Annihilation of Cohomology over Curve Singularities

Transcription

1 Annihilation of Cohomology over Curve Singularities Maurice Auslander International Conference Özgür Esentepe University of Toronto April 29, 2018 Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

2 Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

3 Definition and Motivation Let R be a commutative Noetherian ring. The nth cohomology annihilator ideal of R is defined to be M for n 1. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

4 Definition and Motivation Let R be a commutative Noetherian ring. The nth cohomology annihilator ideal of R is defined to be M N for n 1. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

5 Definition and Motivation Let R be a commutative Noetherian ring. The nth cohomology annihilator ideal of R is defined to be Ext n R (M, N) for n 1. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

6 Definition and Motivation Let R be a commutative Noetherian ring. The nth cohomology annihilator ideal of R is defined to be ann R Ext n R (M, N) for n 1. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

7 Definition and Motivation Let R be a commutative Noetherian ring. The nth cohomology annihilator ideal of R is defined to be ca n (R) = ann R Ext n R (M, N) for n 1. M,N mod R Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

8 Definition and Motivation Let R be a commutative Noetherian ring. The nth cohomology annihilator ideal of R is defined to be ca n (R) = ann R Ext n R (M, N) for n 1. M,N mod R Definition The cohomology annihilator ideal is the union ca(r) = n 1 ca n (R). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

9 For any n 1 and M, N mod R, we have where ΩM is a syzygy of M. Ext n+1 R (M, N) = Ext n R (ΩM, N) Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

10 For any n 1 and M, N mod R, we have where ΩM is a syzygy of M. Ext n+1 R (M, N) = Ext n R (ΩM, N) Therefore, we have an increasing chain ca 1 (R) ca 2 (R)... ca n (R) ca n+1 (R)... Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

11 For any n 1 and M, N mod R, we have where ΩM is a syzygy of M. Ext n+1 R (M, N) = Ext n R (ΩM, N) Therefore, we have an increasing chain ca 1 (R) ca 2 (R)... ca n (R) ca n+1 (R)... As R is Noetherian, we have for s 0. ca(r) = ca s (R) Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

12 Finite Global Dimension Suppose that R has finite global dimension d. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

13 Finite Global Dimension Suppose that R has finite global dimension d. Ext d+1 R (M, N) = 0 for all M, N mod R Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

14 Finite Global Dimension Suppose that R has finite global dimension d. Ext d+1 R (M, N) = 0 for all M, N mod R ann R Ext d+1 R (M, N) = R for all M, N mod R Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

15 Finite Global Dimension Suppose that R has finite global dimension d. Ext d+1 R (M, N) = 0 for all M, N mod R ann R Ext d+1 R (M, N) = R for all M, N mod R Therefore, ca(r) = R. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

16 Finite Global Dimension Suppose that R has finite global dimension d. Ext d+1 R (M, N) = 0 for all M, N mod R ann R Ext d+1 R (M, N) = R for all M, N mod R Therefore, ca(r) = R. So, this ideal is only interesting in the case of infinite global dimension. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

17 In fact, the cohomology annihilator is closely related to the singularities of the corresponding geometric object. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

18 In fact, the cohomology annihilator is closely related to the singularities of the corresponding geometric object. Theorem (Iyengar-Takahashi) Let R be an equicharacteristic excellent local ring or a localization of a finitely generated algebra over a field - of Krull dimension d. Then, the vanishing locus of ca(r) is equal to the singular locus of R. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

19 In fact, the cohomology annihilator is closely related to the singularities of the corresponding geometric object. Theorem (Iyengar-Takahashi) Let R be an equicharacteristic excellent local ring or a localization of a finitely generated algebra over a field - of Krull dimension d. Then, the vanishing locus of ca(r) is equal to the singular locus of R. Theorem (Iyengar-Takahashi) Let R be an equicharacteristic complete local ring or an affine algebra over a field - of Krull dimension d. Then, the Jacobian ideal of R is contained in ca(r). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

20 In fact, the cohomology annihilator is closely related to the singularities of the corresponding geometric object. Theorem (Iyengar-Takahashi) Let R be an equicharacteristic excellent local ring or a localization of a finitely generated algebra over a field - of Krull dimension d. Then, the vanishing locus of ca(r) is equal to the singular locus of R. Theorem (Iyengar-Takahashi) Let R be an equicharacteristic complete local ring or an affine algebra over a field - of Krull dimension d. Then, the Jacobian ideal of R is contained in ca(r). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

