E ∞ -cells and general linear groups of infinite fields
aa r X i v : . [ m a t h . A T ] J un E ∞ -CELLS AND GENERAL LINEAR GROUPSOF INFINITE FIELDS SØREN GALATIUS, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS
Abstract.
We study the general linear groups of infinite fields (or more gener-ally connected semi-local rings with infinite residue fields) from the perspectiveof E ∞ -algebras. We prove that there is a vanishing line of slope 2 for their E ∞ -homology, and analyse the groups on this line by determining all invariantbilinear forms on Steinberg modules. We deduce from this a number of conse-quences regarding the unstable homology of general linear groups, in particularanswering questions of Rognes, Suslin, Mirzaii, and others.
1. Introduction 22. The Steinberg module and its tensor square 62.1. Pairings on Steinberg modules 62.2. Anisotropy and indecomposability 93. Overview of E k -cells and E k -homology 113.1. Glossary and some notation from [GKRW18a] 144. General linear groups as an E ∞ -algebra 145. Higher-dimensional buildings and their split analogues 165.1. The k -dimensional building 165.2. Split buildings 185.3. The Nesterenko–Suslin property 205.4. Relationship with E k -homology 236. General linear groups over a field 256.1. E -homology 256.2. E -homology 266.3. Products 286.4. E ∞ -homology 317. Local and semi-local rings 337.1. Definitions and statements 347.2. A general position lemma and the contractibility of E ( M ) 367.3. A homotopy colimit decomposition of D k ( M ) 397.4. Proofs of connectivity and resolution 417.5. Coinvariants of the E -Steinberg module 458. Rognes’ conjecture 519. Applications to homological stability 549.1. E ∞ -homology in low degrees 549.2. The Nesterenko–Suslin theorem 599.3. One degree higher 639.4. Rings with vanishing rational K n Date : June 3, 2020.2010
Mathematics Subject Classification.
SØREN GALATIUS, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS Introduction
The category P A whose objects are finitely-generated projective modules over aring A and whose morphisms are A -linear isomorphisms, has a symmetric monoidalstructure given by direct sum. Hence its classifying space B P A has the structure ofan E ∞ -space. This paper arose from our attempts at understanding these structuresfrom the point of view of cellular E ∞ -algebras in the case that A is an infinite field,or more generally a connected semi-local ring with infinite residue fields. Manyof the results we obtained can be stated without reference to E ∞ -algebras, and inSections 1.1–1.3 we shall do so. In Section 1.4 we will then explain the methodwhich underlies these (otherwise seemingly unrelated) results.1.1. An invariant pairing on the Steinberg module.
When A = F is a fieldand V is a non-zero finite dimensional F -vector space, the Solomon–Tits theoremasserts that the nerve of the partially ordered set of proper non-zero linear subspacesof V has the homotopy type of a wedge of spheres of dimension dim( V ) −
2. Thisnerve is the
Tits building T ( V ), and the Steinberg module is the reduced homologygroup St( V ) := ˜ H dim( V ) − ( T ( V ); Z ) . As the top-dimensional homology of a simplicial complex, it is a subgroup of theaugmented simplicial chains e C dim( V ) − ( T ( V )), which is free abelian on the set ofcomplete flags in V . We therefore obtain a pairing on St( V ) by restricting thepairing on simplicial chains in which the set of complete flags form an orthonormalbasis. The resulting pairing h− , −i : St( V ) ⊗ St( V ) −→ Z is symmetric, bilinear, GL( V )-invariant, and positive definite. Our first main resultis that it is universal among bilinear GL( V )-invariant pairings. Theorem A.
The induced map on coinvariants (St( V ) ⊗ St( V )) GL( V ) → Z is anisomorphism. There are natural homomorphisms St( V ) ⊗ St( W ) → St( V ⊕ W ), giving n (St( F n ) ⊗ St( F n )) GL n ( F ) ∼ = Z the structure of a graded-commutative ring; in Sec-tion 6.3 we show that the Z ’s in each degree assemble to a divided power algebra.We prove Theorem A in Section 2.1, and in Section 2.2 we discuss several conse-quences, including the following. Theorem B.
The k [GL( V )] -module k ⊗ Z St( V ) is indecomposable, for any con-nected commutative ring k . Recall a module over a ring is irreducible if it contains no non-zero proper sub-modules, and indecomposable if it contains no non-zero proper summands. Wedo not know whether the Steinberg module is irreducible, but when k is a field ofcharacteristic zero we show k ⊗ St( V ) has no finite-dimensional sub-representations.1.2. Rognes’ connectivity conjecture.
Let us next describe some results relat-ing to conjectures of Rognes about his spectrum-level rank filtration of the algebraic K -theory of a ring A [Rog92]. Let us for notational reasons assume A is connectedand has the property that all projective modules are free. Rognes’ rank filtrationis an ascending exhaustive filtration ∗ ⊂ F K ( A ) ⊂ F K ( A ) ⊂ F K ( A ) ⊂ · · · ⊂ K ( A )by subspectra, and he identified the filtration quotients as F n K ( A ) F n − K ( A ) ≃ D ( A n ) // GL n ( A ) , ∞ -CELLS AND GENERAL LINEAR GROUPS OF INFINITE FIELDS 3 the homotopy orbits of GL n ( A ) acting on a certain spectrum D ( A n ) called the stable building . Based on calculations for n ≤
3, Rognes has conjectured that, for A a Euclidean domain or local ring, D ( A n ) is (2 n − n ( A )-coinvariants of H n − ( D ( A n )) aretorsion for n ≥ n ( A ), which we establish in (i) of the following result, at least in the local ringcase. In Section 8 we prove: Theorem C. (i) If A is a connected semi-local ring with all residue fields infinite then thehomotopy orbit spectrum D ( A n ) // GL n ( A ) is (2 n − -connected.(ii) If in addition A = F is an infinite field, then H n − ( D ( F n ) // GL n ( F )) is givenby Tor Γ Z [ x ]1 ( Z , Z ) n ∼ = Z { x } if n = 1 , Z /p { γ p k ( x ) } if n = p k with p prime , otherwise , and in particular is finite for n ≥ . Part (ii) would follow more generally for any connected semi-local ring A withinfinite residue fields, provided a suitable analogue of Theorem A held for such rings,formulated as Conjecture 7.7. In Section 7.5 we prove that conjecture for projectivemodules of rank ≤
3, which is sufficient for some of our intended applications.1.3.
Homology of general linear groups.
Let us next state some results aboutthe unstable homology of general linear groups, which we will describe in moredetail and prove in Sections 9 and 10. Firstly, in Section 9.2 we will show how touse our methods to recover a result of Suslin [Sus84a, Theorem 3.4], Nesterenkoand Suslin [NS89, Theorems 2.7, 3.25] and Guin [Gui89, Th´eor`eme 2], assertingthat for a connected semi-local ring A with infinite residue fields we have H ∗ (GL n ( A ) , GL n − ( A ); Z ) = 0 for ∗ < n, as well as an isomorphism H n (GL n ( A ) , GL n − ( A ); Z ) ∼ = K Mn ( A )between relative group homology and Milnor K -theory, which we recall is the gradedring generated by K M ( A ) = A × subject to the relations a · b = 0 ∈ K M ( A ) when a, b ∈ A × satisfy a + b = 1.We will then extend Nesterenko and Suslin’s theorem in the following way. InSection 9.3 we explain how M n ≥ H n +1 (GL n ( A ) , GL n − ( A ); Q )may be made into a module over K M ∗ ( A ) Q := Q ⊗ Z K M ∗ ( A ), and we then show howto generate this module efficiently. Our answer is expressed in terms of the thirdHarrison homology (see [Har62]) of the graded-commutative ring K M ∗ ( A ) Q . Theorem D.
