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Infinite symmetric products of rational algebras and spaces

We show that the infinite symmetric product of a connected graded-commutative algebra over the rationals is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular, the infinite symmetric product of a connected commutative (in the usual sense) graded algebra over the rationals is a polynomial algebra. Applied to topology, we obtain a quick proof of the Dold-Thom theorem in rational homotopy theory for connected spaces of finite type. We also show that finite symmetric products of certain simple free graded commutative algebras are free; this allows us to determine minimal Sullivan models for finite symmetric products of complex projective spaces.


Introduction
The construction of symmetric products naturally appears in both algebra and topology. In algebra, given a commutative ring R, its n-fold symmetric product is the invariant subring (R ⊗n ) Σn of its n-fold tensor power under the natural action of the n th symmetric group Σ n by permuting the factors. The infinite symmetric product is defined as the inverse limit of these objects, in an appropriate sense. The fundamental theorem of symmetric polynomials states that the n-fold symmetric product (including the case of taking n to infinity) of the single-variable polynomial ring Z[x] is a free polynomial ring generated by the elementary symmetric functions. In contrast, the n-fold symmetric product of the polynomial ring Z[x 1 , . . . , x m ] in more than one variable is not smooth after tensoring with C, let alone free. Geometrically speaking, (R ⊗n ) Σn is the ring of regular functions on the affine variety Spec(R) ×n /Σ n obtained by quotienting the n-fold product of Spec(R) by the natural permutation action of Σ n . Consequently the n-fold symmetric product of an m-dimensional smooth complex algebraic variety is smooth if and only if at least one of m, n is one. Nevertheless, should one tensor Z[x 1 , . . . , x m ] with Q and let n pass to infinity, then surprisingly the singularities are resolved [V05], [Da99]: the infinite symmetric product of Q[x 1 , . . . , x m ] is a free polynomial algebra.
On the topological side, for any pointed connected topological space X, one can form its n-fold symmetric product X ×n /Σ n , as well as its infinite symmetric product which is the direct limit of these spaces. Through the latter, Dold and Thom revealed a stunning relation between homology and homotopy in the 1950's [DT58]: the homotopy groups of the infinite symmetric product of a connected topological space are naturally isomorphic to the reduced homology groups of the original space.
In the two decades to follow, a bridge between algebra and topology was built by Quillen and Sullivan giving us that the rational homotopy theoretic information of a space is encoded in a corresponding differential graded (Lie) algebra over the rationals.
It is our motivation to further understand the relation between the algebraic and topological sides of the (infinite) symmetric product construction through the lens of rational homotopy theory; see [FT10] for work on the rational homotopy types of finite symmetric products, which led the authors to the present note. We give a general structural theorem for the infinite symmetric product of a connected graded-commutative algebra over the rationals (Section 2); in particular, we show this algebra is free. In contrast, finite symmetric products are generally not free. However, we show that the finite symmetric products of a free graded-commutative algebra on two generators, one in even degree and one in odd degree, are free. We then give a quick conceptual proof of the Dold-Thom theorem in rational homotopy theory for spaces of finite type (Section 3). Further, we compute minimal Sullivan models (in particular obtaining the rational homotopy groups) for finite symmetric products of complex projective spaces.

Symmetric products of rational algebras
Henceforth, unless otherwise noted, by an algebra we will always mean a gradedcommutative algebra over Q concentrated in non-negative degrees, denoted by A = ⊕ d≥0 A d . If a, b are two homogeneous elements, then ab = (−1) deg a·deg b ba. We will denote the subspace of positive degree (resp. degree ≤ n) elements by A + (resp. A ≤n ). We say A is connected if A 0 is generated by 1 ∈ A (i.e. A 0 ∼ = Q). A connected algebra has a canonical augmentation ε : A → Q by projection onto A 0 . We say A is of finite type if β d := dim A d is finite for all d.
For V a graded vector space, by Λ(V ) we denote the free algebra on V . For notational convenience, we will sometimes replace V in the notation Λ(V ) by a homogeneous basis for V . For instance, Λ(x) denotes the free algebra on a one-dimensional graded vector space, and it is a polynomial algebra if deg x is even, and an exterior algebra if deg x is odd.