21 An Example Let k be an algebraically closed field of characteristic zero, f = x 3 y 5 and R = k[[x, y]]. (f ) Then, the Jacobian and the cohomology annihilator ideals are Jac(R) = (x 2, y 4 ) (= ( x f, y f )) ca(r) = (x 2, xy, y 3 ) (Details on computations later.) Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

22 Ragnar s Observation Jac(R) = (x 2, y 4 ) ca(r) = (x 2, xy, y 3 ) 1 x y xy y 2 xy 2 y 3 xy 3 This is how the Jacobian algebra R/ Jac(R) looks like. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

23 Ragnar s Observation Jac(R) = (x 2, y 4 ) ca(r) = (x 2, xy, y 3 ) 1 x y xy y 2 xy 2 y 3 xy 3 This is how R/ ca(r) looks like inside the Jacobian algebra. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

24 Ragnar s Observation Jac(R) = (x 2, y 4 ) ca(r) = (x 2, xy, y 3 ) 1 x y xy y 2 xy 2 y 3 xy 3 This is how R/ ca(r) looks like inside the Jacobian algebra. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

25 Ragnar s Observation Jac(R) = (x 2, y 4 ) ca(r) = (x 2, xy, y 3 ) 1 x y xy y 2 xy 2 y 3 xy 3 This is how R/ ca(r) looks like inside the Jacobian algebra. 8 = 2 4. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

26 Ragnar s Observation Jac(R) = (x 2, y 4 ) ca(r) = (x 2, xy, y 3 ) 1 x y xy y 2 xy 2 y 3 xy 3 This is how R/ ca(r) looks like inside the Jacobian algebra. 8 = 2 4. That is, the vector space dimensions have the relation dim k (R/ Jac(R)) = 2 dim k (R/ ca(r)). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

27 Ragnar s Observation The same phenomenon can also be seen in the following examples: k[[x, y]]/(x 2 y n ) (n odd), k[[x, y]]/(x 3 y 4 ), k[[x, y, z]]/(x 3 + y 3 + z 3 λxyz) (λ 3 27). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

28 Ragnar s Observation The same phenomenon can also be seen in the following examples: k[[x, y]]/(x 2 y n ) (n odd), k[[x, y]]/(x 3 y 4 ), k[[x, y, z]]/(x 3 + y 3 + z 3 λxyz) (λ 3 27). Why? This was the question Ragnar asked me. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

29 Figure: (Up to isomorphism of pictures) From left to right: Ragnar Buchweitz, Louis-Philippe Thibault, Vincent Gelinas, Ben Briggs and me. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

30 The Singularity Category From now on, we will assume that R is a commutative Gorenstein ring of Krull dimension d. MCM(R) : stable category of maximal Cohen-Macaulay modules. D sg (R) : the singularity category - the bounded derived category modulo perfect complexes. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

31 The Singularity Category From now on, we will assume that R is a commutative Gorenstein ring of Krull dimension d. MCM(R) : stable category of maximal Cohen-Macaulay modules. D sg (R) : the singularity category - the bounded derived category modulo perfect complexes. Theorem (Buchweitz) MCM(R) = D sg (R) as triangulated categories. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

32 The Singularity Category From now on, we will assume that R is a commutative Gorenstein ring of Krull dimension d. MCM(R) : stable category of maximal Cohen-Macaulay modules. D sg (R) : the singularity category - the bounded derived category modulo perfect complexes. Theorem (Buchweitz) MCM(R) = D sg (R) as triangulated categories. For M mod R, we denote by M st the maximal Cohen-Macaulay approximation of M. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

33 Stable Ext The usual Ext groups: Ext n R (M, N) = Hom D b (R)(M, N[n]). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

34 Stable Ext The usual Ext groups: Ext n R (M, N) = Hom D b (R)(M, N[n]). The stable Ext groups are defined similarly: Ext n R (M, N) = Hom D sg(r)(m, N[n]). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

35 Stable Ext The usual Ext groups: Ext n R (M, N) = Hom D b (R)(M, N[n]). The stable Ext groups are defined similarly: Ext n R (M, N) = Hom D sg(r)(m, N[n]). For any M, N mod(r), one has Ext n R (M, N) = Extn R (M, N) for any n > d, Ext n R (M, N) = Hom R(Ω n M st, N st ) for any n Z.

