KOSZUL COMPLEX OVER SKEW POLYNOMIAL RINGS

Transcription

1 KOSZUL COMPLEX OVER SKEW POLYNOMIAL RINGS JOSEP ÀLVAREZ MONTANER, ALBERTO F. BOIX, AND SANTIAGO ZARZUELA Dedicated to Professor Gennady Lyubeznik on the occasion of his 60th birthday Abstract. We construct a Koszu compex in the category of eft skew poynomia rings associated to a fat endomorphism that provides a finite free resoution of an idea generated by a Koszu reguar sequence. 1. Introduction Let A be a commutative Noetherian ring of characteristic p > 0 and F : A A the associated Frobenius map for which F (a) = a p for a a A. The study of A-modues with an action of the Frobenius map has received a ot of attention over the ast decades and is at the core of the ceebrated theory of tight cosure deveoped by M. Hochster and C. Huneke in [HH90] and the theory of F -modues introduced by G. Lyubeznik [Lyu97]. To provide an action of the Frobenius on an A-modue M is equivaent to give a eft A[Θ; F ]- modue structure on M. Here A[Θ; F ] stands for the eft skew poynomia ring associated to F, which is an associative, N-graded, not necessariy commutative ring extension of A. More generay, we may aso consider the skew poynomia rings A[Θ; F e ] associated to the e-th iterated Frobenius map and the graded ring F M = e 0 F e M introduced by G. Lyubeznik and K. E. Smith in [LS01] that coects a the A[Θ; F e ]-modue structures on M or equivaenty, a possibe actions of F e on M. In the case that F M is principay generated then it is isomorphic to A[Θ; F ]. We want aso to singe out here that, under the terminoogy skew poynomia ring of Frobenius type, Y. Yoshino [Yos94] studied A[Θ; F ] and eft modues over it, and used this study to obtain some new resuts about tight cosure theory; foowing the same spirit, R. Y. Sharp [Sha09] aso studied the idea theory of A[Θ; F ]. The case when A is a fied was studied in detai by F. Enescu [Ene12]. One may aso deveop a dua notion of right skew poynomia ring associated to F that we denote A[ε; F ]. A eft A[ε; F ]-modue structure on M is then equivaent to provide an action of a Cartier operator as considered by M. Bicke and G. Böcke in [BB11]. The corresponding ring coecting a the A[ε; F e ]-modue structures on M, that we denote C M = e 0 CM e, was deveoped by K. Schwede [Sch11] and M. Bicke [Bi13] (see [BS13] for a nice survey) and has been a hot topic in recent years because of its roe in the theory of test ideas. Athough the theory of skew poynomia rings is cassica (see [GW04, Chapter 2] and [MR01, Chapter 1, Sections 2 and 6]), the aim of this work is to go back to its basics and deveop new toos that shoud be potentiay usefu in the study of modues with a Frobenius or Cartier action. Our approach wi be in a sighty more genera setting as we wi consider a Noetherian commutative ring A (not necessariy of positive characteristic) and a ring homomorphism ϕ : A A. Athough 2010 Mathematics Subject Cassification. Primary 13A35; Secondary 13B10, 14B25, 16S36. Key words and phrases. Cartier agebras, Frobenius agebras, skew poynomia rings. The first author is partiay supported by Generaitat de Cataunya 2014SGR-634 project and Spanish Ministerio de Economía y Competitividad MTM P. The second and third authors are supported by Spanish Ministerio de Economía y Competitividad MTM P. The second author is partiay supported by the Israe Science Foundation (grant No. 844/14). 1

2 2 J. ÀLVAREZ MONTANER, A. F. BOIX, AND S. ZARZUELA most of the resuts are mid generaizations of the case of the Frobenius morphism we point out that this genera approach has aready been fruitfu (see [SW07]). Now, we overview the contents of this paper for the convenience of the reader. In Section 2 we introduce a the basics on eft and right skew poynomia rings associated to a morphism ϕ that wi be denoted by A[Θ; ϕ] and A[ε; ϕ]. More generay, given an A-modue M, we wi consider the corresponding rings F M,ϕ and C M,ϕ. The main resut of this Section is Theorem 2.11 which states that, whenever ϕ is a finite morphism, we have an isomorphism between the graded pieces F E R,ϕ e and Ce R,ϕ induced by Matis duaity. Here, E R is the injective hu of the residue fied of R = A/I with I A := K[x 1,..., x n ] being an idea. In Section 3 we present the main resut of this work. Here we consider a fat morphism ϕ satisfying some extra condition (which is naturay satisfied in the case of the Frobenius morphism). First we introduce the ϕ-koszu compex which is a Koszu-type compex associated to x 1,..., x n A in the category of eft A[Θ; ϕ]-modues. In Theorem 3.14 we prove that, whenever x 1,..., x n is an A-Koszu reguar sequence (see Definition 3.13), the ϕ-koszu compex provides a free resoution of A/I n in the category of eft A[Θ; ϕ]-modues, where I n is the idea generated by x 1,..., x n. In the case where A is a reguar ring of positive characteristic p > 0 and ϕ = F e is the e-th iteration of the Frobenius, we obtain a free resoution, in the category of eft A[Θ; F e ]-modues, of the idea I n (see Coroary 3.17). To the best of our knowedge this is one of the first expicit exampes of free resoutions of A- modues as modues over a skew poynomia ring. We hope that a deveopment of this theory of free resoutions woud provide, in the case of the Frobenius morphism, more insight in the study of modues with a Frobenius action. 2. Preiminaries Let A be a commutative Noetherian ring and ϕ : A A a ring homomorphism. Associated to ϕ we may consider non necessariy commutative agebra extensions of A that are usefu in the sense that, some non finitey generated A-modues become finitey generated when viewed over these extensions. In this section we wi coect the basic facts on this theory keeping aways an eye on the case of the Frobenius morphism. First we reca that ϕ aows to describe a covariant functor, the pushforward or restriction of scaars functor ϕ, from the category of eft A-modues to the category of eft A-modues. Namey, given an A-modue M, ϕ M is the (A, A)-bimodue having the usua A-modue structure on the right but the eft structure is twisted by ϕ; that is, if we denote by ϕ m (m M) an arbitrary eement of ϕ M then, for any a A a ϕ m := ϕ (ϕ(a)m). Moreover, given a map g : M N of eft A-modues we can define a map ϕ g : ϕ M ϕ N of eft A-modues by setting, for any m M, ϕ g(ϕ m) := ϕ g(m) Left and right skew-poynomia rings. The most basic ring extensions given by a ring homomorphism are the so-caed skew-poynomia rings or Ore extensions. This is a cassica object of study, and we refer either to [GW04, Chapter 2] or [MR01, Chapter 1, Sections 2 and 6] for further detais. Definition 2.1. Let A be a commutative Noetherian ring and ϕ : A A a ring homomorphism.

