Corrigendum to: "A spectral sequence for stratified spaces and configuration spaces of points"
aa r X i v : . [ m a t h . A T ] J u l CORRIGENDUM TO: “A SPECTRAL SEQUENCE FORSTRATIFIED SPACES AND CONFIGURATION SPACES OFPOINTS”
NIR GADISH AND DAN PETERSEN
The goal of this note is to correct some oversights in the paper [1] by the secondnamed author.In general, Section 4 of [1] is unfortunately rather carelessly written. As anexample, at the top of p. 2548 it is asserted that L • ( S ) ⊠ L • ( T ) is a subcomplex of L • ( S ⊔ T ), when it is in fact a quotient complex. Thus there is a natural map D L • ( S ) ⊠ D L • ( T ) → D L • ( S ⊔ T ) , even though the paper claims that the natural map goes the other way. This erroris fortunately cancelled by an equal and opposite error in the following sentence: amap in this direction is exactly what is needed to make S H −• ( M S , D L • ( S )) atwisted commutative algebra, contrary to what the paper claims.A more subtle issue is the repeated assertion that the chain level construction S R Γ −• ( M S , D L • ( S )) is a twisted commutative algebra in chain complexes; thisclaim is used in Subsection 4.6 to extend the proof of finite generation to integralcoefficients. This is not true as stated since the functor R Γ is not symmetricmonoidal (it is only symmetric monoidal in the sense of ‘higher algebra’, i.e. up tocoherent homotopy); similar remarks apply to the functor D . There is however atrick that one can apply to obtain a strict twisted commutative algebra. Let G( R )be the Godement resolution of the constant sheaf R on M . Instead of applying theconstruction L • to the constant sheaf R on M S , apply it to its resolution G( R ) ⊠ S .Then S Γ c ( M S , L • (G( R ) ⊠ S ))is a twisted cocommutative coalgebra in cochain complexes whose cohomology is S H • c ( F A ( M, S ) , R ) — the point being that the complex L • (G( R ) ⊠ S ) is flasque,so we can apply Γ c instead of the derived functor R Γ c , and the underived functorΓ c is strictly symmetric monoidal. To get a version of Borel–Moore homologyone needs only to apply Hom R ( − , I( R )) to the above complex, where I( R ) is aninjective resolution of the R -module R . This trick of taking tensor powers of afixed resolution to get a strictly commutative object is used more systematically in[2].However, the main subject of this note is an incorrect claim at the bottom of p.2547, that the natural injection(1) P A ( S ) × P A ( T ) ֒ → P A ( S ⊔ T )identifies the left hand side with an order ideal in the poset on the right hand side.Although this holds in many naturally occurring examples, it is not true in general, and we do not see a non-tautological hypothesis that one could add to make thistrue in general.The fact that (1) is not necessarily an order ideal invalidates the proof of Theorem4.15, which is the main result of Section 4 of the paper. Indeed, the proof ofTheorem 4.15 relies on Lemma 4.14, which uses the claim that (1) is an order ideal,specifically in its first line claiming that one needs only keep track of indecomposableelements. This implicitly assumes that the induced multiplication M i + j = n H i − (ˆ0 , β ) ⊗ H j − (ˆ0 , β ′ ) → H n − (ˆ0 , β × β ′ )is surjective, as would follow from (1) being an order ideal: that would alreadyimply that the product (ˆ0 , β ) × (ˆ0 , β ′ ) → (ˆ0 , β × β ′ ) is an isomorphism of posets.However, one can run the proof of Theorem 4.15 under a weaker hypothesisthan (1) being an order ideal, as we will now explain. The key observation is thatthe only part of product structure that comes into the proof of Theorem 4.15 ismultiplication by the trivial element ˆ0 ∈ P A (1), where 1 denotes a one-element set.We will require an additional hypothesis concerning the arrangement of subspaces A : specifically, we must strengthen the hypothesis mentioned in the paragraphbelow Example 4.8. This hypothesis asks that no chosen configuration A i ∈ A is equal to the preimage of some set A ′ i under the coordinate projection maps. Thefollowing stronger assumption turns out to be sufficient for representation stabilityof A -avoiding configuration spaces: Hypothesis 1.
We suppose that A is a finite collection of closed subsets A i ⊆ X S i that do not contain any ‘coordinate axis’: that is, identifying X S i with X S i −{ s } × X ,we must have { ¯ x } × X A i for any ¯ x ∈ X S i −{ s } and any s ∈ S i . Claim 2.
Under Hypothesis 1, the multiplication (2) P A ( S ) × { ˆ0 } ⊂ P A ( S ) × P A (1) → P A ( S ⊔ identifies P A ( S ) with an order ideal of P A ( S ⊔ .Proof. Note that injectivity is obvious. Now suppose B ∈ P A ( S ⊔
1) lies below A × ˆ0. We must show that B is already in the image of the multiplication by ˆ0,that is B = B ′ × ˆ0. Recall that the ‘bottom stratum’ ˆ0 ∈ P A (1) represents theentire space X = X . Thus the hypothesis is that B ⊇ A × X as subsets of X S × X .Next, recall that the poset P A ( T ) is formed by intersecting subspaces of the form (cid:0) π TS i (cid:1) − ( A i ) ∼ = A i × X T \ j ( S i ) inside X S i × X T \ j ( S i ) ∼ = X T for all injections j : S i ֒ → T . In particular, B is the intersection of a collection ofsuch subspaces where T = S ⊔ (cid:0) π TS i (cid:1) − ( A i ) ⊇ B mustcome from injections j : S i ֒ → S ⊔ j ′ : S i ֒ → S .Indeed, it would then follow that B = B ′ × X where B ′ is the intersection of those (cid:0) π SS i (cid:1) − ( A i ) ⊆ X S . EFERENCES 3
To see that the said factorization holds, for every such inclusion j : S i ֒ → S ⊔ A × X ⊆ B ⊆ (cid:0) π TS i (cid:1) − ( A i )If it were the case that 1 ∈ j ( S i ) then A i would already contain a coordinate axis { ¯ a } × X , contradicting Hypothesis 1. It follows that j factors through S . (cid:3) Let us call an element β ∈ P A ( S ) decomposable if it of the form β ′ × ˆ0 for some β ′ ∈ P A ( S ′ ), where S ′ ⊂ S is a proper subset. If no such decompositions exist, call β indecomposable . Note that this is not the same definition of indecomposabilitythat is introduced just before Lemma 4.13 of [1].With this definition of decomposability, the proof of Lemma 4.13 applies verba-tim (in fact, the present definition of decomposability relates to that proof morenaturally). Furthermore using the above Claim 2 to guarantee the order ideal as-sumption, the original proof of Theorem 4.15 using Lemma 4.14 is now valid asstated. To summarize, Theorem 4.15 remains valid if we assume in addition thatthe arrangement of subspaces A satisfies Hypothesis 1. References [1] Dan Petersen. “A spectral sequence for stratified spaces and configurationspaces of points”. In:
Geom. Topol. issn : 1465-3060.[2] Dan Petersen. “Cohomology of generalized configuration spaces”. In:
Compos.Math. issnissn