36 Stable Ext The usual Ext groups: Ext n R (M, N) = Hom D b (R)(M, N[n]). The stable Ext groups are defined similarly: Ext n R (M, N) = Hom D sg(r)(m, N[n]). For any M, N mod(r), one has Ext n R (M, N) = Extn R (M, N) for any n > d, Ext n R (M, N) = Hom R(Ω n M st, N st ) for any n Z. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

37 Back to the Problem Lemma (?) Let R be a commutative Gorenstein ring. Then, ca(r) = ann R (M) M MCM(R) where ann R (M) = ann End R (M) is the stable annihilator of M. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

38 Proof. We know that ca(r) = ca s (R) for s 0. Pick s > d. We have Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

39 Proof. We know that ca(r) = ca s (R) for s 0. Pick s > d. We have Ext s R (M, N) = Exts R (M, N) = Hom R(Ω s M st, N st ) Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

40 Proof. We know that ca(r) = ca s (R) for s 0. Pick s > d. We have Ext s R (M, N) = Exts R (M, N) = Hom R(Ω s M st, N st ) So, ca(r) = ann R Hom R (Ω s M, N) = = M,N MCM(R) M,N MCM(R) M MCM(R) ann R Hom R (M, N) ann R End R (M) Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

41 Remarks Note that r ann R (M) iff multiplication with r factors through a projective R-module. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

42 Remarks Note that r ann R (M) iff multiplication with r factors through a projective R-module. Notation ann R (M) := ann R End R (M). Because for any commutative ring A and any module X one has ann A (X ) = ann A End A (X ). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

43 Remarks Note that r ann R (M) iff multiplication with r factors through a projective R-module. Notation ann R (M) := ann R End R (M). Because for any commutative ring A and any module X one has ann A (X ) = ann A End A (X ). This description is useful in computations. Indeed, one can compute cohomology annihilator over a hypersurface ring using matrix factorizations. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

44 Remarks Note that r ann R (M) iff multiplication with r factors through a projective R-module. Notation ann R (M) := ann R End R (M). Because for any commutative ring A and any module X one has ann A (X ) = ann A End A (X ). This description is useful in computations. Indeed, one can compute cohomology annihilator over a hypersurface ring using matrix factorizations. In terms of matrix factorizations, r is in the cohomology annihilator if and only if multiplication with r is null-homotopic for every matrix factorization. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

45 Remarks - II With this description, it is easy to show that if r stably annihilates M, then it annihilates every object in the smallest subcategory of MCM(R) containing M and closed under finite direct sums, direct summands, syzygies, cosyzygies and duals. We will revisit this later. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

46 Remarks - II With this description, it is easy to show that if r stably annihilates M, then it annihilates every object in the smallest subcategory of MCM(R) containing M and closed under finite direct sums, direct summands, syzygies, cosyzygies and duals. We will revisit this later. So, ann R (M) = ann R Hom(Ω n M, M) n Z Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

47 The Coin Problem Question. We know that we can write any integer as an integer linear combination of 3 and 5(Euclidean algorithm). Which numbers can we write using only non-negative linear combinations? Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

48 The Coin Problem Question. We know that we can write any integer as an integer linear combination of 3 and 5(Euclidean algorithm). Which numbers can we write using only non-negative linear combinations? any number after 7 can be obtained this way. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

49 The Coin Problem Question. We know that we can write any integer as an integer linear combination of 3 and 5(Euclidean algorithm). Which numbers can we write using only non-negative linear combinations? any number after 7 can be obtained this way. On the other hand, notice that we have Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

50 For any two integers a, b with a + b = 7; we have exactly one of a and b in the semigroup generated by 3 and 5. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

51 For any two integers a, b with a + b = 7; we have exactly one of a and b in the semigroup generated by 3 and 5. Every number after 7 is in the semigroup and every number before 0 is not. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

52 For any two integers a, b with a + b = 7; we have exactly one of a and b in the semigroup generated by 3 and 5. Every number after 7 is in the semigroup and every number before 0 is not. The only interesting part is between 0 and 7. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

53 For any two integers a, b with a + b = 7; we have exactly one of a and b in the semigroup generated by 3 and 5. Every number after 7 is in the semigroup and every number before 0 is not. The only interesting part is between 0 and 7. There are 8 numbers here and exactly 4 of them are in the semigroup. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

54 For any two integers a, b with a + b = 7; we have exactly one of a and b in the semigroup generated by 3 and 5. Every number after 7 is in the semigroup and every number before 0 is not. The only interesting part is between 0 and 7. There are 8 numbers here and exactly 4 of them are in the semigroup. 8 = 2 4. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

