Differential Graded Algebras model

differential graded lie algebras and formal deformation theory and on differential systems graded lie algebras and pseudo-groups, differential systems associated with simple graded lie algebras
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Published Date:26-07-2017
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Math 7350: Differential Graded Algebras and Differential Graded Categories Taught by Yuri Berest Notes by David Mehrle dmehrlemath.cornell.edu Cornell University Spring 2017 Last updated June 5, 2017. The latest version is online here.Contents 1 Introduction: “Handwaving” 4 1.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The Role of Derived Categories . . . . . . . . . . . . . . . . . . . 5 I Quivers and Gabriel’s Theorem 8 2 Quivers 9 2.1 Path Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Representations of Quivers . . . . . . . . . . . . . . . . . . . . . 13 2.3 Homological Properties of Path Algebras . . . . . . . . . . . . . 15 3 Gabriel’s Theorem 23 3.1 Representation Varieties . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Algebraic Group Actions on a Variety . . . . . . . . . . . . . . . 27 3.3 (Practical) Algebraic Geometry . . . . . . . . . . . . . . . . . . . 29 3.4 Back to quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.5 Modules over Hereditary Algebras . . . . . . . . . . . . . . . . . 36 3.6 Classification of graphs . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.7 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.8 Proof of Gabriel’s Theorem . . . . . . . . . . . . . . . . . . . . . 49 4 Generalizations of Gabriel’s Theorem 52 4.1 Tame Representation Type . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Kac’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Hall Algebra of a Quiver . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5 Multilocular Categories . . . . . . . . . . . . . . . . . . . . . . . 59 1II Differential Graded Algebras and Hochschild Homol- ogy 61 5 Differential Graded Algebras 62 5.1 Differential Graded Algebras . . . . . . . . . . . . . . . . . . . . 62 5.2 Algebraic de-Rham theory . . . . . . . . . . . . . . . . . . . . . . 67 6 Hochschild Homology 72 6.1 Hochschild Homology . . . . . . . . . . . . . . . . . . . . . . . . 72 6.2 Tor interpretation of Hochschild homology . . . . . . . . . . . . 73 6.3 Koszul Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.4 (Formal) Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.4.1 Grothendieck’s notion of smoothness . . . . . . . . . . . 80 6.4.2 Quillen’s notion of smoothness . . . . . . . . . . . . . . . 82 6.5 Proof of Hochschild-Kostant-Rosenberg . . . . . . . . . . . . . . 83 6.5.1 The antisymmetrizer map . . . . . . . . . . . . . . . . . . 83 6.5.2 The casen =1 . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.5.3 The general case . . . . . . . . . . . . . . . . . . . . . . . 87 6.6 Noncommutative Differential Forms . . . . . . . . . . . . . . . . 88 7 Higher Hochschild Homology 95 7.1 Some Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . 95 7.2 PROPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.3 Higher Hochschild Homology . . . . . . . . . . . . . . . . . . . . 100 7.4 Homotopy Theory of Simplicial groups . . . . . . . . . . . . . . . 101 7.5 Representation Homology . . . . . . . . . . . . . . . . . . . . . . 