Chapter 2 Projective and Injective Representations

Transcription

1 Chapter Projective and Injective Representations Projective representations and injective representations are key concepts in representation theory. A representation P is called projective i the unctor Hom.P; / maps surjective morphisms to surjective morphisms. Dually a representation I is called injective i the unctor Hom. ;I/ maps injective morphisms to injective morphisms. The terminology comes rom the property that or any representation M there is a projective representation P such that there exists a surjective morphism (a projection ): p W P M: Dually, or any representation M, there is an injective representation I such that there exists an injective morphism: i W M,! I : I M is not projective itsel, then the morphism p above will have a kernel, and we can ind another projective P such that there exists a surjective morphism p rom P to the kernel o p. Iterating this procedure yields an exact sequence P 3 p 3 P p P p P p M where each P i is a projective representation. Such a sequence is called a projective resolution. We think o projective resolutions as a way to approximate the representation M by projective representations. Oten it is possible to deduce properties o M rom a projective resolution o M. Springer International Publishing Switzerland 4 R. Schiler, Quiver Representations, CMS Books in Mathematics, DOI.7/

2 34 Projective and Injective Representations Dually, we have injective resolutions, thus exact sequences o the orm M i I i I i I i 3 I 3 where each I i is an injective representation. For representations o quivers without oriented cycles the situation is very simple. We will see that every quiver representation has a projective resolution o the orm P P M, and an injective resolution o the orm M I I. We will use this result to show that every subrepresentation o a projective representation is projective. Categories with this property are called hereditary. Moreover, it is very easy to write down all indecomposable projective representations o a quiver Q without oriented cycles. There is exactly one indecomposable projective representation P.i/ or each vertex i Q, and this representation P.i/ is given by the paths in Q starting at the vertex i. Dually, there is exactly one indecomposable injective representation I.i/ or each vertex i Q, and I.i/ is given by the paths ending at the vertex i. We will see later in Chap. 4 that each quiver deines an algebra A, thepath algebra o the quiver, whose basis consist o the set o all paths in Q. We will also see that we can consider the algebra A as a representation o Q and that this representation is isomorphic to the direct sum o the indecomposable projective representations, thus A Š iq P.i/ as representations o Q. In the current chapter, we also introduce the Auslander Reiten translation, which is crucial to Auslander Reiten theory and Auslander Reiten quivers. It is deined in a rather curious way by taking the beginning o a projective resolution o M P p P M and then setting M D ker p, where is the so-called Nakayama unctor. This unctor maps projective representations to injective representations, and thereore we obtain the beginning o an injective resolution: τm νp ν p νp.

3 Projective and Injective Representations 35 Throughout this chapter, and the rest o the book, the very natural notion o paths in a quiver will be essential. Here is a ormal deinition. Deinition.. Let Q D.Q ;Q ;s;t/be a quiver, i;j Q.Apath c rom i to j o length ` in Q is a sequence with h Q such that c D.ij ; ;:::; `jj/ s. / D i; s. h/ D t. h /; or h D ;3;:::;`, t. `/ D j: Thus a path rom i to j is a way to go rom vertex i to vertex j in the quiver Q, where we are only allowed to walk along an arrow in the direction to which it is pointing. Example.. In the quiver α β γ 3, we have that.j j/,.j ; ˇj/,.j ; ; ˇj/ are paths, but.j ; ˇ; j/ is not. Example... The constant path (or lazy path).ijji/ at vertex i is the path o length ` D which never leaves the vertex i. We denote this path e i.. An arrow i α j is a path.ij jj/o length one. I i D j then i α is called a loop. 3. A path o the orm i α α α 3 α α given by.ij ; ;:::; `ji/ is called an oriented cycle. Thus a loop is an oriented cycle o length one.

4 36 Projective and Injective Representations. Simple, Projective, and Injective Representations Let Q be a quiver without oriented cycles. For every vertex i in Q, we will now deine three representations: the simple, the projective, and the injective representation at i. We will show that in the category rep Q, these representations are respectively simple, projective, or injective objects in the categorical sense. Deinition.. Let i be a vertex o Q. Deine representations S.i/;P.i/, and I.i/ as ollows: (a) S.i/ is o dimension one at vertex i, and zero at every other vertex; thus S.i/ D.S.i/ j ;' / j Q ; Q ; where k i i D j, S.i/ j D and otherwise, ' D or all arrows : (b) S.i/ is called the simple representation at vertex i. P.i/ D.P.i/ j ;' / j Q ; Q where P.i/ j is the k-vector space with basis the set o all paths rom i to j in Q; so the elements o P.i/ j are o the orm P c c c, where c runs over all paths rom i to j, and c k; and i j! ` is an arrow in Q, then ' W P.i/ j! P.i/` is the linear map deined on the basis by composing the paths rom i to j with the arrow j! `. More precisely, the arrow induces an injective map between the bases basis o P.i/ j! basis o P.i/` c D.ijˇ;ˇ;:::;ˇsjj/ 7! c D.ijˇ;ˇ;:::;ˇs; j`/ and ' is deined by '! X c c D X c c c c : (c) P.i/ is called the projective representation at vertex i. I.i/ D.I.i/ j ;' / j Q ; Q

