23.6 Tate resolutions
In this section we briefly discuss the resolutions constructed in [Tate-homology] and [AH] which combine divided power structures with differential graded algebras. In this section we will use homological notation for differential graded algebras. Our differential graded algebras will sit in nonnegative homological degrees. Thus our differential graded algebras $(A, \text{d})$ will be given as chain complexes
\[ \ldots \to A_2 \to A_1 \to A_0 \to 0 \to \ldots \]
endowed with a multiplication.
Let $R$ be a ring (commutative, as usual). In this section we will often consider graded $R$-algebras $A = \bigoplus _{d \geq 0} A_ d$ whose components are zero in negative degrees. We will set $A_+ = \bigoplus _{d > 0} A_ d$. We will write $A_{even} = \bigoplus _{d \geq 0} A_{2d}$ and $A_{odd} = \bigoplus _{d \geq 0} A_{2d + 1}$. Recall that $A$ is graded commutative if $x y = (-1)^{\deg (x)\deg (y)} y x$ for homogeneous elements $x, y$. Recall that $A$ is strictly graded commutative if in addition $x^2 = 0$ for homogeneous elements $x$ of odd degree. Finally, to understand the following definition, keep in mind that $\gamma _ n(x) = x^ n/n!$ if $A$ is a $\mathbf{Q}$-algebra.
Definition 23.6.1. Let $R$ be a ring. Let $A = \bigoplus _{d \geq 0} A_ d$ be a graded $R$-algebra which is strictly graded commutative. A collection of maps $\gamma _ n : A_{even, +} \to A_{even, +}$ defined for all $n > 0$ is called a divided power structure on $A$ if we have
$\gamma _ n(x) \in A_{2nd}$ if $x \in A_{2d}$,
$\gamma _1(x) = x$ for any $x$, we also set $\gamma _0(x) = 1$,
$\gamma _ n(x)\gamma _ m(x) = \frac{(n + m)!}{n! m!} \gamma _{n + m}(x)$,
$\gamma _ n(xy) = x^ n \gamma _ n(y)$ for all $x \in A_{even}$ and $y \in A_{even, +}$,
$\gamma _ n(xy) = 0$ if $x, y \in A_{odd}$ homogeneous and $n > 1$
if $x, y \in A_{even, +}$ then $\gamma _ n(x + y) = \sum _{i = 0, \ldots , n} \gamma _ i(x)\gamma _{n - i}(y)$,
$\gamma _ n(\gamma _ m(x)) = \frac{(nm)!}{n! (m!)^ n} \gamma _{nm}(x)$ for $x \in A_{even, +}$.
Observe that conditions (2), (3), (4), (6), and (7) imply that $\gamma $ is a “usual” divided power structure on the ideal $A_{even, +}$ of the (commutative) ring $A_{even}$, see Sections 23.2, 23.3, 23.4, and 23.5. In particular, we have $n! \gamma _ n(x) = x^ n$ for all $x \in A_{even, +}$. Condition (1) states that $\gamma $ is compatible with grading and condition (5) tells us $\gamma _ n$ for $n > 1$ vanishes on products of homogeneous elements of odd degree. But note that it may happen that
\[ \gamma _2(z_1 z_2 + z_3 z_4) = z_1z_2z_3z_4 \]
is nonzero if $z_1, z_2, z_3, z_4$ are homogeneous elements of odd degree.
Example 23.6.2 (Adjoining odd variable). Let $R$ be a ring. Let $(A, \gamma )$ be a strictly graded commutative graded $R$-algebra endowed with a divided power structure as in the definition above. Let $d > 0$ be an odd integer. In this setting we can adjoin a variable $T$ of degree $d$ to $A$. Namely, set
\[ A\langle T \rangle = A \oplus AT \]
with grading given by $A\langle T \rangle _ m = A_ m \oplus A_{m - d}T$. We claim there is a unique divided power structure on $A\langle T \rangle $ compatible with the given divided power structure on $A$. Namely, we set
\[ \gamma _ n(x + yT) = \gamma _ n(x) + \gamma _{n - 1}(x)yT \]
for $x \in A_{even, +}$ and $y \in A_{odd}$.
