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` onoma 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 M n 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 = End R ( M ). So one needs to study projective modules over S . If X is a summand of M (Λ) then P = Hom R ( M , X ) is a direct summand of 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 of 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 Add ℵ 0 ( M )

The stable category Let C be a full subcategory of modules closed by finite direct sums. Fix an object X in C . For any pair of modules M , N in C consider the following subgroup of Hom R ( M , N ): J X ( M , N ) = { f : M → N | f factors through X (Λ) for some set Λ } J X is an ideal of C and we can consider the quotient category C / J X which has the same objects as C and Hom C / J X ( M , N ): = Hom C ( M , N ) / J X ( 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 Add Λ ( 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 J R 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 = End R ( M ) . Let X be an object of Add ( M ) . Let P S = Hom R ( M , X ) , and let I = Tr S ( P ) = � f ∈ Hom S ( P , S ) f ( P ) . Then, the functor Hom R ( M , − ) ⊗ S S / I induces an equivalence between the categories Add ( M ) / J X 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 Add ℵ 0 ( 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 Add ℵ 0 ( M ) Let M be a finitely generated module over a commutative noetherian ring R . Then: (1) Compute S = End R ( 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 out 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 Add ℵ 0 ( M ): Two points of view Let R be a local commutative noetherian ring, with completion ˆ R . Let M R be a finitely generated right R -module with endomorphism ring S . Then M ⊗ R ˆ R = ˆ M ∼ = L n 1 1 ⊕ · · · ⊕ L n k k with L 1 , . . . , L k 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 ′ i s ! R ( ˆ M ) ∼ = S ⊗ R ˆ R ∼ = ˆ Moreover, End ˆ S and S ) ∼ = M n 1 ( D 1 ) × · · · × M n k ( D k ) ∼ S / J ( ˆ ˆ = S / J ( S ). Where D 1 , . . . , D k are division rings. Therefore if P S projective, P / PJ ( S ) is a direct sum of simple right S / J ( S )-modules.

Monoids of modules If N is in Add ℵ 0 ( M ) we set dim ( � N � ) = ( a 1 , . . . , a k ) = L ( a 1 ) ⊕ · · · ⊕ L ( a k ) where N ⊗ R ˆ R ∼ and a i ∈ N 0 ∪ {∞} = N ∗ 1 0 k = V ( a 1 ) ⊕ · · · ⊕ V ( a k ) Equivalently, Hom R ( M , N ) / Hom R ( M , N ) J ( S ) ∼ 1 k where V i is a simple right S -module with endomorphism ring D i . dim ( N 1 ⊕ N 2 ) = dim ( N 1 ) + dim ( N 2 ) Hence, dim ( Add ℵ 0 ( M )) is a submonoid of ( N 0 ∪ {∞} ) k that we denote by V ∗ ( M ). V ( M ) is the submonoid given by the f.g. summands, that is, V ∗ ( M ) ∩ N k 0 .

We know a nice upper bound for V ∗ ( M )!!! Theorem (H- P. Pˇ r´ ıhoda 2010) 0 ) k containing ( n 1 , . . . , n k ) ∈ N k . Then the Let A be a submonoid of ( N ∗ 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  x 1   x 1   x 1   m 1 N ∗  0 . . . .         E 1  = E 2 and D . . .  ∈ .         . . . .       x k x k x k m ℓ N ∗ 0 where D ∈ M ℓ × k ( N 0 ) , E 1 and E 2 ∈ M m × k ( N 0 ) , and m 1 , . . . , m ℓ are elements of N 0 and m i > 1 . (2) There exist a noetherian semilocal ring S, such that S / J ( S ) ∼ = M n 1 ( D 1 ) × · · · × M n k ( D k ) where D i ’s are division rings, with V ∗ ( S ) = A. In the above situation, V ( S ) = A ∩ N k 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 = L 1 denote the completion. The integral clouse is � x � R = R y R ∼ which has two maximal ideals. So, R ⊗ R ˆ = L 2 ⊕ L 3 Consider M = R ⊕ R . Hence, M ∼ ˆ = L 1 ⊕ L 2 ⊕ L 3 The finitely generated modules in Add ( M ) are all direct sums of R and R . dim � R � = (1 , 0 , 0) and dim � R � = (0 , 1 , 1)

Example: Still R = C [ X , Y ] ( X , Y ) / ( X 2 − ( Y 3 − Y 2 )) and M = R ⊕ R � R � M S = End R ( M ) = R R � 1 � 0 If I is the trace ideal of P = Hom R ( M , R ) ∼ = S . 0 0 S / I ∼ = R / M ∼ = L 2 / ˆ M × L 3 / ˆ M This decomposition lifts to Add ( M )!!!! More precisely R ( ω ) ⊕ R ∼ = X 1 ⊕ X 2 X 1 and X 2 are not a direct sum of f. g. modules and R ( ω ) ⊕ X i ∼ = X i . dim � X 1 � = ( ∞ , 1 , 0) and dim � X 2 � = ( ∞ , 0 , 1) 0 ) 3 are the solutions in N ∗ dim ( V ∗ ( M )) ⊆ ( N ∗ 0 of the equation x + y = x + z

A realization Theorem Theorem Consider submonoid of N k 0 , containing ( n 1 , . . . , n k ) ∈ N k , defined by a system of equations E 1 X = E 2 X where E i ∈ M m × k ( N 0 ) . Set F ∈ M m × k ( N 0 ) 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 1 ⊕ · · · ⊕ L n k M = L n 1 k with L i indecomposable, such that V ∗ ( M ) is the monoid of solutions in N ∗ 0 of the system ( E 1 + F ) X = ( E 2 + F ) X . Hence, any module of the form L ( a 1 ) ⊕ · · · ⊕ L ( a k ) is extended from an 1 k R-module provided that at least one a i 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 M R 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; R K ( ˆ R ) ∼ = K ( ˆ R ) ( I ) ; (ii) there exists a set I such that L ⊗ ˆ (iii) L is extended from a module in Add ( M R ) .