3 KOSZUL COMPLEX OVER SKEW POLYNOMIAL RINGS 3 Left skew poynomia ring of A determined by ϕ: A[Θ; ϕ] := A Θ Θa ϕ(a)θ a A, that is, A[Θ; ϕ] is a free eft A-modue with basis {θ i } i N0 and θ a = ϕ(a)θ for a a A. Right skew poynomia ring of A determined by ϕ: A[ε; ϕ] := A ε aε εϕ(a) a A, that is, A[ε; ϕ] is a free right A-modue with basis {ε i } i N0 and a ε = εϕ(a) for a a A. Remark 2.2. The reader wi easiy note that A[Θ; ϕ] op = A[ε; ϕ] and A[ε; ϕ] op = A[Θ; ϕ], where ( ) op denotes the opposite ring. Both ring extensions are determined by the foowing universa property. Proposition 2.3. Let B = A[Θ; ϕ] be the eft (resp. B = A[ε; ϕ] be the right) skew poynomia ring of A determined by ϕ. Suppose that we have a ring T, a ring homomorphism φ : A T and an eement y T such that, for each a A, yφ(a) = φ(ϕ(a))y (resp. φ(a)y = yφ(ϕ(a))). Then, there is a unique ring homomorphism B ψ T such that ψ(θ) = y (resp. ψ(ε) = y) which makes the triange A B commutative. φ T This universa property aow us to provide more genera exampes of skew poynomia rings. Exampe 2.4. Let u A. As uθ is an eement of A[Θ; ϕ] such that, for any a A, (uθ)a = uθa = uϕ(a)θ = ϕ(a)(uθ), it foows from the universa property for eft skew poynomia rings that there is a unique A agebra homomorphism A[Θ ; ϕ] A[Θ; ϕ] sending Θ to uθ; we sha denote the image of this map by A[uΘ; ϕ]. The previous argument shows that A[uΘ; ϕ] can be aso regarded as a eft skew poynomia ring. Anaogousy, A[εu; ϕ] can be aso regarded as right skew poynomia ring since we have, for any a A, a(εu) = εϕ(a)u = (εu)ϕ(a) The case of Frobenius homomorphism. We singe out the case where A is a ring of positive characteristic p > 0 and ϕ = F is the Frobenius homomorphism. The speciaization of Definition 2.1 to this case is the foowing: The Frobenius skew poynomia ring of A is the non-commutative graded ring A[Θ; F ]; that is, the free eft A-modue with basis {Θ e } e N0 and right mutipication given by ψ Θ a = a p Θ. The Cartier skew poynomia ring of A is the non-commutative graded ring A[ε; F ]; that is, the free right A-modue with basis {ε e } e N0 and eft mutipication given by a ε := εa p.

4 4 J. ÀLVAREZ MONTANER, A. F. BOIX, AND S. ZARZUELA It turns out that A[Θ; F ] is rarey eft or right Noetherian, as it is expicity described by Y. Yoshino in [Yos94, Theorem (1.3)]. He aso provided effective descriptions of basic exampes of eft modues over A[Θ; F ]. Namey, the ring A, the ocaizatons A a at eements a A and oca cohomoogy modues HI i (A), where I is any idea of A, have an abstract structure as finitey generated eft A[Θ; F ]-modues that we coect in the foowing resut. We refer to [Yos94, pp ] for further detais. Proposition 2.5. Let A be a commutative Noetherian ring of characteristic p, et a A be any eement and et J(a) denote the eft idea of A[Θ; F ] generated by the infinite set {a pe 1 Θ e 1 e N}. Then, the foowing statements hod. (i) A has a natura structure as eft A[Θ; F ]-modue given by A = A[Θ; F ]/A[Θ; F ] Θ 1 = A[Θ; F ]/J(1). Moreover, if A is a oca ring: (ii) The ocaization A a has a natura structure as eft A[Θ; F ]-modue given by A a = A[Θ; F ]/J(a). (iii) The Čech compex of A with respect to a 1,..., a t t 0 A A ai A ai a j... i=1 1 i<j t t A a1 â i a t A a1 a t 0 is a compex of eft A[Θ; F ]-modues which is isomorphic to the foowing compex: t 0 A[Θ; F ]/J(1) A[Θ; F ]/J(a i ) A[Θ; F ]/J(a i a j ) i=1 i=1 1 i<j t t A[Θ; F ]/J(a 1 â i a t ) A[Θ; F ]/J(a 1 a t ) 0. i=1 (iv) Any oca cohomoogy modue HI i (A) has an abstract structure as a finitey generated eft A[Θ; F ]-modue. For exampe, if (A, m, K) is a oca ring of characteristic p of dimension d 1, and a 1,..., a d is a system of parameters for A, then H d m(a) = A[Θ; F ]/ (J(a 1 a d ) + A[Θ; F ] a 1,..., a d ). In this section we are going to give a natura generaization of skew poynomia rings associated to a ring homomorphism ϕ. Before doing so, we briefy reca how to give a modue structure over a skew poynomia ring by means of the so-caed ϕ and ϕ 1 -inear maps. Definition 2.6. Let A be a commutative Noetherian ring, ϕ : A A a ring homomorphism and M be an A-modue. Given ψ, φ End A (M) : (i) We say that ψ is ϕ-inear provided ψ(am) = ϕ(a)ψ(m) for any (a, m) A M. (ii) We say that φ is ϕ 1 -inear provided φ(ϕ(a)m) = aφ(m) for any (a, m) A M. We denote by End ϕ (M) and End ϕ 1(M) the A-endomorphisms of M which are ϕ-inear and ϕ 1 -inear respectivey. These endomorphisms can be interpreted in terms of the pushforward functor since we have the foowing bijections: End ϕ (M) Hom A (M, ϕ M), End ϕ 1(M) Hom A (ϕ M, M) ψ [m ϕ (ψ(m))] ψ [ϕ m ψ(m)]

