Algorithmic homological algebra

Algorithmic homological algebra

Exercise sheet 5

Prof. Dr. Mohamed Barakat, Sebastian Posur

Exercise 1. (Computable rings)

Given a constructive field k equipped with the Gaussian normal form algorithm, more precisely, an algorithm to compute the row reduced echelon form (RREF). Apply such an algorithm to the block matrix (^{1 B 0}_{0 A 1}) and obtain_{1 B}⁰ _−X

0 A⁰ Y 0 0 S

. Show that setting

• DecideZeroRows(B,A) := B⁰,

• DecideZeroRowsEffectively(B,A) := (B⁰,−X),

• SyzygiesOfRows(A) :=S,

turns k into a computable ring.

Exercise 2. (Category of matrices)

Letk be a constructive field. We construct the categoryk-vec as follows.

1. Obj_k-vec:=N0.

2. For m, n∈N0, Hom_k-vec(m, n) := k^m×n.

3. For n ∈N⁰, idn is the n×n identity matrix In.

4. For m, n, o ∈ N0, A ∈ k^m×n, B ∈k^n×o, composition is given by the matrix product A·B.

Show thatk-vec is a constructively abelian category.

Exercise 3. (Gr¨obner basis)

(a) Prove Gordan’s lemma: LetR =k[x₁, . . . , x_n] be the polynomial ring innindetermi- nates over a field k. Then any nonempty set X of monomials in R has only finitely many minimal elements in the partial order given by divisibility.

(b) Let f1 = x²y − y³, f2 = x³ ∈ k[x, y]. Compute a Gr¨obner basis for the ideal I = hf₁, f₂i with respect to >_lex. Visualize the monomials in L(I), and compute a multiplication table for k[x, y]/I.

Exercise 4. (Generalized inverses)

Prove that for any generalized morphism γ, we have (γ⁻¹)⁻¹ =γ.

Algorithmic homological algebra

Exercise 5. (Generalized morphisms and additive relations)

Given a ring R and R-modules A and B, we call a submodule S ⊆ A⊕B an additive relationfromA toB. Given two additive relations S ⊆A⊕B and T ⊆B⊕C, we define their composite as

S·T :={(a, c)∈A⊕C | ∃b ∈B : (a, b)∈S∧(b, c)∈T}. Now, let A⁰ ⊆A and B⁰ ⊆B be submodules. Given a generalized morphism

ψ :A←- A⁰ ψ

−→B/B⁰ B,

where A ←- A⁰ and B/B⁰ B are the canoncial inclusion and projection, we define the additive relation

F(ψ) :=

(a, b)∈A⁰⊕B | ψ(a) = b+B⁰ ⊆A⊕B.

(a) Show thatF defines a bijection from generalized morphisms fromAtoB to additive relations from A toB.

(b) Show that F is compatible with composition of generalized morphisms and compo- sition of additive relations, i.e., given another generalized morphism φ fromB to C, we have

F(ψ·φ) = F(ψ)·F(φ).

This exercise sheet will be discussed on 19.01.2017.