Idealization of Modules in Computer Algebra

Martin Kreuzer and Lorenzo Robbiano Chisinau, June 10, 2004

1. What is Idealization of Modules?

Let R be a ring and M an R-module. We turn the set R × M into a ring by componentwise addition and

(r, m) · (r′, m′) = (rr′, rm′ + r′m)

This ring is called the idealization of M. Remarks. a) The image of M in R × M under the injective map ı : M ⟶ R × M defined by m ↦ (0, m) is an ideal.

b) The ideal ı(M) satisfies ı(M)^{2} = 0.

c) Given an R-submodule N ⊆ M, the inclusion R×N ⊆ R × M is an injective ring homomorphism.

d) Let Γ be a monoid, let R be Γ-graded, and let M be a Γ-graded R-module. Then R × M is a Γ-graded ring via (R×M)_{γ} = R_{γ}×M_{γ} for all γ ∈ Γ, and ı(M) is a homogeneous ideal in this ring.

2. Idealization of Graded Submodules

K field P = K[x_{1}, …, x_{n}] polynomial ring P is graded by a positive matrix W ∈ Mat_{m,n}(ℤ) deg_{W} (x_{i}) >_{Lex} 0 is the i^{th} column of W d_{01}, …, d_{0r} ∈ ℤ^{m} F_{0} = ⨁_{i=1}^{r} P (−d_{0i}) graded free P -module M ⊆ F_{0} graded submodule V = (v_{1}, …, v_{s}) deg-ordered tuple of non-zero homoge-

neous vectors which generate M Proposition 1. (Idealization of a Free Module) The map φ : P × F_{0} ⟶ \overline{P} /𝔢 which sends (f, (g_{1}, …, g_{r})) to the residue class of f + g_{1}e_{1} + ⋯ + g_{r}e_{r} is an isomorphism of graded rings. Here we equip \overline{P} = K[x_{1}, …, x_{n}, e_{1}, …, e_{r}] with the grading given by \overline{W} = (W | d_{01} ⋯ d_{0r}) and 𝔢 is the ideal generated by

E = \{e_{i}e_{j} | 1 ≤ i ≤ j ≤ r\} ⊆ \overline{P}

Proposition 2. (Idealization of a Graded Submodule) Under the composition M ⟶^{ı} P × M ⟶ P × F_{0} ⟶^{φ} \overline{P} /𝔢, the module M is identified with the residue class ideal of I_{M} = (v_{1}, …, v_{s}) + 𝔢. The ideal I_{M} is called the (idealization) ideal of M.

Questions. a) How do Gröbner bases behave under this process?

b) How are minimal homogeneous systems of generators of M and I_{M} related to each other?

c) Can the syzygy module Syz_{P} (V) be computed using idealization?

d) What are the advantages of using the idealization ideal I_{M} instead of M?

3. Gröbner Bases and Idealization

Let τ be a term ordering on 𝕋^{n}, the monoid of terms in P.

Let σ be a module term ordering on 𝕋^{n}⟨e_{1}, …, e_{r}⟩ which is compatible with τ, i.e. such that t ≥_{τ} t′ implies t e_{i} ≥_{σ} t′ e_{i} for i = 1, …, r.

Let \overline{σ} be a term ordering on 𝕋(x_{1}, …, x_{n}, e_{1}, …, e_{r}) which extends both σ and τ. Proposition 3. (Gröbner Bases and Idealization)

a) Let G be a σ-Gröbner basis of M. Then G ∪ E is a \overline{σ}-Gröbner basis of I_{M}.

b) Let G be the reduced σ-Gröbner basis of M. Then the reduced \overline{σ}-Gröbner basis of I_{M} is

G ∪ {e_{i}e_{j} ∈ E | e_{i}, e_{j} ∉ LT_{σ}(M)}

In particular, the reduced \overline{σ}-Gröbner basis of I_{M} is G∪E if M ⊆ (x_{1}, …, x_{n}) F_{0}.

