Commutative Algebra - summer term 2017
Assignment sheet 9
Prof. Dr. Mohamed Barakat, M.Sc. Kamal Saleh
Exercise 1. (Computing intersection and sum categorically, 6 points)
Let R be a computable commutative ring. In the lecture we showed that the essentially surjective functor
coker:
R-fpres →R-fpmod,
M 7→M = cokerM,
κ = (M,A,N) 7→ϕA :M →N = cokerN
(*)
is fully faithful and is hence an equivalence of categories R-fpres∼=R-fpmod. Define N to be the 0×1matrix in R-fpres and identify coker(N) =R.
1. Let I = hf1, f2, . . . , fmiR be an ideal in R. Construct a mono κI = (MI,AI,N) ∈ R-fpres which maps to the subobject I ,→ R under the above equivalence of cate- gories. Prove:
• For any such mono the entries of the column matrix AI generate the idealI.
• AI can be set to
f1
... fm
.
Hint: κI can be computed as an image embedding.
2. Let J =hg1, g2, . . . , gniRbe another ideal inRand κJ = (MJ,AJ,N)as in 1. Moreover, let
MI N
MJ
PI,J
AI
AJ BI
BJ
be the pullback diagram of κI, κJ, i.e., PI,J is the categorical intersection of the subobjects κI, κJ. We know from the previous exercise sheet that (PI,J,BIAI,N) is the mono corresponding to the embedding I ∩J ,→ R under the above equivalence (*). Now let
Commutative Algebra - summer term 2017
MI QI,J
MJ
PI,J CI
CJ BI
BJ
be the pushout diagram of (PI,J,BI,MI),(PI,J,BJ,MJ). We know that (PI,J,BIAI,N)∼(PI,J,BJAJ,N), hence there is a unique morphism d:= (QI,J,D,N) such that (MI,CID,N)∼(MI,AI,N) and (MJ,CJD,N)∼(MJ,AJ,N). Show that d is a mono corresponding to the embedding I +J ,→ R under the above equivalence of categories.
3. Compute using the above categorical algorithms I∩J and I+J for (a) R:=Z, I :=h4iZ= 4Z, and J :=h6iZ = 6Z.
(b) R :=Q[x, y, z], I :=hx2−y2, xz−y,−x5 +yzi, and J :=h−x2y+y3+xz2 − yz, y3−xyz+xz2 −yzi.
4. Construct a presentation matrix for the R-module (I+J)/I.
5. Generalize the above to submodules of finitely presented modules over a left com- putable ringR.
Exercise 2. (Computing annihilators categorically, 4 points)
1. Let R be a left computable ring, N ∈ R-fpres the zero matrix of dimensions 0×1, and M ∈ R-fpres of dimensions m×n. For every 1 ≤ i ≤ n, denote by Ei the i-th standard basis row vector withncolumns, and byi the morphism(N,Ei,M). Suppose S:=Tn
i=1ker(i)and κ := (S,A,N)is the embedding of S as a subobject of N. Prove that κ corresponds under the equivalence (*) to the embedding AnnR(M) ,→ R.
In particular, the entries of the column matrix A generate the annihilator of M :=
coker(M).
2. Let R :=Q[x, y, z]. Compute a generating set for the annihilator of M := coker(M), where
M :=
x y
z y
x2−y2 0 0 x2−y2
xz−y 0
0 xz−y
−x5+yz 0 0 −x5+yz
.
Commutative Algebra - summer term 2017
Exercise 3. (Computing ideal quotient categorically, 6 points)
Let R be a computable ring and I, J be two finitely generated ideals in R. Show the following
1. I :J = AnnR((I+J)/I).
2. J ⊂I then I :J =R.
3. I : (J+K) = (I :J)∩(I :K).
4. If R is integral domain, then I : (r) = 1r(I∩(r)).
5. Use 3. and 4. to derived a Gröbner-basis based algorithm to compute I :J when R is a polynomial ring over a field.
6. Give a categorical method to computeI :J over any computable commutative unitial ring.
7. Compute I :J for the ideals in Exercise 1.
Remark. For5., use elimination theory. For 6., use Exercises 1 and 2.
Hand in until Juli 11th 12:15 in the class or in the Box in ENC, 2nd floor, at the entrance of the building part D.
