Contributions to the study of Cartier algebras and local cohomology modules

Transcription

1 Contributions to the study of Cartier algebras and local cohomology modules Alberto F. Boix Advised by J. Àlvarez Montaner and Santiago Zarzuela Thesis lecture

2 Cartier algebras (prime characteristic) Cartier algebras of Stanley-Reisner rings An algorithm for producing F-pure ideals Some local cohomology spectral sequences (characteristic free) Homological spectral sequences Cohomological spectral sequences Degeneration of spectral sequences Extension problems

3 CARTIER ALGEBRAS

4 Unless otherwise is specified: p will always denote a prime number. K is a field of characteristic p. S is the polynomial ring K [x 1,..., x d ]. A is the formal power series ring K[[x 1,..., x d ]]. B is a commutative Noetherian ring of characteristic p.

5 Homogeneous functions Definition Given k Q, and given a B-module M, it is said that a function M g M is homogeneous of degree k provided g(bm) = b k g(m) for any b B and for any m M.

6 Homogeneous functions Example The eth Frobenius map B F e B, which raises an element b B to its p e th power b pe, is a homogeneous function of degree p e. Remark In this talk, we are mostly interested in homogeneous functions of degree p e and, overall, 1/p e = p e, where e runs through N.

7 Homogeneous functions K is now a perfect field (e.g. finite). Denote by A pe for a A. the ring made up by elements of the form a pe A is a free left A pe -module with basis with 0 b i p e 1 for all i. x b 1 1 x b d d,

8 The trace map Consider the unique homogeneous function A πe A of degree p e which acts on a monomial in the following manner: x β := x b 1 1 x b d d (with 0 b i p e 1) π e (x β) := { 1, if b i = p e 1 for all i, 0, otherwise.

9 PACKING HOMOGENEOUS FUNCTIONS

10 Frobenius and Cartier algebras of operators Given a B-module M: End p e (M): homogeneous functions on M of degree p e. End p e (M): homogeneous functions on M of degree p e.

11 Frobenius and Cartier algebras of operators Definition (Lyubeznik,Smith (2001)) The Frobenius algebra of operators attached to M is: F M := e 0 End p e (M). Definition (Schwede (2009), Blickle (2013)) The Cartier algebra of operators attached to M is: C M := e 0 End p e (M).

12 Basic example of principal Frobenius algebra Set Lemma One has that B[Θ; F ] := B θ b p θ θb b B B. F B = B[Θ; F ].

13 Basic example of principal Cartier algebra Set Lemma One has that B[ε; F ] := B ε bε εb p b B B. C B = B[ε; F ].

14 FOR WHAT THE PREVIOUS ALGEBRAS ARE USEFUL?

15 FOR STUDYING SINGULARITIES IN POSITIVE CHARACTERISTIC

16 TEST IDEALS IN FULL GENERALITY

17 Cartier algebras in full generality Definition (Blickle (2013)) An (abstract) B-Cartier algebra (namely, C) is C = e 0 C e, where: C 0 is a B-algebra. For any b B and for any ψ e C e, b ψ e = ψ e b pe. Set C + := e 1 C e.

18 F-pure ideals b B a non-zero ideal. C a B-Cartier algebra. Definition (Blickle (2013)) It is said that b is F -pure (with respect to C) provided C + b = b.

19 Test ideals through Cartier algebras C a B-Cartier algebra. Definition (Blickle (2013)) Define τ(b, C) as the smallest F -pure ideal, provided it exists.

20 F-JUMPING NUMBERS IN FULL GENERALITY

21 F-jumping numbers in full generality b B a non-zero ideal. c [0, + ). Definition (Blickle (2013)) One defines C bc := End p e (B) b c(pe 1), e 0 τ(b, b c ) := τ(b, C bc ).

