--  This is code related to HW 2 for Math 615, Winter 2013
--  By: Daniel Erman

-- Exercise 1 doesn't need Macaualay2, but it might be helpful to look at an example.
S = QQ[a,b,c]
Kab = koszul matrix{{a,b}}
Kc = koszul matrix {{c}}
Kabc = koszul matrix {{a,b,c}}

-- here is how you take the tensor product of two complexes in Macaulay2
Kab ** Kc


--  For Exercise 2, you can compute these groups by hand.  But
--  it might also be useful to try adapting this code to compute Koszul homology groups.
S = QQ[x,y]
K = koszul matrix {{x^2,x*y}}
--  note that Macaulay2's indexing is different than Eisenbud's!
--  For a Koszul complex on n elements, the correspondence Eisenbed <--> M2
--  is given by the formula H^i <--> HH_(n-i).
HH_0 K
HH_1 K
HH_2 K

-- For Exercise 3, you should use the Koszul complex to compute the depth of an ideal.
-- The M2 code ``depth'' is quite incomplete.
--  Here is an example that does not work.
S = QQ[x,y]
I = ideal(x^2,y^2)
-- first we compute depth(I,S) as follows:
K = koszul matrix {{x^2,y^2}}
HH_0 K == 0
HH_1 K == 0
HH_2 K == 0
--  thus the first nonzero group is HH_0.  After adjusting for Eisenbud's different
--  indexing conventions (see above) and applying Eisenbud 17.4, we see that depth(I,S)=2.

-- computing codim(I,S) is easy:
codim(I)
--  codim(I) = 2 = depth(I,S), this example does not work.
--  Try to find an ideal I where these two invariants diverge.
--  If you don't know where to start in this search, feel free to ask me for a hint.



--  There is M2 code this week for helping with Exercise 4.


--  Exercise 5.  Here is how to get K(1,1,1) in M2:
K = koszul matrix {{1,1,1}}
--  This might help you think about the correspondence.