4. Idealization and Minimal Homogeneous Generators Assume that M ⊆ (x_{1}, …, x_{n}) F_{0}. Let d_{0i} >_{Lex} 0 for i = 1, …, r. (Then \overline{P} is positively graded.) Let E = (e_{1}e_{1}, e_{1}e_{2}, …, e_{1}e_{r}, e_{2}e_{2}, e_{2}e_{3}, …, e_{2}e_{r}, …, e_{r}e_{r}) Proposition 4. (Idealization and Minimal Generators)

a) Let V be a minimal set of generators of M. Then the set {v_{1}, …, v_{s}} ∪ E is a minimal set of generators of I_{M}.

b) If we apply Buchberger’s Algorithm with Minimalization (BAM) to the tuple (V | E), it computes a minimal system of generators of the ideal I_{M} ⊆ \overline{P} of the form (V_{min} | E), where V_{min} is a minimal set of generators of M. In particular, the elements of E are minimal generators of I_{M}.

c) If we apply the variant of BAM to the tuple (V | E), it computes a homogeneous \overline{σ}-Gröbner basis of I_{M} of the form (G | E) and a minimal set of generators of the form (G_{min} | E), where G is a σ-Gröbner basis of M and G_{min} is a minimal set of generators of M which is contained in G.

The Buchberger Algorithm with Minimalization 1) Let B = ∅, W = V, G = ∅, s′ = 0, and V_{min} = ∅.

2) Let d be the smallest degree with respect to Lex of an element in B or in W. Form the subset B_{d} = {(i, j) ∈ B | deg_{W} (σ_{ij}) = d} and the subtuple W_{d} of elements of degree d in W, and delete their entries from B and W, respectively.

3) If B_{d} = ∅, continue with step 6). Otherwise, choose a pair (i, j) ∈ B_{d} and remove it from B_{d}.

4) Compute the S-vector S_{ij} and its normal remainder S′_{ij} = NR_{σ,G}(S_{ij}). If S′_{ij} = 0, continue with step 3).

5) Increase s′ by one, append g_{s′} = S′_{ij} to the tuple G, and append {(i, s′) | 1 ≤ i < s′, γ_{i} = γ_{s′}} to the set B. Continue with step 3).

6) If W_{d} = ∅, continue with step 9). Otherwise, choose a vector v ∈ W_{d} and remove it from W_{d}.

7) Compute v′ = NR_{σ,G}(v). If v′ = 0, continue with step 6).

8) Increase s′ by one, append g_{s′} = v′ to the tuple G, append v to the tuple V_{min}, and append {(i, s′) | 1 ≤ i < s′, γ_{i} = γ_{s′}} to the set B. Continue with step 6).

9) If B = ∅ and W = ∅, return the pair (G, V_{min}) and stop. Otherwise, continue with step 2). This is an algorithm which returns a deg-ordered tuple G = (g_{1}, …, g_{s′}) whose elements are a homogeneous σ-Gröbner basis of M and a deg-ordered minimal system of generators V_{min} of M which is a subtuple of V.

The Variant of BAM 1) Let B = ∅, W = V, G = ∅, s′ = 0, and G_{min} = ∅.

2) Let d be the smallest degree with respect to Lex of an element in B or in W. Form the subset B_{d} = {(i, j) ∈ B | deg_{W} (σ_{ij}) = d} and the subtuple W_{d} of elements of degree d in W, and delete their entries from B and W, respectively.

3) If B_{d} = ∅, continue with step 6). Otherwise, choose a pair (i, j) ∈ B_{d} and remove it from B_{d}.

4) Compute the S-vector S_{ij} and its normal remainder S′_{ij} = NR_{σ,G}(S_{ij}). If S′_{ij} = 0, continue with step 3).

5) Increase s′ by one, append g_{s′} = S′_{ij} to the tuple G, and append {(i, s′) | 1 ≤ i < s′, γ_{i} = γ_{s′}} to the set B. Continue with step 3).

6) If W_{d} = ∅, continue with step 9). Otherwise, choose a vector v ∈ W_{d} and remove it from W_{d}.

7) Compute v′ = NR_{σ,G}(v). If v′ = 0, continue with step 6).

8) Increase s′ by one, append g_{s′} = v′ to the tuples G and G_{min}, and append {(i, s′) | 1 ≤ i < s′, γ_{i} = γ_{s′}} to the set B. Continue with step 6).

9) If B = ∅ and W = ∅, return the pair (G, G_{min}) and stop. Otherwise, continue with step 2). This is an algorithm which returns a deg-ordered homogeneous σ-Gröbner basis G = (g_{1}, …, g_{s′}) of M and a subtuple G_{min} of G which generates M minimally.

5. Idealization of a Homogeneous Presentation If F_{2} ⟶^{ψ} F_{1} ⟶^{φ} M ⟶ 0 is a homogeneous presenta-

tion of M and d ∈ ℤ^{m}, then

F_{2}(d) ⟶^{ψ̃} F_{1}(d) ⟶^{φ̃} M(d) ⟶ 0

is a homogeneous presentation of M(d). Therefore we shall from now on assume d_{0i} >_{Lex} 0 for i = 1, …, r. In particular, \overline{P} is then positively graded by \overline{W}.

