Algebraic K -theory of planar cuspidal curves
aa r X i v : . [ m a t h . K T ] J u l Contemporary Mathematics
Algebraic K -theory of planar cuspidal curves Lars Hesselholt and Thomas Nikolaus
Introduction
The purpose of this paper is to evaluate the algebraic K -groups of a planarcuspidal curve over a perfect F p -algebra relative to the cusp point. A conditionalcalculation of these groups was given in [ , Theorem A], assuming a conjecture onthe structure of certain polytopes. Our calculation here, however, is unconditionaland illustrates the advantage of the new setup for topological cyclic homology byNikolaus–Scholze [ ], which we will be using throughout. The only input necessaryfor our calculation is the evaluation by the Buenos Aires Cyclic Homology group [ ]and Larsen [ ] of the structure of the Hochschild complex of the coordinate ring asa mixed complex, that is, as an object of the ∞ -category of chain complexes withcircle action.We consider the planar cuspidal curve “ y a = x b ,” where a, b ≥ m ≥
0, we define ℓ ( a, b, m ) to be the number of pairs ( i, j ) ofpositive integers such that ai + bj = m , and for r ≥
0, define S ( a, b, r ) to be the setof positive integers m such that ℓ ( a, b, m ) ≤ r . The subset S = S ( a, b, r ) ⊂ Z > isa truncation set in the sense that if m ∈ S and d divides m , then d ∈ S , and hence,the ring of (big) Witt vectors W S ( k ) with underlying set k S is defined. We referto [ , Section 1] for a detailed introduction to Witt vectors. Theorem A . Let k be a perfect F p -algebra, and let a, b ≥ be relatively primeintegers. There is a canonical isomorphism K j ( k [ x, y ] / ( y a − x b ) , ( x, y )) ≃ W S ( k ) / ( V a W S/a ( k ) + V b W S/b ( k )) , if j = 2 r ≥ with S = S ( a, b, r ) , and the remaining K -groups are zero. We remark that recently Angeltveit [ ] has given a different proof of this resultthat, unlike ours, employs equivariant homotopy theory. We also remark that thestrategy employed in this paper was used by Speirs [ ] to significantly simplifythe calculation in [ ] of the relative algebraic K -groups of a truncated polynomialalgebra over a perfect F p -algebra.We recall from [ , Section 1] that the group in the statement is a module overthe ring W ( k ) of big Witt vectors in k of finite length (2 r + 1)( a − b − ]. Moreover, it admits a Mathematics Subject Classification.
Primary 19D55; Secondary 55P43. p -typical product decomposition indexed by positive integers m ′ not divisible by p .To state this, we let s = s ( a, b, r, p, m ′ ) be the unique integer such that ℓ ( a, b, p s − m ′ ) ≤ r < ℓ ( a, b, p s m ′ ) , if such an integer exists, and 0, otherwise, and assume (without loss of generality)that p does not divide b and write a = p u a ′ with a ′ not divisible by p . Then W S ( k ) / ( V a W S/a ( k ) + V b W S/b ( k )) ≃ Q m ′ ∈ N ′ W h ( k )by a canonical isomorphism, where h = h ( a, b, r, p, m ′ ) = s, if neither a ′ nor b divides m ′ ,min { s, u } , if a ′ but not b divides m ′ ,0 , if b divides m ′ .Here we write N ′ for the set of positive integers not divisible by p .It is a pleasure to acknowledge the generous support that we have received whileworking on this paper. Hesselholt was funded in part by the Isaac Newton Instituteas a Rothschild Distinguished Visiting Fellow and by the Mathematical SciencesResearch Institute as a Simons Visiting Professor. Nikolaus was funded in part bythe Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy EXC2044 390685587, Mathematics M¨unster: Dynamics–Geometry–Structure. Finally,we are grateful to Tyler Lawson for pointing out that our arguments in an earlierversion of this paper could be simplified significantly.