5 . Simple, Projective, and Injective Representations 37 where I.i/ j is the k-vector space with basis the set o all paths rom j to i in Q; so the elements o I.i/ j are o the orm P c c c, where c runs over all paths rom j to i, and c k; and i j! ` is an arrow in Q, then ' W I.i/ j! I.i/` is the linear map deined on the basis by deleting the arrow j! ` rom those paths rom j to i which start with and sending to zero the paths that do not start with. More precisely, the arrow induces a surjective map between the bases basis o I.i/ j! c D.j jˇ;ˇ;:::;ˇsji/ 7! basis o I.i/`.`jˇ;:::;ˇsji/ i ˇ D ; otherwise; and ' is deined by '! X c c D X c c c.c/: I.i/ is called the injective representation at vertex i. Note that we need the hypothesis that Q has no oriented cycles, because, otherwise, there would be a vertex i such that P.i/ is ininite-dimensional, and hence not a representation in rep Q. For example, i Q is the quiver then P./ and P./ would be ininite-dimensional. The ollowing remark will be very useul later on. Remark.. Let P.i/ D.P.i/ j ;' / be the projective representation at vertex i and let c be a path starting at i,say Then we can deine the map c D.ijˇ;ˇ;:::;ˇ`jj/: ' c W P.i/ i! P.i/ j ' c D 'ˇ` 'ˇ'ˇ as the composition o the maps in the representation P.i/along the path c. Then, i e i denotes the constant path at vertex i, it ollows rom the deinition o P.i/ that Remark.. Simple projectives and simple injectives: ' c.e i / D c: (.) () The projective representation at vertex i is the simple representation at vertex i i and only i there is no arrow in Q such that s. / D i. Such vertices are called sinks o the quiver Q. Thus

6 38 Projective and Injective Representations S.i/ D P.i/ i is a sink in Q. () The injective representation at vertex i is the simple representation at vertex i i and only i there is no arrow in Q such that t. / D i. Such vertices are called sources o the quiver Q. Thus S.i/ D I.i/ i is a source in Q. In the ollowing two examples, we use matrix notation to describe projective and injective representations. We use the isomorphism and not the equality symbol since there are many other possible descriptions or these representations; see Exercise.. Example.3. Let Q be the quiver Then S(3) = k, P(3) = k k, k I(3) = k k. The quiver in the next example contains parallel paths. As a result the indecomposable projective modules can be o dimension greater than at a single vertex.

7 . Simple, Projective, and Injective Representations 39 Example.4. Let Q be the quiver 3 4. Then P() = k k k, k I(4) = k k k. k In category theory, a projective object is an object P such that the Hom unctor Hom.P; / maps surjective morphisms to surjective morphisms. The ollowing proposition shows that the representations P.i/ satisy this condition, which is the reason why we call them projective. Proposition.3. Let gw M! N be a surjective morphism between representations o Q, and let P.i/ be the projective representation at vertex i. Then the map g W Hom.P.i/; M /! Hom.P.i/; N / is surjective. In other words, i W P.i/! N is any morphism, then there exists a morphism hw P.i/! M such that the diagram P(i) h M g N commutes, that is, D g ı h D g.h/.

8 4 Projective and Injective Representations Proo. Exercise.4. Corollary.4. I P is projective, then any exact sequence o the orm ut L M g P splits. Proo. Use Proposition.3 with the identity morphism D P to get the commutative diagram: P h M g P Thereore D g ı h, and g is a retraction. ut A dual statement holds or the injective representations. In category theory, an injective object is an object I such that the Hom unctor Hom. ;I/ maps injective morphisms to surjective morphisms. The next proposition shows that the representations I.i/ satisy this condition, which is the reason why we call them injective. Proposition.5. Let gw L! M be an injective morphism between representations o Q, and let I.i/ be the injective representation at vertex i. Then the map g W Hom.M; I.i//! Hom.L; I.i// is surjective. In other words, i W L! I.i/ is any morphism, then there exists a morphism hw M! I.i/ such that the diagram L g M h I(i) commutes, that is, D h ı g D g.h/. Proo. Exercise.5. ut

9 . Simple, Projective, and Injective Representations 4 Corollary.6. I I is injective then any exact sequence o the orm I g M N splits. Proo. Use Proposition.5 with the identity morphism D I to get a commutative diagram: I g M. h I Thus I D h ı g, and g is a section. ut Finally, a simple object in a category is a nonzero object S that has no proper subobjects. The representations S.i/ have this property, hence their name. The next proposition states that sums o projective objects are projective and that summands o projective objects are projective. We state the result or the category rep Q, but the proo holds in any additive category. Proposition.7. () Let P and P be representations o Q. Then P P is projective P and P are projective. () Let I and I be representations o Q. Then I I is injective I and I are injective. Proo. We only show () since the proo o () is similar. ()) Letg W M! N be surjective in rep Q and let W P! N be any morphism in rep Q. Consider the ollowing diagram: P P pr i h P M g N