5 KOSZUL COMPLEX OVER SKEW POLYNOMIAL RINGS 5 An A[Θ; ϕ]-modue is simpy an A-modue M together with a suitabe action of Θ on M. Actuay, one ony needs to consider a ϕ-inear map ψ : M M and define Θ m := ψ(m) for a m M. Anaogousy, an A[ε; ϕ]-modue is an A-modue M together with an action of ε given by a ϕ 1 -inear map φ : M M; we record a these simpe remarks into the foowing: Proposition 2.7. Let M be an A-modue. (i) There is a bijective correspondence between End ϕ (M) and the eft A[Θ; ϕ]-modue structures which can be attached to M. (ii) There is a bijective correspondence between End ϕ 1(M) and the eft A[ε; ϕ]-modue structures which can be attached to M. More generay, we may consider the e-th powers ϕ e of the ring homomorphism ϕ and define the corresponding notion of ϕ e and ϕ e -inear maps. We may coect a these morphisms in a suitabe agebra that woud provide a generaization of the Frobenius agebra introduced by G. Lyubeznik and K. E. Smith in [LS01, Definition 3.5] and the Cartier agebra considered by K. Schwede [Sch11] and generaized by M. Bicke [Bi13] (see aso [BS13]). Definition 2.8. Let A be a commutative Noetherian ring and ϕ : A A a ring homomorphism. Let M be an A-modue. Then we define: Ring of ϕ-inear operators on M: Is the associative, N-graded, not necessariy commutative ring where F M,ϕ e for any given ψ e F M,ϕ e F M,ϕ := e 0 F M,ϕ e, := Hom A (M, ϕ e M) = End ϕ e(m). A product is defined as ψ e ψ e := ϕ e (ψ e ) ψ e F M,ϕ e+e. and ψ e F M,ϕ e. Ring of ϕ 1 -inear operators on M: Is the associative, N-graded, not necessariy commutative ring C M,ϕ := e 0 where Ce M,ϕ := Hom A (ϕ e M, M) = End ϕ e(m). A product is defined as for any given φ e C M,ϕ e and φ e C M,ϕ e. C M,ϕ e, φ e φ e := φ e ϕ e (φ e ) C M,ϕ e+e Actuay, the generaization of Cartier agebra given by M. Bicke in [Bi13] woud be interpreted in our context as foows: Definition 2.9. An A-Cartier agebra with respect to ϕ is an N-graded A-agebra C ϕ := e 0 C ϕ e such that, for any (a, φ e ) A C ϕ e, we have that a φ e = φ e ϕ e (a). The A-agebra structure of C ϕ is given by the natura map from A to C ϕ 0. We aso assume that the structura map A C ϕ 0 is surjective.

6 6 J. ÀLVAREZ MONTANER, A. F. BOIX, AND S. ZARZUELA We point out that C M,ϕ is generay NOT an A-Cartier agebra with respect to ϕ, since C M,ϕ 0 = End A (M) and therefore the natura map A End A (M) is, in genera, not surjective. Nevertheess, if M = A/I (where I is any idea of A) then End A (M) = A/I and therefore it foows that C A/I,ϕ is an A-Cartier agebra. Whenever the ring of ϕ-inear (resp. ϕ 1 -inear) operators is principay generated it is isomorphic to a eft (resp. right) skew poynomia ring. This is the case of the ring of ϕ-inear operators of the ring A; notice that, when A is of prime characteristic and ϕ = F is the Frobenius map, this was aready observed by G. Lyubeznik and K. E. Smith [LS01, Exampe 3.6]. Exampe We have that F A,ϕ = A[Θ; ϕ]. Indeed, fix e N and et ψ e Fe A,ϕ. We point out that, for any a A, In this way, set ψ e (a) = ψ e (a 1) = ϕ e (a)ψ e (1) = ψ e (1)ϕ e (a). F A,ϕ e b e AΘ e ψ e ψ e (1)Θ e. The previous straightforward cacuation shows the injectivity of this map. In fact, it is a bijective map with inverse AΘ e aθ e aϕ e. F A,ϕ e In this way, setting F A,ϕ b A[Θ; ϕ] as the unique map of rings given in degree e by b e it foows that b is an isomorphism of graded agebras Duaity between Cartier agebras and Frobenius agebras. Let A = K[x 1,..., x d ] be a forma power series ring with d indeterminates over a fied K, I A an idea and R := A/I. Let ϕ : A A be a ring homomorphism such that ϕ e A is finitey generated as A-modue. The aim of this subsection is to estabish an expicit correspondence, at the eve of graded pieces, between C R,ϕ and F ER,ϕ given by Matis duaity. This woud extend the correspondence given in the case of the Frobenius map (cf. [BB11, Proposition 5.2] and [SY11, Theorem 1.20 and Coroary 1.21]). Theorem Let ϕ : A A be a ring homomorphism such that ϕ e A is finitey generated as A-modue. Then we have that Hom A (ϕ e R, R) = HomA (E R, ϕ e E R ) and Hom A (E R, ϕ e E R ) = HomA (ϕ e R, R). Before proving this theorem we have to show a previous statement which we sha need during its proof; abeit the beow resut was obtained by F. Enescu and M. Hochster in [EH08, Discussion (3.4)] (see aso [Yos94, Lemma (3.6)]), we review here their proof for the convenience of the reader. Lemma Let (A, m, K) (B, n, L) be a oca homomorphism of oca rings, and suppose that mb is n-primary and that L is finite agebraic over K (both these conditions hod if B is moduefinite over A). Let E := E A (K) and E B (L) denote choices of injective hus for K over A and for L over B, respectivey. Then, the functor Hom A (, E), on B-modues, is isomorphic with the functor Hom B (, E B (L)). Proof. First of a, we underine that Hom A (, E), on B-modues, can be identified via adjunction with Hom A (( ) A B, E) = Hom B (, Hom A (B, E))