1. Some recollections necessary for the proof
We first recall the Nikolaus–Scholze formula for topological cyclic homologyfrom [ ]; see also [ ]. We write T for the circle group and C p ⊂ T for the subgroupof order p . If R is a ring, then we writeTC − ( R ) TP( R ) can / / for the canonical map from the homotopy fixed points to the Tate construction ofthe spectrum with T -action THH( R ). The Frobenius mapTHH( R ) ϕ / / THH( R ) tC p is T -equivariant, provided that we let T act on the target through the isomorphism ρ : T → T /C p given by the p th root, and therefore, it induces a mapTC − ( R ) = THH( R ) h T (THH( R ) tC p ) h ( T /C p ) . ϕ h T / / The Tate-orbit lemma [ , I.2.1, II.4.2] identifies the p -completion of the target ofthis map with that of TP( R ), and Nikolaus–Scholze show that, after p -completion,the topological cyclic homology of R is the equalizerTC( R ) TC − ( R ) TP( R ) / / ϕ / / can / / of these two parallel maps. If p is nilpotent in R , as is the case in the situation thatwe consider, then the spectra in question are already p -complete. LGEBRAIC K -THEORY OF PLANAR CUSPIDAL CURVES 3 The normalization of A = k [ x, y ] / ( y a − x b ) is the k -algebra homomorphism to B = k [ t ] that to x and y assigns t a and t b , respectively, and this homomorphismidentifies A with sub- k -algebra k [ t a , t b ] ⊂ k [ t ] = B . In this situation, the square K ( A ) TC( A ) K ( B ) TC( B ) / / (cid:15) (cid:15) (cid:15) (cid:15) / / is cartesian. This follows from the birelative theorem, which has now been givena very satisfying conceptual proof by Land–Tamme [ ]. (By contrast, the originalproof in the rational case by Corti˜nas [ ] and the subsequent proof in the p -adic caseby Geisser–Hesselholt both required rather elaborate calculational input.) Now, themap K ( B ) → K ( k ) induced by the k -algebra homomorphism that to t assigns 0is an equivalence, and hence, the relative K -groups that we wish to determine arecanonically identified with the homotopy groups of the common fibers of the verticalmaps in the diagram above.The k -algebras A and B are both monoid algebras. In general, if k [Π] is themonoid algebra of an E -monoid Π in spaces, then, as cyclotomic spectra,THH( k [Π]) ≃ THH( k ⊗ S [Π]) ≃ THH( k ) ⊗ B cy (Π) + , where B cy (Π) denotes the unstable cyclic bar-construction of Π. In addition, onthe right-hand side, the Frobenius map factors as a compositionTHH( k ) ⊗ B cy (Π) + ϕ ⊗ ˜ ϕ / / THH( k ) tC p ⊗ B cy (Π) hC p +can / / (THH( k ) ⊗ B cy (Π) + ) tC p of the map induced by the Frobenius ϕ : THH( k ) → THH( k ) tC p and the unstableFrobenius ˜ ϕ : B cy (Π) → B cy (Π) hC p and a canonical map. Importantly, we have amap of spectra with T -action Z / / Z p ≃ τ ≥ TC( k ) / / THH( k )from Z with (necessarily) trivial T -action, and therefore, we can rewriteTHH( k ) ⊗ B cy (Π) + ≃ THH( k ) ⊗ Z Z ⊗ B cy (Π) + . Accordingly, we do not need to understand the homotopy type of the space with T -action B cy (Π). It suffices to understand the homotopy type of the chain complexwith T -action Z ⊗ B cy (Π) + , which, in the case at hand, is exactly what the BuenosAires Cyclic Homology group [ ] and Larsen [ ] have done for us.To state their result, we let h t a , t b i ⊂ h t i be the free monoid on a generator t and the submonoid generated by t a and t b , respectively, and set B cy ( h t i , h t a , t b i ) = B cy ( h t i ) /B cy ( h t a , t b i ) . Counting powers of t gives a T -equivariant decomposition of pointed spaces B cy ( h t i , h t a , t b i ) ≃ W m ∈ Z > B cy ( h t i , h t a , t b i ; m ) . We can now state the result of the calculation by the Buenos Aires Cyclic Homologygroup [ ] and by Larsen [ ] as follows. LARS HESSELHOLT AND THOMAS NIKOLAUS
