The stable category and big pure projective modules joint work with - - PowerPoint PPT Presentation

The stable category and big pure projective modules

joint work with Pavel Pˇ r´ ıhoda and Roger Wiegand Dolors Herbera Universitat Aut`

• noma de Barcelona

P¨ arnu, July 16, 2019

• Overview. Basic definitions. Some references.

General aim: Study pure-projective modules over a commutative noetherian ring R, that is, direct summands of a direct sum of finitely generated R-modules. Too Wild problem!!! More modest aim for this talk: R will be also local. Study the category Add(M) of direct summands of M(Λ), for some set Λ, where M is a finitely generated. Want to enlarge our view from previous work on the study of add(M), the category of direct summands of Mn for some finite n (Facchini-H, Wiegand 2000) plugging in results on big projective modules. (Pˇ r´ ıhoda and H.-Pˇ r´ ıhoda 2010) These results are based on exploiting the fact that two projective modules are isomorphic if and only if they are isomorphic modulo the Jacobson radical. (Pˇ r´ ıhoda 2007)

Reducing to the case of projective modules.

M any finitely generated right module over a general ring R. Tool: (Dress 70’s) The category Add(M) is equivalent to the cate- gory of projective right modules over S = EndR(M). So one needs to study projective modules over S. If X is a summand of M(Λ) then P = HomR(M, X) is a direct summand

• f S(Λ), hence a projective right module over S.

If P is a direct summand of S(Λ) then X = P ⊗S M is a direct summand

• f M(Λ).

This equivalence takes finitely generate/countably generated objects in Add (M) to finitely generated projective/countably generated projective modules. By a result of Kaplansky we can reduce to the case of countably generated projective modules that is to Addℵ0 (M)

The stable category

Let C be a full subcategory of modules closed by finite direct sums. Fix an

• bject X in C. For any pair of modules M, N in C consider the following

subgroup of HomR(M, N): JX(M, N) = {f : M → N | f factors through X (Λ) for some set Λ} JX is an ideal of C and we can consider the quotient category C/JX which has the same objects as C and HomC/JX (M, N): = HomC(M, N)/JX(M, N) Two objects M and N are isomorphic in this quotient category if and only if there exist a set Λ, P and Q in AddΛ (X) such that P ⊕ M ∼ = Q ⊕ N If M and N are countably generated, Λ can be taken to be countable. For example: If X = R and C consists of finitely generated modules then the quotient by JR is the stable category of C.

We extend the equivalence to the stable category

Theorem

Let M be a finitely generated right module over ANY ring R, and let S = EndR(M). Let X be an object of Add(M). Let PS = HomR(M, X), and let I = TrS(P) =

f ∈HomS(P,S) f (P).

Then, the functor HomR(M, −) ⊗S S/I induces an equivalence between the categories Add(M)/JX and Add(S/I). The equivalence restricts well to countably generated objects. Main result to prove this equivalence: Projective modules can be lifted modulo a trace ideal of a projective module. If S is a module finite algebra over a commutative noetherian ring the objects in Addℵ0(S) can be reconstructed from the objects in add(S/I) where I runs through trace ideals of (countably generated) projective right S-modules.

Recipe to compute the objects in Addℵ0(M)

Let M be a finitely generated module over a commutative noetherian ring

• R. Then:

(1) Compute S = EndR(M). S is a finitely generated algebra over R. (2) Compute the idempotent ideals of S. The set idempotent ideals coincides with the set of trace ideals of projective modules. (3) For any idempotent ideal I of S, compute the objects in add (S/I). (4) “Glue” everything together. We compute all the information on infintiely generated modules

• ut of finitely generated data!!

Passing to the local case: The relation with the completion is very good

If T is a commutative notherian complete ring, then finitely generated modules are direct sum of modules with a local endomorphism ring. By a result of Warfield and the Krull-Remak-Schmidt theorem, all pure-projective modules are direct sum of finitely generated modules with a local endomorphism ring.

Theorem

Let R be a commutative local noetherian ring. Let X, Y be direct summands of an arbitrary direct sum of finitely generated modules over R. Then X ≃ Y if and only if ˆ X = X ⊗R ˆ R ≃ ˆ Y .

Theorem

Let R be a commutative local noetherian ring. Let X be an R-module. Then X is pure projective as R-module if and only if ˆ X is a pure projectitive ˆ R-module.

Description of objects in Addℵ0(M): Two points of view

Let R be a local commutative noetherian ring, with completion ˆ

• R. Let MR

be a finitely generated right R-module with endomorphism ring S. Then M ⊗R ˆ R = ˆ M ∼ = Ln1

1 ⊕ · · · ⊕ Lnk k

with L1, . . . , Lk indecomposable ˆ R-modules (hence, with local endomorphism ring). Therefore, by a result of Warfield, every module in Add ( ˆ M) is a direct sum of L′

is!