22 F-jumping numbers in full generality B an F -finite ring. Theorem/Definition (Blickle (2013)) The following assertions hold. For all ε > 0, τ (B, b c+ε ) τ (B, b c ) with equality for all 0 < ε 1. c is an F -jumping number in case for all ε > 0. τ ( B, b c ε) τ (B, b c )

23 F-jumping numbers in full generality Question Are F -jumping numbers discrete and rational?

24 END OF INTRODUCTION

25 CARTIER ALGEBRAS AND THEIR GENERATION

26 A motivating question Remind that F M = e 0 End p e (M). Question (Lyubeznik, Smith (2001)) Is F M finitely generated as End B (M)-algebra?

27 Some positive partial answers Lemma (Lyubeznik, Smith (2001)) One has that F B = B[Θ; F ].

28 Some positive partial answers (R, m) a complete local ring. Lemma (Lyubeznik, Smith (2001)) If R verifies Serre s S 2 condition, then F Hdim(R) m (R) = R[Θ; F ].

29 Some positive partial answers A = K[[x 1,..., x d ]], and I A an ideal. R := A/I. E = K[[x 1,..., x d ]], and E R := (0 : E I ). Theorem (Fedder (1983)) One has that F E R (( I [p e] : A I ) ) = F e. e 0 I [pe ]

30 Some positive partial answers Corollary If ( I [pe] : A I ) = I [pe] + u pe 1 for some u A, then F E R = R[u p 1 Θ; F ].

31 DUALITY BETWEEN FROBENIUS AND CARTIER ALGEBRAS

32 Frobenius and Cartier algebras K is a finite dimensional K p -vector space. A = K[[x 1,..., x d ]], and I A an ideal. R := A/I. E = K[[x 1,..., x d ]], and E R := (0 : E I ). ( ) := Hom A (, E). Proposition One has that End p e (E R ) = Endp e (R).

33 Frobenius and Cartier algebras Roughly speaking: ( ) F E R = C R. Warning This correspondence through Matlis duality is just at the level of homogeneous elements. Remark This correspondence maps F e to π e up to a unit.

34 Some positive partial answers Theorem (Schwede (2009), Blickle (2013)) The following assertions hold. If R is quasi Gorenstein, then C R is principal. If R is a normal domain, then C R is principal if and only if R is Gorenstein.

35 Some positive partial answers Theorem (Katzman, Schwede, Singh, Zhang (2014); Enescu, Yao (2014)) If R is a normal, Q-Gorenstein domain with index i, then C R is finitely generated if and only if i is not a multiple of p.

36 Cartier algebras of non-normal rings A = Z/2Z[[x, y, z]], and I = xy, xz. R := A/I. E = Z/2Z[[x, y, z ]], and E R := (0 : E I ). Example (Katzman (2010)) F E R (equivalently, C R ) is NOT finitely generated as R-algebra. Remark R is a non-normal, complete Stanley-Reisner ring.

37 Dissertation aim Goal Study the generation of C R, where R is a complete Stanley-Reisner ring.

38 CARTIER ALGEBRAS OF COMPLETE STANLEY-REISNER RINGS

39 Fixing notation A = K[[x 1,..., x d ]]. I = I A a squarefree monomial ideal. R := A/I. E = K[[x 1,..., x d ]], and E R := (0 : E I ). I = I β1... I βs, where β i = (b 1,i,..., b d,i ) {0, 1} d. I βi = x j b j,i = 1 (1 j d, 1 i s).

40 Technical facts: some calculations One has that ( I [pe ] β i : A I βi ) = I [pe ] β i + b j,i =1 x pe 1 j ( ) ( ) I [pe] : A I = I [pe ] β 1 : A I β1..., ( I [pe ] β s : A I βs ).

41 Technical facts: some calculations Summing up: ( ) I [pe] : A I = I [pe] + J e + (x 1 x d ) pe 1. Remark If J e I [pe] + (x 1 x d ) pe 1, then there is at least one monomial generator of J e of the form x c 1 1 x c 2 2 x c d d, where (c i, c j, c k ) = (0, p e 1, p e ) for some 1 i, j, k d.

