A sufficient criterion for homotopy cartesianess

A sufficien t criterion for homotop y cartesian es s Alb erto Canona co, Matthi a s K ¨ unzer Octob er 26, 20 18 Abstract Supp ose giv en a comm utativ e qu adrangle in a V erdier triangulated catego ry such that there exists an induced isomorphism on the horizon tally tak en cones. Sup pose that the endomorphism ring of the initial or the terminal corner ob ject of this quadrangle satisfies a finiteness cond it ion. Then this quadrangle is homotop y cartesian. Con ten ts 0 In tro duction 1 1 A ring theoretical l emma 2 2 The criterion for homotopy cartesianes s 3 3 A counterexa mple 4 0 In tro duction In an ab elian category , a comm utativ e quadrangle is called bic artesian if its diagona l sequenc e is short exact, i.e. if it is a pullbac k and a pushout. A comm utativ e quadrangle is bicartesian if and only if w e get induced isomophisms on the horizontal k ernels and on the horizontal cok ernels. In a triangulated category in the sense of Verdier [3, Def. 1-1 ] , a commutativ e quadrangle is c alled homotopy c artesian (or a Mayer-Vietoris sq uar e , or a distinguishe d we ak squar e ), if its diagonal sequence fits in to a distinguished triangle. A homotopy car t es ian sq uare has a (non-uniquely) indu ced isomorphism on the horizontally tak en cones [2, Lem. 1.4.4 ] . W e consider the con v erse question : a comm utativ e quadrangle that ha s an isomorphism induced on the horizon tally tak en cones, is it homotopy cartesian? W e s how this to b e true if the endomorphism ring of the ob ject in the terminal or initial corner satisfies a finiteness condition. This finiteness condition is for instance satisfied for the endomorphism rings o ccurring in D b ( A -mo d), where A is a finite-dimensional algebra ov er some field; or in A -mo d , where A is a finite-dimensional F rob en ius algebra ov er some field. This finiteness condition, ho w ev er, in general fails for the endomorphism rings occurring in K b ( Z -pro j). W e sho w by an example that the conclusion on our commutativ e quadrangle to b e homotopy cartesian fails there as w ell. MSC 2000: 18 E30. 1 2 1 A rin g the oretical lemma Let R b e a ring. Denote by J( R ) its Ja cobs on r a dical. If R/ J( R ) is artinian, w e fix the notat io n R / J( R ) ≃ Q n i =1 D k i × k i i for its W edderburn decomp o- sition, where D i is a sk ewfield for 1 ≤ i ≤ n . A r ing R shall b e called he ad-finite if its head R / J( R ) is artinian and if in the W edderburn de- comp osition of R/ J( R ), the sk ewfie ld D i is finite dimensional o v er its cen tre for eac h 1 ≤ i ≤ n . F or example, finite dimensional a lgebras o v er some field are head-finite. F or ano ther example, a lo cal ring R fo r whic h R/ J( R ) is comm utativ e is head-finite. Lemma 1.1 Su pp ose given a he ad-finite ring R and an element ε ∈ R . (1) Ther e exists α ∈ R such that 1 + ε + α ε 2 is a unit in R . (2) Ther e exists β ∈ R such that 1 + ε + ε 2 β is a unit in R . Pr o of. As sertion (2) follows b y an application of (1) to R ◦ , so it remains to prov e (1) . Since an elemen t ρ ∈ R is a unit in R if and only if ρ + J( R ) is a unit in R/ J( R ), w e ma y assume J( R ) = 0 and R to b e a pro duct of matrix rings ov e r sk ew fields whic h are finite-dimensional o v er their