For any connected semi-local ring A with all residue fields infinite,there is a natural homomorphism of graded Q -vector spaces M n ≥ Harr ( K M ∗ ( A ) Q ) n −→ Q ⊗ K M ∗ ( A ) Q M n ≥ H n,n +1 ( R Q /σ ) , which is an isomorphism for n ≥ . If A is an infinite field then this map is anisomorphism for n ≥ . SØREN GALATIUS, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS
For readers unfamiliar with Harrison homology let us mention that it is asummand of Hochschild homology [Bar68, Theorem 1.1]. Expressing the lat-ter as Tor-groups, our result implies the degree n part of the graded vectorspace Tor K M ∗ ( A ) Q ( Q , Q ) surjects onto the K M ∗ ( A ) Q -module indecomposables of L n ≥ H n +1 (GL n ( A ) , GL n − ( A ); Q ).Under various concrete assumptions on the ring A our methods can give strongerresults about the relative homology H ∗ (GL n ( A ) , GL n − ( A )), going further beyondthe Nesterenko–Suslin line ∗ = n . The following are three examples of such results: Theorem E. (i) If A is a connected semi-local ring with all residue fields infinite and such that K ( A ) Q = 0 then H d (GL n ( A ) , GL n − ( A ); Q ) = 0 in degrees d < n − .(ii) If p is a prime number and A is a connected semi-local ring with all residuefields infinite and such that A × ⊗ Z /p = 0 then H d (GL n ( A ) , GL n − ( A ); Z /p ) = 0 in degrees d < n .(iii) If F is an algebraically closed field then H d (GL n ( F ) , GL n − ( F ); Z /p ) = 0 in degrees d < n , for all primes p . In particular, part (iii) of this theorem implies that for algebraically closed fields H n +1 (GL n ( F ) , GL n − ( F ); Z /p ) = 0 for all n ≥ p . This relates tothe “higher pre-Bloch groups” suggested by Loday [Lod87, Section 4.4] and denoted p n ( F ) by Mirzaii [Mir07]. These groups are expected to be related to H n +1 (GL n ( F ))in a way that specialises to the relationship between H (GL ( F )) and the pre-Blochgroup p ( F ) = p ( F ). As we explain in Section 9.5, our results imply p n ( F ) ⊗ Z /p = ( Z /p n odd,0 n even,which resolves [Mir07, Conjecture 3.5]. We also resolve a closely related conjecturemade earlier by [Yag00, Conjecture 0.2].Finally, in Section 9.6 we prove the following result, which implies a new case ofSuslin’s “injectivity conjecture.” Theorem F. If F is an infinite field and k is a field in which ( n − is invertiblethen the stabilisation map H n (GL n − ( F ); k ) −→ H n (GL n ( F ); k ) is injective. Suslin asked more generally whether the stabilisation map H i (GL n − ( F ); Q ) → H i (GL n ( F ); Q ) might be injective for all infinite fields and all i [Sah89, Problem4.13], [BY94, Remark 7.7], [DJ02, Conjecture 2], [Mir08, Conjecture 1]. Our argu-ment completes an approach of Mirzaii [Mir08].In a different direction, in Section 10.2 apply our methods to analyse the homol-ogy of the Steinberg module, in particular showing it vanishes in low degrees: Theorem G. If A is a connected semi-local ring with infinite residue fields, then H d (GL n ( A ); St( A n )) = 0 for d < ( n − . Analogous results for fields have been obtained by Ash–Putman–Sam [APS18,Theorem 1.1] and Miller–Nagpal–Patzt [MNP20, Theorem 7.1]. ∞ -CELLS AND GENERAL LINEAR GROUPS OF INFINITE FIELDS 5 E ∞ -cells. Let P A denote the groupoid of finitely generated projective A -modules and A -linear isomorphisms between them. Assuming A is connected, thereis a well-defined functor r : P A → N sending a projective module to its rank. As aconsequence the nerve N P A comes with a map to N , and we let R ( n ) denote theinverse image of { n } . Its homotopy type is given by R ( n ) ≃ a [ M ] r ( M )= n B GL( M ) . Direct sum of R -modules induce products R ( n ) × R ( m ) → R ( n + m ) which areassociative and commutative “up to homotopy and higher homotopies,” making thedisjoint union R = ∞ a n =1 R ( n )into a (non-unital) E ∞ -space. For our purposes it is slightly better to work inthe category sMod NZ of functors from N , regarded as a category with only identitymorphisms, to simplicial Z -modules. The relevant object for us is n R Z ( n ) = Z [ N P A ( n )] , the free simplicial Z -module on the nerve of P A ( n ), an E ∞ -algebra in sMod NZ .There is a homology theory for E ∞ -algebras, an E ∞ -version of Andr´e–Quillenhomology for simplicial commutative rings. We shall refer to [GKRW18a] formore details. It associates bigraded abelian groups H E ∞ n,d ( R Z ) to the object R Z ∈ Alg E ∞ ( sMod NZ ) above, where the n -grading comes from the rank function r : P A → N and the d -grading is the homological grading. We have used a similardevice to study an E -algebra constructed from mapping class groups of surfaces in[GKRW19] and to study general linear groups of finite fields in [GKRW18b]. Thepresent paper started from our attempts to understand general linear groups froma similar point of view.The relationship to Rognes’ conjecture comes from an isomorphism H E ∞ n,d ( R Z ) ∼ = H d ( D ( A n ) // GL n ( A )) , valid assuming all finitely-generated projective A -modules are free. We discoveredTheorem C from attempts to estimate H E ∞ n,d ( R Z ).The relationship to the Steinberg module and its tensor square comes from iso-morphisms H E n,d ( R Z ) ∼ = H d − ( n − (GL n ( A ); St( A n )) H E n,d ( R Z ) ∼ = H d − n − (GL n ( A ); St( A n ) ⊗ St( A n )) , valid when A is an infinite field.Finally, the homology groups H ∗ (GL n ( A )) are the homotopy groups of the sim-plicial abelian group R Z ( n ), and the results listed in Section 1.3 are derived asconsequences of what we learned about H E ∞ ∗ , ∗ ( R Z ). Acknowledgements.
We thank John Rognes for comments on an earlier version.AK and SG were supported by the European Research Council (ERC) underthe European Union’s Horizon 2020 research and innovation programme (grantagreement No. 682922). SG was also supported by the EliteForsk prize and by theDanish National Research Foundation through the Copenhagen Centre for Geom-etry and Topology (DNRF151). AK was also supported by the Danish NationalResearch Foundation through the Centre for Symmetry and Deformation (DNRF92)and by NSF grant DMS-1803766. ORW was partially supported by EPSRC grantEP/M027783/1, by the ERC under the European Union’s Horizon 2020 research
SØREN GALATIUS, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS and innovation programme (grant agreement No. 756444), and by a Philip Lever-hulme Prize from the Leverhulme Trust.2.
