Frobenius maps on injective hulls and their applications Moty Katzman, The University of Sheffield.

Frobenius maps on injective hulls and their applications Moty Katzman, The University of Sheffield.

Consider a complete local ring (S, m) of characteristic p > 0 and its injective hull E S = E S (S/m). We study Frobenius maps φ : E S E S : these are additive and φ(sm) = s p m for s S, m E S. Consider the skew-polynomial ring S[T ; f] := i 0 ST i with multiplication T s = s p T for s S. To give a Frobenius map φ : E S E S amounts to giving E S the structure of a left-s[t ; f] module with T m = φ(m) for all m E.

(The) Example: The Frobenius endomorphism f : S S given by f(s) = s p. Iterating f we obtain f e : S S and can define S[T ; f e ]. We can now view S as a bimodule F e S (S) which has the standard left S-module structure and right S-module structure given by m s = s pe m for m F e S (S), s S. We can go on and construct the Frobenius functor which takes an S-module M to F e S (M) = F e S (S) M on which S acts on the left on F e S (S): s(t m) = (st) m and t sm = (t s) m = s pe t m for s S, t F e S (S) and m M. E.g., for an ideal a S, F e S (S/a) = S/a[pe] where a [pe] is the eth Frobenius power of a, i.e., the ideal generated by p e th powers of elements of a.

Example: Let S = K[[x 1,..., x d ]] with K a field of characteristic p. Now E S is the module of inverse polynomials K[x 1,..., x d ] which is the K-span of all monomials with strictly negative exponents and S-module structure extending { x α 1 1... xα d d x β 1 1... x β d d = x α 1 β 1 1... x α d β d d, if α 1 β 1,..., α d β d < 0 0, otherwise We have a standard S[T ; f]-module structure on E S. given by T λx β 1 1... x β d d = λ p x pβ 1 1... x pβ d Notice that for any g S, gt yields a new Frobenius map on E S. For any g S and β > 0, θ = gt β turns E S into an S[θ; f β ]- module. d.

A duality Assume now that S = R/I where (R, m) is a complete regular local ring. Let C e be the category of Artinian S[T ; f e ]-modules. D e be the category of R-linear maps M FR e (M). We define an exact contravariant functor e : C e D e : for M C e we have an R-linear map α M : FR e (M) M given by α(r m) = rt m for all r R and m M. Applying ( ) = Hom(, E R ) to the map α one obtains an R- linear map α M : M FR e (M). The fact that M is Artinian gives the existence of a (functorial) isomorphism γ M : FR e (M) FR e (M ). We now define e (M) to be the map M γ M α M F e R (M ). (This is the first step in the construction of Gennady Lyubeznik s functor H R,S.)

We can retrace our steps! There is an exact contravariant functor Ψ e : D e C e such that Ψ e e ( ) and e Ψ e ( ) are canonically isomorphic to the identity functor.

The main example Let E S have an S[T ; f]-module structure. We have (E S ) = (E S F R(E S )) = (R/I R/I[p] ); this map must be given by multiplication by some u (I [p] : I). Conversely, any such u endows E S with an S[T ; f]-module structure. A short exact sequence of S[T ; f]-modules 0 M E S E S /M 0 yields a commutative diagram L 0 I u 0 L [p] I [p] R I R u I [p] R L u 0 R L [p] 0 where u (L [p] : L). So all S[T ; f]-submodules of E S have the form ann ES L where u (L [p] : L). We call these L s E S ideals.

Application #1: nilpotents and jumping coefficients Let E S have an S[θ; f β ]-module structure so that (E S ) = (R/I u R/I [p] ) with u (I [p] : I). For all e 0 let N e = {m E S θ e m = 0} and Nil(E S ) = e 0 N e. Theorem [Hartshorne-Speiser 77, Lyubeznik 97]: Nil(E S ) = N e for some e 0. For any ideal J R define I e (J) to be the smallest ideal L for which J L [pe]. Applying e and Ψ e to the inclusion N e E S yields the following: Theorem [K]: N e = ann ES I e (u ν e) where ν e = 1 + p + + p e 1. As a consequence, the decreasing sequence I e (u ν e) stabilizes at the first e for which I e (u ν e) = I e+1 (u ν e+1). (This immediately gives an algorithm for computing Frobenius closures of parameter ideals in CM rings.)

Generalized test ideals The generalized test-ideal of a R with coefficient c R is τ(a c ) = e 0 I e ( a cp e ). A jumping coefficient of a is a c R such that τ(a c ) τ(a c ɛ ) for all ɛ > 0. We consider the case where a is generated by g R and write τ(g c ) for τ(a c ). For any a, β > 0 manufacture an S[θ; f β ]-module structure on E R by defining θm = g a T β m for all m E R and where T is the standard Frobenius. Denote the set of nilpotent submodules { Nil a,β a 0, β > 0 } with N. Theorem: [K-Lyubeznik-Zhang] The map τ(g c ) ann E τ(g c ) defined for all c 0 is a bijection between the set of generalized test-ideals of g and N.

