(1) Let A be an abelian category. (In this and the following problems,
it's fine if you want to assume
that A has enough injectives or something like that. At least for (1),
that's not needed.) Let M be an object of the bounded
derived category D^b(A), so in particular M is a complex.
Show that for each integer j, there
is a complex M^{>=j} (a "truncation" of M)
concentrated in degrees >=j and a map
    f: M -> M^{>=j}
of complexes (not just a map in D^b(A))
such that f induces an isomorphism on H^i for i>=j
and H^i(M^{>=j})=0 for i<j. (In working with derived
categories, it's useful to remember the direction of this map f,
or at least to be able to work it out again when needed.
Note that M^{>=j} is not the "stupid truncation" where you just
set the terms of M in degrees < j to zero, although that can sometimes
be useful for other purposes.)

Likewise, check that there is a complex M^{<j}
and a map M^{<j} -> M with the obvious property. Finally, show that
there is an exact triangle in D^b(A),
   M^{<j} -> M -> M^{>=j} -> M^{<j}[1].

(2) Let A be an abelian category. Let M be an object
of the bounded derived category D^b(A). Suppose that H^i(M) in A
is zero for all i not equal to some integer j.
Show that M is isomorphic to H^i(M)[-i] in D^b(A).
(Try showing this as directly as possible from the definition
of D^b(A), perhaps using truncations as in (1). Of course you need
to remember that M[j] means the object M shifted j steps
_to the left_, and that we are writing complexes cohomologically:
   ... -> M^{-1} -> M^0 -> M^1 -> ...)

(3) Now let M be an object of D^b(A) with cohomology in two different
degrees. By shifting, we can assume that these are degree 0 and r
for some r > 0 (right?). Say X = H^0(M) and Y = H^r(M),
in the abelian category A. Show that objects M of this form
are determined up to isomorphism in D^b(A) by an element
of Ext^{r+1}_A(Y,X), and conversely that any such element
determines an object M. (Perhaps use truncations as in (1).
This result gives you a computational handle
on objects of the derived category. It's a classic application
of "rotating the triangle" in a triangulated category.)

(4) Write out the two exact sequences of Hom-groups associated
to an exact triangle A -> B -> C and an object M
in a triangulated category. (There is one exact sequence
for maps from M and another for maps to M.) Here I don't mean
for you to prove anything, just to recall the statements.

(5) The _global dimension_ of a ring R (not necessarily commutative)
means the supremum of the projective dimensions of all left R-modules.
(For some equivalent conditions, see Eisenbud's Commutative Algebra,
Theorem A3.18, p. 650, which I hope is on Springerlink;
or also the Wikipedia page "Global dimension".)
By Auslander-Buchsbaum-Serre, among commutative noetherian rings,
a regular ring of dimension n has global dimension n,
whereas all non-regular rings have infinite global dimension.
(For a simple example, show that the ring k[x]/(x^2)
has infinite global dimension, for a field k.)

Show that for a ring R of global dimension <=1, every object
M of D^b(R) is isomorphic to the direct sum of the objects
H^i(M)[-i] over the integers i. That is, M is just a direct
sum of R-modules placed in different degrees.
(Use problem (3) and perhaps (4). You don't need
to write out all the details, as long as you see the basic
reason this works.)

For example, this result applies to the derived category of Z-modules
(= abelian groups), or to the coherent derived category of the affine
line over a field k. Explicitly, deduce a classification
of all objects in D^b_{coh}(A^1_C) (where C is the complex
numbers), up to isomorphism.