Example 23.6.3 (Adjoining even variable). Let $R$ be a ring. Let $(A, \gamma )$ be a strictly graded commutative graded $R$-algebra endowed with a divided power structure as in the definition above. Let $d > 0$ be an even integer. In this setting we can adjoin a variable $T$ of degree $d$ to $A$. Namely, set
\[ A\langle T \rangle = A \oplus AT \oplus AT^{(2)} \oplus AT^{(3)} \oplus \ldots \]
with multiplication given by
\[ T^{(n)} T^{(m)} = \frac{(n + m)!}{n!m!} T^{(n + m)} \]
and with grading given by
\[ A\langle T \rangle _ m = A_ m \oplus A_{m - d}T \oplus A_{m - 2d}T^{(2)} \oplus \ldots \]
We claim there is a unique divided power structure on $A\langle T \rangle $ compatible with the given divided power structure on $A$ such that $\gamma _ n(T^{(i)}) = T^{(ni)}$. To define the divided power structure we first set
\[ \gamma _ n\left(\sum \nolimits _{i > 0} x_ i T^{(i)}\right) = \sum \prod \nolimits _{n = \sum e_ i} x_ i^{e_ i} T^{(ie_ i)} \]
if $x_ i$ is in $A_{even}$. If $x_0 \in A_{even, +}$ then we take
\[ \gamma _ n\left(\sum \nolimits _{i \geq 0} x_ i T^{(i)}\right) = \sum \nolimits _{a + b = n} \gamma _ a(x_0)\gamma _ b\left(\sum \nolimits _{i > 0} x_ iT^{(i)}\right) \]
where $\gamma _ b$ is as defined above.
At this point we tie in the definition of divided power structures with differentials. To understand the definition note that $\text{d}(x^ n/n!) = \text{d}(x) x^{n - 1}/(n - 1)!$ if $A$ is a $\mathbf{Q}$-algebra and $x \in A_{even, +}$.
Definition 23.6.5. Let $R$ be a ring. Let $A = \bigoplus _{d \geq 0} A_ d$ be a differential graded $R$-algebra which is strictly graded commutative. A divided power structure $\gamma $ on $A$ is compatible with the differential graded structure if $\text{d}(\gamma _ n(x)) = \text{d}(x) \gamma _{n - 1}(x)$ for all $x \in A_{even, +}$.
Warning: Let $(A, \text{d}, \gamma )$ be as in Definition 23.6.5. It may not be true that $\gamma _ n(x)$ is a boundary, if $x$ is a boundary. Thus $\gamma $ in general does not induce a divided power structure on the homology algebra $H(A)$. In some papers the authors put an additional compatibility condition in order to ensure that this is the case, but we elect not to do so.
Lemma 23.6.6. Let $(A, \text{d}, \gamma )$ and $(B, \text{d}, \gamma )$ be as in Definition 23.6.5. Let $f : A \to B$ be a map of differential graded algebras compatible with divided power structures. Assume
$H_ k(A) = 0$ for $k > 0$, and
$f$ is surjective.
Then $\gamma $ induces a divided power structure on the graded $R$-algebra $H(B)$.
Proof.
Suppose that $x$ and $x'$ are homogeneous of the same degree $2d$ and define the same cohomology class in $H(B)$. Say $x' - x = \text{d}(w)$. Choose a lift $y \in A_{2d}$ of $x$ and a lift $z \in A_{2d + 1}$ of $w$. Then $y' = y + \text{d}(z)$ is a lift of $x'$. Hence
\[ \gamma _ n(y') = \sum \gamma _ i(y) \gamma _{n - i}(\text{d}(z)) = \gamma _ n(y) + \sum \nolimits _{i < n} \gamma _ i(y) \gamma _{n - i}(\text{d}(z)) \]
Since $A$ is acyclic in positive degrees and since $\text{d}(\gamma _ j(\text{d}(z))) = 0$ for all $j$ we can write this as
\[ \gamma _ n(y') = \gamma _ n(y) + \sum \nolimits _{i < n} \gamma _ i(y) \text{d}(z_ i) \]
for some $z_ i$ in $A$. Moreover, for $0 < i < n$ we have
\[ \text{d}(\gamma _ i(y) z_ i) = \text{d}(\gamma _ i(y))z_ i + \gamma _ i(y)\text{d}(z_ i) = \text{d}(y) \gamma _{i - 1}(y) z_ i + \gamma _ i(y)\text{d}(z_ i) \]
and the first term maps to zero in $B$ as $\text{d}(y)$ maps to zero in $B$. Hence $\gamma _ n(x')$ and $\gamma _ n(x)$ map to the same element of $H(B)$. Thus we obtain a well defined map $\gamma _ n : H_{2d}(B) \to H_{2nd}(B)$ for all $d > 0$ and $n > 0$. We omit the verification that this defines a divided power structure on $H(B)$.
