Non-unique factorizations in bounded hereditary noetherian prime - PowerPoint PPT Presentation

Non-unique factorizations in bounded hereditary noetherian prime rings Daniel Smertnig University of Waterloo Conference on Rings and Factorizations Graz, Feb 21, 2018

Outline Factorizations in noncommutative rings Non-unique factorizations Bounded hereditary noetherian prime (HNP) rings Beyond bounded HNP rings

Defjnition Factorizations such that R (unital) ring, H = R • its monoid of non-zero-divisors. Assume: R • is divisor-closed in R . A non-unit u ∈ H is an atom if u = ab with a , b ∈ H ⇒ a ∈ H × or b ∈ H × . A ( H ) ... set of all atoms. H is atomic if for every a ∈ H \ H × , there exist atoms u 1 , . . . , u k , a = u 1 · · · u k .

Factorizations Question What is a factorization, precisely? Two problems: 2 Units should have a trivial factorization. First attempt: an element of F ∗ ( A ( H )) ... free monoid on atoms. 1 In H , we have uv = ( u ε )( ε − 1 v ) for ε ∈ H × Note: Cannot reduce H / H × in general.

Factorizations factorizations . Defjnition On H × × F ∗ ( A ( H )) defjne ( ε, u 1 ∗ · · · ∗ u k ) ∼ ( η, v 1 ∗ · · · ∗ v l ) if 1 ε u 1 · · · u k = η v 1 · · · v l in H , 2 k = l , and 3 there exist δ i ∈ H × s.t. u i = δ − 1 u k = δ − 1 ε u 1 = η v 1 δ 1 , i − 1 v i δ i , k − 1 v k . � � H × × F ∗ ( A ( H )) Z ∗ ( H ) = / ∼ is the monoid of (rigid) There is a homomorphism π : Z ∗ ( H ) → H Z ∗ ( a ) = π − 1 ( { a } ) is the set of (rigid) factorizations of a .

Factor posets The Factor poset is Then [ aR , R ] = { bR | b ∈ R • , aR ⊆ bR ⊆ R } Z ∗ ( a ) ← → maximal , fjnite chains in [ aR , R ] . u 1 ∗ · · · ∗ u k corresponds to R � u 1 R � u 1 u 2 R � · · · � u 1 · · · u k R = aR .

Lemma If R satisfjes ACCP, that is ACC on principal left and right ideals, Note: ACC on one side is not suffjcient. ACCP ⇒ atomic By taking cofactors, ACC on the left implies DCC on [ aR , R ] ! then R • is atomic.

Similiarity factoriality Question What should it mean for R to be factorial? We say R is similarity factorial . Suppose R is atomic, and if bR , cR ∈ [ aR , R ] then bR + cR and bR ∩ cR are principal (e.g., R a PID). ⇒ [ aR , R ] is a fjnite length modular lattice ⇒ If u 1 ∗ · · · ∗ u k , v 1 ∗ · · · ∗ v l ∈ Z ∗ ( a ) , then k = l , and there exists a permutation σ s.t. R / u i R ∼ = R / v σ ( i ) R .

Limitations... Remark distributive lattices appear as factor lattices. half-factorial! [ aR , R ] need not be distributive, e.g., R = M 2 ( Z ) . K � x , y � has distributive factor lattices, but all fjnite Z � x , y � is not similarity factorial (but subsimilarity factorial). Let H be the Q -division algebra of Hamilton quaternions. Then H [ x ] is Euclidean ( ⇒ PID), but H [ x , y ] is not

Non-unique factorizations

Arithmetical Invariants Defjnition System of sets of lengths: Elasticity : Let a ∈ R • . The set of lengths of a is L ( a ) = { | z | | z ∈ Z ∗ ( a ) } = { k ∈ N 0 | a = u 1 · · · u k with u 1 , . . . , u k ∈ R • atoms } . L ( R ) = { L ( a ) | a ∈ R • } . R is half-factorial if | L ( a ) | = 1 for all a ∈ R • . | L ( a ) | ≥ 2 ⇒ | L ( a n ) | ≥ n + 1 . ρ ( a ) = sup L ( a ) min L ( a ) ∈ Q ≥ 1 ∪ {∞} , ρ ( R ) = sup { ρ ( a ) | a ∈ R • } ∈ R ≥ 1 ∪ {∞} .

