$ 0 \to \lim B^k(X_n) \to \lim Z^k(X_n) \to \lim H_n^k \to 0 $$
+the last part is because $B_n^k$ are ML in $n$, hence the proj lim SES is also right exact.
+  * we have snake lemma
+  * middle column isom implies right column surjective
+  * however, $B^k(X_\infty) =Im(X^{k-1}_\infty) \to \lim_n B^k(X_n)$ may not be surjective without further assumption. This is because $B^k$ is like taking cokernel of a map, and $\lim$ is taking projective lim, they don't commute. If we assume $H^{k-1}(X_n)$ is ML, then we know $Z^{k-1}(X_n)$ is ML. Then, we have
+$$ 0 \to Z^{k-1}(X_n) \to X^{k-1}_n \to B^k(X_n) \to 0 $$
+then we use the proposition that the first term is ML, then the projective limit preserve the SES, and we get
+$ B^k(X_\infty) \cong \lim B^k(X_n) $
+----
+But why do we care about these ML stuff? Why do we care about the non-characteristic deformation lemma?