7 KOSZUL COMPLEX OVER SKEW POLYNOMIAL RINGS 7 and therefore Hom A (B, E) is injective as B-modue. Moreover, as mb is n-primary any eement of Hom A (B, E) is kied by a power of n and therefore Hom A (B, E) = E B (L). In this way, it ony remains to check that = 1. Indeed, we note that Hom A (L, E) = Hom A (L, K). However, as A-modue, Hom A (L, K) is abstracty isomorphic to L (here we are using the assumption that L is finite agebraic over K). Thus, a these foregoing facts impy that E B (L) = Hom A (B, E), hence Hom A (( ) A B, E) = Hom B (, E B (L)) and we get the desired concusion. Proof of Theorem First of a, we underine that Hom A (ϕ e R, R) = HomA (E R, ϕ e (R) ). Now, et E be the injective hu of the residue fied of ϕ e R. In this way, from Lemma 2.12 we deduce that Hom A (, E) = Hom ϕ e A(, E ) as functors of ϕ e A-modues. Therefore, combining a these facts joint with the exactness of ϕ e it foows that ϕ e (R) = HomA (ϕ e R, E) = Hom ϕ e A(ϕ e R, E ) = ϕ e Hom A (R, E) = ϕ e E R. Thus, taking into account this ast chain of isomorphisms one obtains the first desired concusion. On the other hand, using once more Lemma 2.12 it turns out that ϕ e (E R ) = HomA (ϕ e E R, E) = Hom ϕ e A(ϕ e E R, ϕ e E) = ϕ e (E R) = ϕ e R. Thus, bearing in mind this ast chain of isomorphisms it foows that Hom A (E R, ϕ e E R ) = HomA (ϕ e (E R ), E R) = Hom A (ϕ e R, R), just what we finay wanted to show The case of Frobenius homomorphism. Once again we singe out the case where A is a ring of positive characteristic p > 0 and ϕ = F is the Frobenius homomorphism. In this case we adopt the terminoogy of p e and p e -inear maps or, foowing [And00], Frobenius and Cartier inear maps. Definition Let M be an A-modue and ψ, φ End A (M). (i) We say that ψ is p e -inear provided ψ(am) = a pe ψ(m) for any (a, m) A M. Equivaenty, ψ Hom A (M, F e M). (ii) We say that φ is p e -inear provided φ(a pe m) = aφ(m) for any (a, m) A M. Equivaenty, φ Hom A (F e M, M). The corresponding rings of p e and p e -inear maps are defined as foows: Definition Let A be a commutative Noetherian ring of prime characteristic p and et M be an A-modue. (i) The Frobenius agebra attached to M is the associative, N-graded, not necessariy commutative ring F M := e 0 Hom A (M, F e M). (ii) The Cartier agebra attached to M is the associative, N-graded, not necessariy commutative ring C M := e 0 Hom A (F e M, M).

8 8 J. ÀLVAREZ MONTANER, A. F. BOIX, AND S. ZARZUELA In this work we are mainy interested in the case where the Frobenius (resp. Cartier) agebra of a modue is principay generated and thus isomorphic to the eft (resp. right) skew poynomia ring. G. Lyubeznik and K. E. Smith aready carried out such an exampe in [LS01, Exampe 3.7]. Exampe Let (A, m, K) be a oca ring of characteristic p. Then F Hdim(A) m (A) = S[Θ; F ], where S denotes the S 2 -ification of the competion Â. Another source of exampes is given by the foowing resut. Proposition Let (A, m, K) be a compete F -finite oca ring of characteristic p and E A wi stand for a choice of injective hu of K over A. Then, the foowing statements hod. (i) If A is quasi Gorenstein then F E A is principa. (ii) If A is norma then F E A is principa if and ony if A is Gorenstein. (iii) If A is a Q-Gorenstein norma domain then F E A is a finitey generated A-agebra if and ony if p is reativey prime with the index of A. (iv) If A is a Q-Gorenstein norma domain then F E A is principa if and ony if the index of A divides p 1. Proof. If A is quasi Gorenstein then E A dim(a) = H (A). But we have seen in Exampe 2.15 that, m under our assumptions, F Hdim(A) m (A) = A[Θ; F ]; indeed, A is compete and any quasi Gorenstein ring is, in particuar, S 2. The second part is proved in [Bi13, Exampe 2.7]. On the other hand, part (iii) foows combining [KSSZ14, Proposition 4.3] and [EY16, Theorem 4.5]; finay, part (iv) aso foows from [KSSZ14, Proposition 4.3]. We point out that an expicit description of F E A can be obtained using the foowing resut due to R. Fedder (cf. [Fed83, pp. 465]). Theorem Let A = K[x 1,..., x d ] be a forma power series ring over a fied K of prime characteristic p. Let I be an arbitrary idea of A. Then, one has that { F E R } = (I [pe] : A I)/I [pe ] Θ e, e 0 where E denotes a choice of injective hu of K over A, R := A/I, Θ is the standard Frobenius action on E and E R := (0 : E I). In [ÀMBZ12] we used this resut to study Frobenius agebras associated to Staney-Reisner rings. It turns out that, whenever they are principay generated, they are isomorphic to A[uΘ; F ] with u = x p 1 1 x p 1 d. 3. The ϕ-koszu chain compex Let S = K[x 1,..., x n ] be the poynomia ring with coefficients on a commutative ring K, and et ϕ : S S be a fat map of K agebras satisfying the extra condition that for any 1 i n, ϕ(x i ) x i. Thus, there are non zero eements s 1,..., s n of S such that ϕ(x i ) = s i x i, for each 1 i n. Remark 3.1. If K is a fied of positive characteristic p > 0 and ϕ = F e is the iterated Frobenius morphism, then this extra condition is naturay satisfied. Indeed, F e (x i ) = x pe i so s i = x pe 1 i. More generay [SW07, Exampe 2.2], if K is any fied and t 1 is an integer, then the K inear map ϕ on S sending each x i to x t i aso satisfies this condition; indeed, in this case, s i = x t 1 i.