The Steinberg module and its tensor square
Before proving Theorem A, let us recall the definition of the Steinberg moduleof GL( V ) for a finite-dimensional non-zero vector space V over a field F . Definition 2.1.
The
Tits building T ( V ) is the set of proper nontrivial subspaces W ⊂ V , partially ordered by inclusion. We let T • ( V ) = N • ( T ( V ) , ⊂ ) and write T ( V ) = | T • ( V ) | for its thin geometric realisation.The following is proven in [Sol69] for finite fields, and in [Gar73, Theorem 2.2]for all fields. See [KS14, Corollary 1] for a proof using techniques similar to thosein Section 7. Theorem 2.2 (Solomon–Tits) . The space T ( V ) has the homotopy type of a wedgeof (dim( V ) − -spheres. The Steinberg module is the Z [GL( V )] -module St( V ) := e H dim( V ) − ( T ( V ); Z ) , which restricts to a free module of rank 1 over the subring Z [ U ] , where U ⊂ GL( V ) denotes the subgroup of upper unitriangular matrices with respect to a basis of V . (cid:3) Example . As a special case, T ( F ) = ∅ and St( F ) = Z with trivial action ofGL( F ). Similarly, T ( F ) is the set of lines in F and St( F ) is the kernel of theaugmentation Z { T ( F ) } ։ Z with GL( F ) acting by permutation on the set oflines.Let us describe some particular classes in the Steinberg module. For an orderedset L = ( L , L , . . . , L n ) of 1-dimensional subspaces giving a direct sum decom-position of V , there is a map f L : sd( ∂ ∆ n − ) → T ( V ) given by sending a flag ofnonempty proper subsets of { , , . . . , n } to the corresponding flag of nonzero propersubspaces of V . The image a L of the fundamental class of sd( ∂ ∆ n − ) under ( f L ) ∗ is called the apartment associated to L , and is an element of St( V ). It is well-known(e.g. [BS73, Theorem 8.5.2]) that the apartments span St( V ).2.1. Pairings on Steinberg modules.
Let us choose bases and identify V ∼ = F n .In the introduction we have described a positive definite symmetric bilinear form h− , −i : St( F n ) ⊗ St( F n ) −→ Z which is GL n ( F )-invariant, and therefore induces a map(St( F n ) ⊗ St( F n )) GL n ( F ) −→ Z on coinvariants. We must show that this is an isomorphism as long as n ≥
1. Let usassume for the moment that the coinvariants are cyclic, and explain how to deducethe rest of the theorem.
Proof of Theorem A assuming cyclicity.
It remains to show that h− , −i : St( F n ) ⊗ St( F n ) → Z is surjective for n ≥
1, which we do using the theory of apartmentsas described above. Consider the decomposition L = ( L , L , . . . , L n ) with L i =span( e i ), and the decomposition L ′ = ( L ′ , L ′ , . . . , L ′ n ) with L ′ = L and L ′ i =span( e i + e i − ) for i = 2 , , . . . , n . The corresponding maps f L , f L ′ : sd( ∂ ∆ n − ) → T ( F n ) defining the apartments a L and a L ′ share precisely one ( n − h a L , a L ′ i = 1. (cid:3) ∞ -CELLS AND GENERAL LINEAR GROUPS OF INFINITE FIELDS 7 To prove that the coinvariants are indeed cyclic we shall the Lee–Szczarba pre-sentation ([LS76, Section 3], see also Section 7 below): Z X ∂ −→ Z X −→ St( F n ) −→ . Here X is the set of bases for F n , where we shall identify φ = ( φ , . . . , φ n ) withthe matrix φ ∈ GL n ( F ) whose i th column is φ i . Similarly, X is the set of spanning( n + 1)-tuples of non-zero vectors in F n . Thus this presentation says that St( F n ) isgenerated over Z by symbols [ φ ] with φ ∈ X subject to certain relations arisingfrom X . The generator [ φ ] corresponds to the apartment class from the direct sumdecomposition of F n into the spans of the columns of the matrix φ .In particular, for any φ = ( φ , . . . , φ n ) ∈ X we have elements of X given by( φ , . . . , φ i , φ i +1 , φ i , φ i +2 , . . . , φ n )( φ , . . . , φ i , αφ i , φ i +1 , . . . , φ n )( φ , . . . , φ i , φ i + φ i +1 , φ i +1 , . . . , φ n ) . These give the following three relations:(I) Column permutations: the first of these three elements of X gives rise tothe relation that we may swap adjacent columns of φ if we simultaneouslychange the sign of the generator [ φ ]. By induction we get the relation[ φ σ (1) , . . . , φ σ ( n ) ] = sign( σ )[ φ , . . . , φ n ]for any permutation σ .(II) Column scalings: the second element of X , in which α ∈ F × , allows us tomultiply any column of the matrix φ by a non-zero element.(III) Column addition: the third element of X gives the relation[ φ , . . . , φ n ] = [ φ , . . . , φ i − , φ i + φ i +1 , φ i +1 , φ i +2 , . . . , φ n ]+ [ φ , . . . , φ i − , φ i , φ i + φ i +1 , φ i +2 , . . . , φ n ] . Combining these, we see that the symbol [ φ ] ∈ St( F n ) is subject to the usualcolumn operations from linear algebra applied to the matrix φ ∈ GL n ( F ), exceptthat “column addition” between the i th and j th column is symmetric in i and j :we must add the i th column to the j th and simultaneously the j th to the i th, andthen take the formal sum of the two resulting matrices. Proof of cyclicity.
By tensoring together two copies of the Lee–Szczarba resolution,we obtain an exact sequence Z [( X × X ) ∐ ( X × X )] ∂ ⊗ ⊗ ∂ −−−−−−→ Z [ X × X ] → St( F n ) ⊗ St( F n ) → Z [GL n ( F )]-modules. Hence St( F n ) ⊗ St( F n ) is generated over Z by symbols[ φ ] ⊗ [ ψ ] with φ, ψ ∈ GL n ( F ), each subject to the “column operation” relationsabove. In the coinvariants we additionally have the relation[ φ ] ⊗ [ ψ ] = [1] ⊗ [ φ − ψ ] , using [1] as shorthand for [id n ], with id n the identity ( n × n )-matrix.We deduce from this that (St( F n ) ⊗ St( F n )) GL n ( F ) is generated by symbols [1] ⊗ [ φ ]for matrices φ ∈ GL n ( F ), subject to the following two types of relations. The first isthe column operations described above. The second is that we allow row operationsin a similar fashion (where “row addition” is symmetrised in the same way as before,resulting in the formal sum of the results of adding the i th row to the j th and the j th to the i th). Actually, column addition in φ becomes row subtraction in φ − ψ ,but row addition may be achieved by combining row subtraction with scaling ofrows by −
1. The proof of cyclicity of the coinvariants is finished in the followingtwo steps.
SØREN GALATIUS, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS
Claim 1. (St( F n ) ⊗ St( F n )) GL n ( F ) is generated as an abelian group by symbols[1] ⊗ [ φ ] where φ ∈ GL n ( F ) is a matrix with entries φ i,i = 1 for all i = 1 , . . . , n , φ i,i +1 ∈ { , } for all i = 1 , . . . , n −
1, and all other entries zero. That is, it is amatrix in Jordan form with 1’s on the diagonal, such as . Claim 2. If φ ∈ GL n ( F ) is a matrix in Jordan form with 1’s on the diagonal, thenthe class [1] ⊗ [ φ ] ∈ (St( F n ) ⊗ St( F n )) GL n ( F ) is an integer multiple of [1] ⊗ [ J n ],where J n is the matrix with entries φ i,i = 1 for all i = 1 , . . . , n , φ i,i +1 = 1 for all i = 1 , . . . , n −
1, and all other entries zero. That is, J n is a single Jordan block ofsize n × n , such as J = . These claims together imply cyclicity. They are proved separately below. (cid:3)
Proof of Claim 1.