Theorem . Let a, b ≥ be relatively prime integers, and let m ≥ be aninteger. In the ∞ -category D ( Z ) B T of chain complexes with T -action, there is acanonical equivalence between Z ⊗ B cy ( h t i , h t a , t b i ; m ) and the total cofiber of the square Z ⊗ ( T /C m/ab ) + [2 ℓ ( a, b, m )] / / (cid:15) (cid:15) Z ⊗ ( T /C m/a ) + [2 ℓ ( a, b, m )] (cid:15) (cid:15) Z ⊗ ( T /C m/b ) + [2 ℓ ( a, b, m )] / / Z ⊗ ( T /C m ) + [2 ℓ ( a, b, m )] . Here, all maps in the square are induced by the respective canonical projections,and if c does not divide m , then T /C m/c is understood to be the empty space. The result is not stated in this from in op. cit., and therefore, some explanationis in order. The ∞ -category D ( Z ) B T is equivalent to the ∞ -category of modulesover the E -algebra C ∗ ( T , Z ) given by the singular chains on the circle. This E -algebra, in turn, is a Postnikov section of a free E -algebra over Z , and therefore,it is formal in the sense that, as an E -algebra over Z , it is equivalent to thePontryagin ring H ∗ ( T , Z ) given by its homology. As a model for the ∞ -category ofmodules over the latter, we may use the dg-category of dg-modules over H ∗ ( T , Z ).But a dg-module over H ∗ ( T , Z ) = Z [ d ] / ( d ) is precisely what is called a mixedcomplex in op. cit. This shows that D ( Z ) B T is equivalent to the ∞ -category ofmixed complexes. Some additional translation is necessary to bring the results inthe above form for which we refer to [ , Section 5].In the following, we will use the abbreviation B m = B cy ( h t i , h t a , t b i ; m ) . Theorem 1 shows, in particular, that the connectivity of B m tends to infinity with m . Hence, tensoring with THH( k ), we obtain an equivalenceTHH( k ) ⊗ B cy ( h t i , h t a , t b i ) ≃ L m ∈ Z > THH( k ) ⊗ B m ≃ Q m ∈ Z > THH( k ) ⊗ B m , which, in turn, implies a product decomposition(THH( k ) ⊗ B cy ( h t i , h t a , t b i )) tC p ≃ Q m ∈ Z > (THH( k ) ⊗ B m ) tC p . We will use the following result repeatedly below.
Lemma . The unstable Frobenius induces an equivalence
THH( k ) tC p ⊗ B m/p (THH( k ) ⊗ B m ) tC p , id ⊗ ˜ ϕ / / where the left-hand side is understood to be zero if p does not divide m . Proof.
We recall two facts from [ , Section 3]. The first is that the pointedspace B m/p is finite, and the second is that the cofiber of the composition B m/p ˜ ϕ / / ( B m ) hC p / / B m of the unstable Frobenius map and the canonical “inclusion” is a finite colimit offree pointed C p -cells. Here C p acts trivially on the left-hand and middle terms. LGEBRAIC K -THEORY OF PLANAR CUSPIDAL CURVES 5 Now, the map in the statement factors as the compositionTHH( k ) tC p ⊗ B m/p / / (THH( k ) ⊗ B m/p ) tC p / / (THH( k ) ⊗ B m ) tC p of the canonical colimit interchange map, where B m/p is equipped with the trivial C p -action, and the map of Tate spectra induced from the composite map above.The first fact implies that the left-hand map is an equivalence, and the second factimplies that the right-hand map is an equivalence. (cid:3) Proposition . Let G be a compact Lie group, let H ⊂ G be a closed subgroup,let λ = T H ( G/H ) be the tangent space at H = eH with the adjoint left H -action,and let S λ the one-point compactification of λ . For every spectrum with G -action X , there are canonical natural equivalences ( X ⊗ ( G/H ) + ) hG ≃ ( X ⊗ S λ ) hH , ( X ⊗ ( G/H ) + ) tG ≃ ( X ⊗ S λ ) tH . Proof.