cen tres. F urthermore, w e ma y assume R to b e a single matrix ring o v er a sk ewfield whic h is finite-dimensional ov er its cen tre K . In particular, we ma y assume R to b e a finite dimensional K -algebra. Let m ≥ 0 and s ( X ) ∈ K [ X ] b e suc h that ε is a ro ot of the p olynomial X m + X m +1 s ( X ). Let α := s ( ε ). Then ( ε + αε 2 ) m +1 =  ε m (1 + s ( ε ) ε )  ε (1 + s ( ε ) ε ) m  = 0 , and th us 1 + ε + α ε 2 is a unit in R . Remark 1.2 (1) The conclusions of Lemma 1 .1 do not hold for all rings, as the example R = Z a nd ε = 3 sho ws. (2) The conclusions of Lemma 1.1 hold for a lo cal ring R , regardless whether R / J( R ) is finite-dimensional o v er its cen tre or not. (3) In Lemma 1.1.(1), w e do not claim that α ε = εα . Whereas this pro perty can b e ac hiev ed if R is a finite dimensional algebra o v er some field, as w e ha v e seen in t he pro of of lo c. cit., the fir st reduction step at the b eginning of this pro of p ossibly migh t not resp ec t this prop ert y . 3 2 The crite rion for homotop y carte sianess Let C be a triangulated categor y in the sense o f Verdier [3, Def. 1-1 ] . A comm utativ e quadrangle B b   g / / C c   B ′ g ′ / / C ′ in C is said to be homotopy c artesian if there exists a distinguished triangle containing the sequence B “ b g ” / / B ′ ⊕ C ( g ′ − c ) / / C ′ , cf. [2, D ef. 1.4 .1 ] . W e r emark that by [2, Lem. 1.4.4 ] , suc h a homotop y cartesian square fits in to a morphism o f distinguished triangles of the form ( b, c, 1) (and, by symmetry , also in one of t he fo rm ( g , g ′ , 1)). Prop osition 2.1 Supp ose given a c ommutative diagr am in C A f / / a ≀   B g / / b   C h / / c   A [1] a [1] ≀   A ′ f ′ / / B ′ g ′ / / C ′ h ′ / / A ′ [1] whose r ows ar e distinguishe d triangles. (1) Supp ose End C C ′ to b e he ad-finite. Then the quadr angle ( g , g ′ , b, c ) is homotopy c artesian. (2) Supp ose End C B to b e he ad-finite. Then the quadr a ngle ( g , g ′ , b, c ) is homotopy c artesian. Pr o of. By duality , it suffices to pro v e (1). By isomorphic replacemen t at A ′ , w e may a s sume that A = A ′ and a = 1. By [2, Lem. 1.4 .3 ] , there exists C ✲ ˜ c C ′ suc h that t he quadrangle ( g , g ′ , b, ˜ c ) is homotopy cartesian and such that h ′ ◦ ˜ c = h . It suffices to show that the quadrang les ( g , g ′ , b, c ) a nd ( g , g ′ , b, ˜ c ) are isomorphic. Since ( ˜ c − c ) ◦ g = 0 , there exists A [1] ✲ ψ C ′ suc h that ψ ◦ h = ˜ c − c . L et ε := ψ ◦ h ′ ∈ End C C ′ . By a ssumption on End C C ′ , w e ma y apply Lemma 1.1 t o find a n elemen t α ∈ End C C ′ suc h that 1 + ε + α ◦ ε 2 is a unit in End C C ′ , i.e. an auto morphism of C ′ . 4 W e claim that w e hav e the f ollo wing isomorphism of comm utativ e quadrangles. B g / / b A A A A A A A C c A A A A A A A B ′ g ′ / / C ′ 1+ ε + α ◦ ε 2 ≀   B g / / b A A A A A A A C ˜ c A A A A A A A B ′ g ′ / / C ′ In fact, (1 + ε + α ◦ ε 2 ) ◦ c = c + ψ ◦ h ′ ◦ c + α ◦ ψ ◦ h ′ ◦ ψ ◦ h ′ ◦ c = c + ψ ◦ h + α ◦ ψ ◦ h ′ ◦ ψ ◦ h = ˜ c + α ◦ ψ ◦ h ′ ◦ ( ˜ c − c ) = ˜ c + α ◦ ψ ◦ ( h − h ) = ˜ c , and (1 + ε + α ◦ ε 2 ) ◦ g ′ = g ′ + ψ ◦ h ′ ◦ g ′ + α ◦ ψ ◦ h ′ ◦ ψ ◦ h ′ ◦ g ′ = g ′ . Question 2.2 (op en) Supp ose given a diagr am as in Pr op os ition 2.1 . Is the c one o f b iso- morphic to the c one of c ? Note that the isomorphism in