42 Contribution Theorem (Àlvarez Montaner, B., Zarzuela (2012)) The following statements are equivalent: F E R is principal; in such case, F E R = R[(x1 x d ) p 1 Θ; F ]. C R is principal; in such case, C R = R[ε(x 1 x d ) p 1 ( ; F ]. I [pe] : A I ) = I [pe] + (x 1 x d ) pe 1 ( for any e 0. I [2] : A I ) = I [2] + x 1 x d. Otherwise, F E R R-algebra. (equivalently, C R ) is an infinitely generated

43 Contribution: sketch of proof If ( I [pe] : A I ) = I [pe] + (x 1 x d ) pe 1 then we are done. Otherwise, ( I [pe] : A I ) = I [pe] + J e + (x 1 x d ) pe 1, where J e I [pe] + (x 1 x d ) pe 1.

44 Technical facts: Katzman criterion Set ( K e := I [pe] : A I L e := ), 1 c 1,...,c t e 1 c c t=e 1 K c1 K [pc 1] c 2 K [pc 1 +c 2] c 3 K [pc c t 1 ] c t. Moreover, denote by F <e the R-subalgebra of F E R F E R 0,..., F E R e 1. Theorem (Katzman (2010)) One has that F E R e F <e = L e. generated by

45 Contribution: sketch of proof Pick a minimal monomial generator of J e which does not belong to I [pe] + (x 1 x d ) pe 1 ; namely, m. m F E R ; however, m L e and therefore m F <e.

46 APPLICATIONS OF OUR CONTRIBUTION

47 DISCRETENESS OF F-JUMPING NUMBERS

48 Fixing notation S = Z/pZ[x 1,..., x d ]. x α = x a 1 1 x a d d, α = (a 1,..., a d ) N d. x α := max{a 1,..., a d }. Given set where s = s := γ N d s γ x γ S, max γ supp(s) xγ, supp(s) := {γ N d s γ 0}.

49 Key technical tool: gauge boundedness I S an ideal and R := S/I. Definition (Anderson (2000), Blickle (2013)) C R is gauge bounded provided there is a set {ψ i ψ i C ei, e i 1}, which generates C + as right R-module, such that ψ i (r) r p e i + K for some constant K and for any r R.

50 Why gauge boundedness is introduced? Theorem (Blickle (2013)) The following assertions hold. If C R is gauge bounded, then the F -jumping numbers of any ideal a of R form a discrete subset inside [0, + ). If C R is finitely generated as R-algebra, then C R is gauge bounded.

51 CONTRIBUTION

52 Contribution A = K[[x 1,..., x d ]]. I = I squarefree monomial ideal and R := A/I. a R an ideal. Theorem (Àlvarez Montaner, B., Zarzuela (2012)) The following statements hold. C R is gauge bounded. The set of F -jumping numbers of a is discrete.

53 Contribution Remark The previous result is not covered by Blickle s one because, under our assumptions, C R is, in general, not finitely generated as R-algebra.

54 DIFFERENTIAL OPERATORS IN POSITIVE CHARACTERISTIC

55 Differential operators in positive characteristic B any commutative Noetherian ring of prime characteristic. There is, for any e, a pairing End p e (B) B End p e (B) End B p e (B) := D (e) B which sends ϕ e ψ e ϕ e ψ e.

56 Differential operators in positive characteristic In general: Theorem (Àlvarez Montaner, Blickle, Lyubeznik (2005)) If B is regular and F -finite, then the previous pairing is an isomorphism. Question What happens if B is not regular?

57 CONTRIBUTION

58 Contribution B = K[[x 1,..., x d ]]/I. Theorem (Àlvarez Montaner, B., Zarzuela (2012)) The pairing End p e (B) B End p e (B) End B p e (B) is not surjective.

59 Contribution: sketch of proof I = I β1... I βs. Pick 1 i d such that x i / I βj for some 1 j s. x i pe 1 i pairing. D (e) B that does not belong to the image of the

60 AN ALGORITHM FOR PRODUCING F-PURE IDEALS

61 The trace map K perfect field. Consider the unique homogeneous function S πe S of degree p e which acts on a monomial in the following manner: x α := x a 1 1 x a d d (with 0 a i p e 1) π e (x α ) := { 1, if a i = p e 1 for all i, 0, otherwise.

