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A coprime factorisation lifted modulo three successive powers

Example

Over $A = k ⟦ t ⟧$ , consider $f (T) = T^{2} - T - t .$ Modulo $t$ one has $\overline{f} = T (T - 1),$ and the two residue factors are coprime.

Facts & Assumptions

Given: The polynomial $f (T) = T^{2} - T - t$ over $k ⟦ t ⟧$ with residue factorization $\overline{f} = T (T - 1)$ .

[L1]

Coprime residue factors admit a lifted Bezout identity (Lift a Bezout identity for coprime residue factors).

[L2]

One Hensel correction step raises the factorization by one power of the ideal (One correction step raises factor lifting by one ideal power).

Verification

technique · compute the correction terms explicitly

1.1

Start with $g_{1} = T$ and $h_{1} = T - 1$ . Then $f - g_{1} h_{1} = - t$ . A residue Bezout identity is $1 = 1 \cdot T + (- 1) \cdot (T - 1) (m o d t),$ so [L1] applies.

L1givenalgebra

2.1

Choose constant corrections $u_{1} = t$ and $v_{1} = - t$ . Then $u_{1} h_{1} + v_{1} g_{1} = t (T - 1) - t T = - t = f - g_{1} h_{1},$ so $g_{2} = T + t, h_{2} = T - 1 - t$ satisfies $g_{2} h_{2} = T^{2} - T - t - t^{2} \equiv f (m o d t^{2}) .$ This is the first explicit correction step from [L2].

L2step 1.1algebra

3.1

Now $f - g_{2} h_{2} = t^{2}$ . Choose $u_{2} = - t^{2}$ and $v_{2} = t^{2}$ . Then $u_{2} h_{2} + v_{2} g_{2} \equiv - t^{2} (T - 1) + t^{2} T = t^{2} (m o d t^{3}),$ so $g_{3} = T + t - t^{2}, h_{3} = T - 1 - t + t^{2}$ satisfies $g_{3} h_{3} \equiv f (m o d t^{3}) .$

L2step 2.1algebra

4.1

Thus the factorization is lifted explicitly modulo $t$ , modulo $t^{2}$ , and modulo $t^{3}$ . The computation makes the abstract correction lemma concrete.

step 2.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Melvin Hochster, The structure theory of complete local rings (standard reference, not scraped)