question is not required to satisfy an y comm utativities. In § 3, w e giv e an example in whic h there is no isomorphism b et w een these cones that is compatible with t he diagram. Note that the middle quadrangle of suc h a diag ram is a w eak square b y the k ernel-cok ernel criterion applied in the F reyd category of C ; cf. e.g. [1, § A.6.3, Def. A.9, Lem. A.11 ] . Question 2.3 (op en) Supp ose given a c ommutative quadr a n gle that has an isomorphism in- duc e d on the horizontal ly and on the v ertic al ly taken c ones. Is it homotopy c artesian? Of course, pro vided t he endomorphism ring of its ob ject in the initial or terminal corner is head-finite, suc h a quadrangle is homotopy cartesian by Prop osition 2.1 . 3 A co un t erexample W e shall giv e a n example of a commutativ e quadrangle in a triangulated category that ho r i- zon tally fits in to a morphisms of tr iangles con taining an isomorphism as in Prop osition 2.1, but that is not homotopy cartesian; somewhat w orse still, v ertically , it do es not fit into a morphism of triangles con taining an isomorphism. This will sho w that the head-finiteness conditions in Prop osition 2.1 cannot b e entirely dropp ed. 5 Let C := K b ( Z -pro j). As to sign conv entions, the standard distinguished triangle on a morphism X ✲ f Y in C is given as follow s. . . .   . . .   . . .   . . .   X i f i / / δ i   Y i “ 1 0 ” / / ∂ i   Y i ⊕ X i +1 ( 0 1 ) / / „ ∂ i f i +1 0 − δ i +1 «   X i +1 − δ i +1   X i +1 f i +1 / /   Y i +1 “ 1 0 ” / /   Y i +1 ⊕ X i +2 ( 0 1 ) / /   X i +2   . . . . . . . . . . . . W e shall allo w ourselv es to omit zero ob ject en tries when displa ying complexes. By Z ⊕ m w e denote t he direct sum of m copies of Z , where m ≥ 2. Lemma 3.1 The f o l lowing triangle s ar e distinguishe d in C for every a, b ∈ Z . (1) 0 / /   0 / /   Z 1 / / „ b a 2 «   Z a 2   Z − a 2   b / / Z   “ 1 0 ” / / Z ⊕ 2   ( 0 1 ) / / Z   Z / / 0 / / 0 / / 0 (2) 0 / /   Z 1 / / „ − a a 2 «   Z a / / „ − a 3 a 2 «   Z a 2   Z − a 2   “ 1 0 ” / / Z ⊕ 2   „ a 2 0 0 1 « / / Z ⊕ 2   ( − 1 0 ) / / Z   Z / / 0 / / 0 / / 0 (3) 0 / /   Z “ 1 a ” / / „ − a 3 a 2 «   Z ⊕ 2 ( a − 1 ) / / „ a 2 0 0 a 2 «   Z   Z „ a 2 0 « / / Z ⊕ 2 “ 0 1 − 1 0 ” / / Z ⊕ 2 / / 0 (4) 0 / /   0 / /   Z 1+ a / / a 2   Z   Z a 2 / / Z 1 − a / / Z / / 0 6 Pr o of. It is enough to sho w that eac h triangle is of the form X f / / Y u ◦ g / / Z ′ h ◦ u − 1 / / X [1] with X ✲ f Y ✲ g Z ✲ h X [1] a standard distinguished triangle and Z ✲ u Z ′ an isomorphism in C . Indeed, (1) is already standard, and it is straightforw ard to c hec k that in cases (2), (3) and (4) one can take resp ectiv ely the follow ing morphisms f o r u . Z ⊕ 2 ( 1 0 ) / / − a 1 a 2 0 0 a 2 !   Z „ − a 3 a 2 «   Z ⊕ 3 „ a 2 0 − 1 0 1 0 « / / Z ⊕ 2 Z ⊕ 2 “ 1 0 a − 1 ” / / „ − a 3 a 2 a 2 0 «   Z ⊕ 2 „ a 2 0 0 a 2 «   Z ⊕ 2 “ 0 1 − 1 0 ” / / Z ⊕ 2 Z 1 − a / / a 2   Z a 2   Z 1 − a / / Z Let a ∈ Z b e suc h that a ≥ 3. C onsider the follow ing mor phis m of distinguished triangles. The differen tials of the complexes are displa y ed from lo w er left to upp er right, and the triangles are displa y ed from left to righ t. Notice that the tria ngles are distinguished b ecause they can b e obtained b y applying axiom (TR 2) to the triangles (2) and (1) (with b = − a ) of