Distances Defjnition E.g. d sim , compare factors up to similarity, ... Let D = { ( z , z ′ ) ∈ Z ∗ ( H ) × Z ∗ ( H ) : π ( z ) = π ( z ′ ) } . A distance on R • is a map d : D → N 0 s.t. 1 d ( z , z ) = 0 2 d ( z , z ′ ) = d ( z ′ , z ) 3 d ( z , z ′ ) ≤ d ( z , z ′′ ) + d ( z ′′ , z ′ ) 4 d ( x ∗ z , x ∗ z ′ ) = d ( z , z ′ ) = d ( z ∗ x , z ′ ∗ x ) 5 || z | − | z ′ || ≤ d ( z , z ′ ) ≤ max {| z | , | z ′ | , 1 } .

Catenary degrees that Defjnition Fix a distance d; let z , z ′ ∈ Z ∗ ( a ) . An N -chain is a sequence z = z 0 , z 1 , . . . , z l = z ′ in Z ∗ ( a ) , such d ( z i − 1 , z i ) ≤ N for i ∈ [1 , l ] . The catenary degree c d ( a ) is the smallest N such that for all z , z ′ ∈ Z ∗ ( a ) , there exists an N -chain between z and z ′ . c d ( H ) = sup { c d ( a ) | a ∈ H } .

Transfer homomorphisms Defjnition and Let H , T be cancellative monoids, T × = { 1 } . A homomorphism θ : H → T is a transfer homomorphism if 1 θ ( H ) = T and θ − 1 ( { 1 } ) = H × . 2 Whenever θ ( a ) = st , there exist b , c ∈ H such that a = bc , θ ( b ) = s , θ ( c ) = t .

Transfer homomorphisms Theorem H If T is commutative If θ : H → T is a transfer homomorphism, it induces a homomorphism θ ∗ , θ ∗ Z ∗ ( H ) Z ∗ ( T ) θ T , with θ ∗ ( Z ∗ ( a )) = Z ∗ ( θ ( a )) . L ( H ) = L ( T ) . c d ( H ) ≤ max { c p ( T ) , c ( θ ) } .

Defjnition Monoid of zero-sum sequences The submonoid (fjnitely many atoms, arithmetical invariants fjnite, ...) Let ( G , +) be an abelian group, G 0 ⊆ G , ( F ( G 0 ) , · ) the free abelian monoid with basis G 0 . S = g 1 · · · g l ∈ F ( G 0 ) is called a sequence (formal product!). σ ( S ) = g 1 + · · · + g l ∈ G is its sum. S is a zero-sum sequence if σ ( S ) = 0 . B ( G 0 ) = { S ∈ F ( G 0 ) | σ ( S ) = 0 G } ⊂ F ( G 0 ) is the monoid of zero-sum sequences over G 0 . If G 0 is fjnite, then B ( G 0 ) is a fjnitely generated Krull monoid

Reminder: Commutative Dedekind domains Theorem a aR Let R be a commutative Dedekind domain, ( G , +) its class group, G 0 = { [ p ] | p ∈ spec ( R ) } . There is a transfer homomorphism θ : R • → B ( G 0 ) : R • F ( spec ( R )) p 1 · · · p r θ B ( G 0 ) F ( G 0 ) [ p 1 ] · · · [ p r ] Moreover, c ( θ ) ≤ 2 .