V = (v_{1}, …, v_{s}) is a deg-ordered homogeneous tuple which generates M d_{1i} = deg_{W} (v_{i}) >_{Lex} 0 for i = 1, …, s F_{1} = ⨁_{i=1}^{s} P (−d_{1i}) graded free P -module {ϵ_{1}, …, ϵ_{s}} canonical basis of F_{1} Syz_{P} (V) is a graded submodule of F_{1} P̃ = K[x_{1}, …, x_{n}, e_{1}, …, e_{r}, ϵ_{1}, …, ϵ_{s}] is positively graded by W̃ = (W | d_{01} ⋯ d_{0r} | d_{11} ⋯ d_{1s}).

Proposition 5. (Idealization of a Presentation)

a) The idealization of F_{0} ⊕ F_{1} the ring P̃/𝔢̃, where 𝔢̃ is the ideal generated by {e_{i}e_{j}} ∪ {e_{i}ϵ_{j}} ∪ {ϵ_{i}ϵ_{j}}.

b) If (f_{1}, …, f_{s}) ∈ Syz_{P} (V) is a homogeneous syzygy of V, then the corresponding element f_{1}ϵ_{1} + ⋯ + f_{s}ϵ_{s} of P̃ is contained in the ideal (v_{1} − ϵ_{1}, …, v_{s} − ϵ_{s}).

The ideal Ĩ_{M} = (v_{1} − ϵ_{1}, …, v_{s} − ϵ_{s}) + 𝔢̃ in P̃ is called the ideal of the presentation of M.

c) There exists a unique P -algebra homomorphism

Φ : P [ϵ_{1}, …, ϵ_{s}]/(ϵ_{i}ϵ_{j})_{i,j=1,...,s} ⟶ \overline{P} /𝔢

which maps the residue class of ϵ_{i} to v_{i} + 𝔢 for i = 1, …, s.

d) The image of Φ is the residue class ideal of the ideal of M.

e) The kernel of Φ is the residue class ideal of the ideal of Syz_{P} (V).

f) The ideal of Syz_{P} (V) is given by Ĩ_{M} ∩ P [ϵ_{1}, …, ϵ_{s}].

6. Computing Minimal Presentations Vertically

1) Choose a term ordering σ on 𝕋^{n}(e_{1}, …, e_{r}). Let B = ∅, W = V, G = ∅, s′ = 0, G_{min} = ∅, μ = 0, and S = ∅.

2) Let d be the smallest degree with respect to Lex of an element in B or in W. Form the subset B_{d} and the subtuple W_{d}, and delete their entries from B and W, respectively.

3) If B_{d} = ∅, continue with step 6). Otherwise, chose a pair (i, j) ∈ B_{d} and remove it from B_{d}.

4) Compute the S-vector S_{ij} of g_{i} and g_{j}. Then compute S′_{ij} = NR_{σ,G}(S_{ij}). If S′_{ij} = 0 continue with step 3). If S′_{ij} ≠ 0 and it does not involve the indeterminates e_{1}, …, e_{r}, then append it to S and continue with step 3).

5) Increase s′ by one, append g_{s′} = S′_{ij} to the tuple G, and append the set {(i, s′) | 1 ≤ i < s′, γ_{i} = γ_{s′}} to B. Then continue with step 3).

6) If W_{d} = ∅, continue with step 9). Otherwise, choose a vector v ∈ W_{d} and remove it from W_{d}.

7) Compute v′ = NR_{σ,G}(v) and v̄ = v′|_{ϵ_{j}↦0}. If v̄ = 0, continue with step 6).

8) Increase s′ and μ by one. Adjoin a new indeterminate ϵ_{μ} to \overline{P} and extend the grading to this new ring by defining deg\overline{W} (ϵ_{μ}) = deg\overline{W} (v̄). Extend the term ordering σ to the new ring in such a way that the extension is an elimination ordering for {e_{1}, …, e_{r}}. Append g_{s′} = v̄ − ϵ_{μ} to G and v̄ to G_{min}. Append the set {(i, s′) | 1 ≤ i < s′, γ_{i} = γ_{s′}} to B. Continue with step 6).

9) If B ≠ ∅ or W ≠ ∅, continue with step 2).