Lemma 3.1 . ( ∗ ) Z ⊕ 2 „ a 2 0 0 1 « / / Z ⊕ 2 ( − 1 0 ) / / ( 0 1 )   Z / /   0 Z 1 / /  − a a 2  B B          Z a / /  − a 3 a 2  B B          1   Z “ − 1 0 ” / / a 2 B B          1+ a   Z ⊕ 2 B B          0 / / B B          0 / / B B            0 / / B B            Z  a − a 2  B B          Z ⊕ 2 ( 0 1 ) / / Z / / 0 / / 0 Z 1 / /  − a a 2  B B          Z a / / a 2 B B          Z “ − 1 0 ” / / B B          Z ⊕ 2 B B          0 / / B B          0 / / B B          0 / / B B          Z  a − a 2  B B          7 As a n aside, we remark that the second morphism fro m the left in ( ∗ ) is split epimorphic. W e claim that the middle quadrangle of ( ∗ ) is not homotopy cartesian. W e assume the con trary . The diagonal sequence of the middle quadrangle is giv en as follow s. Z “ 1 a ” / / „ − a 3 a 2 «   Z ⊕ 2 ( a − 1 − a ) / / „ a 2 0 0 a 2 «   Z   Z ⊕ 2 “ 0 1 − 1 0 ” / / Z ⊕ 2 / / 0 In con trast, the follo wing sequence fits in to the distinguished tria ng le (3) of Lemma 3 .1. Z “ 1 a ” / / „ − a 3 a 2 «   Z ⊕ 2 ( a − 1 ) / / „ a 2 0 0 a 2 «   Z   Z ⊕ 2 “ 0 1 − 1 0 ” / / Z ⊕ 2 / / 0 By uniqueness of the cone up to isomorphism in C , there exists a comm utativ e triangle in C Z   Z ⊕ 2 „ a 2 0 0 a 2 «   ( a − 1 − a ) / / ( a − 1 ) 3 3 h h h h h h h h h h h h h h h h h h h h h h h h h h Z s ? ?          0 Z ⊕ 2 / / h h h h h h h h h h h h h h h h h h h 3 3 h h h h h h h 0 @ @         with s ∈ {− 1 , +1 } . W e conclude that sa ≡ a 2 a and − s − sa ≡ a 2 − 1. If s = 1 , then the second congruence gives a ≡ a 2 0, whic h is imp ossible since a ≥ 2 . If s = − 1, then the first congruence giv es 2 a ≡ a 2 0, whic h is imp ossible since a ≥ 3. W e hav e ar r ived at a c ontr adiction . W e claim that v ertically , the middle quadrangle of ( ∗ ) do es not fit in to a morphism of distin- guished triangles that contains a n isomorphism. W e note that this claim implies the preceding claim (cf. [2, Lem. 1.4.4 ] ), which we will thu s ha v e prov en t wice. W e assume the con trary . Inserting triangles (1) (with b = − a 3 ) and (4) of Lemma 3 .1 v ertically , w e obtain the following comm utativ e quadrangle ab o v e our giv e n one, where t ∈ {− 1 , +1 } . Differen tials are displa y ed from low er left to upp er right. Z t / /  1 0    Z 1 − a   0 : : u u u u u u u u u u / /   0 < < x x x x x x x x x   Z ⊕ 2 ( − 1 0 ) / / Z Z  − a 3 a 2  ; ; v v v v v v v v v a / / Z a 2 < < y y y y y y y y y Comm utativit y of this quadrangle in C means that t (1 − a ) ≡ a 2 − 1. If t = − 1, then a ≡ a 2 0 ensues, whic h is imp ossible since a ≥ 2. If t = 1, then 2 ≡ a 2 a and hence 2 ≡ a 0 ensues, whic h is imp ossible since a ≥ 3. W e ha v e arriv ed at a c ontr adiction . 8 References [1] K ¨ unzer, M., Hel ler triangulate d c ate gori es, prepr in t, math.CT/050 8565, 2005. [2] Neeman, A. , T riangulate d Cate gories, Ann. Math. Stud. 148, 2001 . [3] Verdier, J.L. , Cat´ egories Deriv ´ ees , published in SGA 4 1/2 , SLN 56 9 , p. 26 2–311, 1977 (written 1 963). Alber to Canona co Univ ersit` a degli studi di Pa via Dipartimento di matematica F. Casora ti Via F er rata, 1 I-2710 0 Pavia alb erto.canonaco@unipv.it Matthias K ¨ unzer Lehrstuhl D f ¨ ur Ma thema tik R WTH Aa c hen T emplergra ben 64 D-52062 Aa chen kuenzer@math.r w th-a ac hen.de www.math.rwth-aachen.de/ ∼ kuenzer