103 8 Quillen Homology 105 8.1 Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.2 Quillen Homology and Cyclic Formalism . . . . . . . . . . . . . 108 8.2.1 Quillen Homology of Simplicial Sets . . . . . . . . . . . . 110 8.2.2 Quillen Homology of Algebras . . . . . . . . . . . . . . . 112 8.3 Andre-Quillen ´ Homology . . . . . . . . . . . . . . . . . . . . . . 116 8.4 Automorphic Sets and Quandles . . . . . . . . . . . . . . . . . . 119 8.4.1 Topological Interpretation . . . . . . . . . . . . . . . . . . 123 8.4.2 Quillen Homology of Racks . . . . . . . . . . . . . . . . . 125 8.4.3 Rack (co)homology . . . . . . . . . . . . . . . . . . . . . . 129 III Differential Graded Categories 130 9 Differential Graded Categories 131 9.1 DG functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 29.2 The DG category of small DG categories . . . . . . . . . . . . . . 136 9.3 DG-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.3.1 Examples of DG-modules . . . . . . . . . . . . . . . . . . 137 9.4 Cofibratly generated model categories . . . . . . . . . . . . . . . 140 9.5 Quillen’s Small Object Argument . . . . . . . . . . . . . . . . . . 143 9.6 Tabuada’s Model Structure on dgCat . . . . . . . . . . . . . . . 147 k 10 Bibliography 150 3Chapter 1 Introduction: “Handwaving” 1.1 References All of these are linked to on the course webpage. We will frequently refer to the notes from two previous courses, Homological Algebra HA1 and Homotopical Algebra HA2. For references on DG categories, see Kel06, Kel93, Dri04, Toe11 ¨ , Toe11 ¨ , Tab05b, Tab05a, Kon98. 1.2 Some Definitions LetA be a unital associative algebra over a fieldk. Let A = Mod(A), the category of (left or right) modules overA. Definition 1.2.1. Let Com(A) be the category of complexes in A, that is, an  object inCom(A) is a complexC , that is, a diagram in A n n+1 n+2 d d d n n+1 n+2  C C C  n+1 n    withd2 Mor(A) andd d =0 for alln. A morphismf :C D of n complexes is a collection off 2 Mor(A) such that the following commutes. n n+1 d d n n+1 n+2  C C C  n n+1 n+2 f f f n n+1 d d n n+1 n+2  D D D    Definition 1.2.2. The cohomology of a complexC is the complexH (C), with n n n-1 H (C) = ker(d )/ im(d ). Cohomology defines a functor on complexes. 4The Role of Derived Categories 25 January, 2016 We want to study cohomology of complexes, rather than the complexes themselves. So we need to get rid of the “irrelevant information.”    Definition 1.2.3. A morphismf : C D is called a quasi-isomorphism  =    (weak equivalence) ifH(f ):H (C) -H (D). Definition 1.2.4. The derived categoryD(Mod(A)) is the localization of Com(A) at the quasi-isomorphisms, -1 D(Mod(A)) := Com(A)Qis . Here, Qis is the class of all quasi-isomorphisms. Remark 1.2.5 (Universal property of the derived category). The pair   D(Mod(A)),Q: Com(A)D(Mod(A)) is universal among all pairs (D,F: Com(A) D) where D is an additive category andF(f) is an isomorphism for allf2 Com(A). F Com(A) D F Q D(Mod(A)) Example 1.2.6. H(-) Com(A) Com(A) Q H(-) D(Mod(A)) 1.3 The Role of Derived Categories Derived categories have appeared in. . . (1) Algebraic Geometry (due to Grothendieck, Verdier) (i) Grothendieck duality theory (rigid dualizing complexes, see HA I) (ii) “Tilting” theory. Beilinson’s Derived Equivalence. n LetX be a projective algebraic variety overC, for exampleX = P . Let C VB(X) be