Moreover, End ˆ

R( ˆ

M) ∼ = S ⊗R ˆ R ∼ = ˆ S and ˆ S/J( ˆ S) ∼ = Mn1(D1) × · · · × Mnk(Dk) ∼ = S/J(S). Where D1, . . . , Dk are division rings. Therefore if PS projective, P/PJ(S) is a direct sum of simple right S/J(S)-modules.

Monoids of modules

If N is in Addℵ0 (M) we set dim(N) = (a1, . . . , ak) where N ⊗R ˆ R ∼ = L(a1)

1

⊕ · · · ⊕ L(ak)

k

and ai ∈ N0 ∪ {∞} = N∗ Equivalently, HomR(M, N)/HomR(M, N)J(S) ∼ = V (a1)

1

⊕ · · · ⊕ V (ak)

k

where Vi is a simple right S-module with endomorphism ring Di. dim(N1 ⊕ N2) = dim(N1) + dim(N2) Hence, dim (Addℵ0 (M)) is a submonoid of (N0 ∪ {∞})k that we denote by V ∗(M). V (M) is the submonoid given by the f.g. summands, that is, V ∗(M) ∩ Nk

0.

We know a nice upper bound for V ∗(M)!!!

Theorem (H- P. Pˇ r´ ıhoda 2010)

Let A be a submonoid of (N∗

0)k containing (n1, . . . , nk) ∈ Nk. Then the

following statements are equivalent: (1) A is is the set of solutions in N∗

0 of a system of homogeneous

diophantic linear equations and of congruences

E1     x1 . . . xk     = E2     x1 . . . xk     and D     x1 . . . xk     ∈     m1N∗ . . . mℓN∗    

where D ∈ Mℓ×k(N0), E1 and E2 ∈ Mm×k(N0), and m1, . . . , mℓ are elements of N0 and mi > 1. (2) There exist a noetherian semilocal ring S, such that S/J(S) ∼ = Mn1(D1) × · · · × Mnk(Dk) where Di’s are division rings, with V ∗(S) = A. In the above situation, V (S) = A ∩ Nk

0.

Example:

R = C[X, Y ](X,Y )/(X 2 − (Y 3 − Y 2)). Let x, y denote the classes of X and Y . The ring R has maximal ideal m = (x, y). Let ˆ R = L1 denote the completion. The integral clouse is R = R x y

• which has two maximal ideals. So, R ⊗R ˆ

R ∼ = L2 ⊕ L3 Consider M = R ⊕ R. Hence, ˆ M ∼ = L1 ⊕ L2 ⊕ L3 The finitely generated modules in Add(M) are all direct sums of R and R. dimR = (1, 0, 0) and dimR = (0, 1, 1)

Example:

Still R = C[X, Y ](X,Y )/(X 2 − (Y 3 − Y 2)) and M = R ⊕ R S = EndR(M) = R M R R

• If I is the trace ideal of P = HomR(M, R) ∼

= 1

• S.

S/I ∼ = R/M ∼ = L2/ ˆ M × L3/ ˆ M This decomposition lifts to Add(M)!!!! More precisely R(ω) ⊕ R ∼ = X1 ⊕ X2 X1 and X2 are not a direct sum of f. g. modules and R(ω) ⊕ Xi ∼ = Xi. dimX1 = (∞, 1, 0) and dimX2 = (∞, 0, 1) dim(V ∗(M)) ⊆ (N∗

0)3 are the solutions in N∗ 0 of the equation

x + y = x + z

A realization Theorem

Theorem

Consider submonoid of Nk

0, containing (n1, . . . , nk) ∈ Nk, defined by a

system of equations E1X = E2X where Ei ∈ Mm×k(N0). Set F ∈ Mm×k(N0) be such that all its entries are 1. Then there exists a local noetherian domain R of Krull dimension 1 with reduced completion ˆ R, and a finitely generated torsion free R-module M, with ˆ M = Ln1

1 ⊕ · · · ⊕ Lnk k with Li indecomposable, such that V ∗(M) is the

monoid of solutions in N∗

0 of the system

(E1 + F)X = (E2 + F)X. Hence, any module of the form L(a1)

1

⊕ · · · ⊕ L(ak)

k

is extended from an R-module provided that at least one ai is infinite.

Infinite version of the Levy-Odenthal criteria

If R is a domain of Krull dimension 1, and M is f.g. torsion free (=maximal Cohen-Macaulay) then S/I is artinian for any nonzero two-sided ideal I. Therefore V (S/I) is a free commutative monoid.

Theorem

Let R be a local noetherian domain of Krull dimension 1 with reduced completion ˆ

• R. Let K( ˆ

R) denote the localization of ˆ R at the complementary of the union of the set of minimal primes of ˆ

• R. (Hence ˆ

R is a product of fields and K( ˆ R) ∼ = ˆ R ⊗R K(R)). Let MR be a finitely generated torsion free R-module. Then the following statements are equivalent for an ˆ R-module L ∈ Add( ˆ M ˆ

R),

(i) L is extended from an R-module; (ii) there exists a set I such that L ⊗ ˆ

R K( ˆ

R) ∼ = K( ˆ R)(I); (iii) L is extended from a module in Add(MR).