We first explain how rewrite an arbitrary generator [1] ⊗ [ φ ] asan integral linear combination of generators in which the last row of φ is of theform (0 , . . . , ,
1) and the last column is of the form (0 , . . . , , ε,
1) with ε ∈ { , } ,such as ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ε , using all available column operations but only row operations which do not involvethe last row.If the last row has a single non-zero entry, we may scale its column and thenpermute columns so that the last row is of the desired form. Otherwise we may usecolumn scalings to arrange that the last row has one entry 1 and one entry −
1, sothat the sum of the corresponding columns has last entry 0. Using column additionbetween these two columns we obtain a relation [1] ⊗ [ φ ] = [1] ⊗ [ φ ′ ] + [1] ⊗ [ φ ′′ ]where φ ′ and φ ′′ have strictly more zeros in the last row than φ did. By inductionwe obtain a relation [1] ⊗ [ φ ] = P i [1] ⊗ [ φ i ] where the last row of each φ i hasprecisely one non-zero entry. Proceeding as in the first sentence of this paragraphwe see that the coinvariants are generated by those [1] ⊗ [ φ ] where the last row of φ is (0 , . . . , , φ is of this form we may apply the same argument with the roles of columnsand rows swapped: by induction on the number of zeros in the last column wemay use row operations among the first ( n −
1) rows to decrease the number ofnon-zero entries in the last column. Without using the last row we achieve that nomore than two entries are non-zero, i.e. we have a relation [1] ⊗ [ φ ] = P i [1] ⊗ [ φ i ]where the last column of each φ i has at most two non-zero entries. One of thesemust be the last entry, so (after scaling and swapping among the first ( n −
1) rows)the last column of each φ i is either (0 , . . . , ,
1) or (after scaling rows) of the form(0 , . . . , , n − × ( n −
1) block of φ ∈ GL n ( F ) without making ∞ -CELLS AND GENERAL LINEAR GROUPS OF INFINITE FIELDS 9 modifications to the last row and column. By induction we rewrite an arbitrarygenerator [1] ⊗ [ φ ] as a linear combination of generators where φ is in the desiredJordan block form. (cid:3) Proof of Claim 2.
If we write J a,b = diag( J a , J b ) with a + b = n , it suffices toexplain how to use the relations to rewrite [1] ⊗ [ J a,b ] as a multiple of [1] ⊗ [ J a + b ].In the case where φ has more than two Jordan blocks we just repeatedly combinetwo of them into one.To this end, let us write J a,b ( i ) for the matrix obtained from J a,b by adding a 1in the ( a + 1)st column and i th row, for i = 1 , . . . , a , such as J , = , and J , (1) = , J , (2) = = J . Then clearly J a,b ( a ) = J a + b and we shall also write J a,b (0) = J a,b .We claim the relation[1] ⊗ [ J a,b ( i )] = [1] ⊗ [ J a,b ( i + 1)] + [1] ⊗ [ J b + i,a − i ( i + 1)] (2.1)holds for all i = 0 , . . . , a −
1. To see this, first scale rows i + 1 , . . . , a and columns i + 1 , . . . , a of J a,b ( i ) by −
1, with the effect of changing the sign of the entry in the i th row and ( i + 1)st column. Then perform a column addition operation betweenthe i th and the ( a +1)st columns. The result is two terms, the first of which becomes J a,b ( i +1) after scaling rows 1 , . . . , i and columns 1 , . . . , i by −
1, the second of whichbecomes J b + i,a − i ( i + 1) after conjugating by the matrix id i a − i b , where id k denotes the identity ( k × k )-matrix. Since this is a permutation matrixthe conjugation may be achieved by permuting rows and columns.We leave it to the reader to verify that these arguments apply also in the case i = 0 (for example by taking the above proof for J a +1 ,b (1) and deleting the first rowand column, which are not moved during the proof). We have finished the proofof the relation (2.1), which by induction implies that [1] ⊗ [ J a,b ] is a multiple of[1] ⊗ [ J a + b ]. (In fact the multiple can easily be seen to be the binomial coefficient (cid:0) a + ba (cid:1) .) (cid:3) Anisotropy and indecomposability.
For any commutative ring k we havean induced k [GL( V )]-module St k ( V ) = k ⊗ Z St( V ), and we extend the pairingabove to a k -bilinear pairing h− , −i k : St k ( V ) × St k ( V ) −→ k . Let us record some properties of this pairing.
Theorem 2.4. (i) The pairing induces an isomorphism (St k ( V ) ⊗ k St k ( V )) GL( V ) ∼ → k .(ii) The set of GL( V ) -invariant k -bilinear pairings b : St k ( V ) × St k ( V ) → k formsa free k -module of rank 1, with the pairing h− , −i k as basis. In particular allinvariant forms are symmetric. (iii) If k is an ordered field the pairing is positive definite. More generally, if k is a ring in which 0 is not a non-trivial sum of squares, then the pairing isanisotropic. In either case, the restriction to any k -linear submodule A ⊂ St k ( V ) also induces an injection A → A ∨ .(iv) If F is infinite then the adjoint St k ( V ) → St k ( V ) ∨ = Hom k (St k ( V ) , k ) isinjective.Remark . In (iv) the assumption that F is infinite cannot be removed: one mayverify by hand that the conclusion is false for F = F , V = F , and k = Z / k -bilinear pairing b : M × M → k on a k -module M is called non-degenerate if the adjoint M → M ∨ = Hom k ( M, k ) is injective. Itis called anisotropic if b ( x, x ) = 0 for any x ∈ M \ { } . Anisotropy implies non-degeneracy, but the stronger notion has the advantage of passing to submodules: infact, anisotropy is equivalent to the restrictions M ′ × M ′ → k being non-degeneratefor all submodules M ′ ⊂ M . If k is an ordered field, then any positive definite formis anisotropic (but anisotropy is more general, e.g., the quadratic form x − y over the ordered field Q is anisotropic but not positive definite). Proof of Theorem 2.4.