Definition 2.1. Let A be a connected algebra. We define its n-fold symmetric product SP n (A) to be (A ⊗n ) Σn , i.e. the subalgebra of A ⊗n consisting of elements invariant under the action of the n th symmetric group Σ n by permuting the factors with the usual Koszul signs (for example, the permutation (12) takes a⊗b to (−1) deg a·deg b b⊗a). The projection A ⊗n+1 → A ⊗n , a 1 ⊗ · · · ⊗ a n ⊗ a n+1 → ε(a n+1 )a 1 ⊗ · · · ⊗ a n induces a morphism π n+1 n : SP n+1 (A) → SP n (A). We define the infinite symmetric product SP(A) of A to be the inverse limit of the system {π n+1 n } n∈N , that is SP(A) := lim ←n SP n (A). For each m > n, we denote π m n := π n+1 n • · · · • π m m−1 : SP m (A) → SP n (A) and π n := lim ←m π m n : SP(A) → SP n (A). It is clear that π n = π m n • π m . We emphasize that this inverse limit is taken in the category of graded-commutative algebras. As such, it can be constructed as follows: first we take the degree-wise inverse limits lim ←n SP n (A) d , and then we form the direct sum d lim ←n SP n (A) d with its inherited algebra structure.
Remark 2.2. Consider the truncated polynomial algebras Q[x]/(x n ), which form an inverse system by quotienting. In the category of commutative algebras, the inverse limit of this system is the formal power series algebra Q [[x]]. Likewise, if we set deg x = 0, the inverse limit of this system in the category of graded-commutative algebras will be Q [[x]]. However, if we set deg x = 2, thus placing our system in the category of connected graded-commutative algebras, the inverse limit will be the polynomial algebra Q[x]. Now let A be a fixed connected algebra. We consider the algebra map It is clear from construction that φ n = π n+1 n • φ n+1 . We set Proof. For simplicity of computation we will use the isomorphism of algebras Λ n A ∼ = SP n (A) given by [FT10,§2] a 1 ∧ · · · ∧ a n → σ∈Σn ±a σ(1) ⊗ · · · ⊗ a σ(n) , where the multiplication on Λ n A is determined by (a 1 ∧ · · · ∧ a n ) * (b 1 ∧ · · · ∧ b n ) = σ∈Σn ±(a 1 b σ(1) ) ∧ · · · ∧ (a n b σ(n) ).
Proof. Assume first that A is of finite type. Then by Lemma 2.3 it suffices to show SP n (A) ≤n and Λ(A + ) ≤n have the same dimension. To do so, we examine the Poincaré series of SP n (A) and Λ(A + ). Recall the Poincaré series of a graded algebra of finite type is a formal power series in a formal variable z whose coefficient of z d is the dimension β d of its degree d subspace.
By [M62] the Poincaré series for SP n (A) is the coefficient of t n in Then the Poincaré series for SP n (A) is Notice that in each factor of G, t is always multiplied with a positive power of z, On the other hand, we claim the Poincaré series of Λ(A + ) is Indeed, since the free algebra on a direct sum of vector spaces is the tensor product of the free algebras on each direct summand, and the Poincaré series of a tensor product is the product of the Poincaré series of each factor, we are reduced to considering the Poincaré series of the free algebra on a one-dimensional graded vector space concentrated, say, in degree d. If d is even, then this free algebra is a polynomial algebra whose Poincaré series is (1 − z d ) −1 . If d is odd, then it is an exterior algebra whose Poincaré series is 1 + z d .
Hence the dimensions of SP n (A) ≤n and Λ(A + ) ≤n are the same, namely n i=0 G i (z) reduced modulo z n+1 and then evaluated at z = 1. This completes the proof for finite type algebras. The general case follows from the lemma below and from the fact that taking direct limits is exact.
Remark 2.5. We can define the n-fold symmetric product of the augmentation ideal A + by equipping the permutation invariants of the n-fold tensor power of the underlying graded vector space of A + with the induced multiplication. Since φ n−1 factors over π n n−1 , it follows from Lemma 2.3 that π n n−1 : SP n (A) → SP n−1 (A) is surjective. The kernel of this map is precisely SP n (A + ), whose Poincaré series is G n (z). As graded vector spaces, Lemma 2.6. Let A be a connected algebra and I = {B | B ⊆ A connected of finite type} be the direct system formed by all connected finite type subalgebras of A and the corresponding inclusions. We have the following induced isomorphisms: Proof.
(1) The inclusion B + ֒→ A + clearly induces an injection Λ(B + ) ֒→ Λ(A + ). Since taking direct limits is exact, we have lim I Λ(B + ) ֒→ Λ(A + ). To prove this map is also surjective, it suffices to show that A + is in the image. For each a ∈ A + , a is contained in some finite type subalgebra B (for instance, the subalgebra generated by a), hence a is in the image of Λ(B + ) → Λ(A + ) and therefore in the image of (2) Similarly, the inclusion B ֒→ A induces an injection B ⊗n ֒→ A ⊗n . Then from the commutative diagram All of these a j i 's are contained in some finite type subalgebra B (for instance, the subalgebra generated by these a j i 's) so the symmetrization of α is in fact contained in SP n (B). This proves lim I SP n (B) → SP n (A) is onto, and therefore an isomorphism. By Lemma 2.4, the maps φ m and φ n are isomorphisms, and so π m n is also an isomorphism. It follows that π n = lim ←m π m n is an isomorphism. Then since π n and φ n are isomorphisms, we see that φ is an isomorphism.