We recall that for every map of spaces f : S → T , the restrictionfunctor f ∗ : Sp T → Sp S has both a left adjoint f ! and a right adjoint f ∗ . In thecase of the unique map p : BG → pt, we have p ! ( X ) ≃ X hG and p ∗ ( X ) ≃ X hG . Wenow consider the following diagram of spaces. BH BG pt f / / q (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ p (cid:0) (cid:0) ✁✁✁✁✁✁✁✁ The top horizontal map is the map induced by the inclusion of H in G . It is a fiberbundle, whose fibers are compact manifolds. Therefore, by parametrized Atiyahduality, its relative dualizing spectrum Df ∈ Sp BH is given by the sphere bundleassociated with the fiberwise normal bundle, which is Df ≃ S − λ . By definition ofthe dualizing spectrum, we have for all Y ∈ Sp BH , a natural equivalence f ! ( Y ⊗ S − λ ) ≃ f ∗ ( Y )in Sp BG . It follows that for all Y ∈ Sp BH , we have a natural equivalence p ∗ f ! ( Y ⊗ S − λ ) ≃ p ∗ f ∗ ( Y ) ≃ q ∗ ( Y )in Sp, which we also write as(( Y ⊗ S − λ ) ⊗ H G + ) hG ≃ Y hH . By [ , Theorem I.4.1 (3)], we further deduce a natural equivalence(( Y ⊗ S − λ ) ⊗ H G + ) tG ≃ Y tH . Indeed, the left-hand side vanishes for Y = Σ ∞ H + , and the fiber of the map(( Y ⊗ S − λ ) ⊗ H G + ) hG (( Y ⊗ S − λ ) ⊗ H G + ) tG can / / preserves colimits in Y .Finally, given X ∈ Sp BG , we set Y = f ∗ ( X ) ⊗ S λ to obtain the equivalences inthe statement. (cid:3) Strictly speaking this statement does not make sense, since BH and BG are only definedas homotopy types. What we mean is that the map is classified by a map BG → B Diff(
G/H ),where the latter is the diffeomorphism group of the compact manifold
G/H . LARS HESSELHOLT AND THOMAS NIKOLAUS
2. Proof of Theorem A
We consider the diagram with horizontal equalizersTC( A ) TC − ( A ) TP( A )TC( B ) TC − ( B ) TP( B ) / / ϕ / / can / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) ϕ / / can / / and wish to evaluate the cofiber of the left-hand vertical map. We havecofiber(THH( A ) → THH( B )) ≃ L m ≥ THH( k ) ⊗ B m ≃ Q m ≥ THH( k ) ⊗ B m . The right-hand equivalence follows from the fact that the connectivity of B m tendsto infinity with m . Therefore, the cofiber of TC( A ) → TC( B ) is identified with theequalizer of the induced maps Q m ≥ (THH( k ) ⊗ B m ) h T ϕ / / can / / Q m ≥ (THH( k ) ⊗ B m ) t T . While the canonical map “can” preserves this product decomposition, the Frobeniusmap “ ϕ ” takes the factor indexed by m to the factor factor indexed pm . Therefore,we write m = p v m ′ with m ′ not divisible by p and rewrite the diagram as Q m ′ Q v ≥ (THH( k ) ⊗ B p v m ′ ) h T ϕ / / can / / Q m ′ Q v ≥ (THH( k ) ⊗ B m ) t T , where both maps now preserve the outer product decomposition indexed by positiveintegers m ′ not divisible by p . We will abbreviate and writeTC( m ′ ) / / TC − ( m ′ ) ϕ / / can / / TP( m ′ )for the