Hereditary noetherian prime (HNP) rings

Hereditary orders Let order. R is a maximal order if it is not contained in a strictly larger Defjnition K Maximal orders are hereditary (right ideals are projective). K be a number fjeld, A a central simple K -algebra, A ∼ = M n ( D ) O its ring of algebraic integers, . . . R . . . O ⊂ R ⊂ A an order in A (subring, R O fjnitely generated, O KR = A ).

Examples... Hurwitz quaternions With p a prime, � � 1 , i , j , 1 + i + j + k Z 2 with i 2 = j 2 = k 2 = − 1 , ij = − ji = k . � Z � p Z . Z Z

HNP rings (Noncommutative) hereditary noetherian prime (HNP) rings are analogues of commutative Dedekind domains. Structure theory for f. g. projective modules and for fjnite-length modules (Levy–Robson 2011). Examples: Hereditary orders over commutative Dedekind domains. Endomorphism rings of f. g. projective modules over Dedekind domains. Some skew polynomial rings over commutative Dedekind domains, e.g., K [ x ± 1 ][ y ± 1 ; σ ] with yx = qxy . A = A 1 ( K ) = K [ y ][ x ; d dy ] , R is right bounded , if for every a ∈ R • , there exists a nonzero ideal I ⊆ R with I ⊆ aR .

From factor lattices to modules r Problem! I is stably free . R aR R I There can be non-principal, stably free right ideals I . r Z ∗ ( a ) [ aR , R ] ? R / aR . How to go from R / aR back to [ aR , R ] ? Commutative: ann ( R / I ) = I ; if R is a Dedekind domain: � � i ∼ R / = R / p e i i . p e i i =1 i =1 Noncommutative: R / aR ∼ = R / I ⇒ ? 0 R / aR 0 0 R / aR 0 ⇒ I ⊕ R ∼ = aR ⊕ R .

Hermite rings Defjnition R is a (right) Hermite ring if every stably free right R -module is free. Commutative Dedekind domains are Hermite. Indefjnite hereditary orders over rings of algebraic integers are Hermite (by strong approximation ). Defjnite (quaternion) orders over rings of algebraic integers are usually not Hermite. HNP rings R with udim R ≥ 2 are Hermite. A 1 ( K ) is not Hermite.

Modules over HNP rings Let V , W be simple modules. Defjnition Isomorphism classes of simple modules are organized into cycle towers and faithful towers . successor. In a bounded HNP ring, all simple modules are unfaithful. W is a successor of V if Ext 1 R ( V , W ) � = 0 . W 1 , . . . , W n pairwise non-isomorphic simple modules. Cycle tower : All W i are unfaithful. W i +1 is a successor of W i , and W 1 is a successor of W n . Faithful tower : W 1 is faithful, W 2 , . . . , W n are unfaithful. W i is a successor of W i − 1 , and W n has no unfaithful

Modules over HNP rings If R is bounded, every fjnite length module M is a direct sum of uniserial modules, modules of a cycle tower T . If a ∈ R • , then R / aR has fjnite length. M ∼ = U 1 ⊕ · · · ⊕ U n . The composition factors of U i form a slice of a repetition of the

A class group Proposition S ( R ) ... isomorphism classes of simple modules. T ( R ) ⊂ F ( S ( R )) ... towers (as sums of their simple modules), K 0 mod fl ( R ) = q F ( S ( R )) ⊇ q F ( T ( R )) For M a module of fjnite length with composition factors W 1 , . . . , W n , have ( M ) = ( W 1 ) + · · · + ( W n ) ∈ F ( S ( R )) . If a ∈ R • , then ( R / aR ) ∈ F ( T ( R ))

A class group Defjnition The class group of R is Set P ( R ) = { ( R / aR ) | a ∈ R • } ⊆ F ( T ( R )) . C ( R ) = q F ( T ( R )) / �P ( R ) � . Set C max ( R ) = { [ T ] ∈ C ( R ) | T ∈ T ( R ) } . C ( R ) ∼ = G ( R ) = ker (Ψ + ) . C ( R ) and C max ( R ) are Morita invariant.