55 Conductor Let T = k[[t 3, t 5 ]] be the corresponding semigroup algebra - which is isomorphic to R = k[[x, y]]/(x 3 y 5 ). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

56 Conductor Let T = k[[t 3, t 5 ]] be the corresponding semigroup algebra - which is isomorphic to R = k[[x, y]]/(x 3 y 5 ). Ragnar! I think the cohomology annihilator of R is the image of the ideal in T generated by the powers of t which come after the last number that we can not write as a linear combination of 3 and 5!. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

57 Conductor Let T = k[[t 3, t 5 ]] be the corresponding semigroup algebra - which is isomorphic to R = k[[x, y]]/(x 3 y 5 ). Ragnar! I think the cohomology annihilator of R is the image of the ideal in T generated by the powers of t which come after the last number that we can not write as a linear combination of 3 and 5!. That ideal has a name: the conductor ideal! And you should look up Wang s paper. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

58 Conductor Let T = k[[t 3, t 5 ]] be the corresponding semigroup algebra - which is isomorphic to R = k[[x, y]]/(x 3 y 5 ). Ragnar! I think the cohomology annihilator of R is the image of the ideal in T generated by the powers of t which come after the last number that we can not write as a linear combination of 3 and 5!. That ideal has a name: the conductor ideal! And you should look up Wang s paper. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

59 Conductor Let R be a commutative Noetherian ring and let R be its normalization: the integral closure of R inside its total quotient ring. Then, the conductor ideal co(r) is co(r) = { r R : r R R } co(r) is the largest subset of R which is both an ideal of R and R. Theorem (Wang) Let R be a one-dimensional reduced complete Noetherian local ring. Then, co(r) ann R Ext 2 (M, N) for any M, N mod R. In other words, for a one-dimensional reduced complete Noetherian local ring R ; co(r) ca(r). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

60 Conductor Lemma Let R be any Japanese ring. Then, End R ( R) = R cor as R-modules via the isomorphism f f (1). Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

61 Conductor Lemma Let R be any Japanese ring. Then, End R ( R) = R cor as R-modules via the isomorphism f f (1). If R is a one or two dimensional Gorenstein ring, then R is maximal Cohen-Macaulay over R. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

62 Conductor Lemma Let R be any Japanese ring. Then, End R ( R) = R cor as R-modules via the isomorphism f f (1). If R is a one or two dimensional Gorenstein ring, then R is maximal Cohen-Macaulay over R. Hence, ca(r) ann R R = ann R R cor = co(r) Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

63 Theorem (E.) Let R be a one-dimensional complete local Gorenstein ring. Then, ca(r) co(r). If R is also reduced then there is equality. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

64 Back to Ragnar s Observation Ragnar s observation is now explained via the famous Milnor-Jung formula for algebraic curves: Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

65 Back to Ragnar s Observation Ragnar s observation is now explained via the famous Milnor-Jung formula for algebraic curves: Theorem (Milnor-Jung Formula) Let C be a reduced irreducible curve with an isolated singular point. Let R be the coordinate ring of C - localized at this singular point. Then, dim k R Jac(R) = 2 dim k R co(r) r + 1 where r is the number of branches at the singular point. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

66 Stably Annihilating an Algebra of Finite Global Dimension Are there other examples? Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

67 Stably Annihilating an Algebra of Finite Global Dimension Example Are there other examples? Let G be a finite subgroup of SL(2, C), S = k[[x, y]] and R = S G be the invariant ring. Then, ca(r) = ann R (S) = ann R (S G) Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

68 Stably Annihilating an Algebra of Finite Global Dimension Example Are there other examples? Let G be a finite subgroup of SL(2, C), S = k[[x, y]] and R = S G be the invariant ring. Then, In general? ca(r) = ann R (S) = ann R (S G) Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

69 Theorem (E.) Let R be a commutative Gorenstein ring and Λ be a finite R-algebra of finite global dimension. Suppose that R is a direct summand in Λ. Then, ann R End Dsg (R)(Λ) gldim Λ+1 ca(r) ann R End Dsg (R)(Λ) If, in addition, Λ MCM(R) then ann R (Λ) gldim Λ+1 ca(r) ann R (Λ) In particular, up to radicals, in order to annihilate the singularity category, it is enough to stably annihilate a noncommutative resolution. Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33

70 Thank you! Özgür Esentepe (University of Toronto) Annihilation of Cohomology over Curve Singularities April 29, / 33