3.2 Regular Extensions

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

Assume the complex $K_{i-1}$ to be a free resolution of $R/I_{i-1}$:

$$K_{i-1}: 0 \rightarrow R^{n_l} \rightarrow \ldots \rightarrow R^{n_2} \rightarrow R^{i-1} \rightarrow R^1 \rightarrow 0.$$

The multiplication by $f_i$ induces a complex

$$K: 0\rightarrow R^1 \rightarrow R^1 \rightarrow 0.$$

Now, the tensor product (see Chapter 17.3 in [3]) of the two complexes is a double complex $\hat{K_i}$:

$$\begin{array}{lllllllllllll} 0 & \rightarrow & R^{n_l} & \r... ... & R^{i-1} & \rightarrow & R^1 & \rightarrow & 0\\ \end{array}$$

where the vertical mappings are the multiplications with $(-1)^jf_i$ for $j=l,\ldots,0$.

Lemma 3.5   The total complex $K_i$ associated with the double complex $\hat{K_i}$ is a free resolution for $R/I_i$.

PROOF: This is easily deduced from the spectral sequence $E^{p,q}_1=H^q(\hat{K_i}^{*,p})$ converging to the homology of the total complex $K_i$ (see Chapter A3.13.4, [3]).$\Box$
According to this lemma we construct the extension $K_i$ from 2 copies of the resolution $K_{i-1}$ and the homomorphism of complexes induced by $f_i$. As arithmetical operations this procedure requires only duplication and addition of polynomials. The number of them depends on the size of $K_{i-1}$. Thus, the involved operations are of polynomial complexity w.r.t. to the input of $K_{i-1}$ and $f_i$.