the category of vector bundles overX, and let Coh(X) be the category of coherent sheaves onX. 5The Role of Derived Categories 27 January, 2016 n Example 1.3.1. IfX is affine, or maybe evenX = A , andA = CX, then C fg fg Coh(X) = Mod (A) and VB(X) = Proj (A). Think of coherent sheaves as a generalization of vector bundles where we may allow singular points. We might want to classify algebraic vector bundles overX; this was a classical question. To answer this question, we can instead classify algebraic coherent sheaves (see Example 1.3.1) overX. This is still a very difficult question, but we enlarge the category again and consider the bounded derived category b D (Coh(X)). This last question we can answer, with Theorem 1.3.2. b VB(X), Coh(X),D (Coh(X)) 2 Theorem 1.3.2 (Beilinson 1982). Let X = P (or any n). There is a natural C equivalence of triangulated categories     b n b fd D Coh(P ) D Rep (Q) , C whereQ is the quiver (0) (1) (2) x x x 1 1 1 . . . Q =      . . . . . . (0) (1) (2) x x x n n n So the derived category of coherent sheaves on projective spaces can be described by some complicated linear algebra.    b 2 b fd  Example 1.3.3. For example, ifn =2, thenD Coh(P ) =D Rep (Q) C whereQ is the quiver Q =    Ifn =1, thenQ is the Kronecker Quiver Q =   Example 1.3.4 (More motivation). Forn =2,A =kx,y, withk an algebraically closed field of characteristic zero. We want to classify ideals inA (asA-modules) in terms of linear algebra. 2 J  A () J2 Coh(A ) k ideal 2 2 Then we can embedA insideP , which contains a line` at infinity. 1 k k 2 2 A P ` 1 k k e e J J Jj =O 1 ` 1 P 6The Role of Derived Categories 27 January, 2016 2 We can embed coherent sheaves insideP as 2 b 2 Coh(P ) D (Coh(P ))   e e J 0J0 And then apply Theorem 1.3.2. In general, ifX is a smooth projective variety, we have the following theorem. Theorem 1.3.5 (Bondal, Kapranov, Van der Bergh, et. al.). b b  D (Coh(X)) =D (DGMod(A)), where DGMod(A) is dg-modules over the dg-algebraA. The problem with derived categories is that most invariants ofX are deter- b b mined byD (Coh(X)) but they cannot be computed directly fromD (Coh(X)). To understand the derived category, we need to “represent” the derived cate- gory in the same way that differential forms “represent” de Rham cohomology. b Therefore, we need to “enhance”D (Coh(X)) by replacing it by a dg-category b  D(Coh(X)) such thatH(D(Coh(X))) =D (Coh(X)). b There are many different dg-models forD (Coh(X)). we need a way to get rid of the irrelevant information carried byD(Coh(X)). The best way is to put a Quillen model structure on the category dgCat of all (small) dg- categories, making it into a model category. This is referred to as the study of noncommutative motives. Remark 1.3.6 (Goal). Our goal is to understand this model structure on dgCat. 7Part I Quivers and Gabriel’s Theorem 8Chapter 2 Quivers Definition 2.0.1. A quiver is a quadrupleQ = (Q ,Q ,s,t), where 0 1  Q is the set of vertices; 0  Q is the set of arrows; 1  s:Q Q is the source map; and 1 0  t:Q Q is the target map. 1 0 j i   a s(a) =j, t(a) =i Together,s,t are called the incidence maps. Definition 2.0.2. A quiver is called finite if and only ifjQ j1 andjQ j1. 