9 KOSZUL COMPLEX OVER SKEW POLYNOMIAL RINGS 9 Our aim is to construct a Koszu compex in the category of S[Θ; ϕ] -modues that we wi denote as ϕ-koszu compex. To begin with, et us fix some notations. Let K(x 1,..., x n ) := 0 K n d n K n 1... K 2 d 2 K 1 d 1 K 0 0 be the Koszu chain compex of S with respect to x 1,..., x n (regarded as a chain compex in the category of eft S-modues) and suppose that each differentia d is represented by right mutipication by matrix M. Moreover, for each 0 M [ϕ] denotes the matrix obtained by appying to each entry of M the map ϕ. In particuar, for each entry m, the sign of m is equa to the sign of ϕ(m). Definition 3.2. We define the ϕ-koszu chain compex with respect to x 1,..., x n as the chain compex FK (x 1,..., x n ) := 0 FK n+1 n+1 FK n n... Here, for each 0 n + 1, FK := S[Θ; ϕ](e i1... e i ) 1 i 1 <...<i n 1 j 1 <...<j 1 n 1 FK 0 0. S[Θ; ϕ](e j1... e j 1 u), where e 1,..., e n corresponds respectivey to x 1,..., x n and u corresponds to Θ 1. Here, we are adopting the convention that FK 0 := S[Θ; ϕ]. Moreover, one defines FK FK 1 as the unique homomorphism of eft S[Θ; ϕ]-modues which, on basic eements, acts in the foowing manner: Given 1 i 1 <... < i n, set (e i1... e i ) := ( 1) r 1 ( ) x ir ei1... e ir 1 e ir+1 e ir+2... e i. r=1 Given 1 j 1 <... < j 1 n, set ( ej1... e j 1 u ) := ( 1) 1 ( Θ (s j1 s j 1 ) ) ( e j1... e j 1 ) 1 + ( 1) r 1 ϕ(x jr ) ( e j1... e jr 1 e jr+1 e jr+2... e j 1 u ). r=1 Before going on, we make the foowing usefu: Discussion 3.3. Given a free, finitey generated eft S-modue M (whence M is abstracty isomorphic to S r for some r N), we denote by S[Θ; ϕ] S S r λ M S[Θ; ϕ] r the natura isomorphism of eft S[Θ; ϕ]-modues given by the assignment s m sm; the reader wi easiy note that, for each 0 n, we have the foowing commutative diagram: 1 S[Θ;ϕ] d +1 S[Θ; ϕ] S K +1 S[Θ; ϕ] S K λ K+1 1 i 1 <...<i +1 n S[Θ; ϕ](e i 1... e i+1 ) d +1 λ K 1 i 1 <...<i n S[Θ; ϕ](e i 1... e i ). Here, d +1 denotes the map +1 restricted to the direct summand S[Θ; ϕ](e i1... e i+1 ). 1 i 1 <...<i +1 n As we sha see quicky, this fact turns out to be very usefu in what foows.

10 10 J. ÀLVAREZ MONTANER, A. F. BOIX, AND S. ZARZUELA The first thing we have to check out is that FK (x 1,..., x n ) defines a chain compex in the category of eft S[Θ; ϕ]-modues. This fact foows from the next: Proposition 3.4. For any 0 n, one has that +1 = 0. Proof. Regarding the very definition of the s, we ony have to distinguish two cases. Given 1 i 1 <... < i +1 n one has, keeping in mind Discussion 3.3, that ( ( )) ( ( )) +1 ei1... e i+1 = d d +1 ei1... e i+1 ( = λ K 1 ( ) ) ( 1 S[Θ;ϕ] d 1 λ 1 K λ K ( ) ) (ei1 ) 1 S[Θ;ϕ] d λ 1 K e i+1 ( = λ K 1 ( 1 S[Θ;ϕ] (d d +1 ) ) ) λ 1 (ei1 ) K e i+1 ( = λ K 1 ( 1 S[Θ;ϕ] 0 ) ) λ 1 (ei1 ) K e i+1 = 0; indeed, notice that d d +1 = 0 because they are the usua chain differentias in the Koszu chain compex of S with respect to x 1,..., x n. Given 1 j 1 <... < j n, one has that ( +1 (e j1... e j u)) = (( 1) (Θ (s j1 s j )) (e j1... e j ) + ( 1) r 1 ϕ(x jr ) ( e j1... e jr 1 e jr+1 e jr+2... e j u )) = r=1 ( 1) r+ 1 (ϕ(x jr )Θ (s j1 s j )x jr ) ( ) e j1... e jr 1 e jr+1 e jr+2... e j + r=1 r 1 ( 1) r+k 2 ϕ(x jk x jr ) ( e j1... e jk 1 e jk+1 e jk+2... e jr 1 e jr+1 e jr+2... e j u ) + r=1 k=1 r=1 k=r ( 1) r+k 2 ϕ(x jk x jr ) ( e j1... e jr 1 e jr+1 e jr+2... e jk 1 e jk+1 e jk+2... e j u ) + ( 1) +r 2 ( ϕ(x jr )Θ (s j1 s jr 1 s jr+1 s jr+2 s j )ϕ(x jr ) ) ( ) e j1... e jr 1 e jr+1 e jr+2... e j. r=1 Starting from the top, the first summand (respectivey, the second summand) cances out the fourth summand (respectivey, the third summand) and therefore the whoe expression vanishes, just what we finay wanted to check. Remark 3.5. The differentias of the ϕ-koszu compex FK (x 1,..., x n ) := 0 FK n+1 n+1 FK n n... 1 FK 0 0. are described as foows: ( (1) n+1 is represented by right mutipication by matrix ( 1) n (Θ (s 1 s n )) (2) For each 1 n 1, +1 is represented by right mutipication by matrix ( ) M+1 0, ( 1) D M [ϕ] ) M n [ϕ]. where D is a diagona matrix with non-zero entries Θ (s i1 s i ), 1 i 1 <... < i n. (3) 1 is represented by right mutipication by matrix ( x 1... x n Θ 1 ) T.