equalizer diagram given by the factors indexed by m ′ . To complete theproof, we evaluate the induced diagram on homotopy groups.We fix m = p v m ′ . It follows from Theorem 1 thatTHH( k ) ⊗ B m ≃ THH( k ) ⊗ Z ( Z ⊗ B m )agrees, up to canonical equivalence, with the total cofiber of the square THH( k ) ⊗ ( T /C m/ab ) + [2 ℓ ( a, b, m )] / / (cid:15) (cid:15) THH( k ) ⊗ ( T /C m/a ) + [2 ℓ ( a, b, m )] (cid:15) (cid:15) THH( k ) ⊗ ( T /C m/b ) + [2 ℓ ( a, b, m )] / / THH( k ) ⊗ ( T /C m ) + [2 ℓ ( a, b, m )] . By Proposition 3, the induced square of T -homotopy fixed points takes the formTHH( k ) hC m/ab [2 ℓ ( a, b, m ) + 1] / / (cid:15) (cid:15) THH( k ) hC m/a [2 ℓ ( a, b, m ) + 1] (cid:15) (cid:15) THH( k ) hC m/b [2 ℓ ( a, b, m ) + 1] / / THH( k ) hC m [2 ℓ ( a, b, m ) + 1]with the maps in the diagram given by the corestriction maps on homotopy fixedpoints. Indeed, the adjoint representation λ = T C r ( T /C r ) is a trivial one-dimensional Here we use in an essential way that, as a spectrum with T -action, THH( k ) is a Z -module. LGEBRAIC K -THEORY OF PLANAR CUSPIDAL CURVES 7 real C r -representation. We now write a = p u a ′ with a ′ not divisible by p and assumethat p does not divide b . If a and b both do not divide m , then(THH( k ) ⊗ B m ) h T ≃ THH( k ) hC m [2 ℓ ( a, b, m ) + 1] ≃ THH( k ) hC pv [2 ℓ ( a, b, m ) + 1] , and if a = p u a ′ but not b divides m , then(THH( k ) ⊗ B m ) h T ≃ cofiber(THH( k ) hC m/a → THH( k ) hC m )[2 ℓ ( a, b, m ) + 1] ≃ cofiber(THH( k ) hC pv − u → THH( k ) hC pv )[2 ℓ ( a, b, m ) + 1] . Similarly, if b but not a divides m , then(THH( k ) ⊗ B m ) h T ≃ cofiber(THH( k ) hC m/b → THH( k ) hC m )[2 ℓ ( a, b, m ) + 1] ≃ cofiber(THH( k ) hC pv → THH( k ) hC pv )[2 ℓ ( a, b, m ) + 1] ≃ , and if a and b both divide m , then(THH( k ) ⊗ B m ) h T ≃ , since B m ≃
0. By the same reasoning, we find that(THH( k ) ⊗ B m ) t T ≃ THH( k ) tC pv [2 ℓ ( a, b, m ) + 1] , if a and b both do not divide m , that(THH( k ) ⊗ B m ) t T ≃ cofiber(THH( k ) tC pv − u → THH( k ) tC pv )[2 ℓ ( a, b, m ) + 1] , if a = p u a ′ divides m but b does not divide m , and that(THH( k ) ⊗ B m ) t T ≃ , otherwise.It follows from [ , Section IV.4] that, on homotopy groups, the diagramTHH( k ) hC pu THH( k ) tC pu THH( k ) hC pv THH( k ) tC pv can / / can / / cor (cid:15) (cid:15) cor (cid:15) (cid:15) becomes W ( k )[ t, x ] / ( tx − p, p u t ) W ( k )[ t ± , x ] / ( tx − p, p u t ) W ( k )[ t, x ] / ( tx − p, p v t ) W ( k )[ t ± , x ] / ( tx − p, p v t ) , / / / / (cid:15) (cid:15) (cid:15) (cid:15) where deg( t ) = − x ) = 2, where the horizontal maps are the uniquegraded W ( k )-algebra homomorphisms that take t to t and x to x = pt − , andwhere the vertical maps are the unique maps of graded W ( k )[ t, x ]-modules thattake 1 to p v − u .We have now determined the