10 4 Projective and Injective Representations where pr denotes the projection on the irst summand and i is the canonical injection. Clearly, pr ı i D P. Since P P is projective, there exists a map h W P P! M such that ghd pr. Thereore ghi D pr i D P D : Now we can deine h W P! M as h D hi, and we have gh D.This shows that P is projective. One can show in a similar way that P is projective. (() Letg W M! N be a surjective morphism in rep Q and let W P P! N be any morphism in rep Q. Consider the ollowing commutative diagram: P i h P P M g N where i denotes the canonical injection and, since P is projective, there exists a morphism h such that gh D i. By symmetry, there also exists a morphism h W P! M such that gh D i. Deine h D.h ;h / W P P! M by h.p C p / D h.p/ C h.p /. Then gh.pc p / D gh.p/ C gh.p / D i.p/ C i.p / D.pC p /; which shows that P P is projective. ut Proposition.7 implies that i we know the indecomposable projective, respectively injective, representations, then we know all projective, respectively injective, representations. The next step is to show that the representations P.i/ and I.i/ are in act indecomposable. We will see later in Corollary. that there are no other indecomposable projective or injective representations. Proposition.8. The representations S.i/;P.i/, and I.i/ are indecomposable. Proo. For S.i/ this ollows directly rom the act that S.i/ is simple. Let us prove the result or the projective representation P.i/ D.P.i/ j ;' / j Q ; Q : Since Q has no oriented cycles, we have P.i/ i D k. Suppose that P.i/ D M N or some M; N rep Q. Then we may suppose without loss o generality that P.i/ i D M i and N i D. Let` be a vertex o Q such that N`. Now,P.i/` has a basis consisting o the paths rom i to ` in Q. Letc D.ijˇ;:::;ˇsj`/ be such a path.

11 . Simple, Projective, and Injective Representations 43 Let ' c D 'ˇs 'ˇ denote the composition o the linear maps o the representation P.i/ along the path c. Then, since P.i/ is the direct sum o M and N,themap ' c W M i! M` N` sends the unique basis element e i o M i to an element ' c.e i / o M`. But rom Remark. we know that ' c.e i / D c; thus every basis element c o P.i/` lies in M`, a contradiction. The proo or I.i/ is similar. ut The ollowing proposition shows that the simple representations S.i/ orm a complete set o simple representations in rep Q, up to isomorphism. Proposition.9. A representation o Q is simple i and only i it is isomorphic to S.i/,orsomei Q : Proo. It is clear that the S.i/ are simple representations. Conversely, let M D.M i ;' / be any representation o Q. We want to show that there is a vertex i such that S.i/ is a subrepresentation o M, and we have to choose this vertex i careully. We do not want to have a nonzero map in the representation M that starts at the vertex i. For example, i i is a sink in the quiver, we have what we want. But on the other hand, we also need the representation M to be nonzero at the vertex i. This leads us to pick i as ollows. Let i Q such that M i and M j D, whenever there is an arrow i! j in Q. Note that such a vertex exists since Q has no oriented cycles. Choose any injective linear map i W S.i/ i Š k! M i, and extend it trivially to a morphism W S.i/! M by letting j D i i j. Note that actually is a morphism since the diagram S(i) i i M ϕ α M i ϕ β commutes, or all arrows `! ˇ i and i! j in Q. Since is injective this shows that S.i/ is a subrepresentation o M, and thereore, either M Š S.i/ or M is not simple. ut Remark.. Proposition.9 does not hold i the quiver has oriented cycles. For example, i Q is the quiver then or each k, there is a simple representation k, where is given by multiplication by.

12 44 Projective and Injective Representations The vector space at vertex i o any representation can be described as a space o morphisms using the projective representation P.i/ as ollows: Theorem.. Let M D.M i ; / be a representation o Q. Then, or any vertex i in Q, there is an isomorphism o vector spaces: Hom.P.i/; M / Š M i : Proo. Let e i D.ijji/ be the constant path at i. Then e i g is a basis o the vector space P.i/ i. Deine a map W Hom.P.i/; M /! M i D. j / j Q 7! i.e i /: I is a morphism rom P.i/ to M, then its component i is a linear map rom P.i/ i to M i, which shows that the map is well deined, since e i P.i/ i. We will show that is an isomorphism o vector spaces. Let us use the notation P.i/ D.P.i/ j ;' /. First, we show that is linear. I ; g Hom.P.i/; M / are two morphisms, then. C g/ D. C g/ i.e i / D i.e i / C g i.e i / D. /C.g/, and i k then. / D. / i.e i / D i.e i / D. /. Next, we show that is injective. I D. / D i.e i /, then the linear map i sends the basis e i g to zero, and thus i is the zero map. We will now show that j W P.i/ j! M j is the zero map, or any vertex j, and this will show that is injective. By deinition o P.i/, the vector space P.i/ j has a basis consisting o all paths rom i to j.letc D.ij ;:::; tjj/ be such a basis element, and consider the maps ' c D ' t ıı' and ' c D ' t ıı' deined as the composition o the maps along the path c o the representation P.i/ and M, respectively. It ollows rom the deinition o P.i/ that ' c.e i / D c. Since is a morphism o representations, we have j ' c D ' c i, and, since i.e i / D, this implies that j maps c to zero. As c is an arbitrary basis element o P.i/ j,itollowsthat j D. It remains to show that is surjective. Let m i M i. We want to construct a morphism W P.i/! M such that i.e i / D m i. Let us start by ixing its component i W P.i/ i! M i by requiring the condition we need, that is, i.e i / D m i. Since e i g is a basis o P.i/ i, this condition deines the linear map i in a unique way. We can extend the map i to a morphism D. j / j Q by ollowing the paths in Q. More precisely, or any path c rom i toavertexj in Q, put j.c/ D ' c.m i/. This deines each map j on a basis o P.i/ j, and we extend this map linearly to the whole vector space P.i/ j. It ollows rom our construction that is a morphism o representations, thus Hom.P.i/; M / and. / D m i,so is surjective. ut Note that in the proo o Theorem., the particular structure o the representation P.i/ is the essential ingredient to show the injectivity and surjectivity o the map. As an immediate consequence o Theorem., we can describe the morphisms between projective representations as ollows:

13 . Projective Resolutions and Radicals o Projectives 45 Corollary.. Let i and j be vertices in Q. () The vector space Hom.P.i/; P.j // has a basis consisting o all paths rom j to i in Q. In particular, End.P.i// D Hom.P.i/; P.i// Š k: () I A D iq P.i/, then the vector space End.A/ D Hom.A; A/ has a basis consisting o all paths in Q. Remark.3. In Chap. 4, we will see that the so-called path algebra o the quiver Q is isomorphic to End.A/ as a vector space and as an algebra. Proo. Theorem. implies that Hom.P.i/; P.j // is isomorphic to P.j/ i, and this vector space has a basis consisting o all paths rom j to i in Q. The act that Q has no oriented cycles implies that End.P.i// is o dimension one; hence End.P.i// Š k. This proves (), and () is a direct consequence. ut Corollary.4. The representation P.j/ is a simple representation i and only i Hom.P.i/; P.j // D or all i j. Proo. The representation P.j/ is simple i and only i j is a sink, which means that there are no paths rom j to any other vertex i. The statement now ollows rom Corollary.. ut. Projective Resolutions and Radicals o Projectives We have seen above that the projective representations can be used to describe the vector spaces o an arbitrary representation using the Hom unctor. Now we will introduce another way o describing arbitrary representations by means o projective representations: the projective resolutions. As usual, there is a dual notion, the injective resolutions. Deinition.3. Let M be a representation o Q. () A projective resolution o M is an exact sequence P 3 P P P M, where each P i is a projective representation. () An injective resolution o M is an exact sequence M I I I I 3, where each I i is an injective representation.

14 46 Projective and Injective Representations Theorem.5. Let M be a representation o Q. () There exists a projective resolution o M o the orm P P M. () There exists an injective resolution o M o the orm M I I. Proo. We will show only (), and to achieve this, we will construct the so-called standard projective resolution o M.LetM D.M i ;' i /, and denote by d i the dimension o M i. Deine P D M Q d s. / P.t. // P D M iq d i P.i/; where d i P.i/ stands or the direct sum o d i copies o P.i/. Beore deining the morphisms o the projective resolution, let us examine the representations P and P. For every vector space M i,wehaved i D dim M i copies o P.i/ in P. The natural map g rom P to M will send the d i copies o the constant path e i in P to a basis o M i. Now in each copy o P.i/, the kernel o the map g contains a copy o P.t. // or every that starts at i. So, or every arrow with s. / D i, wehaved s. / copies o P.t. // in the kernel o g, which justiies the deinition o P. To deine the morphisms o the projective resolution, we introduce speciic bases or each o the representations P ;P and M as ollows: For each i Q,let m i ;:::;m idi g be a basis or M i, and thus B Dm ij j i Q ;j D ;;:::;d i g is a basis or M. Taking the standard bases or the projective representations, the set B Dc ij j i Q ;c i a path with s.c i / D i; and j D ;:::;d i g is a basis or P ; and the set B Db j j Q ;b a path with s.b / D t. /; and j D ;:::;d s. / g is a basis or P. Deine a map g on the basis B by g.c ij / D ' ci.m ij / M t.ci / and extend g linearly to P. Deine the map on the basis B by.b j / D. b / j b M ;