10) Apply the Buchberger Algorithm with Minimalization to the module generated by S and obtain a subtuple S_{min} of S which minimally generates that module. Return the pair (G_{min}, S_{min}) and stop.

This is an algorithm which computes a pair (G_{min}, S_{min}) such that G_{min} is a deg-ordered tuple of homogeneous vectors in F_{0} which generate M minimally, and such that S_{min} is a degordered tuple of homogeneous vectors in ⨁_{i=1}^{μ} P (−d_{1i}) which generate Syz_{P} (G_{min}) minimally.

Here μ is the number of elements in G_{min} and d_{1i} is the degree of the i^{th} element in G_{min} for i = 1, …, μ.

7. Computing Minimal Presentations Horizontally

1) Choose a term ordering σ on 𝕋^{n}(e_{1}, …, e_{r}). Let B = ∅, B′ = ∅, W = V, G = ∅, s′ = 0, G_{min} = ∅, μ = 0, S = ∅, s′′ = 0, and S_{min} = ∅.

2) Let d be the smallest degree with respect to Lex of an element in B ∪ B′ or in W. Form the subset B_{d} of B, the subset B′_{d} of B′, the subtuple W_{d} of W, and delete their entries from B, B′, and W, respectively.

3) If B′_{d} = ∅, continue with step 6). Otherwise, choose a pair (i, j) ∈ B′_{d} and remove it from B′_{d}.

4) Compute the S-vector of h_{i} and h_{j} and call it S_{ij}. Then compute the normal remainder S′_{ij} = NR_{σ,S}(S_{ij}). If S′_{ij} = 0, continue with step 3).

5) Increase s′′ by one, append h_{s′′} = S′_{ij} to the tuple S, append the set {(i, s′′) | 1 ≤ i < s′′, η_{i} = η_{s′′}} to B′, and continue with step 3).

6) If B_{d} = ∅, continue with step 10). Otherwise, choose a pair (i, j) ∈ B_{d} and remove it from B_{d}.

7) Compute the S-vector of g_{i} and g_{j} and call it S_{ij}. Then compute the normal remainder S′_{ij} = NR_{σ,G}(S_{ij}). If S′_{ij} involves one of the indeterminates e_{1}, …, e_{r}, then increase s′ by one, append g_{s′} = S′_{ij} to G, append {(i, s′) | 1 ≤ i < s′, γ_{i} = γ_{s′}} to B, and continue with step 6).

8) Compute the normal remainder S′′_{ij} = NR_{σ,S}(S′_{ij}). If S′′_{ij} = 0, continue with step 6).

9) Increase s′′ by one. Append h_{s′′} = S′′_{ij} to the tuples S and S_{min}. Append {(i, s′′) | 1 ≤ i < s′′, η_{i} = η_{s′′}} to B′. Continue with step 6).

10) If W_{d} = ∅, continue with step 13). Otherwise, choose a vector v ∈ W_{d} and remove it from W_{d}.

11) Compute v′ = NR_{σ,G}(v) and v̄ = v′|_{ϵ_{j}↦0}. If v̄ = 0, continue with step 10).

12) Increase s′ and μ by one. Adjoin a new indeterminate ϵ_{μ} to \overline{P} and extend the grading to this new ring by defining deg\overline{W} (ϵ_{μ}) = deg\overline{W} (v′). Extend the term ordering σ to the new ring in such a way that the extension is an elimination ordering for {e_{1}, …, e_{r}}. Append g_{s′} = v̄ − ϵ_{μ} to the tuple G and v̄ to G_{min}. Append {(i, s′) | 1 ≤ i < s′, γ_{i} = γ_{s′}} to B. Continue with step 10).

13) If B = B′ = ∅ and W = ∅, then return the pair (G_{min}, S_{min}) and stop. Otherwise, continue with step 2).

This is an algorithm which computes a pair (G_{min}, S_{min}) such that G_{min} is a deg-ordered tuple of homogeneous vectors in F_{0} which generate M minimally, and such that S_{min} is a degordered tuple of homogeneous vectors in ⨁_{i=1}^{μ} P (−d_{1i}) which generate Syz_{P} (G_{min}) minimally.

Here μ is the number of elements in G_{min} and d_{1i} is the degree of the i^{th} element in G_{min} for i = 1, …, μ.