62 Why the previous homogeneous function is distinguished? Answer It is known that any homogeneous function of degree p e can be written as π e u for some u S, where S u S is nothing but the multiplication by the polynomial u. Remark The previous fact will play a crucial role in what follows.

63 What is an F-pure ideal? Definition Let C be a graded subalgebra of C S. An ideal I of S is called F-pure (with respect to C) provided C + I = I.

64 Why F-pure ideals are of some interest? Theorem (Blickle (2013)) S itself is an F-pure ideal with respect to C S if and only if C S + contains a splitting of a certain power of the Frobenius map on S. Theorem (Blickle (2013)) The following assertions hold. 1. The set of F-pure ideals is finite. 2. The test ideal is the minimal element of such set.

65 What we want to do with the previous stuff? Problem Compute all the F-pure ideals with respect to C S. We shall be less ambitious. Aim Provide a procedure to calculate all the F-pure ideals of S contained in m with respect to the subalgebra of C S generated by a single homogeneous function of degree p e.

66 COMPATIBLE AND FIXED IDEALS

67 Compatible and fixed ideals φ denotes a fixed homogeneous function of degree p e. C denotes the subalgebra of C S generated by φ. Remark The problem of finding the F-pure ideals of S with respect to C amounts to finding all ideals I S such that φ(i ) = I.

68 Compatible and fixed ideals Definition It is said that an ideal I of S is φ-compatible (respectively, φ-fixed) if φ(i ) I (respectively, φ(i ) = I ). Question How to recognize algorithmically whether an ideal is either φ-compatible or φ-fixed?

69 The ideal of p e th roots Definition (Blickle, Mustațǎ, Smith (2008)) Let J be an ideal of S. We set I e (J) as the smallest ideal (namely, I ) such that I [pe] J.

70 Why the ideal of p e th roots is interesting for our purposes? Question How to recognize algorithmically whether an ideal is either φ-compatible or φ-fixed? The answer is given in the next: Lemma An ideal J of S is φ-compatible (respectively, φ-fixed) if and only if I e (uj) J (respectively, I e (uj) = J).

71 How to calculate the ideal of p e th roots? Proposition (Blickle,Mustațǎ, Smith (2008)) Let J 1,..., J r be ideals of S and let g S. (i) I e (J J r ) = I e (J 1 ) I e (J r ). (ii) If g = α=(a 1,...,a d ) 0 a i p e 1 for all i g pe α x α, then I e (gs) is the ideal of S generated by the g α s.

72 THE HASH OPERATION

73 The hash operation Let J be any ideal of S. Set S De D e := u p e. 1 is the K-vector space generated by monomials x α with x α D e.

74 The hash operation J 0 := J and J i+1 := ( J i (J [pe ] i : S u) I e (uj i ) S De ) S. Definition (Hash operation) Set J #e := i 0 J i.

75 Why we introduce the hash operation? Theorem (B., Katzman (2014)) For any ideal J of S, J #e is the greatest φ-fixed ideal contained in J.

76 THE ALGORITHM

77 Contribution: the statement of the algorithm The input data are: a prime number p, a finite field K, S = K[x 1,..., x d ], and u S. We initialize I = S and L as the empty list.

78 Contribution: the statement of the algorithm Algorithm (B., Katzman (2014)) From now on, execute the following commands: (i) Compute I #e. Assign to I the value of I #e. (ii) If I is not in the list L, then add it. (iii) If I = 0, then stop and output the list L. (iv) If I 0 but principal, assign to I the value of mi and come back to step (i). (v) If I 0 and not principal, then compute { V ideal mi V I, dimz/pz I /V = 1 }. For each element V of such set, come back to step (i).

79 Output The list L of all the φ-fixed ideals of S contained in m, where φ = π e u. Remark The previous program has been implemented in Macaulay2.