15 . Projective Resolutions and Radicals o Projectives 47 where b is the path rom s. / to t.b / given by the composition o and b, and b M D P d t. / `D `b `, where the ` are the scalars that occur when writing '.m s. /j / in the basis m t. /` j ` D ;:::;d t. / g o M t. / ; thus We will now prove that the sequence! M Q d s. / P.t. // d t. / X '.m s. /j / D `m t. /`: (.) `D! M g d i P.i/! M! (.3) iq is exact. g is surjective, because or any basis vector m ij o M,wehavem ij D g.e ij /, where e i is the constant path at vertex i. ker g im : It suices to show that g ı.b j / D or any b j in the basis B. We compute g..b j // D g.. b / j b M / D ' b.m s. /j / ' b. P` `m t. /`/ D ' b '.m s. /j / t. /` P` `m D ' b./ D ; where the next to last equation ollows rom (.). ker g im : First note that any x L iq d i P.i/ can be written as a linear combination o the basis B; thus x D X cij c ij D x C X cij c ij ; c ij B c ij BnB where B is the subset o B consisting o constant paths (together with a choice o j ), B De ij j i Q ;j D ;:::;d i g; and x D P e ij B eij e ij. Any nonconstant path is the product o an arrow and another path; thus x D x C X c ij Wc i D b cij. b / j ;

16 48 Projective and Injective Representations and using the deinition o, we get x D x C X c ij Wc i D b cij.b j / C cij b M : (.4) Let x D x C P c i D b cij b M : Note that x x im. Deine the degree o a linear combination o paths to be the length o the longest path that appears in it with nonzero coeicient. Note that deg x < deg x and deg x D. Now let us suppose that x ker g. We want to show that x im.using(.4) and the act that g ı D, we get D g.x/ D g.x /: Summarizing, we have x ker g,degx < deg x and x x im. Now we repeat the argument with x instead o x. We get x ker g, degx < deg x < deg x, and x x im. Continuing like this, we will eventually get x h ker g, x x h im, and deg x h D ; thus x h is a linear combination o constant paths, say x h D P i;j ij e ij ; or some ij k. By deinition o g,wehave D g.x h / D X i;j ij m ij ; and since the m ij orm a basis o M, this implies that all ij are zero. Hence x h D and thus x im. is injective. Suppose that D P b h b h D P b h. b / h b M : Then P b h. b / h D P b h b M D P b h P `b `: Let i be a source in M, that is, i is such that there is no arrow j! i with d j. Note that such an i exists since Q has no oriented cycles. Then, since each o the paths b starts at the endpoint o the arrow, none o the paths b can go through i, and this shows that b h must be zero, or all arrows with s. / D i.now let i be a source in M n i, that is, i is such that there is no arrow j! i with j i and d j. Since b h D or all arrows with s. / D i, there is no path b h with b h that goes through i. Continuing in this way, we show that every b h D, since Q has only initely many arrows. Thus is injective. This concludes the proo o the exactness o the standard resolution (.3). ut Remark.6. There are other projective resolutions than the standard resolution.

17 . Projective Resolutions and Radicals o Projectives 49 Example.5. Let Q be the quiver 3 and consider the representations M D S.3/ D 3 and M D 3. Then we have the standard projective resolutions: The second resolution is not minimal in the sense that one can eliminate a direct summand S./ D in each o the projective modules and still have a projective resolution: 3 3. Example.6. Let Q be the quiver β 3 α γ 4 and consider the representation M D given by 34 k k k. k Then we have the standard projective resolution: 34 (3 3) (4 4) Again, this resolution is not minimal in the sense that one can eliminate three direct summands in each o the projective modules and still have a projective resolution:

18 5 Projective and Injective Representations To make this notion o minimality more precise, we need the deinitions o projective covers and injective envelopes. Deinition.4. Let M rep Q. A projective cover o M is a projective representation P together with a surjective morphism gw P! M with the property that, whenever g W P! M is a surjective morphism with P projective, then there exists a surjective morphism h W P P such that the diagram P h g P g M commutes, that is, gh D g. An injective envelope o M is an injective representation I together with an injective morphism W M! I with the property that, whenever W M! I is an injective morphism into an injective representation I, then there exists an injective morphism h W I,! I such that the diagram M I h I commutes, that is, h D.

19 . Projective Resolutions and Radicals o Projectives 5 Deinition.5. A projective resolution P 3 3 P P P M is called minimal i W P! M is a projective cover and i W P i! ker i is a projective cover, or every i>. An injective resolution M I I I 3 I 3 is called minimal i W M! I is an injective envelope and, or every i >, i W coker i! I i is an injective envelope. The next two propositions show that projective covers are unique up to isomorphism. Proposition.7. Let gw P! M be a projective cover o M and let g W P! M be a surjective morphism with P projective. Then P is isomorphic to a direct summand o P. Proo. From the deinition o projective covers, we see that there exists a surjective morphism hw P! P. This morphism gives rise to an exact sequence: ker h P h P. Since P is projective, Corollary.4 implies that this sequence splits, and then the result ollows rom Proposition.8. ut Proposition.8. Let gw P! M and g W P! M be projective covers o M. Then P is isomorphic to P. Proo. From Proposition.7, we conclude that P is isomorphic to a direct summand o P, and P is isomorphic to a direct summand o P. Thus P Š P. ut Remark.9. The dual statements to the propositions.7 and.8 about injective envelopes hold too. We leave the statements and their proos as an exercise. We introduce now the concept o a ree representation. The prototype o a ree representation is the direct sum o the indecomposable projective representations, and in general, ree representations are direct sums o this prototype. Deinition.6. Let A D iq P.i/. A representation F rep Q is called ree i F Š A A. Proposition.. A representation M rep Q is projective i and only i there exists a ree representation F rep Q such that M is isomorphic to a direct summand o F.