(1) As a consequence, we have that every (connected) free gradedcommutative algebra Λ(V ) is the infinite symmetric product of some algebra. Indeed, consider Q ⊕ V with Q in degree zero, with product given by (c + v) · (d + w) = cd + (cw + dv) for c, d ∈ Q, v, w ∈ V (i.e. all products in V are trivial) and units given by c + 0. Then SP(Q ⊕ V ) ∼ = Λ(V ).
(2) Notice that Theorem 2.7 holds over any field of characteristic zero, e.g. it applies to the ℓ-adic cohomology of a smooth projective variety over a finite field of characteristic different from ℓ.
Every commutative graded algebra (i.e., a commutative algebra in the ordinary sense whose multiplication respects the grading) can be viewed as a graded-commutative algebra by doubling the grading. This way, the category of commutative graded algebras over Q embeds into the category of graded-commutative algebras over Q as the full subcategory consisting of algebras concentrated in even degrees. Therefore, Theorem 2.7 applies to connected commutative graded algebras as well:  We apply Corollary 2.9 to the theory of multisymmetric functions. Recall that the theory of symmetric and multisymmetric functions concerns the symmetric products of the commutative graded rings Z[x] and Z[x 1 , . . . , x m ] (m > 1) where generators are put in degree one [Da99], [V05]. As mentioned before, the fundamental theorem of symmetric functions asserts that SP n (Z[x]) is a (free) polynomial ring. Meanwhile SP n (Z[x 1 , . . . , x m ]) is not free. However, Corollary 2.9 implies the following generalization of the fundamental theorem of symmetric functions over the rationals: We end this section with another generalization of the fundamental theorem of symmetric functions in the category of graded-commutative algebras over the rationals. Let x i denote 1 ⊗ · · · ⊗ x ⊗ · · · 1, where x is in the i th factor, and define y i in a similar fashion. Then recall by construction [x k ] = x k ⊗ 1 ⊗ · · · ⊗ 1 + · · · + 1 ⊗ · · · ⊗ 1 ⊗ x k = (x ⊗ 1 ⊗ · · · ⊗ 1) k + · · · + (1 ⊗ · · · ⊗ 1 ⊗ x) k = x k 1 + · · · + x k n .
Similarly [x k−1 y] = x k−1 1 y 1 + · · · + x k−1 n y n . Let be the elementary symmetric polynomials in x 1 , . . . , x n . It is well-known (from Newton's identities) that e j (x) is contained in the subalgebra generated by Therefore for 1 ≤ i ≤ n we have

Since the Poincaré series of
From [M79, pp.19 Example 5] for variables u, v, q we have So, by taking u = 1, v = −z 2s−1 , q = z 2r , we see that the Poincaré series of SP n (A) is as desired.

Symmetric products of spaces and the rational Dold-Thom theorem
Given a (pointed) connected cell complex X, one can form the n-fold symmetric product SP n (X) by quotienting the n-fold product X ×n by the natural action of the symmetric group Σ n on n elements. The space SP n (X) naturally includes into SP n+1 (X), and in the limit of this directed system over n one obtains the infinite symmetric product SP(X).
To study the rational homotopy type of a space X, one can associate to it its commutative differential graded algebra (cdga) of piecewise-linear forms A PL (X). By a model for a space X we mean any cdga which is quasi-isomorphic to A PL (X). The rational cohomology of X is that of any of its models, and if X is furthermore a nilpotent space of finite type, i.e. H n (X; Q) is finite dimensional for all n, then its rational (co)homotopy groups are obtained from any free model (i.e. one whose underlying graded-commutative algebra is free) by considering the cohomology of the differential restricted to the subspace of indecomposable elements. We refer the reader to [BG76], [S77].
We remark that SP(X) has a natural H-space structure, and as such its fundamental group is abelian and acts trivially on the higher homotopy groups; in particular, SP(X) is a nilpotent space. If X is of finite type, then so is SP(X), as will be evident from the proof of the following theorem combined with the observation that each SP n (X) is of finite type as well (e.g. we can apply Macdonald's formula for the Poincaré polynomials of finite symmetric products [M62]). This result, under the additional assumptions that X is simply connected and that A is a free algebra, has already been obtained in [ScSt99, Proposition 3]. We remark that the proof therein makes use of the Dold-Thom theorem. Proof. A model for the symmetric product SP n (X) is given by SP n (A) = (A ⊗n ) Σn equipped with the induced differential [FT10,§2]. Note that the map Λ(A + ) φn − − → SP n (A) considered in Section 2 commutes with the differentials. Namely, for a ∈ A + we have Da = da ∈ A + and hence φ n (Da) = da ⊗ 1 ⊗ · · · ⊗ 1 + · · · + 1 ⊗ · · · ⊗ 1 ⊗ da = dφ(a); since D on Λ(A + ) is obtained by extending d as a derivation, the claim follows.