0 1 Definition 2.0.3. A path inQ is a sequencea = (a ,::: ,a ) such thatt(a ) = 1 m i s(a ) for alli. i-1 1 2 m+1     a a a 1 2 m Definition 2.0.4. WriteP for the set of all paths inQ. Notice thats,t extend Q to mapss,t:P Q bys(a) =s(a ) andt(a) =t(a ). Q 0 m 1 Definition 2.0.5. The path algebrakQ for a quiverQ over a fieldk is defined by kQ = Span (P ), Q k with a product defined by concatenating paths:  ab ift(b) =s(a) ab = 0 otherwise 9Path Algebras 30 January, 2016 a e b      Paths of length zero are by convention the verticese fori2Q . This product i 0 is associative, and satisfies relations (for example) 2 ae =a,eb =b,ea =0,be =0,e =e What kind of algebras are the path algebras of quivers? Example 2.0.6. IfjQ j =1 andjQ j =r, then the path algebra is the free onr 0 1   variables,kQ khx ,x ,::: ,xi. Ifr =1, thenkQ kx. = = 1 2 r Example 2.0.7. If   1 2 n Q =  , thenkQ is isomorphic to the algebra of lower triangularnn matrices overk. Exercise 2.0.8. IfQ has at most one path between any two vertices, show that   kQ A2M (k)jA =0 if there is no pathji = n ij Example 2.0.9.   1 0 kx kx v  k   X M (kx) = 2 0 k The generators ofkQ aree ,e ,v,x, with relations 1 0 2 e v =ve =v, e x =xe =x, e =e , 0 1 0 0 1 1 2 ve =e v =0, e x =xe 1 =0, e =e . 0 1 1 1 0 0 The isomorphism is given on generators by     1 0 0 0 e 7 e 7 1 0 0 0 0 1     x 0 0 1 X7 v7 0 0 0 0 2.1 Path Algebras LetQ be a quiver and letA =kQ. Proposition 2.1.1.fe g is a complete set of orthogonal idempotents: i i2Q 0 10Path Algebras 30 January, 2016 2  e =e (idempotent), i i  e e =0 fori6=j (orthogonal), i j X  e =1 (complete). i A i2Q 0 Proposition 2.1.2. Fori,j2Q , the spaceAe ,e A ande Ae have the follow- 0 i j j i ing bases:  Ae = all paths starting ati i  e A = all paths ending atj j  e Ae = all paths starting ati and ending atj. j i Proposition 2.1.3. Decompositions ofA into direct sums of projective ideals. M (a) A = Ae as a leftA-module =) Ae is a projective leftA-module. i i i2Q 0 M (b) A = e A as a rightA-module=)e A is a projective rightA-module. j j j2Q 0 Proposition 2.1.4. For any leftA-moduleM and rightA-moduleN:  (a) Hom (Ae ,M) =e M A i i  (b) Hom (e A,N) =Ne A j j Proof of (a). Anyf2 Hom (Ae ,M) is determined byf(e ) = x2 M, byA- A i i 2 linearity. On the other hand,e =e , so i i 2 e f(e ) =f(e ) =f(e ) i i i i Hence, for anyx2M, iff(e ) =x, thene x =x. The map is then i i Hom (Ae ,M) - e M A i i f 7- f(e ) i Proposition 2.1.5. If06=a2Ae and06=b2e A thenab6=0 inA. i i Proof. Write a =cx+::: b =ecy+::: wherex is the longest path starting ati, andy is the longest path ending ati, withC6=06=ec. ab =cecxy+::: This is nonzero becausecec6=0. 11Path Algebras 30 January, 2016 Proposition 2.1.6. Eache is aprimitive idempotent, meaning that eachAe is i i an indecomposable leftA-module. Proof. IfM is decomposable, then there is some submoduleNM such that  M =NK. In this case, End (M) has at least one idempotent A pr i e: M N M 2 End (M). A Thus, we need to check that End (Ae ) has no nontrivial idempotents. A i  End (Ae ) = Hom (Ae ,Ae ) = e Ae A i A i i i i (4) 2 Iff:Ae Ae is idempotent in End (Ae ) =e Ae , thenf =f =fe , so i i A i i i i f(f-e ) =0. This implies by Proposition 2.1.1(5) thatf =0 orf-e =0. Hence, i i End (Ae ) has no nontrivial idempotents. A i Definition 2.1.7. Let M  kQ = ke =k:::k 0 i z i2Q 0 jQ j 0 M kQ = ka 1 a2Q 1 Notice thatkQ is naturally ankQ -bimodule. 