80 LOCAL COHOMOLOGY SPECTRAL SEQUENCES

81 Motivation: Mayer-Vietoris long exact sequence A commutative Noetherian ring. M an A-module and a, b A ideals. Theorem There is a natural long exact sequence 0 H 0 a+b (M) H 0 a (M) H 0 b (M) H 0 a b (M) H 1 a+b (M) H i a+b (M) H i a(m) H i b (M) H i a b (M) H i+1 a+b (M)...

82 The Mayer-Vietoris spectral sequence A commutative Noetherian ring, I A ideal and M an A-module. I = I 1... I n its minimal primary decomposition. P: poset given by all the possible sums of the ideals I i s ordered by reverse inclusion. Theorem (Àlvarez Montaner, García López and Zarzuela (2003), Lyubeznik (2007)) There is a homological spectral sequence 2 = L i lim H j I p (M) E i,j p P i H j i I (M).

83 Motivation: long exact sequence of local cohomology A commutative Noetherian ring. M an A-module and J, a, b A ideals. Short exact sequence 0 M/ (a b) M M/aM M/bM M/ (a + b) M 0.

84 Motivation: long exact sequence of local cohomology Long exact sequence 0 HJ 0 (M/ (a b) M) HJ 0 (M/aM) H0 J (M/bM) HJ 0 (M/ (a + b) M) H 1 J (M/ (a b) M) HJ i (M/ (a b) M) HJ i (M/aM) Hi J (M/bM) HJ i (M/ (a + b) M) H i+1 J (M/ (a b) M)...

85 Again in prime characteristic K field of prime characteristic. A = K[[x 1,..., x d ]] and I A an ideal. R := A/I Theorem (Lyubeznik (1997)) There is a contravariant functor (namely, H R,A ) such that ( H R,A Hm (R)) d i = H i I (A). This leads to:

86 Again in prime characteristic Question Is there a cohomological spectral sequence 2 = R i lim Hm j (A/I p ) E i,j p P i H i+j m (A/I ) that corresponds under H R,A to the Mayer-Vietoris spectral sequence? Answer Such spectral sequence does not exist!!

87 Why the previous spectral sequence does not exist? If so, then one could conclude that H j m(a/i ) = p P H j m(a/i p ). Problem The previous (false) isomorphism would contradict Hochster s decomposition of local cohomology modules!!

88 Purposes of this work Homological aim Establish the Mayer-Vietoris spectral sequence for several functors. Cohomological aim Construct cohomological spectral sequences.

89 CONSTRUCTION OF SPECTRAL SEQUENCES

90 Fixing notation A is a commutative Noetherian ring containing a field. A is the category of A-modules. I A an ideal and I = I 1... I n its primary decomposition. P: poset given by all the possible sums of the ideals I i s ordered by reverse inclusion. P := P {0 P, 1 P}. Dir(P, A) is the category of direct systems valued on A. Inv(P, A) is the category of inverse systems valued on A.

91 Important remark Notice that, under the previous assumptions, 1 P can only be the ideal I.

92 A picture I I 1 I 2 I 1 + I 2 A

93 Key construction in the homological case T [ ] Let A Dir( P, A) be an additive univariate functor which verifies the following requirements: (i) For any p P, T p is a covariant, left exact, univariate functor which commutes with arbitrary direct sums. (ii) If p q then there exists a natural transformation of derived functors R i T p R i T q.

94 Key construction in the homological case Assume that T [ ] verifies one (and only one) of the following two assumptions. (a) For any p Spec(A) and for any maximal ideal m of A, there exists an A-module X such that, for any p P, { X, if p W(I p, J) and p m, T p (E(A/p)) m = 0, otherwise. It must be mentioned that X may depend on p and m, but not on p. Moreover, here W(I p, J) := { q Spec(A) I n p q + J for some integer n 1 }, and J is an ideal of A which does not depend on any of the previous choices.