It is clear that(St k ( V ) ⊗ k St k ( V )) GL( V ) = k ⊗ Z (St( V ) ⊗ St( V )) GL( V ) , so (i) and (ii) follow from Theorem A.For (iii), recall that St( V ) = ker[ ∂ : e C dim( V ) − ( T ( V )) → e C dim( V ) − ( T ( V ))] andthat the pairing on St k ( V ) is the restriction of the chain level pairing in which theset of full flags forms an orthonormal basis. If x = P F a F · F is a finite sum offull flags F with coefficients a F ∈ k , then h x, x i k = P F a F . If x = 0 then thisself-pairing is positive if k is an ordered field, and non-zero if no non-trivial sum ofsquares is zero in k .For (iv) we first establish the following claim. Recall that top-dimensionalsimplices of T ( V ) correspond to full flags in V ; we write F for a full flag0 ⊂ F ⊂ F ⊂ · · · ⊂ F n = V where dim( F i ) = i . The apartment correspond-ing to a splitting of V into 1-dimensional subspaces L , L , . . . , L n consists of thosefull flags which maybe obtained as partial sums of the L i in some order. Claim.
Assume F is infinite. For a full flag F and a finite set of full flags { F α } α ∈ I distinct from F , there is an apartment of T ( V ) having F as a face and having no F α as a face. Proof of claim.
For each r consider the set V r := F r \ F r − ∪ [ α ∈ I F αr − ! . This is the complement in F r of finitely-many proper subspaces, so is non-emptyunder our assumption that F is infinite.Choose elements v r ∈ V r , and let L r := span( v r ). As v r F r − we have F r = L ⊕ · · · ⊕ L r , so the apartment given by this splitting has the flag F as aface. If this apartment contained F α as a face then, for some permutation σ , wewould have F αr = L σ (1) ⊕ L σ (2) ⊕ · · · ⊕ L σ ( r ) for each r . Thus v σ ( r ) ∈ F αr for each r , so by definition of the sets V r we have r ≥ σ ( r ) for each r , but then σ must bethe identity permutation and so F α = F , a contradiction. (cid:3) To finish the proof of (iv) let x ∈ St k ( V ) be nonzero: it is then a finite non-trivial k -linear sum of full flags, and we let F be a full flag having non-zero coefficient k ∈ k and { F α } α ∈ I be the remaining full flags arising in this sum. The claim providesan apartment a containing F but not containing any F α , but then the definition ∞ -CELLS AND GENERAL LINEAR GROUPS OF INFINITE FIELDS 11 of the pairing shows that h x, a i = k = 0 so x is not in the kernel of the adjointmap. (cid:3) Let us explain how to use the bilinear pairing to prove that the k [GL( V )]-moduleSt k ( V ) is indecomposable for any connected commutative ring k and any finite-dimensional vector space V over a field F . The irreducibility/indecomposabilityquestion for Steinberg modules of infinite fields was also asked by A. Putman [Put].The connection between Theorem A and irreducibility was pointed out to us byA. Venkatesh. Proof of Theorem B.
The k -linear map St k ( V ) → St k ( V ) ∨ = Hom k (St k ( V ) , k )adjoint to the bilinear form is injective by Theorem 2.4 (ii). Applying the left exactfunctor Hom k (St k ( V ) , − ) gives an injective homomorphismHom k (St k ( V ) , St k ( V )) ֒ → Hom k (St k ( V ) , St k ( V ) ∨ ) ∼ = Hom k (St k ( V ) ⊗ k St k ( V ) , k ) , sending an endomorphism A to the bilinear form x ⊗ y
7→ h
Ax, y i . This injectivehomomorphism is GL( V )-equivariant when that group acts by conjugation in thedomain and by dualising the diagonal action on St k ( V ) ⊗ k St k ( V ) in the codomain.Passing to fixed points we get an injective k -linear homomorphismEnd k (St k ( V )) GL( V ) → Hom k (St k ( V ) ⊗ k St k ( V ) , k ) GL( V ) . We have seen that the codomain is a free k -module of rank one, and since1 ∈ End k [GL( V )] (St k ( V )) is sent to h− , −i k , it follows that the identity gives anisomorphism k → End k (St k ( V )) GL( V ) . In particular if the ring k is connected (i.e. has no non-trivial idempotents) thenthere are no non-trivial GL( V )-equivariant idempotent endomorphisms of St k ( V ). (cid:3) We do not know whether St k ( V ) is irreducible, but when k is a field in whichno non-trivial sum of squares is zero, then it cannot contain any submodules whichare finite-dimensional over k . Indeed, suppose for contradiction that A ⊂ St k ( V )were such a submodule and consider the composition A ֒ → St k ( V ) → St k ( V ) ∨ → A ∨ . It is injective because the pairing is anisotropic, but A and A ∨ are vector spacesof the same finite dimension so it is an isomorphism, so A is a k [GL( V )]-linearsummand of St k ( V ). Remark . A similar conclusion holds when k admits an involution whose fixedfield k + ⊂ k satisfies that no non-trivial sum of squares is zero, e.g., for k = C .The proof is similar, except the pairing should be extended to a sesquilinear pairingon St k ( V ). 3. Overview of E k -cells and E k -homology Let us briefly outline the theory developed [GKRW18a], which will play a role inthe rest of the paper. We refer there for further details and for proofs and references,and we shall refer to things labelled X in [GKRW18a] as E k .X here.This theory is designed to analyse the homology of (non-unital) E k -algebras like a n ≥ B GL n ( F ) . (3.1)In fact, as we are only interested in the k -homology we may as well take the k -linear singular simplices on this space. Furthermore, as we wish to distinguish the contributions to homology of the different path components, we shall work withan additional N -grading, called rank : we therefore work in the category sMod N k of N -graded simplicial k -modules, i.e. the category of functors M : N → sMod k . Thisis given a symmetric monoidal structure by Day convolution, with p simplices inrank n given by ( M ⊗ N ) p ( n ) = M a + b = n M p ( a ) ⊗ k N p ( b ) . The homology of a simplicial k -module means the homology of the associated chaincomplex, or equivalently the homotopy groups of the underlying simplicial set. Anobject X ∈ sMod N k hence has bigraded homology groups via H n,d ( X ) := H d ( X ( n )).The little k -cubes operad has a space of r -ary operations given by r -tuples ofrectilinear embeddings of a k -cube into k -cube with disjoint interior, except when r = 0 in which case it is empty. We may import this into sMod N k by first taking its k -linear singular simplicial set, and then placing the result in rank 0: we denotethe result by C k .A (non-unital) E k -algebra in sMod N k is then an algebra over the operad C k ; wedenote these by bold letters such as R and write Alg E k ( sMod N k ) for the category of E k -algebras in sMod N k . We denote by E k ( − ) the monad on sMod N k associated to C k ,and by E k ( X ) the free (non-unital) E k -algebra on X . We will mainly be concernedwith considering the k -linearisation of (3.1) as an E -, E -, or E ∞ -algebra in thecategory sMod N k , and in particular describing cell structures on it.To explain what we mean by this, let us denote by ∂D n,d ∈ sMod N k the objectgiven at n ∈ N by k [ ∂ ∆ d ], the k -linearisation of the simplicial set given by theboundary of the d -simplex, and by 0 otherwise; we write D n,d for the analogousconstruction with k [∆ d ]. By adjunction, a morphism ∂D n,d → R in sMod N k extendsto a morphism E k ( ∂D n,d ) → R of E k -algebras, using which we may form thepushout E k ( ∂D n,d ) RE k ( D n,d ) R ∪ E k D n,d . in the category Alg E k ( sMod N k ). This is what it means to attach a cell to R . A cellular E k -algebra is one obtained by iterated cell attachments starting with