diagramTC( m ′ ) / / TC − ( m ′ ) ϕ / / can / / TP( m ′ ) LARS HESSELHOLT AND THOMAS NIKOLAUS at the level of homotopy groups, the Frobenius given by Lemma 2. Hence, it ismerely a matter of bookkeeping to see that the statement of the theorem ensues.We recall the functions s = s ( a, b, r, p, m ′ ) and h = h ( a, b, r, p, m ′ ) from the p -typicaldecomposition recalled in the introduction, W S ( k ) / ( V a W S/a ( k ) + V b W S/b ( k )) ≃ Q m ′ ∈ N ′ W h ( k ) . Suppose first that neither a ′ nor b divides m ′ . Then π r +1 ((THH( k ) ⊗ B p v m ′ ) h T ) ≃ ( W v +1 ( k ) , if 0 ≤ v < s , W v ( k ) , if s ≤ v , π r +1 ((THH( k ) ⊗ B p v m ′ ) t T ) ≃ W v ( k ) , with s = s ( a, b, r, p, m ′ ). Also, we remark that the corresponding homotopy groupsin even degree 2 r are zero. The Frobenius map π r +1 ((THH( k ) ⊗ B p v m ′ ) h T ) π r +1 ((THH( k ) ⊗ B p v +1 m ′ ) t T ) ϕ / / is an isomorphism for 0 ≤ v < s , and the canonical map π r +1 ((THH( k ) ⊗ B p v m ′ ) h T ) π r +1 ((THH( k ) ⊗ B p v m ′ ) t T ) can / / is an isomorphism for s ≤ v . Hence, we have a map of exact sequences0 / / Q s ≤ v W v ( k ) / / ϕ − can (cid:15) (cid:15) TC − r +1 ( m ′ ) / / ϕ − can (cid:15) (cid:15) Q ≤ v
1. V. Angeltveit. Picard groups and the K -theory of curves with cuspidal singularities.arXiv:1901.00264.2. G. Corti˜nas. The obstruction to excision in K -theory and in cyclic homology. Invent. Math. ,164:143–173, 2006.3. J. A. Guccione, J. J. Guccione, M. J. Redondo, and O. E. Villamayor. Hochschild and cyclichomology of hypersurfaces.
Adv. Math. , 95:18–60, 1992.4. L. Hesselholt. On the K -theory of planar cuspical curves and a new family of polytopes. In Algebraic Topology: Applications and New Directions , volume 620 of
Contemp. Math. , pages145–182, Stanford, CA, July 23-27, 2012, 2014. Amer. Math. Soc., Providence, RI.5. L. Hesselholt. The big de Rham–Witt complex.
Acta Math. , 214:135–207, 2015.6. L. Hesselholt and I. Madsen. Cyclic polytopes and the K -theory of truncated polynomialalgebras. Invent. Math. , 130:73–97, 1997.7. L. Hesselholt and T. Nikolaus. Topological cyclic homology. Handbook of Homotopy Theory,Chap. 15 (to appear).8. M. Land and G. Tamme. On the K -theory of pullbacks. Ann. of Math. (to appear).9. M. Larsen. Filtrations, mixed complexes, and cyclic homology in mixed characteristic. K -Theory , 9:173–198, 1995.10. T. Nikolaus and P. Scholze. On topological cyclic homology. Acta Math. , 221:203–409, 2018.11. M. Speirs. On the K -theory of truncated polynomial algebras, revisited. arXiv:1901.10602.12. J. J. Sylvester. On subvariants, i.e. semi-invariants to binary quantics of an unlimited order. Amer. J. Math. , 5:79–136, 1882.
Nagoya University, Japan, and University of Copenhagen, Denmark
E-mail address : [email protected] Universit¨at M¨unster, Germany
E-mail address ::