95 Key construction in the homological case (b) For any p Spec(A) and for any maximal ideal m of A, there exists an A-module Y such that, for any p P, { Y, if p / W(I p, J) and p m, T p (E(A/p)) m = 0, otherwise. It must be mentioned that Y may depend on p and m, but not on p. Once again, J is an ideal of A which does not depend on any of the previous choices.

96 EXAMPLES OF FUNCTORS VERIFYING THE FOREGOING ASSUMPTIONS

97 Examples Let N be a finitely generated A-module with finite projective dimension. The generalized torsion functor ( ) Γ Ip (N, ) := lim Hom j A k N N/Ip k N,. The generalized ideal transform ) D Ip (N, ) := lim Hom A (I p k N,. k N

98 Examples The local cohomology module with respect to pairs of ideals: { } Γ Ip,J(M) := m M Ipm l Jm for some l N. It is worth noting that Γ Ip,J(M) = {x M Supp A (Ax) W (I p, J)}.

99 Warning If T [ ] = Hom A (A/[ ], ), then the previous assumptions are not fulfilled!

100 CONTRIBUTION

101 Main result in the homological framework Theorem (Àlvarez Montaner, B., Zarzuela) Given any A-module M, there is an spectral sequence E i,j where T := T 1 P. 2 = L i lim R j T p (M) p P i R j i T (M),

102 Generalized local cohomology If T [ ] = Γ [ ] (N, ), then E i,j 2 = L i lim H j I p (N, M) p P i H j i I (N, M).

103 Generalized ideal transforms If T [ ] = D [ ] (N, ), then E i,j 2 = L i lim R j D Ip (N, M) p P i R j i D I (N, M).

104 Local cohomology of pairs of ideals If T [ ] = Γ [ ],J ( ), then E i,j 2 = L i lim H j I (M) p,j p P i H j i I,J (M).

105 CONSTRUCTION OF COHOMOLOGICAL SPECTRAL SEQUENCES

106 Key construction in the cohomological framework P: poset given by all the possible quotients M/JM, where J runs over all the possible sums of I 1,..., I n. Let A T A be a covariant, left exact, univariate functor. Set Inv(P, A) T Inv(P, A) G = (G p ) p P T (G) := (T (G p )) p P.

107 A picture M/IM M/I 1 M M/I 2 M M/I 3 M M/(I 1 + I 2 )M M/(I 1 + I 3 )M M/(I 2 + I 3 )M M/(I 1 + I 2 + I 3 )M 0

108 Requiring more restrictions on T In addition, we suppose that T commutes with arbitrary direct sums and that T verifies one (and only one) of the following two assumptions. (a) For any p Spec(A) and for any maximal ideal m of A, there exists an A-module X such that { X, if p W(J, K) and p m, T (E (A/p)) m = 0, otherwise. It is worth noting that X only depends on p and m. Here, J and K are ideals of A which do not depend on any of the previous choices.

109 Requiring more restrictions on T (b) For any p Spec(A) and for any maximal ideal m of A, there exists an A-module Y such that { Y, if p / W(J, K) and p m, T (E (A/p)) m = 0, otherwise. It is worth noting that Y only depends on p and m. Here, J and K are ideals of A which do not depend on any of the previous choices.

110 WHEN THESE ASSUMPTIONS ARE FULFILLED?

111 Examples in the cohomological framework J and K are arbitrary ideals of A. N is a finitely generated A-module with finite projective dimension. Here, we present our list of examples: Hom A (N, ). Γ J (N, ). D J (N, ). Γ J,K.

112 CONTRIBUTION

113 Contribution Theorem (Àlvarez Montaner, B., Zarzuela) The following statements hold. (i) There is a first quadrant spectral sequence ( ) E i,j 2 = R i lim R j T (A/[ ]) R i+j lim i T (A/[ ]). p P p P

114 Contribution Theorem (Àlvarez Montaner, B., Zarzuela) (ii) Moreover, if then E i,j 2 = R i lim p P lim p P R j T (A/[ ]) T = T lim, i p P R i+j (T lim p P ) (A/[ ]).