thezero object, and a cellular approximation to R is a weak equivalence C ∼ → R froma cellular object.In order to control cell structures we will use a homology theory for E k -algebras.The indecomposables of R ∈ Alg E k ( sMod N k ) are defined by the exact sequence M n ≥ C k ( n ) ⊗ R ⊗ n −→ R −→ Q E k ( R ) −→ sMod N k , with the leftmost map given by the E k -algebra structure of R . Thatis, we collapse all elements of R which can be obtained by applying at least 2-ary operations. The construction Q E k : Alg E k ( sMod N k ) → sMod N k is not homotopyinvariant, but has a left derived functor Q E k L called the derived indecomposables . (Inthe notation of [GKRW18a] we have actually defined the relative indecomposables,but because C ∞ (1) ≃ ∗ at the level of derived functors there is no difference betweenthis and the absolute indecomposables, cf. equation (11.2) in Section E k .11.3.) Wethen define the E k -homology groups of R by the formula H E k n,d ( R ) := H n,d ( Q E k L ( R )) = H d ( Q E k L ( R )( n )) . ∞ -CELLS AND GENERAL LINEAR GROUPS OF INFINITE FIELDS 13 One does not typically try to compute E k -homology directly from the defini-tion. Instead, one uses a description in terms of a k -fold bar construction (in-stances have been given by Getzler–Jones, Basterra–Mandell, Francis, and Fresse).More precisely, if R + denotes the unitalisation of the non-unital E k -algebra R ,and ε : R + → k its corresponding augmentation, then in Section E k .13.1 we havedescribed an equivalence k ⊕ Σ k Q E k L ( R ) ≃ B E k ( R + , ε ) , where the right-hand side denotes a certain model for the k -fold bar constructionof the augmented E k -algebra ε : R + → k . An advantage of this point of view isthat the k -fold bar construction can be calculated iteratively, which makes it easyto relate information about E k − -homology to information about E k -homology.Derived indecomposables are related to cellular approximation as follows. Beinga left adjoint, Q E k preserves pushouts, and by direct calculation Q E k ( E k ( X )) ∼ = X .A cellular E k -algebra C is cofibrant, so we may consider its underived indecom-posables Q E k ( C ): this is then a cellular object in sMod N k , with one cell for each E k -cell of C . The groups H E k ∗ , ∗ ( R ) therefore give a lower bound on the cells neededfor any cellular approximation of R . In fact there is a Hurewicz theorem for E k -homology (Corollary E k .11.12), which means it can be used to construct minimal E k -cell structures. In particular if R (0) ≃ H E k n,d ( R ) = 0 for d < f ( n ), then R admits a cellular approximation C ∼ −→ R built only using cells of bidegrees ( n, d ) such that d ≥ f ( n ). This C may be chosento be CW rather than just cellular, meaning it comes with a lift to an E k -algebra infiltered objects of sMod N k with good properties (not a filtered object in E k -algebras),see Section E k .6.3.An overview of how we shall use these ideas in the rest of the paper is as follows.In Section 4 we will explain how to construct an E ∞ -algebra R in sSet N with R ( n ) ≃ B GL n ( A ) , which for any commutative ring k we may k -linearise to give R k ∈ Alg E ∞ ( sMod N k )having H n,d ( R k ) = H d (GL n ( A ); k ). Our first goal is to establish vanishing linesfor H E k n,d ( R k ) for k ∈ { , , ∞} . To do so, in Section 5 we will describe the k -foldsimplicial sets which arise in the k -fold bar construction model for derived E k -indecomposables of this R , as well as Rognes’ k -fold analogue of the Tits building,and conditions under which these are related. The main goal is to show that if thedata ( A, k ) satisfies the “Nesterenko–Suslin property” then Rognes’ k -fold analogueof the Tits building may be used to calculate H E k n,d ( R k ).After this general theory, in Section 6 we will specialise to the case that A isan infinite field (in which case ( A, k ) does indeed satisfy the Nesterenko–Suslinproperty for any coefficients k ). In this case we will simply observe that Rognes’ 2-fold analogue of the Tits building is simply the smash-square of the 1-fold analogue,so that it is twice as highly-connected as the Tits building. This means that there isa much steeper vanishing line for the E -homology (and hence E ∞ -homology) of R k than one can formally deduce from the vanishing line for its E -homology. Finally,we analyse completely the E - and E ∞ -homology along this vanishing line, whichin particular will prove Theorem 6.9. In Section 7 we will explain the analoguesof most of these results in the case that A is a connected semi-local ring with allresidue fields infinite.We then come to applications. Here we shall use most of the technical toolsdeveloped in [GKRW18a], and will not try to summarise them all here. Glossary and some notation from [GKRW18a] . For the reader’s conve-nience we collect some notation from op.cit., but we refer there for more details. • Many arguments take place in the categories sSet N , sSet N ∗ , and sMod N k , of functorsfrom N (regarded as a category with only identity morphisms) to the category ofsimplicial sets, pointed simplicial sets, and simplicial k -modules, respectively. • Given an algebra R ∈ Alg E ∞ ( sSet N ) and a commutative ring k , we denote by R k ∈ Alg E ∞ ( sMod N k ) the k -linearisation of R . • Objects X ∈ sMod N k have homology groups which are bigraded k -modules, definedby H n,d ( X ) = π d ( X ( n )). • The object D n,d ∈ sSet N is given at n ∈ N by ∆ d and ∅ otherwise. Similarly, ∂D n,d ∈ sSet N is given at n ∈ N by ∂ ∆ d and ∅ otherwise. Finally, S n,d = D n,d /∂D n,d ∈ sSet N ∗ . • On homology, smashing with S n,d has the effect of shifting bidegrees: for X ∈ sMod N k , H n + n ′ ,d + d ′ ( S n ′ ,d ′ ∧ X ) = H n,d ( X ) if n ≥ d ≥ • The unitalisation of an E k -algebra R is denoted R + , see Section E k .4.4. By“ E k -algebra” we always mean the non-unital type, unless otherwise specified. • If R is an E -algebra, then R denotes a unital and associative algebra naturallyweakly equivalent to R + as a unital E -algebra, see Proposition E k .12.9 and theconstruction preceding it. If R is an E k -algebra, we regard it is an E -algebrabefore applying these constructions. • Filtered objects in a category C are functors Z ≤ → C , where Z ≤ denotes thecategory with object set Z and morphism set m → n either a singleton or empty,depending on whether m ≤ n or m > n . We often consider filtered objects infunctor categories such as C = sMod N k , in which case filtered objects are functors N × Z ≤ → sMod k . • To an unfiltered object X is associated a filtered object a ∗ X for each a ∈ Z ,making the functor X a ∗ X left adjoint to evaluation at the object a ∈ Z ≤ .Explicitly, ( a ∗ X )( n ) is X for n ≥ a and is the initial object for n < a . See Section E k . . . • There is a spectral sequence associated to a filtered object X ∈ sMod N × Z ≤ k . Inour grading conventions, see Theorem E k .10.10, it has E n,p,q = H n,p + q ( X ( q ) , X ( q − , d r : E rn,p,q → E rn,p + r − ,q − r , and if it conditionally converges, it does so to H n,p + q (colim X ). This does notrelate the different n at all (disregarding any multiplicative structures) so may beregarded as one spectral sequence for each n ∈ N . Then ( E rn, ∗ , ∗ , d r ) r ≥ is gradedas the usual homological Serre spectral sequence, but with p and q swapped.4. General linear groups as an E ∞ -algebra Convention 4.1.
