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## 3.2 Regular Extensions

These are the computationally important extensions since they could be computed without Groebner techniques.

Assume the complex to be a free resolution of :

The multiplication by induces a complex

Now, the tensor product (see Chapter 17.3 in [3]) of the two complexes is a double complex :

where the vertical mappings are the multiplications with for .

Lemma 3.5   The total complex associated with the double complex is a free resolution for .

PROOF: This is easily deduced from the spectral sequence converging to the homology of the total complex (see Chapter A3.13.4, [3]).
According to this lemma we construct the extension from 2 copies of the resolution and the homomorphism of complexes induced by . As arithmetical operations this procedure requires only duplication and addition of polynomials. The number of them depends on the size of . Thus, the involved operations are of polynomial complexity w.r.t. to the input of and .

Next: 3.3 Non-regular Extensions Up: 3. The Extension of Previous: 3.1 Criteria for Regularity
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