8. Idealization of Minimal Graded Free Resolutions Let V = (v_{1}, …, v_{s}) be a deg-ordered tuple of homogeneous vectors generating a P -submodule M of F_{0} = ⨁_{i=1}^{r_{0}} P (−d_{0i}). The graded free resolution of M has the shape

0 ⟶ F_{ℓ} ⟶^{φ_{ℓ}} F_{ℓ−1} ⟶ ⋯ ⟶ F_{1} ⟶^{φ_{1}} F_{0} ⟶^{φ_{0}} M ⟶ 0

where F_{0}, …, F_{ℓ} are finitely generated graded free P -modules and for every i ∈ {1, …, ℓ} the homogeneous homomorphism φ_{i} maps the canonical basis of F_{i} to a minimal homogeneous system of generators of Ker(φ_{i−1}).

Recall that ℓ ≤ n.

We may assume that d_{0i} >_{Lex} 0 for i = 1, …, r_{0}.

We write F_{i} = ⨁_{j=1}^{r_{i}} P (−d_{ij}).

The canonical basis {ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}} of F_{i} is always kept deg-ordered. Remarks. a) The idealization of F_{0} ⊕ F_{1} ⊕ ⋯ ⊕ F_{n} is a residue class ring of the polynomial ring

P̃ = K[{ϵ_{j}^{(i)} | i ∈ {0, …, n}, j ∈ {1, …, r_{i}}}]

b) The elements of F_{i} can be identified with their images in P̃.

9. Computing Minimal Resolutions Vertically

1) Let i = 0. Equip \overline{P} = P [ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}] with the grading defined by \overline{W} = (W | d_{i1} ⋯ d_{ir_{i}}). Choose a term ordering σ on 𝕋^{n}(ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}). Let B = ∅, W = V, G = ∅, s′ = 0, G_{min}^{(i)} = ∅, r_{i+1} = 0, and S = ∅.

2) Let d be the smallest degree with respect to Lex of an element in B or in W. Form the subset B_{d} and the subtuple W_{d}, and delete their entries from B and W, respectively.

3) If B_{d} = ∅, continue with step 6). Otherwise, chose a pair (j, k) ∈ B_{d} and remove it from B_{d}.

4) Form the S-vector S_{jk} of g_{j} and g_{k}. Then compute S′_{jk} = NR_{σ,G}(S_{jk}). If S′_{jk} = 0 continue with step 3). If S′_{jk} ≠ 0 and it does not involve the indeterminates ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}, then append it to S and continue with step 3).

5) Increase s′ by one, append g_{s′} = S′_{jk} to the tuple G, and append the set {(j, s′) | 1 ≤ j < s′, γ_{j} = γ_{s′}} to B. Then continue with step 3).

6) If W_{d} = ∅, continue with step 9). Otherwise, choose a vector v ∈ W_{d} and remove it from W_{d}.

7) Compute v′ = NR_{σ,G}(v) and v̄ = v′|_{ϵ_{j}^{(i+1)}↦0}. If v̄ = 0, continue with step 6).

8) Increase s′ and r_{i+1} by one. Adjoin a new indeterminate ϵ_{r_{i+1}}^{(i+1)} to \overline{P} and extend the grading to this new ring by defining deg\overline{W} (ϵ_{r_{i+1}}^{(i+1)}) = deg\overline{W} (v̄). Extend the term ordering σ to the new ring in such a way that the extension is an elimination ordering for {ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}}. Append the element g_{s′} = v̄ − ϵ_{r_{i+1}}^{(i+1)} to G and the element v̄ to G_{min}^{(i)}. Append the set {(j, s′) | 1 ≤ j < s′, γ_{j} = γ_{s′}} to B. Continue with step 6).

9) If B ≠ ∅ or W ≠ ∅, continue with step 2).

10) If S ≠ 0, then increase i by one and equip \overline{P} = P [ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}] with the grading defined by the matrix \overline{W} = (W | d_{i1} ⋯ d_{ir_{i}}). Restrict σ to 𝕋^{n}(ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}). Let B = ∅, W = S, G = ∅, s′ = 0, G_{min}^{(i)} = ∅, r_{i+1} = 0, and S = ∅. Then continue with step 2).

11) Let ℓ = i + 1. Return the list (G_{min}^{(0)}, …, G_{min}^{(ℓ−1)}) and stop.