Further, note that the composition of φ n and the map SP n (A) → SP n−1 (A) induced by the natural inclusion SP n−1 (X) ֒→ SP n (X) is the map φ n−1 (see e.g. [FT10, Remark 2.7]). We thus have the commutative diagram is an isomorphism in degrees ≤ n − 1. (In particular, the inclusion SP n−1 (X) ֒→ SP n (X) is an isomorphism on rational cohomology in degrees ≤ n − 2.) Hence from our description of (the algebra underlying) the inverse limit in Section 2, we conclude that φ is an isomorphism of cdga's. Now let us confirm that the inverse limit lim induced by the inclusions of symmetric products, where the vertical arrows are quasiisomorphisms, we obtain a map of inverse limits lim ←n (SP n (A), d) → A PL (SP(X)). Since the inclusion SP n−1 (X) ֒→ SP n (X) is an isomorphism on rational cohomology in degrees ≤ n − 2 as we saw, it follows that this map of inverse limits is a quasi-isomorphism. We conclude that lim ←n (SP n (A), d) maps quasi-isomorphically to A PL (SP(X)).
Proof. The rational (co)homotopy groups of SP(X) are given by the linearized cohomology of a free model for SP(X). Concretely, by Theorem 3.1 we take our model (Λ(A + ), D) for SP(X), where (A, d) is a model for X, and consider the cohomology of the linear part of the differential D. By construction this is the same as the cohomology of (A + , d), i.e. the reduced cohomology of X.
One can also see with this line of reasoning that SP(X) has the rational homotopy type of i K(H i (X; Q), i). Indeed, consider the quotient map of complexes Λ(A + ) → Λ(A + )/Λ(A + ) + · Λ(A + ) + ∼ = A + to the indecomposables. This is the dual of the rational Hurewicz map, and is evidently surjective. It is well known that a space has the homotopy type of a product of Eilenberg-MacLane spaces if and only if the Hurewicz map is a split injection; the claim now follows by rationalizing and dualizing.
Remark 3.3. One can also use the rational Dold-Thom theorem to prove the algebraic Theorem 2.7 in the finite type case; see also [ScSt99,Proposition 3]. Indeed, take a finite type connected algebra A. Equipping it with the trivial differential, we can realize this cdga as the cohomology of a (formal) rational space X. From Dold-Thom, we know that SP(X) ≃ i K( H i (X; Q), i), so the (rational) cohomology of SP(X) is a free algebra on the reduced cohomology of X, which is A + . We have SP(A) = SP(H * (X)) = lim ←n SP n (H * (X)), and using the finite type assumption and that the cohomology of SP n (X) stablizes, this is isomorphic to lim ←n H * (SP n (X)) ∼ = H * (lim →n SP n (X)) = H * (SP(X)). We conclude that SP(A) ∼ = Λ(A + ).
Example 3.4. Take the model (Λ(x, y), d) for the two-sphere S 2 , where deg x = 2, deg y = 3, and dx = 0, dy = x 2 . By Theorem 3.1, a model for SP(S 2 ) is given by where k ≥ 0, and the differential D is given by From here we see that the cohomology of the linear part of D, restricted to the indecomposables [x k ], [x k y], is spanned by [x]. That is, π 2 SP(S 2 ) ⊗ Q ∼ = Q and all other rational homotopy groups are trivial. This is as expected, since SP(S 2 ) is homeomorphic to CP ∞ .
Alternatively, since S 2 is formal, we can take Q[x]/(x 2 ), d = 0 with deg x = 2 as a model for S 2 , and immediately reach the above conclusion; see Example 2.10. with the induced differential. In either case, the differential d takes generators of the form [x k ] to zero and generators of the form [x k y] to elements in the subalgebra generated by the [x k ]'s, which must be sums of decomposables for degree reasons. So the linearized differential vanishes on indecomposables, and we obtain π * (SP n (CP m )) ⊗ Q ∼ =      Q if * = 2, . . . , 2 min{m, n}; Q if * = 2 max{m, n} + 1, . . . , 2n + 2m − 1; 0 otherwise.
We sketch how to obtain the rational cohomology of SP n (CP m ) from our model. Let This is as expected, since SP n (CP 1 ) is homeomorphic to CP n . Algebraically, this corresponds to SP n Q[x]/(x 2 ) ∼ = Q[x]/(x n+1 ). which is isomorphic to (Λ(x, z), d) with dx = 0 and dz = x n+1 .