Finite generation of Andre-Quillen (co-)homology of F-finite algebras
aa r X i v : . [ m a t h . A C ] S e p Finite generation of Andr´e-Quillen (co-)homologyof F-finite algebras
Cristodor IonescuSimion Stoilow Institute of Mathematicsof the Romanian AcademyP.O. Box 1-764RO 014700 BucharestRomaniaemail: [email protected] 30, 2019
Abstract
We prove that the Andr´e-Quillen homology and cohomology modulesof F-finite Z ( p ) -algebras are finitely generated.MSC 2010: 13D03 Andr´e-Quillen homology was invented independently by M. Andr´e and D. Quillen(see [1], [8] and [2]). The theory proved its importance over the years, for exam-ple by characterizing several classes of noetherian rings and morphisms betweensuch rings (see [2], [8] and [6]). The finite generation of certain Andr´e-Quillenhomology modules appears in many situations, maybe the most well-known be-ing the proof of the celebrated Andr´e’s Theorem on the localization of formalsmoothness (see [3]). In a recent paper, Dundas and Morrow [4] proved thatfor a F-finite algebra A , the modules H i ( C, A, E ) , i ≥ C is any of the rings Z , Z ( p ) or Z /p e Z and E is a finitely generated A -module. Actually they proved more, namely that also the higher Andr´e-Quillenhomology modules are finitely generated, as their final goal was to prove thefinite generation of the Hochschild homology modules. Our purpose is to giveanother proof of the finite generation of H i ( C, A, E ), where C is as above, aswell as some similar results concerning the Andr´e-Quillen cohomology modules.All the rings are commutative, with unit and noetherian. All over the paper p > F p the prime field ofcharacteristic p and by Z ( p ) the ring of fractions of Z with denominators notdivisible by p. For a ring A containing the field F p , the Frobenius morphism of A will mean the ring morphism F : A → A, F ( a ) = a p . Remark 1.1
Let A be a Z ( p ) -algebra. We have a canonical commutative dia-gram of ring morphisms ✲ Z ( p ) ✲ A Z /p e Z ❄ ✲ Z /p Z ≃ Z ( p ) /p Z ( p ) ≃ F p ❄ ✲ A/pA. ❄ There are two possibilities:1) pA = 0 . Then A = A/pA contains F p . pA = 0 . Then A does not contain F p , but A/pA does.
Thus in both cases we can say that
A/pA contains F p and the followingdefinition makes sense and generalizes the usual one (see [7] and [4]): Definition 1.2
A noetherian Z ( p ) -algebra A is called F-finite, if the Frobeniusmorphism of A/pA is a finite morphism.
We will heavily rely on the following result of Gabber:
Theorem 1.3 [5, Rem. 13.6]
Let k be a field of characteristic p > and Abe a noetherian F-finite k-algebra. Then A is the quotient of a regular F-finitek-algebra. Remark 1.4
Theorem 1.3 does not extend to the case 2) of remark 1.1, that isto algebras not containing a field of characteristic p . The next lemma is well-known, but we couldn’t find a precise reference.
Lemma 2.1
Let A be a noetherian ring and E u → F v → G be an exact sequenceof A-modules. If E and G are finitely generated, then F is finitely generated.Proof: We have the exact sequences0 → ker( u ) → E → E/ ker( u ) ∼ = im( u ) → → ker( v ) → F → F/ ker( v ) ∼ = im( v ) → . From the first sequence it follows that im( u ) = ker( v ) is finitely generated andsince im( v ) is finitely generated, from the second one we obtain the conclusion. Proposition 2.2
Let k be a perfect field of characteristic p > and A be anoetherian F-finite k-algebra. Then H i ( k, A, E ) and H i ( k, A, E ) are finitelygenerated A -modules for any i ≥ and for any finitely generated A -module E. Proof:
From 1.3 it follows that there exists a regular ring R and a surjectivemorphism R → A. Consider the Jacobi-Zariski exact sequence attached to k → R → A. For any i ≥ H i ( k, R, E ) → H i ( k, A, E ) → H i ( R, A, E ) . H i ( R, A, E ) is finitely generated. As k isperfect and R is regular, we get that k → R is a regular morphism and from [6,Thm. 9.5], we have that H i ( k, R, E ) = 0 , ∀ i ≥ , whence H i ( k, A, E ) is finitelygenerated. For i = 0 we have that H ( k, A, E ) ∼ = Ω A/k = Ω
A/k [ A p ] which isfinitely generated.As R is F-finite, the module of differentials Ω R/k is finitely generated. Since H i ( k, R, R ) = 0 for all i ≥ , by [2, Prop. 3.19] it follows that H j ( k, R, E ) ≃ Ext jR (Ω R/k , E ) , j ≥ , hence by [9, Th. 7.36] we obtain that H j ( k, R, E ) isfinitely generated. Now the Jacobi-Zariski exact sequence associated to k → R → A is H i ( R, A, E ) → H i ( k, A, E ) → H i ( k, R, E )and 2.1 ends the proof. Remark 2.3
Along the same lines one can prove that, if k is a perfect field(possibly of characteristic zero) and A is a k -algebra which is a quotient of aregular ring, then H i ( k, A, E ) is a finitely generated A -module for any i ≥ andany finitely generated A -module E. Proposition 2.4