$\square$
Lemma 23.6.7. Let $(A, \text{d}, \gamma )$ be as in Definition 23.6.5. Let $R \to R'$ be a ring map. Then $\text{d}$ and $\gamma $ induce similar structures on $A' = A \otimes _ R R'$ such that $(A', \text{d}, \gamma )$ is as in Definition 23.6.5.
Proof.
Observe that $A'_{even} = A_{even} \otimes _ R R'$ and $A'_{even, +} = A_{even, +} \otimes _ R R'$. Hence we are trying to show that the divided powers $\gamma $ extend to $A'_{even}$ (terminology as in Definition 23.4.1). Once we have shown $\gamma $ extends it follows easily that this extension has all the desired properties.
Choose a polynomial $R$-algebra $P$ (on any set of generators) and a surjection of $R$-algebras $P \to R'$. The ring map $A_{even} \to A_{even} \otimes _ R P$ is flat, hence the divided powers $\gamma $ extend to $A_{even} \otimes _ R P$ uniquely by Lemma 23.4.2. Let $J = \mathop{\mathrm{Ker}}(P \to R')$. To show that $\gamma $ extends to $A \otimes _ R R'$ it suffices to show that $I' = \mathop{\mathrm{Ker}}(A_{even, +} \otimes _ R P \to A_{even, +} \otimes _ R R')$ is generated by elements $z$ such that $\gamma _ n(z) \in I'$ for all $n > 0$. This is clear as $I'$ is generated by elements of the form $x \otimes f$ with $x \in A_{even, +}$ and $f \in \mathop{\mathrm{Ker}}(P \to R')$.
$\square$
Lemma 23.6.8. Let $(A, \text{d}, \gamma )$ be as in Definition 23.6.5. Let $d \geq 1$ be an integer. Let $A\langle T \rangle $ be the graded divided power polynomial algebra on $T$ with $\deg (T) = d$ constructed in Example 23.6.2 or 23.6.3. Let $f \in A_{d - 1}$ be an element with $\text{d}(f) = 0$. There exists a unique differential $\text{d}$ on $A\langle T\rangle $ such that $\text{d}(T) = f$ and such that $\text{d}$ is compatible with the divided power structure on $A\langle T \rangle $.
Proof.
This is proved by a direct computation which is omitted.
$\square$
In Lemma 23.12.3 we will compute the cohomology of $A\langle T \rangle $ in some special cases. Here is Tate's construction, as extended by Avramov and Halperin.
Lemma 23.6.9. Let $R \to S$ be a homomorphism of commutative rings. There exists a factorization
\[ R \to A \to S \]
with the following properties:
$(A, \text{d}, \gamma )$ is as in Definition 23.6.5,
$A \to S$ is a quasi-isomorphism (if we endow $S$ with the zero differential),
$A_0 = R[x_ j: j\in J] \to S$ is any surjection of a polynomial ring onto $S$, and
$A$ is a graded divided power polynomial algebra over $R$.
The last condition means that $A$ is constructed out of $A_0$ by successively adjoining a set of variables $T$ in each degree $> 0$ as in Example 23.6.2 or 23.6.3. Moreover, if $R$ is Noetherian and $R\to S$ is of finite type, then $A$ can be taken to have only finitely many generators in each degree.
