Divided-Power Tate Resolutions

• Divided-power Tate resolutions are self-dual acyclic free resolutions of length four over commutative Noetherian rings, structured as DGT-algebras that exhibit intrinsic Poincaré duality.
• They are built using differential graded algebras enriched with divided power operations, which remove the need for locality, the Gorenstein hypothesis, or minimality in the resolution.
• The construction leverages canonical perfect pairings and specialized maps to resolve cohomological obstructions, thereby generalizing classical Tate resolution methods in a robust algebraic framework.

A divided-power Tate resolution is a specific type of self-dual acyclic free resolution of length four over a commutative Noetherian ring, structured as a differential graded algebra (DGA) with additional divided power operations. These structures refine the classical perspective on Tate resolutions and encode compatibility with Poincaré duality at the algebraic and module levels. This framework generalizes earlier results by eliminating assumptions of locality, the Gorenstein property, or minimality for the underlying ring or resolution. The essential insight is that any self-dual free resolution of length four, with the highest and lowest terms both of rank one, can be canonically endowed with the structure of a DGA with divided powers (DGT-algebra), thereby exhibiting Poincaré duality purely algebraically (Kustin, 2019).

1. Structure of Length-Four Tate Resolutions

Let $P$ be a commutative Noetherian ring. A length-four Tate resolution of a cyclic $P$-module is an acyclic complex of finitely generated free $P$-modules

$$0 \rightarrow F_4 \xrightarrow{d_4} F_3 \xrightarrow{d_3} F_2 \xrightarrow{d_2} F_1 \xrightarrow{d_1} F_0 \cong P \rightarrow 0$$

subject to:

• $F$ is acyclic except possibly at $H_0(F)$, with $H_0(F)=P/I$,
• $F$ is self-dual: an isomorphism of complexes $F \cong \mathrm{Hom}_P(F, F_4)$,
• The ranks of $F_0$ and $F_4$ are both one.

Self-duality imposes canonical perfect pairings $F_i \otimes_P F_{4-i} \rightarrow F_4 \cong P$ for $i=0,\ldots,4$. It also guarantees

$$F_i \cong (F_{4-i})^{\vee} := \mathrm{Hom}_P(F_{4-i},F_4),$$

notably $F_1 \cong (F_3)^{\vee}$ and $F_2 \cong (F_2)^{\vee}$, embodying Poincaré duality at the level of modules [(Kustin, 2019), Lemma 3.2].

2. Differential Graded Algebras with Divided Powers

A differential graded $P$-algebra (DGA) is a complex $(F,d)$ of free $P$-modules, graded by nonnegative integers, equipped with a unitary associative product $\cdot$ such that:

• $F_i \cdot F_j \subset F_{i+j}$,
• Graded-commutativity: $x \cdot y = (-1)^{|x| \cdot |y|} y \cdot x$,
• The Leibniz rule: $$d(x \cdot y) = d(x) \cdot y + (-1)^{|x|} x \cdot d(y)$$,
• For odd-degree $x$, $x^2 = 0$.

A DGA is a DGT-algebra (DGA with divided powers) if for every even-degree homogeneous $x$ (particularly $x\in F_{2j}$), there exist operations $\gamma^n(x)\in F_{2jn}$ for all $n\geq 0$, meeting classical divided power axioms:

• $\gamma^0(x) = 1$, $\gamma^1(x) = x$,
• Multiplicativity: $$\gamma^n(x)\cdot\gamma^m(x) = \binom{n+m}{n} \gamma^{n+m}(x)$$,
• Linear scaling: $\gamma^n(f x) = f^n \gamma^n(x)$ for $f\in P$,
• Additivity: $$\gamma^n(x+y) = \sum_{i+j=n} \gamma^i(x)\gamma^j(y)$$,
• Differential compatibility: $d(\gamma^n(x)) = d(x) \cdot \gamma^{n-1}(x)$.

Poincaré duality in a DGA appears when $F_i=0$ for $i>m$, $F_m\cong P$, and all pairings $F_i \otimes_P F_{m-i} \rightarrow F_m\cong P$ are perfect; in these resolutions, $m=4$ [(Kustin, 2019), Section 2].