This is an algorithm which computes a list of deg-ordered homogeneous matrices (G_{min}^{(0)}, …, G_{min}^{(ℓ−1)}) such that the P - linear maps φ_{j} : F_{j} ⟶ F_{j−1} given by G_{min}^{(j−1)} for j = 1, …, ℓ yield a minimal graded free resolution

0 ⟶ F_{ℓ} ⟶^{φ_{ℓ}} F_{ℓ−1} ⟶^{φ_{ℓ−1}} ⋯ ⟶^{φ_{3}} F_{2} ⟶^{φ_{2}} F_{1} ⟶^{φ_{1}} M ⟶ 0

9. Computing Minimal Resolutions Horizontally

1) Let σ be a term ordering on 𝕋^{n}(ϵ_{1}^{(0)}, …, ϵ_{r_{0}}^{(0)}), let \overline{P} = P [ϵ_{1}^{(0)}, …, ϵ_{r_{0}}^{(0)}] be graded by \overline{W} = (W | d_{01} ⋯ d_{0r_{0}}), let r_{1} = ⋯ = r_{n} = 0, let B = {v_{1}, …, v_{s}}, let G = ∅, and let G_{min} = ∅.

2) Let d be the smallest degree with respect to Lex of an element of B. Form the subset B_{d} of B and remove it from B.

3) If B_{d} = ∅, continue with step 7). Otherwise, let i be the largest upper index of an indeterminate ϵ_{k}^{(j)} occuring in a polynomial of B_{d}. Let f ∈ B_{d} be a polynomial which involves that indeterminate. Remove f from B_{d}.

4) Compute f′ = HR_{σ,G}(f). If one of the indeterminates ϵ_{1}^{(i−1)}, …, ϵ_{r_{i−1}}^{(i−1)} occurs in f′, append f′ to G, append to B all S-polynomials of f′ and a polynomial g in G such that LT_{σ}(f′) and LT_{σ}(g) involve the same ϵ_{j}^{(i−1)}, and continue with step 3).

5) If none of the indeterminates {ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}} occurs in f′, continue with step 3).

6) Increase r_{i+1} by one. Adjoin an indeterminate ϵ_{r_{i+1}}^{(i+1)} to \overline{P} and extend the grading to this new ring by defining deg\overline{W} (ϵ_{r_{i+1}}^{(i+1)}) = d. Extend the term ordering σ to the new ring in such a way that the extension is an elimination ordering for {ϵ_{1}^{(0)}, …, ϵ_{r_{i}}^{(i)}}. Compute the polynomial

\overline{f} = f′(x_{1}, …, x_{n}, ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}, 0, …, 0)

Append g = \overline{f} −ϵ_{r_{i+1}}^{(i+1)} to G and \overline{f} to G_{min}. For all h ∈ B such that LT_{σ}(g) and LT_{σ}(h) involve the same indeterminate ϵ_{j}^{(i)}, compute the S-polynomial of g and h and append it to B. Then continue with step 3).

7) If B = ∅, return the tuple G_{min} and stop. Otherwise, continue with step 2).

This is an algorithm which computes a deg-ordered tuple G_{min} of homogeneous polynomials in

\overline{P} = P [ϵ_{1}^{(0)}, …, ϵ_{r_{0}}^{(0)}, …, ϵ_{1}^{(ℓ)}, …, ϵ_{r_{ℓ}}^{(ℓ)}]

such that the homogeneous maps of graded free P -modules φ_{i} : F_{i} ⟶ F_{i−1} defined by the elements of G∩P [ϵ_{1}^{(i)}, …, ϵ_{r_{i}}^{(i)}] yield a minimal graded free resolution

0 ⟶ F_{ℓ} ⟶^{φ_{ℓ}} F_{ℓ−1} ⟶^{φ_{ℓ−1}} ⋯ ⟶^{φ_{3}} F_{2} ⟶^{φ_{2}} F_{1} ⟶^{φ_{1}} M ⟶ 0

Summary and Conclusions

• For submodules of graded free modules the presentation of the idealization was described explicitly.

• We have determined the relation between the module and its idealization with respect to their Gröbner bases and with respect to minimal homogeneous sets of generators.

• We can idealize a homogeneous presentation and even a graded free resolution of the module.

• The computation of a minimal homogeneous presentation or a graded free resolution is then nothing but the computation of one Gröbner basis for the ideal representing the idealization.

• Classical strategies for computing resolutions correspond to different selection strategies for this Gröbner basis computation.

• The horizontal strategy corresponds to a particularly brief algorithm.

• All standard optimizations (avoiding unnecessary critical pairs, Hilbert driven, ...) can be applied.

• It is easy to analyze which operations have to be implemented efficiently to speed up the computation.

If you can’t realize your ideal,

idealize the real. (Marriage Counsel)