Proof.
We write out the construction for the case that $R$ is Noetherian and $R\to S$ is of finite type. Without those assumptions, the proof is the same, except that we have to use some set (possibly infinite) of generators in each degree.
Start of the construction: Let $A(0) = R[x_1, \ldots , x_ n]$ be a (usual) polynomial ring and let $A(0) \to S$ be a surjection. As grading we take $A(0)_0 = A(0)$ and $A(0)_ d = 0$ for $d \not= 0$. Thus $\text{d} = 0$ and $\gamma _ n$, $n > 0$, is zero as well.
Choose generators $f_1, \ldots , f_ m \in R[x_1, \ldots , x_ n]$ for the kernel of the given map $A(0) = R[x_1, \ldots , x_ n] \to S$. We apply Example 23.6.2 $m$ times to get
\[ A(1) = A(0)\langle T_1, \ldots , T_ m\rangle \]
with $\deg (T_ i) = 1$ as a graded divided power polynomial algebra. We set $\text{d}(T_ i) = f_ i$. Since $A(1)$ is a divided power polynomial algebra over $A(0)$ and since $\text{d}(f_ i) = 0$ this extends uniquely to a differential on $A(1)$ by Lemma 23.6.8.
Induction hypothesis: Assume we are given factorizations
\[ R \to A(0) \to A(1) \to \ldots \to A(m) \to S \]
where $A(0)$ and $A(1)$ are as above and each $R \to A(m') \to S$ for $2 \leq m' \leq m$ satisfies properties (1) and (4) of the statement of the lemma and (2) replaced by the condition that $H_ i(A(m')) \to H_ i(S)$ is an isomorphism for $m' > i \geq 0$. The base case is $m = 1$.
Induction step: Assume we have $R \to A(m) \to S$ as in the induction hypothesis. Consider the group $H_ m(A(m))$. This is a module over $H_0(A(m)) = S$. In fact, it is a subquotient of $A(m)_ m$ which is a finite type module over $A(m)_0 = R[x_1, \ldots , x_ n]$. Thus we can pick finitely many elements
\[ e_1, \ldots , e_ t \in \mathop{\mathrm{Ker}}(\text{d} : A(m)_ m \to A(m)_{m - 1}) \]
which map to generators of this module. Applying Example 23.6.2 or 23.6.3 $t$ times we get
\[ A(m + 1) = A(m)\langle T_1, \ldots , T_ t\rangle \]
with $\deg (T_ i) = m + 1$ as a graded divided power algebra. We set $\text{d}(T_ i) = e_ i$. Since $A(m+1)$ is a divided power polynomial algebra over $A(m)$ and since $\text{d}(e_ i) = 0$ this extends uniquely to a differential on $A(m + 1)$ compatible with the divided power structure. Since we've added only material in degree $m + 1$ and higher we see that $H_ i(A(m + 1)) = H_ i(A(m))$ for $i < m$. Moreover, it is clear that $H_ m(A(m + 1)) = 0$ by construction.
To finish the proof we observe that we have shown there exists a sequence of maps
\[ R \to A(0) \to A(1) \to \ldots \to A(m) \to A(m + 1) \to \ldots \to S \]
and to finish the proof we set $A = \mathop{\mathrm{colim}}\nolimits A(m)$.
$\square$
Lemma 23.6.10. Let $R \to S$ be a pseudo-coherent ring map (More on Algebra, Definition 15.82.1). Then Lemma 23.6.9 holds, with the resolution $A$ of $S$ having finitely many generators in each degree.
Proof.
This is proved in exactly the same way as Lemma 23.6.9. The only additional twist is that, given $A(m) \to S$ we have to show that $H_ m = H_ m(A(m))$ is a finite $R[x_1, \ldots , x_ m]$-module (so that in the next step we need only add finitely many variables). Consider the complex
\[ \ldots \to A(m)_{m - 1} \to A(m)_ m \to A(m)_{m - 1} \to \ldots \to A(m)_0 \to S \to 0 \]
Since $S$ is a pseudo-coherent $R[x_1, \ldots , x_ n]$-module and since $A(m)_ i$ is a finite free $R[x_1, \ldots , x_ n]$-module we conclude that this is a pseudo-coherent complex, see More on Algebra, Lemma 15.64.9. Since the complex is exact in (homological) degrees $> m$ we conclude that $H_ m$ is a finite $R$-module by More on Algebra, Lemma 15.64.3.