All our rings A are commutative.For any ring A there is a groupoid P A with objects the finitely-generated projec-tive A -modules, and morphisms given by the A -module isomorphisms. We writeGL( M ) for the group of automorphisms of an A -module M . Direct sum ⊕ endows P A with a symmetric monoidal structure. Lemma 4.2.
The symmetric monoidal groupoid ( P A , ⊕ , has the following prop-erties:(i) for the monoidal unit we have GL(0) = { e } ,(ii) the homomorphism − ⊕ − : GL( M ) × GL( N ) → GL( M ⊕ N ) induced by themonoidal structure is injective.e. ∞ -CELLS AND GENERAL LINEAR GROUPS OF INFINITE FIELDS 15 A finitely-generated projective A -module M has a rank rk p ( M ) for each primeideal p ⊂ A , and these form a locally constant function on Spec( A ) [Sta18,Tag 00NV]). We will therefore suppose that Spec( A ) is connected (and will usu-ally say that “ A is connected”), so that there is a unique rank associated to eachfinitely-generated projective A -module. This gives a functor r : P A −→ N . We will use this category to construct a non-unital E ∞ -algebra R ∈ Alg E ∞ ( sSet N )with R (0) = ∅ and R ( n ) ≃ a [ M ] r ( M )= n B GL( M ) , (4.1)where the disjoint union is over isomorphism classes of rank n projective A -modules,following Section E k .17.1.We now proceed as in Section E k .17.1, using that by Lemma 4.2 (i) above thegroupoid P A satisfies Assumption E k .17.1. There is a functor ∗ > in sSet P A givenby 0 ∅ and M
7→ ∗ for M = 0, which has a unique non-unital commutativealgebra structure and hence is in particular a non-unital E ∞ -algebra. As describedin Section E k .9.2 the category Alg E k ( sSet P A ) of E k -algebras in sSet P A admits theprojective model structure. By cofibrantly replacing ∗ > we obtain a cofibrantnon-unital E ∞ -algebra T with T ( P ) ≃ ( ∅ if M = 0, ∗ if M = 0.Precomposition by r gives a functor r ∗ : sSet N → sSet P A which admits a leftadjoint r ∗ by left Kan extension, and this is (strong) symmetric monoidal. As inSection E k .4.3 it induces a functor between categories of E ∞ -algebras, i.e. it maybe considered as a functor r ∗ : Alg E ∞ ( sSet P A ) → Alg E ∞ ( sSet N ), which is the leftadjoint in a Quillen adjunction. We may thus take its left derived functor L r ∗ , anddefine R := r ∗ ( T ) ≃ L r ∗ ( ∗ > ) ∈ Alg E ∞ ( sSet N ) , which satisfies R (0) = ∅ and (4.1) above.As we have mentioned we will often only be interested in homology groups, inwhich case there is no loss in passing from simplicial sets to simplicial k -modulesand considering instead R k := k [ R ] ∈ Alg E ∞ ( sMod N k ). This satisfies H n,d ( R k ) = M [ M ] r ( M )= n H d (GL( M ); k ) . In practice we will usually work under ring-theoretic assumptions on A whichimply that every finitely-generated projective A -module is free (namely that A issemi-local and connected [Sta18, Tag 02M9]). In this case we simply have H n,d ( R k ) = H d (GL n ( A ); k ) . We can mimic this for any ring A by considering the subcategory F A of P A consistingof free modules, which inherits a symmetric monoidality, and which again has arank functor. Repeating the above construction gives a R ∈ Alg E ∞ ( sSet N ) with R (0) = ∅ and R ( n ) ≃ B GL n ( A ). However the general constructions in Sections5 and 7 are most natural from the point of view of projective modules, and this isthe perspective we shall take. Higher-dimensional buildings and their split analogues
In this section we study various buildings of summands or flags of A -modules.In Section 5.3 we relate buildings of summands to buildings of flags using theNesterenko–Suslin property. Both are in Section 5.4 related to E k -homology of R .5.1. The k -dimensional building. As usual we write [ p ] for the finite linearorder 0 < < < · · · < p , considered as a category. Products [ p ] × . . . × [ p k ]denote the product category (or equivalently, product poset), and use the shorternotation [ p , . . . , p k ]. Given a, b ∈ [ p ] with a ≤ b we shall write [ a ≤ b ] for the fullsubcategory on the set { a, b } , isomorphic as a poset to either [0] or [1]. Similarly,given a, b ∈ [ p , . . . , p k ] with a ≤ b we have a subposet[ a ≤ b ] × . . . × [ a k ≤ b k ] ֒ → [ p , . . . , p k ]which is a cube , i.e. isomorphic as a poset to [1] k ′ for some k ′ ≤ k . We shall needto refer to the subposet[ a ≤ b ] × . . . × [ a k ≤ b k ] \ { b } ֒ → [ p , . . . , p k ] , the cube with its terminal element removed, the punctured cube .For a ring A and an A -module M , let Sub ( M ) denote the set of submodules P ⊂ M which are summands. By definition, P is a summand if it admits acomplement, i.e. there exists a submodule Q ⊂ M such that the natural map P ⊕ Q → M is an isomorphism. If P ⊂ M is a summand in M , it is also asummand in any submodule M ′ ⊂ M containing P ; its complement is the kernelof the restriction to M ′ of the projection M → P .We shall regard Sub ( M ) as a partially ordered set with respect to inclusion. Definition 5.1.
A functor φ : [ p , . . . , p k ] → Sub ( M ) is called a lattice if the naturalmap colim [ a ≤ b ] × ... × [ a k ≤ b k ] \{ b } φ −→ φ ( b ) (5.1)is a monomorphism onto a summand (in M , or equivalently in φ ( b )). (Here thecolimit denotes colimit in the category of A -modules, not in the poset Sub ( M ).)A lattice φ : [ p , . . . , p k ] → Sub ( M ) is full if it satisfies(i) φ ( a , . . . , a k ) = 0 if a i = 0 for some i ∈ { , . . . , k } ,(ii) φ ( p , . . . , p k ) = M . Remark . Our definition differs from Rognes’ [Rog92, Definition 2.3], in thathe includes condition (i) in his notion of “lattice”. Nonetheless, our notions of k -dimensional building and stable building are isomorphic to Rognes’ (compareour Definition 5.4 with [Rog92, Definition 3.9] and our Definition 5.5 with [Rog92,Definition 10.8]): they differ only in that Rognes discards those functors not satis-fying (i) by passing to a pointed subset, while passing to a quotient set seems morenatural to us.The following two properties are easily verified. Lemma 5.3. (i) For any lattice φ : [ p , . . . , p k ] → Sub ( M ) and any morphism θ : [ q , . . . , q k ] → [ p , . . . , p k ] in ∆ × k , the functor θ ∗ φ : [ q , . . . , q k ] −→ Sub ( M ) is again a lattice.(ii) If φ is not full, then θ ∗ φ is also not full. (cid:3) ∞ -CELLS AND GENERAL LINEAR GROUPS OF INFINITE FIELDS 17 Definition 5.4.