1 0 Definition 2.1.8. For anyk-algebraS and anyS-bimoduleM, the tensor algebra T M is S T M =SM(M M):::(M  M)::: S S S S is defined by the following universal property. Given anyk-algebraf :SA and anyS-bimodule mapf :MA, there 0 1 is a uniqueS-bimodule mapf:T MA such thatfj =f andfj =f . S S 0 M 1  Proposition 2.1.9.kQ is naturally isomorphic to the tensor algebrakQ =T (V) S Proof. Check the universal property. IfS =kQ ,M =V =kQ , then 0 1 f :kQ ,kQ 0 0 f :kQ ,kQ 1 1 f:T (V)kQ S f is surjective by definition ofkQ, andf is injective by induction on the grading inT (V). S 12Representations of Quivers 1 February, 2016 Corollary 2.1.10.kQ is a gradedS-algebra with grading determined by the length function on paths. Exercise 2.1.11. (a) dim (kQ)1 if and only ifQ has no (oriented) cycles. k (b) kQ is prime (i.e.IJ6=0 for any two 2-sided idealsI,J6=0) if and only if for alli,j2Q , there is a pathij. 0 (c) kQ is left (resp. right) Noetherian () if there is an oriented cycle ati, then at most one arrow starts (resp. ends) ati. Example 2.1.12. Consider the quiver 1 0 v Q =   X The path algebrakQ is left Noetherian but not right Noetherian. 2.2 Representations of Quivers Fix a fieldk, and letQ = (Q ,Q ,s,t) be a quiver. Recall thatP is the set of 0 1 Q all paths inQ, ands,t extend to mapss,t:P Q . Q 0 Definition 2.2.1. The path category Q is the category with objectsQ and 0 Hom (i,j) =fa2P :s(a) =i, t(a) =jg. Q Q Composition is given by concatenating paths. Remark 2.2.2. We can modify this definition in two ways. First, we can make Q into ak-category (a category enriched ink-modules)kQ whose objects are Q and 0   Hom (i,j) =k Hom (i,j) kQ Q Second, we can also make Q ak-linear category (to be defined later). Definition 2.2.3. A representation ofQ is a functorF: Q Vect . The category k of all such representations is a functor category, denoted Rep (Q) := Fun(Q, Vect ). k k A representationX: Q Vect is usually denoted as follows. k Q 3i7-X(i) =X 0 i Q 3i7-X(a) =X 1 a   i j X a a   7- X X i j 13Representations of Quivers 1 February, 2016 Definition 2.2.4. IfQ = (Q ,Q ,s,t) is a quiver, define the opposite quiver 0 1 op   Q := (Q ,Q ,s =t,t =s). 0 1 Theorem 2.2.5. There are natural equivalences of categories Rep (Q)'kQ-Mod (leftkQ-modules) k op Rep (Q )' Mod-kQ (rightkQ-modules) k Proof. The functorF:kQ-Mod Rep (Q) is given on objects by k M7X := (X ,X ) M i a i2Q 0 a2Q 1 whereX =e M andX is the morphism given by i i a X a X X i j a e M e M. i j (recall thatae =a =e a). The functorF is given on morphisms by i j  (f:MN)7- fj :e Me N . eM i i i i2Q 0 Conversely,G: Rep (Q)kQ-Mod is given on objects by k M X = (X ,X ) 7-X = X . i a i2Q i 0 a2Q 1 i2Q 0 Write " : X , X and  : X  X for the canonical maps. Given a path i i i i a = (a ,::: ,a )2P ,x2X, define 1 n Q ax =" X X X  x a a a t(a ) 1 2 n s(a ) 1 n Now check thatFG' id,GF' id. Example 2.2.6. Consider a quiver representation S(i), where i 2 Q . The 0 representationS(i) is defined by  k (j =i) S(i) = j 0 (j6=i) andS(i) =0 for alla2Q . Every irreduciblekQ-module looks likeS(i) for a 1 somei2Q . 0 Example 2.2.7. The indecomposable projectivekQ-modules are of the formAe , i which correspond to the quiver representationX withX =e Ae forj2Q . j j i 0 14Homological Properties of Path Algebras 1 February, 2016 2.3 Homological Properties of Path Algebras 1 Recall Baer’s definition of Ext (V,W). A Definition 2.3.1 (Baer). Let A be a k-algebra and V,W two objects in A = Mod(A).  