Let k be a field of characteristic p > , A be a noetherianF-finite k-algebra and E a finitely generated A-module. Then:a) H i ( k, A, E ) and H i ( k, A, E ) are finitely generated A -modules for any i ≥ , i = 1 ;b) If moreover k is F-finite, then H ( k, A, E ) and H ( k, A, E ) are finitelygenerated A -modules.Proof: a) Let us consider the Jacobi-Zariski exact sequence in homology asso-ciated to the morphisms F p → k → A, namely . . . → H i ( F p , k, E ) → H i ( F p , A, E ) → H i ( k, A, E ) → H i − ( F p , k, E ) → . . . But for i ≥ , by the separability of k over F p and [2, Prop. VII.11 and VII.4]we have that H i ( F p , A, E ) ∼ = H i ( k, A, E ) , hence by 2.2 we obtain the assertion.For i = 0 we have H ( k, A, E ) = Ω A/k ⊗ A E = Ω A/k [ A p ] ⊗ A E, hence it is finitelygenerated.Let us now consider the Jacobi-Zariski exact sequence in cohomology associatedto the same morphisms as above, that is . . . → H i − ( F p , k, E ) → H i ( k, A, E ) → H i ( F p , A, E ) → H i ( F p , k, E ) → . . . Again, for i ≥ , by the separability of k over F p and [2, Prop. VII.4] we get theisomorphism H i ( F p , A, E ) ∼ = H i ( k, A, E ) , hence by 2.2 we obtain the assertion.b) We have the exact sequences0 → H ( F p , A, E ) → H ( k, A, E ) → Ω k ⊗ A E and Der F p ( k, E ) → H ( k, A, E ) → H ( F p , A, E ) → H ( F p , k, E ) = (0) . But the hypothesis implies that Ω k is finitely generated, hence from 2.2 and2.1 we get that H ( k, A, E ) is finitely generated. Moreover Der F p ( k, E ) =Hom k (Ω k/ F p , E ) which is finitely generated, since Ω k/ F p is finitely generated.From the second exact sequence, 2.1 and 2.2 we obtain the desired conclusion.3 xample 2.5 If [ k : k p ] = ∞ the module H ( k, A, E ) is not necessarily finitelygenerated. Indeed, let k be a field such that [ k : k p ] = ∞ and let ¯ k be thealgebraic closure of k. The Jacobi-Zariski sequence associated to F p → k → ¯ k is → H ( k, ¯ k, ¯ k ) → Ω k ⊗ ¯ k → Ω ¯ k → Ω ¯ k/k → . Then Ω ¯ k is finitely generated and if H ( k, ¯ k, ¯ k ) is finitely generated, by 2.1 itfollows that Ω k is finitely generated, contradicting the hypothesis that [ k : k p ] = ∞ . Corollary 2.6
Let p be a prime number, A be a noetherian F-finite F p -algebraand E a finitely generated A-module. Then for any i ≥ and any e ≥ , the A -modules H i ( Z , A, E ) , H i ( Z /p e Z , A, E ) , H i ( Z , A, E ) and H i ( Z /p e Z , A, E ) arefinitely generated.Proof: Considering the Jacobi-Zariski exact sequence associated to the mor-phisms Z → Z /p Z = F p → A we get an exact sequence H i ( Z , F p , E ) → H i ( Z , A, E ) → H i ( F p , A, E ) . By 2.2 we know that H i ( F p , A, E ) is finitely generated and by [2], IV.55 weobtain that H i ( Z , F p , E ) is finitely generated. Now 2.1 tells us that H i ( Z , A, E )is finitely generated. For the second assertion let us remark first that we canassume e ≥ . Consider the morphisms Z /p e Z → Z /p Z = F p → A and theJacobi-Zariski associated sequence H i ( Z /p e Z , F p , E ) → H i ( Z /p e Z , A, E ) → H i ( F p , A, E ) . Then applying 2.2, 2.1 and [2, Prop. IV.55] we get that H i ( Z /p e Z , A, E ) isfinitely generated. In the same way, using the Jacobi-Zariski exact sequence incohomology, we prove that the cohomology modules H i ( Z , A, E ) and H i ( Z /p e Z , A, E )are finitely generated. Remark 2.7
Looking at Remark 2.3 and Corollary 2.6, one can see that if k is a perfect field (possibly of characteristic zero) and A is a k -algebra which is aquotient of a regular ring, then H i ( Z , A, E ) is a finitely generated A -module forany i ≥ and any finitely generated A -module E. Corollary 2.8
Let p be a prime number, A be a noetherian F-finite F p -algebraand E a finitely generated A-module. Then H i ( Z ( p ) , A, E ) and H i ( Z ( p ) , A, E ) are finitely generated A -modules, for any i ≥ .Proof: For i = 0 we have H ( Z ( p ) , A, E ) ≃ Ω A/ Z ( p ) ⊗ E ≃ Ω A/ Z ⊗ E ≃ H ( Z , A, E )which is finitely generated by 2.6. For i ≥ , from the morphisms Z → Z ( p ) → A we have the exact sequence0 = H i ( Z , Z ( p ) , E ) → H i ( Z , A, E ) → H i ( Z ( p ) , A, E ) → H i − ( Z , Z ( p ) , E ) = 0and we apply again 2.6. The proof for the cohomology modules is similar.We shall consider now the case of a Z ( p ) -algebra.4 roposition 2.9 Let A be a noetherian F-finite Z ( p ) -algebra and E a finitelygenerated A-module. Then H i ( Z ( p ) , A, E ) and H i ( Z ( p ) , A, E ) are finitely gener-ated A -modules, for all i ≥ . Proof:
The case pA = 0 , that is A contains F p , was considered above.If pA = 0 , consider first the morphisms Z ( p ) → F p → A/pA.