20 5 Projective and Injective Representations Proo. (() By the Krull Schmidt Theorem. and Proposition.8, every direct summand o F is a direct sum o P.i/ s, hence projective, by Proposition.7..)/ Suppose that M is projective and dim M D.d i / iq. The standard projective resolution o M gives a surjective morphism gw d i P.i/! M. Thus there is a short exact sequence: kerg d i P(i) g M Since M is projective, this sequence splits, and thereore M is isomorphic to a direct summand d i P.i/. ut Corollary.. Any projective representation P rep Q is a direct sum o P.i/ s, that is, P Š P.i / P.i t /; with i ;:::;i t not necessarily distinct. Proo. This ollows directly rom Proposition. ut Our next goal is to show that, in rep Q, subrepresentations o projective representations are projective. We start by introducing a particular subrepresentation o P.i/. Deinition.7. Let P.i/ D.P.i/ j ;' / be the projective representation at vertex i. The radical o P.i/ is the representation rad P.i/ D.R j ;' / deined by R i D ; R j D P.i/ j i i j; and ' D i s. / D i ' otherwise. The next lemma shows that the radical o P.i/ is the maximal proper subrepresentation o P.i/. Lemma.. Any proper subrepresentation o P.i/ is contained in rad P.i/. Proo. Suppose W M,! P.i/ is an injective morphism o representations. Let M D.M i ; / and P.i/ D.P.i/ j ;' /. It is clear that i M i D, we have that.m i / rad P.i/, so let us suppose that M i. We will show that this implies that the morphism is an isomorphism. Since P.i/ i Š k, itollowsthatm i Š k, and there is an element m i M i such that i.m i / D e i.nowletj be any vertex, and let c be a path rom i to j. Then c D ' c.e i / D ' c. i.m i // D j. c.m i // im j ;

21 . Projective Resolutions and Radicals o Projectives 53 where the irst identity is shown in Remark., and the third identity holds because is a morphism o representations. Thus we see that the arbitrary element c o the basis o P.i/ j lies in the image o j, which implies that is surjective, hence an isomorphism, and so M is not a proper subrepresentation o P.i/. ut Lemma.3. I P.i/ is simple, then rad P.i/ D. IP.i/ is not simple, then the radical o P.i/ is projective. Proo. We will show that rad P.i/ is isomorphic to the projective representation P D Ws. /Di P.t. //.Ii j, then.rad P.i// j D P.i/ j has as a basis the set o paths rom i to j. Deine a morphism D. j / j Q W rad P.i/! P on this basis by j.ij ; ˇ;:::ˇsjj/ D.t. /jˇ;:::;ˇsjj/: Then j sends the basis o.rad P.i// j to a basis o P j, or each j Q, and thus is an isomorphism. ut Theorem.4. Subrepresentations o projective representations in rep Q are projective. Remark.5. The subrepresentation inherits the projectivity. Categories with this property are called hereditary. Proo. Suppose that P is a projective representation with dimension vector.d i / iq. We will prove the theorem by induction on d D P iq d i, the dimension o P. I d D, then P is simple and there is nothing to prove. So suppose that d>. Let M be a subrepresentation o P and let u W M! P be the inclusion morphism. By Corollary., wehavep Š P.i / P.i t / or some vertices i ;:::;i t, and thus the inclusion u is o the orm 3 u 6 u D : 4 7 : 5 u t with im u j P.i j /.ItollowsthatM Š im u im u t, and, by Proposition.7, it suices to show that im u j is projective or each j. This is obvious in the case where im u j D P.i j /, so let us suppose that im u j is a proper subrepresentation o P.i j /. Then im u j is a subrepresentation o rad P.i j /, and rad P.i j / is projective, by Lemma.3. Moreover, the dimension o rad P.i j / is strictly smaller than d, and, by induction, we conclude that im u j is projective, which completes the proo. ut As a consequence o Theorem.4, we obtain the ollowing result on morphisms into projective modules:

22 54 Projective and Injective Representations Corollary.6. Let W M! P be a nonzero morphism rom an indecomposable representation M to a projective representation P. Then M is projective, and is injective. Proo. Since the image o is a subrepresentation o P, it is projective, by Theorem.4. Thereore, the short exact sequence ker M im splits, and then Proposition.8 implies that im is isomorphic to a direct summand o M.ButM is indecomposable, so M Š im is projective and ker D. ut Corollary.6 shows that when we construct the Auslander Reiten quiver o Q, we must start with the projective representations and that the projective representations are partially ordered by inclusion..3 Auslander Reiten Translation In this section, we will deine the Auslander Reiten translation, which is undamental or the Auslander Reiten theory and Auslander Reiten quivers. We consider at this point only the Auslander Reiten translation in the category o quiver representations. Later, we will also consider the more general situation o bound quiver representations and modules. We start with another notion rom category theory. Categories 5 I C ; D are two categories. We say that two unctors F ;F W C! D are unctorially isomorphic, and we write F Š F,i or every object M C, there exists an isomorphism M W F.M /! F.M / D such that, or every morphism W M! N in C,the ollowing diagram commutes: F (M) F ( ) F (N) η M F (M) F ( ) η N F (N) A covariant unctor F W C! D is called an equivalence o categories i there exists a unctor G W D! C such that GıF Š C and F ıg Š D. The unctor G is called a quasi-inverse unctor or F. A contravariant unctor F that has a (contravariant) quasi-inverse is called a duality