1 2 Ext (V,W) := ( , )2 Mor(A) 0W -X -V0  A 0 0 0 0 where ( , )  ( , ) if and only if there is some:XX such that = 0  and  = . 0 W X V 0  0 0 0 0 W X V 0 Remark 2.3.2. We can think of a quiver as a kind of finite non-commutative space. We can think of Example 2.0.9 as a kind of non-commutative “extension” 1 of the affine lineA = Spec(kx). k Recall thatX is an affine variety overC, andA =O(X) is a finitely generated commutativeC-algebra, andX = Specm(A); that is, the points ofX correspond to irreducible representations ofA, which have the formA/m, where m is a maximal ideal ofA.. Points ofX are “homologically disjoint” in the sense that    A A Ext / , / =0 (i6=j). m m A i j On the contrary, in the noncommutative case (for quivers), we will see that 1 Ext (S(i),S(j))6=0 kQ if there is an arrowij. Thus, the arrows play a role of “homological links” between the “points” in the quiverQ. Theorem 2.3.3. LetA = kQ. For any (left)A-moduleX, there is an exact se- quence ofA-modules: M M g f 0 Ae e X Ae e X X0 (2.1) t() k s() i k i 2Q i2Q 1 0 whereg(a x) :=ax andf(b x) :=ab x-a bx. Proof. First we show thatg is surjective. This can be seen from the fact that any element ofx can be written as 0 1 0 1 X X X A A x =1x = e x = e x =g e e x . i i i i i2Q i2Q i2Q 0 0 0 15Homological Properties of Path Algebras 1 February, 2016 The fact that im(f) ker(g) is just a direct computation. Indeed, gf(a e e x) =g(a x-a x) =ax-ax =0 t() s() L n To show that ker(g) im(f), we first note that any2 Ae e X can i i i=1 be uniquely written as n X X  = a x a i=1 pathsa s(a)=i where all but finitely many of thex 2e X are zero. Let the degree of be a s(a) the length of the longest patha such thatx 6=0. Ifa is a nontrivial path, we a 0 0 0 can factor it asa = a, withs(a ) = t() anda consisting of only a single edge. We then have that 0 0 0 a x =ae e x =ae e . a s(a) s(a) t() s(a) Then by definition, 0 0 0 0 f(a x ) =a x -a x =a x -a x a a a a a Now claim that for any, the set+ im(f) contains elements of degree zero. For if deg() =d, then 0 1 n B C X X B C 0 -f a x B C a A i=1s(a)=i `(a)=d has degree strictly less thand. The claim then follows by induction on`(a) =d. 0 Now let2 ker(g), and take an element 2 + im(f) of degree zero. In other words, n X 0  = e x . i e i i=1 Ifg() =0, then becausegf =0, we get n n X M 0 g() =g( ) = e x 2 e X. i e i i i=1 i=1 0 This is zero if and only if eache x =0. Butxe =0 implies that =0, or that i e i i 2 im(f). This demonstrates that ker(g) im(f). Finally, let’s show thatf is injective. Supposef() =0, yet6=0. Then we can write X X  = a x =b x +::: , ,a ,b 2Q pathsa 1 s(a)=t() 16Homological Properties of Path Algebras 1 February, 2016 whereb is a path of maximal length. We then get X X f() = a x - a x =b x + lower terms =0 ,a ,a ,b ,a ,a Here the lower terms are of the formc x , wherec is a path shorter than ,c b. Hence, nothing can cancel with theb x term, which contradicts our ,b choice ofb as the path of maximal length. Definition 2.3.4. The resolution (2.1) in Theorem 2.3.3 is called the standard resolution ofA. Remark 2.3.5.  (a) There is a compact way to express this resolution if we identifykQ = L L T (V), whereS = ke ,V = ka asS-bimodules. S i i2Q a2Q 0 i The exact sequence (2.1) can be written for any tensor algebraT and any (left)T-moduleX. g f 0 T V X T X X 0 (2.2) S S S Note that the standard resolution is projective because eachAe e X is i k j a direct summand ofA X, which is a freeA-module based onV. k (b) IfX =A, then the standard resolution becomes anA-bimodule resolution. Exercise 2.3.6. Check that the sequence (2.2) gives the standard resolution (2.1). Definition 2.3.7. An algebraA is a (left or right) hereditary algebra if every submodule of a projective (left or right)A-module is projective. Proposition 2.3.8. For ak-algebraA, the following conditions are equivalent: (a) Every A-module X has projective dimension pdim (X)  1, that is, A i Ext (X,Y) =0 for alli2. A (b) A is a (left and right) hereditary algebra. Proof. Consider the exact sequence P 0XP / 0, X 17Homological Properties of Path Algebras 1 February, 2016 whereX is anA-submodule of the projectiveA-moduleP. Apply the functor Hom (-,Y) to get the long exact sequence A  P 0 Hom / ,Y Hom (P,Y) Hom (X,Y) A X A A  1 P 1 1 Ext / ,Y Ext (P,Y) Ext (X,Y) X A A A  2 2 2 P Ext / ,Y Ext (P,Y) Ext (X,Y)  X A A 1 SinceP is projective, we have that Ext (P,Y) =0, so this long exact sequence shows that   1 2 P  Ext (X,Y) Ext / ,Y . = X A A  2 P If everyA-moduleX has projective dimension at most one, then Ext / ,Y = X A 1 0 and therefore Ext (X,Y) =0. HenceX is projective. A Conversely, ifA is a hereditary algebra, then asX is a submodule of the  1 2 P projective moduleP, we have Ext (X,Y) =0. Hence, Ext / ,Y =0 for any X A A-module of the formP/X. But anyA-module whatsoever is the quotient of a free module, and therefore of the formP/X. So anyA-module has projective dimension at most1. Remark 2.3.9. Suppose that we want to apply Hom (-,Y) to the exact se- A quence :0 W X U 0 to get a the long exact sequence. The connecting homomorphism in the long exact sequence above is defined as follows. Givenf2 Hom (W,Y), let(f) be A 1 the class in Ext (U,Y) such that A :0 W X U 0 f p f :0 Y WX U 0  where the square indicated is a pushout. Note that ifW =Y, andf = id , then W 1 f  = 2 Ext (U,Y).  A Remark 2.3.10. Another way to say Proposition 2.3.8(a) is to say that the global dimension ofkQ is at most1, for any quiverQ. 18Homological Properties of Path Algebras 3 February, 2016 Definition 2.3.11. IfX is a finite-dimensionalA-module, we define the dimen- sion vector ofX to be n n dim (X) := (dim X ,::: , dim X )2N Z , k 1 k n k whereX =e X = Hom (Ae ,X). i i A i Definition 2.3.12. For a finite quiverQ, with jQ j = n, the Euler form is a 0 n n n bilinear formh-,-i :Z Z Z given by Q X X h , i := - Q i i t(a) s(a) i2Q a2Q 0 1 Sometimes we also need a symmetric version of the Euler form, which is n n written(-,-):Z Z Z and given by ( , ) :=h , i +h , i . Q Q n n Definition 2.3.13. The associated quadratic formq:Z Z is called the Tits form. q( ) :=h , i . Q fg Lemma 2.3.14. For any two finite-dimensionalA-modulesX,Y2A-Mod , we have     1 hdimX, dimYi = dim Hom (X,Y) - dim Ext (X,Y) . Q k A k A Proof. Apply the functor Hom (-,Y) to the standard resolution (2.1). Then we A get a long exact sequence M M 0 Hom (X,Y) Hom (Ae e X,Y) Hom (Ae e X,Y) A A i i A t(a) s(a) i2Q a2Q 0 1 1 Ext (X,Y) 0 A  Recall that Hom (Ae ,X) =e X, for anyA-moduleX, so applying this to A i i the above sequence gives     Hom (Ae e X,Y) = Hom (e X,e Y) = (e Y) (e X) =Y X A i i K i i i i i i This then implies that dim(Hom (Ae e X,Y)) = dim(Y ) dim(X ) = (dimX) (dimY) A i i i i i i 19

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