The k -dimensional building D k ( M ) • ,..., • is the k -fold simplicialpointed set with ( p , p , . . . , p k )-simplices defined by the pushout { non-full lattices } { lattices [ p , . . . , p k ] → Sub ( M ) }{∗} D k ( M ) p ,...,p k The simplicial structure is given by the evident functoriality on ∆ k , via Lemma 5.3.We write D k ( M ) for the k -fold thin geometric realisation of this k -fold simplicialpointed set.There is a map of ( k + 1)-fold simplicial pointed spaces S • ∧ D k ( M ) • ,..., • −→ D k +1 ( M ) • ,..., • given on the non-degenerate simplex of S • by the map D k ( M ) p ,p ,...,p k −→ D k +1 ( M ) ,p ,p ,...,p k induced by sending a lattice φ : [ p , . . . , p k ] → Sub ( M ) to the lattice φ ′ : [1 , p , . . . , p k ] → Sub ( M ) given by φ ′ (0 , a , . . . , a k ) = 0 φ ′ (1 , a , . . . , a k ) = φ ( a , . . . , a k )Upon geometric realisation these provide the structure maps for a symmetric spec-trum [Rog92, Proposition 3.8]. Definition 5.5.
The stable building D ( M ) is the spectrum { D k ( M ) } k ∈ N withstructure maps as above. It comes equipped with an action of GL( M ) by naturality.We will use the following criterion to recognise full lattices, using the notation p := { , . . . , p } : Lemma 5.6.
Let φ : [ p , . . . , p k ] → Sub ( M ) be a functor. Then the following areequivalent:(i) φ is a full lattice,(ii) the map (5.1) is a monomorphism onto a summand whenever a i = b i − , φ ( a , . . . , a k ) = 0 if a i = 0 for some i ∈ { , . . . , k } and φ ( p , . . . , p k ) = M ,(iii) there exist M i ∈ Sub ( M ) for each i = ( i , . . . , i k ) ∈ p × . . . × p k such that the M i ’s span M and such that for all a ∈ [ p , . . . , p k ] , the natural map a M i =1 . . . a k M i k =1 M i −→ M is a monomorphism with image φ ( a ) .Proof. That (i) ⇒ (ii) is obvious, as we are requiring condition (5.1) for fewer cubes.For (iii) ⇒ (i), suppose the functor φ satisfies the condition about the existenceof M i ’s as stated. In particular the direct sum of all the M i maps to M by anisomorphism. Then it is easily verified that the natural mapscolim [ b − ≤ b ] × ... × [ b k − ≤ b k ] \{ b } φ −→ X i
There is an injection of pointed sets D k ( M ) p ,...,p k −→ D ( M ) p ∧ . . . ∧ D ( M ) p k , (5.4) assembling to a levelwise injective map of multisimplicial pointed sets. Split buildings.
The k -dimensional split building is a k -fold simplicial pointedset e D k ( M ) where a non-basepoint simplex is a non-basepoint simplex σ of D k ( M )together with choices of submodules M i as in Lemma 5.6. For k = 1, a twicedesuspended version was first defined by Charney [Cha80, p. 3]. It also appearedin [GKRW18b, Section 3.3]. Let us spell out the definition. Definition 5.8.
Let X be a finite pointed set and f : X → Sub ( M ) a function. Wesay that f is a splitting if the natural map M x ∈ X f ( x ) −→ M is an isomorphism.If f is a splitting and θ : X → Y is a map of finite pointed sets, then the map θ ∗ f : Y −→ Sub ( M ) y X x ∈ θ − ( x ) f ( x ) ∞ -CELLS AND GENERAL LINEAR GROUPS OF INFINITE FIELDS 19 is also a splitting. If f ( ∗ ) = 0, then also ( θ ∗ f )( ∗ ) = 0. Hence the association S : X splittings f : X → Sub ( M )splittings with f ( ∗ ) = 0 (5.5)defines a functor from finite pointed sets to pointed sets, i.e. a Γ-set in the sense ofSegal. Definition 5.9.
The k -dimensional split building e D k ( M ) • ,..., • is the k -fold simpli-cial set given as the composition of (5.5) with the functor(∆ op ) × k −→ finite pointed sets([ p ] , . . . , [ p k ]) S p ∧ . . . ∧ S p k , where S • = ∆[1] /∂ ∆[1] denotes the usual model of the simplicial circle: S p is thequotient of the set of maps [ p ] → [1] in ∆ by the subset of constant maps. Write e D k ( M ) for the k -fold geometric realisation of this k -fold simplicial pointed set.A non-basepoint element θ : [ p ] → [1] of S p ] may be identified with the uniquenumber i ∈ { , . . . , p } for which θ − (0) = { , . . . , i − } . Hence non-basepointelements of e D k ( M ) p ,...,p k may be identified with functions (recalling p = { , . . . , p } ) M : p × . . . × p k −→ Sub ( M )for which the natural map L i M ( i ) → M is an isomorphism.There is a forgetful map of multisimplicial pointed sets e D k ( M ) • ,..., • −→ D k ( M ) • ,..., • M 7−→ φ M (5.6)whose value on a non-basepoint M is defined by φ M ( a ) = X i ≤ a M ( i ) . This is a full flag and, comparing with Lemma 5.6, we see that the modules denoted M i there may be chosen as M ( i ). In this way we have set up a bijection betweennon-basepoints of e D k ( M ) p ,...,p k and non-basepoints of D k ( M ) p ,...,p k together witha choice of splitting submodules as in Lemma 5.6. Under this bijection the map (5.6)amounts to forgetting the splittings.Just like in Lemma 5.7 we may compare e D k to a k -fold smash product of e D : Lemma 5.10.
The analogous formula to that of Lemma 5.7 gives (no longer in-jective) maps of pointed sets e D k ( M ) p ,...,p k −→ e D ( M ) p ∧ . . . ∧ e D ( M ) p k , (5.7) which assemble to a map of multisimplicial pointed sets. These are related to thoseof Lemma 5.7 by the maps (5.6) . The group GL( M ) = Aut A ( M ) acts on the set Sub ( M ) by sending a submoduleto its image under an automorphism of M , and hence acts on the multisimplicialpointed sets D k ( M ) • ,..., • and e D k ( M ) • ,..., • and their thin geometric realisations. Proposition 5.11.
Let A be a commutative ring and M a finitely generated pro-jective module. Then the map of GL( M ) -orbit sets e D k ( M ) p ,...,p k / GL( M ) −→ D k ( M ) p ,...,p k / GL( M ) induced by the GL( M ) -equivariant map (5.6) is a bijection. Proof.