23 .3 Auslander Reiten Translation Duality Let Q be a quiver without oriented cycles, and let Q op be the quiver obtained rom Q by reversing each arrow. Thus Q op D Q and Q op D op j Q g with s. op / D t. / and t. op / D s. /. In this section, we need to work with projective and injective representations o both Q and Q op. To distinguish between these, we will oten use the notation P Q.i/; I Q.i/ or the representations o Q and P Q op.i/; I Q op.i/ or the representations o Q op. The duality D D Hom k. ;k/w rep Q! rep Q op is the contravariant unctor deined as ollows: On objects M D.M i ;' /,wehave DM D.DM i ;D' op/ iq ; Q ; where DM i is the dual vector space o the vector space M i, and thus DM i D Hom k.m i ;k/is the space o linear maps M i! k; and i is an arrow in Q then D' op is the pullback o ', and thus D' op W DM t. /! DM s. / u 7! u ı ' : On morphisms W M! N in rep Q, wehaved W DN! DM in rep Q op deined by D.u/ D u ı : M N. u u I we compose the duality o Q with the duality o Q op, we get the identity unctor rep Q ; thus the quasi-inverse o D Q is D Q op. Let proj Q be the category o projective representations o Q and let inj Q be category o injective representations o Q. Thus the objects in proj Q are the projective representations o Q, and the morphisms are the morphisms between projective representations. Proposition.7. We have D.P Q.i// D I Q op.i/, or all vertices i Q,in particular, the duality restricts to a duality proj Q! inj Q op. k

24 56 Projective and Injective Representations Proo. Exercise.. ut Example.7. Let Q be the quiver 3 as in Example.4. The indecomposable representations o the subcategory proj Q are the three projective representations: 3 : The quiver Q op is 3, and the indecomposable representations o the subcategory inj Q op are 3 :.3. Nakayama Functor Let A be the ree representation given as the direct sum o the indecomposable projective representations o Q, that is, A D j Q P.j/. Consider the contravariant unctor Hom. ;A/. We know already rom Sect..4 that the Hom unctors map representations o Q to vector spaces, and thus Hom.X; Y / is a vector space or every pair o representations X; Y. But now, instead o the arbitrary representation Y, we use the special ree representation A, and, in this case, we can give Hom.X; A/ the structure o a representation.m i ;' op/ o the opposite quiver Q op as ollows: Deine the vector space at vertex i as M i D Hom.X; P.i// or every i Q, and or an arrow rom i to j in Q, deine a linear map ' op W Hom.X; P.j //! Hom.X; P.i// as ' op. / D ı ; thus we have the diagram X P( j) ϕ α op ( ) α P(i) where we use the act that, being a path rom i to j, gives a morphism rom P.j/ to P.i/, by Corollary.. Thus Hom.X; A/ is a representation o Q op.

25 .3 Auslander Reiten Translation 57 To show that Hom. ;A/is a unctor rom rep Q to rep Q op, we must check that the image under Hom. ;A/o any morphism g W X! X o representations o Q is a morphism o representations o Q op. This means we must check or every arrow i! j in Q that the ollowing diagram commutes: Hom(X,P( j)) ϕ α op Hom(X,P(i)) g =Hom(g,P( j)) g =Hom(g,P(i)) Hom(X,P( j)) ϕ α op Hom(X,P(i)). To do so, let Hom.X ; P.j //, then g ' op. / D g. ı / D. ı /ı g, whereas ' op ı g. / D ' op. ı g/ D ı. ı g/ D. ı /ıg, so the diagram commutes. We have shown the ollowing: Proposition.8. Hom. ;A/is a unctor rom rep Q to rep Q op. Composing the two contravariant unctors D and Hom. ;A/, we get the ollowing important covariant unctor: Deinition.8. The unctor D DHom. ;A/ W rep Q! rep Q is called the Nakayama unctor: repq Hom(,A) repq op D repq ν Corollary.6 implies that the unctor Hom. ;A/ is zero on all representations which have no projective direct summands. Thereore we must study the behavior o the Nakayama unctor when applied to an indecomposable projective representation, say, P Q.i/. Let us irst consider the unctor Hom. ;A/ only. Let M D Hom.P Q.i/; A/, and use the habitual notation M D.M j ;' op/ j Q ; Q to denote the representation. Then the vector space M j is Hom.P Q.i/; P Q.j //; thus, according to Corollary.4, the space M j has a basis consisting o all paths rom j to i in Q. In terms o Q op this can be rephrased as M j has a basis consisting o all paths rom i to j in Q op. Moreover, or any arrow h! j in Q, the linear map ' op W M j! M h maps the basis element c, which is a path rom i to j in Q op, to the basis element c op, whichisapathromi to h in Q op. This shows that M D Hom.P Q.i/; A/ is the indecomposable projective Q op -representation P Q op.i/ at vertex i.

