I was studying the "GEOMETRY OF MODULI OF SHEAVES" by Daniel Huybrechts and Manfred Lehn. I have come across the following definition of torsion filtration of a coherent sheaf $\mathbb E$ of dimension $\mathbb d$ (definition 1.1.4)
Now my problem is in understanding the explaination given for existence of such a torsion filtration (which is claimed to be unique ). We have to show 3 things
1) for each $ i\le d$ we must show the existence of $ T_i$($E$). For this I go for zorn's lemma.So I consider,
$A_i$={subsheaf $F$ of $E$| $ dim$F$\le i$ }
consider a chain $C$ inside $A_i$.It is non empty as Zero subsheaf is there. Now I choose the upperbound for the chain $C$ as the sum of all the sheaves in $C$ .This is defined as follows
$$\sum F $$ takes an open set U to $\bigcup F(U)$.Which can be justified from the lemma 5 of the following lecture noteenter link description here or by defininig sum of sheaves naturally and observing that since we are working in chain, sum of modules becomes union of modules.
But this candidate must satisfy that its dimension is $\le i$. So we have Supp($\bigcup F$) is contained in $\bigcup Supp F$.From where I don't see how it follows that it's dimension $\le i$.
Question:Has it something to do with Noetherianness and coherence?.Is it true that the number of subsheaves (with conditions on them) is finitely many?Even in that case I am not sure how to arrive at conclusion on its dimension.
2)We also must show the maximal element in each $A_i$ is unique.That can be easily shown once the previous no.1 question is addresed i.e sum of subsheaves(here 2 is enough) of dimensions $\le i$ is again a sheaf of dimension $\le i$.(I somehow assume that coherence will be preserved by addition of sheaves)
3)Once we have 1 and 2 in hand the we are also supposed to show that $T_i$ is contained in $T_ (i+1)$. Which I have no intuitive idea how to go about?
Finally I would like to ask a question which is bit more philosophical.
I am a graduate student who is self studying this book with an aim of grasping the core material within 1 or 2 months.Though I find learning the details gives me a sense of completion and satisfaction ,is it worth spending 1 or 2 days trying to give meticulous attention to the details of each and every definition which is already established probably 100 of years ago?
Any help from anyone is welcome.
(1) Yes it has to do with (local) Noetherian condition, in that you will be using any submodule of a finitely-presented module over a Noetherian ring is finitely-presented and that finitely-generated and finitely-presented are the same thing.
So over an affine $A=\operatorname{Spec R}$, $\bigcup_j F_j(A)$ is finitely presented $R$-module. Look at where the generators sit in the chain and this tells you $\bigcup_j\operatorname{Supp}F_j$ is actually a finite union (locally) and that's enough to show $\dim T_i(E)\leq i$.
(2) Yes, same reason as above.
(3) Note $A_i\subseteq A_{i+1}$ so $C$ is also a chain of dimension $\leq i+1$ supported sheaves, so extend that to a maximal such in $A_{i+1}$ gives $T_i(E)\subset T_{i+1}(E)$
Final answer depends on what you want to do really. If you want to go into algebraic geometry then it is very much worthwhile to make sure you are comfortable with these basics (e.g. work through all the exercises in Hartshorne). On the other hand, if your interests lie elsewhere (e.g. topology) then just learn what the machinery of algebraic geometry could do for you is probably enough for now.