We have theJacobi-Zariski exact sequence H i ( Z ( p ) , F p , E ) → H i ( Z ( p ) , A/pA, E ) → H i ( F p , A/pA, E ) . But H i ( Z ( p ) , F p , E ) is finitely generated by [2], IV.55 and H i ( F p , A/pA, E ) isfinitely generated by 2.2. By 2.1 we obtain that H i ( Z ( p ) , A/pA, E ) is finitelygenerated. Consider now the morphisms Z ( p ) → A → A/pA.
We have the exactsequence H i +1 ( A, A/pA, E ) → H i ( Z ( p ) , A, E ) → H i ( Z ( p ) , A/pA, E ) . By the previous assertion H i ( Z ( p ) , A/pA, E ) is finitely generated and by [2],IV.55 H i +1 ( A, A/pA, E ) is finitely generated. Now apply 2.1 to get the assertion.For the cohomology modules, consider first the morphisms Z ( p ) → F p → A/pA.
We have the exact sequence H i ( F p , A/pA, E ) → H i ( Z ( p ) , A/pA, E ) → H i ( Z ( p ) , F p , E ) . Then H i ( Z ( p ) , F p , E ) is finitely generated by [2], IV.55 and H i ( F p , A/pA, E ) isfinitely generated by 2.2. By 2.1 we obtain that H i ( Z ( p ) , A/pA, E ) is finitelygenerated. Consider now the morphisms Z ( p ) → A → A/pA.
We have theassociated Jacobi-Zariski exact sequence H i ( Z ( p ) , A/pA, E ) → H i ( Z ( p ) , A, E ) → H i +1 ( A, A/pA, E ) . As in the proof of 2.9 it follows that H i ( Z ( p ) , A, E ) is finitely generated. Corollary 2.10
Let A be a noetherian F-finite Z ( p ) -algebra and E a finitelygenerated A-module. Then H i ( Z , A, E ) and H i ( Z , A, E ) are finitely generated A -modules, for all i ≥ . Proof:
It follows at once from 2.9 and the Zariski-Jacobi exact sequence as-sociated to Z → Z ( p ) → A , taking account of the fact that H i ( Z , Z ( p ) , E ) = H i ( Z , Z ( p ) , E ) = (0) , ∀ i ≥ , cf. [2, Prop. V.25]. References [1] M. Andr´e -
M´ethode simpliciale en alg`ebre homologique et alg`ebre com-mutative - Lecture Notes in Math. 32, Springer Verlag, Berlin HeidelbergNew-York, 1967.[2] M. Andr´e -
Homologie des alg`ebres commutatives - Springer Verlag, BerlinHeidelberg New-York, 1974.[3] M. Andr´e -
Localisation de la lissit´e formelle - Manuscripta Math.,13(1974), 297-307. 54] B. I. Dundas, M. Morrow -
Finite generation and continuity of topologicalHochschild and cyclic homology - Ann. Scient. ´Ec. Norm. Sup., 50(2017),201-238.[5] O. Gabber -
Notes on some t-structures - in: Geometric Aspects of DworkTheory, II, A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz, F. Loeser(eds.) - Walter de Gruyter GmbH & Co. KG, Berlin, 2004, 711-734.[6] S. Iyengar -
Andr´e-Quillen homology of commutative algebras - in: Interac-tions between Homotopy Theory and Algebra, L. L. Avramov, J. D. Chris-tensen, W. G. Dwyer, M. A. Mandell, B. E. Shipley (eds.) - ContemporaryMath., 436, 2007, 203-234.[7] A. Langer, T. Zink -
De Rham-Witt cohomology for a proper and smoothmorphism - J. Inst. Math. Jussieu., 3(2004), 231-314.[8] D. Quillen -
On the(co-)homology of commutative rings - in: Applicationsof categorical algebra, New York, 1968, Proc. Symp. Pure Math.17, Amer.Math. Soc., Providence, RI, 1970, 65–87.[9] J. Rotman -