11 KOSZUL COMPLEX OVER SKEW POLYNOMIAL RINGS 11 For exampe, when n = 2, FK (x 1, x 2 ) bois down to the chain compex 0 S[Θ; ϕ] 3 S[Θ; ϕ] 3 2 S[Θ; ϕ] 3 1 S[Θ; ϕ] 0, where 3, 2 and 1 are given by right mutipication by matrices ( Θ (s1 s 2 ) ϕ(x 2 ) ϕ(x 1 ) ) x 2 x 1 0 s 1 Θ 0 ϕ(x 1 ) 0 s 2 Θ ϕ(x 2 ) x 1 x 2 Θ 1 Now, we want to singe out the foowing technica fact because it wi pay a key roe during the proof of the first main resut of this paper (see Theorem 3.10). Lemma 3.6. Preserving the notations introduced in Remark 3.5, one has, for any 0 n, that M [ϕ] D 1 = D M. Proof. Fix 0 n. Proposition 3.4 impies that +1 = 0. Then, according to Remark 3.5, this equaity corresponds to ( ) M+1 0 ( ) M 0 ( 1) D ( 1) 1 = 0. D 1 M [ϕ] M [ϕ] 1 In particuar, we must have ( 1) D M + ( 1) 1 M [ϕ] D 1 = 0, which is equivaent to say that ( 1) ( D M M [ϕ] D 1 ) = 0. Whence M [ϕ] D 1 = D M, just what we finay wanted to show. Before showing our main resut, we want to estabish a certain technica fact, which is interesting in its own right; namey: Proposition 3.7. S[Θ; ϕ] is a fat right S-modue. Proof. By the very definition of eft skew poynomia rings, S[Θ; ϕ] = e 0 SΘ e. Since a direct sum of right S modues is fat if and ony if it is so a its direct summands [Rot09, Proposition 3.46 (ii)], it is enough to check that, for any e 0, SΘ e is a fat right S-modue. Fix e 0. Firsty, abeit the notation SΘ e might suggest that it is just a eft S-modue, this is not the case because SΘ e can be identified with Θ e ϕ e S; from this point of view, it is cear that SΘ e may be aso regarded as a right S-modue. Therefore, keeping in mind the previous identification one has that the map Θ e ϕ e S ϕ e S given by the assignment Θ e ϕ e s ϕ e s defines an abstract isomorphism of right S-modues, whence SΘ e is (abstracty) isomorphic to ϕ e S in the category of right S-modues and then the resut foows from the fact that ϕ e S is a fat right S-modue because of the fatness of ϕ; the proof is therefore competed. Remark 3.8. When S is a commutative Noetherian reguar oca ring of prime characteristic p and F is the Frobenius map on S, the fact that S[Θ; F ] is a fat right S modue was aready observed by Y. Yoshino [Yos94, Proof of Exampe (9.2)]. Next resut provides some usefu properties of FK (x 1,..., x n ). Proposition 3.9. H 0 (FK (x 1,..., x n )) = S/I n as eft S[Θ; ϕ]-modues, where I n = x 1,..., x n ; moreover, H n+1 (FK (x 1,..., x n )) = 0.

12 12 J. ÀLVAREZ MONTANER, A. F. BOIX, AND S. ZARZUELA Proof. First we notice that, as in the case of the Frobenius morphism studied by Y. Yoshino (see S[Θ;ϕ] Proposition 2.5), we have S[Θ;ϕ](Θ 1) = S. Then, using Remark 3.5 it foows that H 0 (FK (x 1,..., x n )) = Now, consider the composition S[Θ; ϕ] Im( 1 ) = S[Θ; ϕ] S[Θ; ϕ]i n + S[Θ; ϕ](θ 1) = S/I n. S[Θ; ϕ] n+1 S[Θ; ϕ] S[Θ; ϕ] n π S[Θ; ϕ] n, where π denotes the corresponding projection. In this way, we have that π n+1 turns out to be, up to isomorphisms, 1 S[Θ;ϕ] d [ϕ] n (indeed, this fact foows directy from the commutative square estabished in Discussion 3.3); regardess, since d [ϕ] n is an injective homomorphism between free eft S-modues, and S[Θ; ϕ] is a fat right S-modue (cf. Proposition 3.7), one has that 1 S[Θ;ϕ] d [ϕ] n is an injective homomorphism between free eft S[Θ; ϕ]-modues. Therefore, π n+1 is aso an injective homomorphism, whence n+1 is so. This fact concudes the proof. Now, we state and prove the first main resut of this paper, which is the foowing: Theorem The ϕ-koszu compex FK (x 1,..., x n ) provides a free resoution of S/I n in the category of eft S[Θ; ϕ]-modues. Proof. By Proposition 3.9, it is enough to check, for any 1 n, that H (FK (x 1,..., x n )) = 0. So, fix 1 n. Our goa is to show that ker( ) Im( +1 ); in other words, we have to prove +1 that the chain compex FK +1 FK FK 1 is midterm exact. First of a, remember that FK = K K, where K := S[Θ; ϕ](e i1... e i ), and K := 1 i 1 <...<i n 1 j 1 <...<j 1 n S[Θ; ϕ](e j1... e j 1 u). Furthermore, Discussion 3.3 impies that the chain compexes K : 0 K n d n K n 1... K 2 d 2 K 1 d 1 K 0 0 and (K ) [ϕ] : 0 K n (d n )[ϕ] K n 1... K 2 (d 2 )[ϕ] K 1 (d 1 )[ϕ] K 0 0 are respectivey canonicay isomorphic to S[Θ; ϕ] S K and S[Θ; ϕ] S K [ϕ] ; in particuar, since S[Θ; ϕ] is a fat right S-modue (cf. Proposition 3.7), K and (K ) [ϕ] are both acycic chain compexes in the category of eft S[Θ; ϕ]-modues. On the other hand, we aso have to keep in mind that d and (d )[ϕ] are respectivey represented by right mutipication by matrix M and M [ϕ] (cf. Proposition 3.9 and its corresponding notation). Now, et P ker( ) FK. Since FK = K K, we may write P = (P, P ) for certain P K and P K ; in this way, as P ker( ) it foows that ( ) ( ( ) 0 0 = P P ) M 0 ( ) ( 1) 1 = P D M + P ( 1) 1 D 1 P M [ϕ], 1 1 M [ϕ] 1 which eads to the foowing system of equations: P M + P ( 1) 1 D 1 = 0, P M [ϕ] 1 = 0.