115 Contribution Theorem (Àlvarez Montaner, B., Zarzuela) (iii) If, furthermore, A/[ ] is flasque, then the previous spectral sequence becomes into the next one: E i,j 2 = R i lim p P R j T (A/[ ]) i R i+j T (A/I ).

116 DEGENERATION OF SPECTRAL SEQUENCES

117 Degeneration of spectral sequences Question When the previous spectral sequences degenerate at the second page?

118 Main result about degeneration (homological framework) A containing a field K. M an A-module. T [ ] as before. For all p P, R j T p (M) = 0 for all j h p. For all p q, Hom A (R hp T p (M), R hq T q (M)) = 0.

119 Main result about degeneration (homological framework) Theorem The spectral sequence E i,j 2 = L i lim R j T p (M) p P degenerates at the E 2 -page. i R j i T (M)

120 Main result about degeneration (homological framework) Theorem (Àlvarez Montaner, B., Zarzuela) For each 0 r cd(t ) there is an increasing filtration {G r k } r k dim(a) of R r T (M) by A-modules such that G r k /G r k 1 = {q P k+r=h q} for all r k dim(a). ( Hk 1 ((q, 1 P ); K) K R hq T q (M))

121 Main result about degeneration (cohomological framework) A containing a field K. A/[ ] is flasque. R j T (A/I p ) = 0 for all j d p. ( For all p q, Hom A R d p T (A/I p ), R dq T (A/I q ) ) = 0.

122 Main result about degeneration (cohomological framework) Theorem The spectral sequence E i,j 2 = R i lim p P R j T (A/[ ]) i R i+j T (A/I ) degenerates at the E 2 -sheet.

123 Main result about degeneration (cohomological framework) Theorem (Àlvarez Montaner, B., Zarzuela) For each 0 r cd(t ) there is an increasing filtration {H r k } r k dim(a) of R r T (A/I ) by A-modules such that H r k /Hr k 1 = {q P r k=d q} ( T dq (A/I q ) K Hk 1 (( q, 1 P) ; K ) ).

124 EXTENSION PROBLEMS

125 Extension problems in the homological framework 0 G r k 1 G r k G r k /G r k G r k G r k+1 G r k+1 /G r k G r dim(a) R r T (M) G r dim(a) /G r dim(a) 1 0. Question How far are these short exact sequences from being splitted?

126 LOCAL COHOMOLOGY EXTENSION PROBLEMS

127 Local cohomology extension problems A regular ring containing a field K. Mayer-Vietoris spectral sequence of local cohomology modules: E i,j 2 = L i lim H j I p (M) p P i H j i I (M). Remark Its extension problems have been extensively studied.

128 Extension problems in the cohomological framework 0 H r k 1 H r k H r k /Hr k H r k H r k+1 H r k+1 /Hr k H r dim(a) R r T (A/I ) H r dim(a) /Hr dim(a) 1 0. Question How far are these short exact sequences from being splitted?

129 Extension problems in the cohomological framework Lemma We assume, in addition, that Ext 1 A ( R d p T (A/I p ), R dq T (A/I q ) ) = 0 provided d p d q + 2. Then, the natural maps Ext 1 A (H k/h k 1, H k 1 ) Ext 1 A (H k/h k 1, H k 1 /H k 2 ) are injective for all k 2.

130 Local cohomology extension problems A is any commutative Noetherian ring containing a field K. E i,j 2 = j=d q H dq m (A/I q ) t i i H i+j m (A/I ). Question What about its extension problems when it degenerates at its second page?

131 CONTRIBUTION

132 Contribution B is a polynomial ring graded by a matrix of positive type W (e.g. B = K[x 1,..., x d ] and W the identity matrix of size d). I a squarefree monomial ideal inside B. Proposition (Àlvarez Montaner, B., Zarzuela) The extension problems are labeled by the multiplication by x i.

133 Contribution Remark When B = K[x 1,..., x d ] and W the identity matrix of size d, this result is exactly what is known as Gräbe s formula.

134 ACKNOWLEDGMENTS

135 WE STOP HERE