There is much interest in deciding whether jumping coefficients are rational and form a discrete set. We can reduce the problem to the following: Theorem: [K-Lyubeznik-Zhang] The set of jumping coefficients of g cannot have an accumulation point of the form with a, β N. a (p β 1) Proof: For all c R there exists an ɛ > 0 such that τ(g c ) = τ(g d ) for all d [c, c + ɛ). Let c = a be such an accumulation point; it is itself a (p β 1) jumping coefficient. Endow E R with an S[θ; f β ]-module structure as above; For all r 0, τ(g r/pe ) is just I e (g r ); τ g a(1+p β + +p β(e 1) ) p eβ = I eβ (g a(1+pβ + +p β(e 1)) ) stabilizes and a(1 + p β + + p β(e 1) ) p eβ = a p eβ p eβ 1 p β 1 e c.

Application #2: parameter test ideals of CM rings Fix an S[T ; f]-module H. For any S[T ; f]-submodule M H let gr-ann M be the largest ideal of the form LS[T ; f] in ann S[T ;f] M. Call an ideal I R an H-special ideal if IS[T ; f] = gr-ann M for an S[T ; f]-submodule M H. Now let S = R/I be d-dimensional Cohen-Macaulay with canonical module ω S. Take H = H d ms (S) with its standard S[T ; f]-module structure; assume it has no-nilpotents. Theorem: [Sharp] The parameter test ideal of S is the intersection of all H-special ideals of positive height. We now compute this parameter test ideal!

The -closure The computation will involve producing small E S ideals (ul L [p] ). Fix an u R. Given an ideal A R construct the following ascending chain: A 0 = A and A i+1 = I 1 (ua i ) + A i for i 0. Define A u to be the stable value of this sequence. Proposition: A u the smallest ideal L R containing A for which ul L [p]. If E S has an S[T ; f]-module structure such that (E s ) = (R/I u R/I [p] ), then for any ideal A R containing I, A u is the smallest E S -ideal containing A.

Lift our calculation to E S as follows. The short exact sequence 0 ω S S/ω 0 gives a surjection of E S = H d ms (ω) H of S[T ; f]-modules whose kernel is ann ES J for some E S -ideal J. Proposition: (1) The set of E S -special ideals coincides with the set of E S - ideals. (2) [Sharp] The set of H ideals is given by {(L : J) L is an E S -ideal, L J}.

Theorem[K]: Assume that E S is T -torsion-free. Let c R be such that its image in S is a parameter test element. The parameter test ideal τ of S is given by ((cj + I) u : R J)S. Proof (cj + I) u is an E S -ideal and ht((cj + I) u : J)S > 0 since c ((cj + I) u : J). Now τ = {K K S is a H-special ideal, ht K > 0} = {(L : R J) L J is an E S -ideal, ht(l : R J)S > 0} = ( {L L J is an E S -ideal, ht(l : J)S > 0} : J ) so τ ((cj + I) u : J). Also, c τ hence cj L for all E S -ideals L for which ht(l : J)S > 0 and the minimality of -closures implies that (cj + I) u L and hence that ((cj + I) u : J) (L : J) for all E S -ideals L for which ht(l : J) > 0. We conclude that ((cj + I) u : J) τ.

To turn the previous theorem into an algorithm we have to exhibit J (the E S ideal for which ann ES J is the kernel of E S H = H d ms (S)) and u R (for which (E s) = (R/I u R/I [p] )). It turns out that J is the lifting of ω to R. To find u we use the following: Proposition: [Lyubeznik and Smith] The S-module of Frobenius maps on H is free of rank 1. Now H = E S / ann ES J, u J[p] I [p] (H) = J I H corresponds to ( I [p] : R I ) I [p] ( ) J [p] : R J and so the S-module of Frobenius maps on and we take u to be the free-generator of this.

An actual calculation. Let K be the field of two elements, R = K[x 1, x 2, x 3, x 4, x 5 ], let I be the ideal of R generated by the 2 2 minors of ( ) x1 x 2 x 2 x 5 x 4 x 4 x 3 x 1 and let S = R/I. This quotient is reduced, 2-dimensional, Cohen-Macaulay and of Cohen-Macaulay type 3; we produce a canonical module by computing Ext 3 R (S, R) = coker x 2 x 1 0 0 x 3 + x 4 x 4 x 5 x 4 0 0 x 3 x 4 0 0 x 1 0 x 5 x 5 x 5 x 5 0 x 2 0 x 1 this is isomorphic to the ideal ω S which is the image in S of the ideal Ω R generated by x 1, x 4, x 5. ;

We now take J = Ω and compute the generator u of the S- module ( I [2] : R I ) ( J [2] : R J ) I [2] ; this turns out to be u = x 3 1 x 2x 3 + x 3 1 x 2x 4 + x 2 1 x 3x 4 x 5 + x 1 x 2 x 3 x 4 x 5 + x 1 x 2 x 2 4 x 5 + x 2 2 x2 4 x 5 + x 3 x 2 4 x2 5 + x3 4 x2 5. We compute I 1 (u ν 1R) = R hence E S is T -torsion free. Now the parameter-test-ideal τ is computed as ((cj + I) u : J) where c is randomly chosen to be in the defining ideal of the singular locus of S and not in a minimal prime of I. This calculation yields τ = (x 1, x 2, x 3 + x 4, x 4 x 5 )R and we deduce that S is not F -regular.