13 KOSZUL COMPLEX OVER SKEW POLYNOMIAL RINGS 13 In particuar, since P M [ϕ] 1 = 0 one has that P ker((d 1 )[ϕ] ) = Im((d )[ϕ] ); therefore, there is Q K such that Q M [ϕ] = P. Using this fact, it foows that P M + Q ( 1) 1 M [ϕ] D 1 = 0. Regardess, Lemma 3.6 tes us that M [ϕ] D 1 = D M, whence P M + Q ( 1) 1 D M = 0, which is equivaent to say that ( P + Q ( 1) 1 D ) M = 0. In this way, one has that P + Q ( 1) 1 D ker(d ) = Im(d +1 ) and therefore there exists Q K +1 such that Q M +1 = P + Q ( 1) 1 D. Summing up, setting Q := (Q, Q ) FK +1, it foows that ( ( ) Q Q ) M +1 0 ( ) ( 1) D M [ϕ] = Q M +1 + Q ( 1) D Q M [ϕ] = ( P + Q ( 1) 1 D + Q ( 1) D P ) = ( P P ) and therefore we can concude that P Im( +1 ), which is exacty what we wanted to show. The reader wi easiy note that, during the proof, we have obtained the beow resut; we speciay thank Rishi Vyas to singe out to us this fact. Proposition There is a short exact sequence of chain compexes 0 S[Θ; ϕ] S K (S; x 1,..., x n ) FK (x 1,..., x n ) (S[Θ; ϕ] S K (S; ϕ(x 1 ),..., ϕ(x n ))) [1] 0 in the category of eft S[Θ; ϕ] modues The ϕ-koszu chain compex in fu generaity. Our next aim is to define the ϕ-koszu chain compex over a more genera setting, and expore some specific situations on which we can ensure that defines a finite free resoution. Let A be a commutative Noetherian ring containing a commutative subring K, and y 1,..., y n denote arbitrary eements of A; in addition, assume that ϕ we fix a fat endomorphism of K agebras S := K[x 1,..., x n ] S satisfying ϕ(x i ) x i for any 1 i n. In this section, we regard A as an S-agebra under S ψ A, where S ψ A is the natura homomorphism of K-agebras which sends each x i to y i. Finay, we suppose that there exists a K agebra homomorphism A Φ A making the square S ψ A ϕ S ψ A Φ commutative. Definition We define the ϕ-koszu chain compex of A with respect to y 1,..., y n as the chain compex FK (y 1,..., y n ; A) := A S FK (x 1,..., x n ). Under sighty different assumptions, we want to show that FK (y 1,..., y n ; A) sti defines a finite free resoution; with this purpose in mind, first of a we review the foowing notion, which was introduced independenty in [Bou07, Definition 2 of page 157] (using a different terminoogy) and by T. Kabee in [Kab71, Definition 2].

14 14 J. ÀLVAREZ MONTANER, A. F. BOIX, AND S. ZARZUELA Definition Let R be a commutative ring, et s N, and et f 1,..., f n be a sequence of eements in R. It is said that f 1,..., f n is a Koszu reguar sequence provided the Koszu chain compex K (f 1,..., f n ; R) provides a free resoution of R/I n, where I n = f 1,..., f n. Next statement may be regarded as a generaization of Theorem 3.10; this is the main resut of this section. Theorem Let A be a commutative Noetherian ring containing a commutative subring K, and et y 1,..., y n be a sequence of eements in A. Moreover, we assume that Φ is fat and y 1,..., y n is an A-Koszu reguar sequence. Then, FK (y 1,..., y n ; A) defines a finite free resoution of A/I n in the category of eft A[Θ; Φ]-modues, where I n = y 1,..., y n. Proof. The proof of this resut is, mutatis mutandi, the same as the one of Theorem 3.10 repacing S by A and x 1,..., x n by y 1,..., y n. Indeed, a simpe inspection of the proof of Theorem 3.10 reveas that we ony used there the fatness of ϕ and the fact that the Koszu chain compex K (x 1,..., x n ) defines a finite free resoution of K; the proof is therefore competed. Remark The goba homoogica dimension of right skew poynomia rings was studied by K. L. Fieds in [Fie69, Fie70]. An upper bound for this goba dimension is n + 1 when ϕ is injective and it is exacty n + 1 in the case that ϕ is an automorphism. Passing to the opposite ring (see Remark 2.2) we woud get anaogous resuts for eft skew poynomia rings. Notice that the ength of the ϕ-koszu compex FK (y 1,..., y n ; A) is n The case of the Frobenius homomorphism. For the convenience of the reader, we wi speciaize the construction of the ϕ-koszu compex to the case where A is a commutative Noetherian reguar ring of positive characteristic p > 0, and ϕ = F e is an e-th iteration of the Frobenius morphism. In this case, we wi denote this compex simpy as Frobenius-Koszu compex. Namey we have: FK (x 1,..., x n ) := 0 FK n+1 n+1 FK n n... 1 FK 0 0. where, for each 0 n + 1, FK := A[Θ; F e ](e i1... e i ) 1 i 1 <...<i n and the differentias FK FK 1 are given by: For 1 i 1 <... < i n, set (e i1... e i ) := 1 j 1 <...<j 1 n A[Θ; ϕ](e j1... e j 1 u), ( 1) r 1 ( ) x ir ei1... e ir 1 e ir+1 e ir+2... e i. r=1 For 1 j 1 <... < j 1 n, set ( ej1... e j 1 u ) ( := ( 1) 1 Θ (x pe 1 j 1 x pe 1 (ej1 ) j 1 ))... e j ( 1) r 1 ( x pe j ej1 r... e jr 1 e jr+1 e jr+2... e j 1 u ). r=1 Remark For n = 2, FK (x 1, x 2 ) is just 0 A[Θ; F e ] 3 A[Θ; F e ] 3 2 A[Θ; F e ] 3 1 A[Θ; F e ] 0,

