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Purity twists and cohomological shifts with explicit premises

PDF p. 7: Motivic purity at a smooth closed point

Statement

Let $T$ be a triangulated category, let $A (c)$ be a specified twist of an object $A$ , and let $E_{j}$ be coefficient objects for specified weights. Fix $c \geq 0$ . Suppose, in the weights being used, there is a supplied natural identification ${Hom}_{T} (A (c), E_{j} [n]) ≅ {Hom}_{T} (A, E_{j - c} [n])$ for every integer $n$ . Then ${Hom}_{T} (A (c) [2 c], E_{j} [a]) ≅ {Hom}_{T} (A, E_{j - c} [a - 2 c]) .$ If the analogous identification is supplied for derived mapping complexes in a stable enhancement, then $R Hom (A (c) [2 c], E_{j}) ≃ R Hom (A, E_{j - c}) [- 2 c],$ and its degree- $a$ cohomology is $H^{a - 2 c} R Hom (A, E_{j - c})$ . No cancellation, invertibility of twists, or definition of negative-weight coefficients is inferred without that supplied identification. In particular the indexing rule applies only within its stated weight range.

Facts & Assumptions

Given: The objects, integer indices, and the displayed Hom identification; the enhanced version is additionally assumed only for the mapping-complex assertion.

Goal: Compute the shift without concealing the required twist comparison.

[F1]

The cochain convention is $C [s]^{n} = C^{n + s}$ and $H^{a} (C [s]) = H^{a + s} (C)$ . Derived category of an abelian category

[F2]

Shift is an autoequivalence, so $Hom (V [s], W [a]) ≅ Hom (V, W [a - s])$ by shifting both source and target by $- s$ . The same rule in mapping complexes gives a shift $[- s]$ when the source is shifted by $s$ .

Proof

1.1F2given

Set $s = 2 c$ in [F2]. It gives $Hom (A (c) [2 c], E_{j} [a]) ≅ Hom (A (c), E_{j} [a - 2 c])$ . Now apply the assumed twist identification with $n = a - 2 c$ , obtaining exactly the displayed group on $A$ and weight $j - c$ .

2.1F1F2givenstep 1.1

If the enhanced twist comparison is supplied, first use the mapping-complex version of [F2], then that comparison, to identify the mapping complex with $R Hom (A, E_{j - c}) [- 2 c]$ . By [F1], its degree- $a$ cohomology is the degree- $a - 2 c$ group of the unshifted complex.

3.1givenstep 1.1step 2.1∎

Both computations explicitly used a premise containing $E_{j - c}$ . They establish neither the existence of that object in a weight outside the given coefficient system nor a twist-adjunction theorem. Thus the conclusions retain precisely the supplied weight bounds and no additional purity assertion has entered their proof.

Depends on

Derived category of an abelian category

Used by · 2

Dependency tree · 3 levels

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