$\square$
Lemma 23.6.11. Let $R$ be a commutative ring. Suppose that $(A, \text{d}, \gamma )$ and $(B, \text{d}, \gamma )$ are as in Definition 23.6.5. Let $\overline{\varphi } : H_0(A) \to H_0(B)$ be an $R$-algebra map. Assume
$A$ is a graded divided power polynomial algebra over $R$.
$H_ k(B) = 0$ for $k > 0$.
Then there exists a map $\varphi : A \to B$ of differential graded $R$-algebras compatible with divided powers that lifts $\overline{\varphi }$.
Proof.
The assumption means that $A$ is obtained from $R$ by successively adjoining some set of polynomial generators in degree zero, exterior generators in positive odd degrees, and divided power generators in positive even degrees. So we have a filtration $R \subset A(0) \subset A(1) \subset \ldots $ of $A$ such that $A(m + 1)$ is obtained from $A(m)$ by adjoining generators of the appropriate type (which we simply call “divided power generators”) in degree $m + 1$. In particular, $A(0) \to H_0(A)$ is a surjection from a (usual) polynomial algebra over $R$ onto $H_0(A)$. Thus we can lift $\overline{\varphi }$ to an $R$-algebra map $\varphi (0) : A(0) \to B_0$.
Write $A(1) = A(0)\langle T_ j:j\in J\rangle $ for some set $J$ of divided power variables $T_ j$ of degree $1$. Let $f_ j \in B_0$ be $f_ j = \varphi (0)(\text{d}(T_ j))$. Observe that $f_ j$ maps to zero in $H_0(B)$ as $\text{d}T_ j$ maps to zero in $H_0(A)$. Thus we can find $b_ j \in B_1$ with $\text{d}(b_ j) = f_ j$. By the universal property of divided power polynomial algebras from Lemma 23.5.1, we find a lift $\varphi (1) : A(1) \to B$ of $\varphi (0)$ mapping $T_ j$ to $f_ j$.
Having constructed $\varphi (m)$ for some $m \geq 1$ we can construct $\varphi (m + 1) : A(m + 1) \to B$ in exactly the same manner. We omit the details.
$\square$
Lemma 23.6.12. Let $R$ be a commutative ring. Let $S$ and $T$ be commutative $R$-algebras. Then there is a canonical structure of a strictly graded commutative $R$-algebra with divided powers on
\[ \operatorname {Tor}_*^ R(S, T). \]
Proof.
Choose a factorization $R \to A \to S$ as above. Since $A \to S$ is a quasi-isomorphism and since $A_ d$ is a free $R$-module, we see that the differential graded algebra $B = A \otimes _ R T$ computes the Tor groups displayed in the lemma. Choose a surjection $R[y_ j:j\in J] \to T$. Then we see that $B$ is a quotient of the differential graded algebra $A[y_ j:j\in J]$ whose homology sits in degree $0$ (it is equal to $S[y_ j:j\in J]$). By Lemma 23.6.7 the differential graded algebras $B$ and $A[y_ j:j\in J]$ have divided power structures compatible with the differentials. Hence we obtain our divided power structure on $H(B)$ by Lemma 23.6.6.
The divided power algebra structure constructed in this way is independent of the choice of $A$. Namely, if $A'$ is a second choice, then Lemma 23.6.11 implies there is a map $A \to A'$ preserving all structure and the augmentations towards $S$. Then the induced map $B = A \otimes _ R T \to A' \otimes _ R T' = B'$ also preserves all structure and is a quasi-isomorphism. The induced isomorphism of Tor algebras is therefore compatible with products and divided powers.
$\square$
Comments (2)
Comment #1664 by Ragnar-Olaf Buchweitz on
Comment #1671 by Johan on