15 KOSZUL COMPLEX OVER SKEW POLYNOMIAL RINGS 15 where 3, 2 and 1 are given by right mutipication by matrices ( ) x 2 x 1 0 Θ (x pe 1 1 x pe 1 2 ) x pe 2 x pe 1 x pe 1 1 Θ 0 x pe 1 0 x pe 1 2 Θ x pe 2 x 1 x 2 Θ 1 As a direct consequence of Theorem 3.14 we obtain the beow: Coroary Let A be a commutative Noetherian reguar ring of prime characteristic p, and et y 1,..., y n be an A Koszu reguar sequence. Then, FK (y 1,..., y n ; A) defines a finite free resoution of A/I n in the category of eft A[Θ; F e ]-modues, where I n = y 1,..., y n. Acknowedgements. The second author thanks Rishi Vyas for fruitfu discussions about the materia presented here. References [ÀMBZ12] J. Àvarez Montaner, A. F. Boix, and S. Zarzuea. Frobenius and Cartier agebras of Staney-Reisner rings. J. Agebra, 358: , [And00] G. W. Anderson. An eementary approach to L-functions mod p. J. Number Theory, 80(2): , [BB11] M. Bicke and G. Böcke. Cartier modues: finiteness resuts. J. Reine Angew. Math., 661:85 123, , 6 [Bi13] M. Bicke. Test ideas via agebras of p e -inear maps. J. Agebraic Geom., 22(1):49 83, , 5, 8 [Bou07] N. Bourbaki. Ééments de mathématique. Agèbre. Chapitre 10. Agèbre homoogique. Springer-Verag, Berin, Reprint of the 1980 origina. 13 [BS13] M. Bicke and K. Schwede. p 1 -inear maps in agebra and geometry. In Commutative agebra, pages Springer, New York, , 5 [EH08] F. Enescu and M. Hochster. The Frobenius structure of oca cohomoogy. Agebra Number Theory, 2(7): , [Ene12] F. Enescu. Finite-dimensiona vector spaces with Frobenius action. In Progress in commutative agebra 2, [EY16] pages Water de Gruyter, Berin, F. Enescu and Y. Yao. The Frobenius compexity of a oca ring of prime characteristic. J. Agebra, 459: , [Fed83] R. Fedder. F -purity and rationa singuarity. Trans. Amer. Math. Soc., 278(2): , [Fie69] K. L. Fieds. On the goba dimension of skew poynomia rings. J. Agebra, 13:1 4, [Fie70] K. L. Fieds. On the goba dimension of skew poynomia rings An addendum. J. Agebra, 14: , [GW04] K. R. Goodear and R. B. Warfied, Jr. An introduction to noncommutative Noetherian rings, voume 61 of London Mathematica Society Student Texts. Cambridge University Press, Cambridge, second edition, , 2 [HH90] M. Hochster and C. Huneke. Tight cosure, invariant theory, and the Briançon-Skoda theorem. J. Amer. Math. Soc., 3(1):31 116, [Kab71] T. Kabee. Reguarity conditions in nonnoetherian rings. Trans. Amer. Math. Soc., 155: , [KSSZ14] M. Katzman, K. Schwede, A. K. Singh, and W. Zhang. Rings of Frobenius operators. Math. Proc. Cambridge Phios. Soc., 157(1): , [LS01] G. Lyubeznik and K. E. Smith. On the commutation of the test idea with ocaization and competion. Trans. Amer. Math. Soc., 353(8): (eectronic), , 5, 6, 8 [Lyu97] G. Lyubeznik. F -modues: appications to oca cohomoogy and D-modues in characteristic p > 0. J. [MR01] [Rot09] Reine Angew. Math., 491:65 130, J. C. McConne and J. C. Robson. Noncommutative Noetherian rings, voume 30 of Graduate Studies in Mathematics. American Mathematica Society, Providence, RI, revised edition, With the cooperation of L. W. Sma. 1, 2 J. J. Rotman. An introduction to homoogica agebra. Universitext. Springer, New York, second edition, [Sch11] K. Schwede. Test ideas in non-q-gorenstein rings. Trans. Amer. Math. Soc., 363(11): , , 5 [Sha09] R. Y. Sharp. Graded annihiators and tight cosure test ideas. J. Agebra, 322(9): ,

16 16 J. ÀLVAREZ MONTANER, A. F. BOIX, AND S. ZARZUELA [SW07] A. K. Singh and U. Wather. Loca cohomoogy and pure morphisms. Iinois J. Math., 51(1): , , 8 [SY11] R. Y. Sharp and Y. Yoshino. Right and eft modues over the Frobenius skew poynomia ring in the F-finite case. Math. Proc. Cambridge Phios. Soc., 150(3): , [Yos94] Y. Yoshino. Skew-poynomia rings of Frobenius type and the theory of tight cosure. Comm. Agebra, 22(7): , , 4, 6, 11 Departament de Matemàtiques, Universitat Poitècnica de Cataunya, Avinguda Diagona 647, Barceona 08028, SPAIN E-mai address: Josep.Avarez@upc.edu Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653 Beer-Sheva 84105, Israe. E-mai address: fernana@post.bgu.ac.i Departament d Àgebra i Geometria, Universitat de Barceona, Gran Via de es Corts Cataanes 585, Barceona 08007, SPAIN E-mai address: szarzuea@ub.edu