3. Multiplication Map Construction and Divided Powers

Establishing the DGT-algebra structure involves constructing the multiplication map and the divided power operations as follows:

• Fix the perfect pairings: $w_3: F_1 \otimes F_3 \rightarrow F_4$, $w_4: D^2(F_2) \rightarrow F_4$, characterizing $F_1 \cong (F_3)^\vee$ and $F_2 \cong (F_2)^\vee$.
• Select a $P$-linear, alternating, 1-compatible map $Y_1: T^2(F_1) \rightarrow F_2$.
• Define products:
  • $x_1,y_1 \in F_1$: $x_1 \cdot y_1 := Y_1(x_1 \otimes y_1)$.
  • $x_1 \in F_1,\ x_2\in F_2$: $$x_1 \cdot x_2 := w_4(Y_1(x_1 \otimes d_2(x_2)) - x_2 \cdot Y_1(x_1 \otimes -)) \in F_3$$.
  • $x_1\in F_1,\ x_3\in F_3$: $x_1\cdot x_3 := w_3(x_1\otimes x_3)\in F_4$.
  • $x_2,y_2\in F_2$: $x_2\cdot y_2 := w_4(x_2 \otimes y_2)\in F_4$.
• Divided powers on $F_2$: for $x_2\in F_2$, $k\geq 0$, $x_2^{[k]} = w_4(x_2^{(k)})$, where $x_2^{(k)}\in D^k(F_2)$ is the $k$-th divided power.

The construction is verified to yield:

• Graded associativity and commutativity,
• The Leibniz rule,
• Correct differential on divided powers: $d(x_2^{[k]}) = d(x_2) \cdot x_2^{[k-1]}$,
• All divided power laws on $F_2$ [(Kustin, 2019), Lemma 4.3].

4. Poincaré Duality in Length-Four Resolutions

The DGT-algebra structure constructed ensures Poincaré duality in degree 4, with the duality pairings:

• $F_1 \otimes F_3 \rightarrow F_4\cong P$,
• $F_2 \otimes F_2 \rightarrow F_4 \cong P$,

given by $w_3$ and $w_4$, are perfect $P$-pairings. This, together with $F_4\cong P$ and triviality above degree 4, realizes full algebraic Poincaré duality in the constructed DGT-algebra [(Kustin, 2019), Section 4].

5. Removal of Locality, Gorenstein, and Minimality Requirements

Earlier constructions of DGT-algebra structures on resolutions relied on $P$ being a local Gorenstein ring and $F$ being minimal, as in results of Tate, Gulliksen, Buchsbaum-Eisenbud, and Kustin–Miller. The fundamental advancement is the removal of these conditions:

• No locality: The construction is entirely intrinsic to $P$, never requiring localization.
• No Gorenstein hypothesis: Self-duality alone suffices to guarantee the required perfect pairings.
• No minimality requirement: Construction of duality pairings and solutions to the cohomological obstructions needed for the multiplication $Y_1$ can be carried out directly using only the structure of $P$ and $F$, avoiding localization or reductions modulo $2$ or $3$.

The technical apparatus involves:

• Lemma 3.2: Existence of perfect pairings for any self-dual resolution.
• Lemmas 4.4 and 4.5: Construction of $3$- and $2^n$-compatible alternating maps by resolving obstructions modulo $3$ and $2^n$.
• Theorem 4.6: Final existence of $Y_1$ over $\mathbb{Z}$ using integer combinations, ensuring a unique (up to homotopy) DGT-algebra structure on $F$ (Kustin, 2019).

6. Comparative Summary with Prior Results

Aspect	Prior Results (Tate, etc.)	Current Advancement (Kustin, 2019)
Ring $P$	Local Gorenstein	Arbitrary commutative Noetherian
Resolution $F$	Minimal	Not necessarily minimal
Construction Method	Often via localization, modulo $2$, $3$ techniques	Entirely intrinsic, no need for localization or divisions
Existence of DGT-algebra	Under restrictive hypotheses	Always for self-dual length-4 $F$ with $F_4\cong P$
Poincaré Duality	Assumed or deduced in special cases	Verified in full generality

The unrestricted applicability of divided-power Tate resolutions allows broader usage in commutative algebra and related homological questions, extending the reach and relevance of DGT-algebra methods.