26 58 Projective and Injective Representations Thus the restriction o Hom. ;A/ to the subcategory proj Q gives a duality o categories proj Q! proj Q op, whose quasi-inverse is given by Hom Q op. ;A op /, where A op is the sum o all indecomposable projective Q op -representations. To obtain the Nakayama unctor, we must now orm the composition with the duality D. Note that DA op D iq I Q.i/. Proposition.9. The restriction o to proj Q is an equivalence o categories proj Q! inj Q whose quasi-inverse is given by Moreover, or any vertex i, D Hom.DA op ; /W inj Q! proj Q: P.i/ D I.i/; and i c is a path rom i to j, and c morphism, then Hom.P.j /; P.i// is the corresponding c W I.j/! I.i/ is the morphism given by the cancellation o the path c. Proo. The unctor is an equivalence because it is the composition o the two dualities D and Hom. ;A/. Its quasi-inverse is the composition o the quasiinverses o D and Hom. ;A/, thus D Hom Q op. ;A op / ı D. Note that since Hom Q op.dx; DY / Š Hom Q.Y; X/ or all X; Y rep Q, we have in particular that Hom Q op.dx; A op / Š Hom Q.DA op ;X/, whence D Hom.DA; /. Finally, P Q.i/ D DHom.P Q.i/; A/ D D.P Q op.i// D I Q.i/: To show the last statement, let c be a path rom i to j, and let c W P Q.j /! P Q.i/ be deined by.x/ D cx as in the proposition. Let c be the image o c under the unctor Hom. ;A/. Thus c W Hom.P Q.i/; A/! Hom.P Q.j /; A/ maps a morphism g to the pullback g ı c. Now using Hom.P Q.x/; A/ Š P Q op.x/, we see that c W P Q op.i/! P Q op.j / is given by.y/ D cop y, where c op denotes the opposite o the path c in Q op. Finally, c D Dc is the map sending D.c op y/ to D.y/; thus the result ollows rom D.c op y/ D D.y/c: ut Example.8. Let Q be the quiver 3. Its Auslander Reiten quiver is

27 .3 Auslander Reiten Translation Note that the Nakayama unctor sends proj Q to inj Q as ollows: P()=νP(3) P() νp() P(3) νp() Example.9. Let Q be the quiver 4 3. Then proj Q and inj Q are as ollows:

28 6 Projective and Injective Representations Beore we can state the next property o the Nakayama unctor, we need to introduce the notion o exactness or unctors. Categories 6 Let C and D be abelian categories. A (covariant or contravariant) unctor F W C! D is called exact i it maps exact sequences in C to exact sequences in D. For example, every equivalence or duality o abelian categories is exact. Many nice unctors, the Homunctors or example, are not exact but have the weaker property o being let exact or right exact, which we deine below. The deinition o these notions is dierent or covariant and contravariant unctors. Let F W C! D be a covariant unctor. F is called let exact i or any exact sequence L M g N the sequence F(L) F( ) F(M) F(g) F(N) is exact. F is called right exact i or any exact sequence L M g N the sequence F(L) F( ) F(M) F(g) F(N) is exact. Let GW C! D be a contravariant unctor. G is called let exact i or any exact sequence L M g N the sequence G(N) G(g) G(M) G( ) G(L) is exact.

29 .3 Auslander Reiten Translation 6 G is called right exact i or any exact sequence L M g N the sequence G(N) G(g) G(M) G( ) G(L) is exact. We have shown in Sect..4 that the Hom unctors Hom.X; / and Hom. ;X/ are let exact. Since the Nakayama unctor is the composition o the let exact unctor Hom. ;A/ and the exact contravariant unctor D, we get the ollowing proposition: Proposition.3. The Nakayama unctor is right exact. Example.. In the setting o Example.8 there is a short exact sequence 3 3 g, and applying yields the exact sequence 3 ν νg. This conirms that is right exact. Since the morphism is clearly not injective, this also shows that is not exact..3.3 The Auslander Reiten Translations ; Let Q be a quiver without oriented cycles, and let M be an indecomposable representation o Q. Deinition.9. Let P p P p M be a minimal projective resolution. Applying the Nakayama unctor, we get an exact sequence τm νp ν p νp ν p νm,

30 6 Projective and Injective Representations where M D ker p is called the Auslander Reiten translate o M and the Auslander Reiten translation. Let M i I i I i be a minimal injective resolution. Applying the inverse Nakayama unctor, we get an exact sequence ν M ν i ν I ν i ν I τ M, where M D coker i is called the inverse Auslander Reiten translate o M and the inverse Auslander Reiten translation. Example.. Continuing Example., we compute. We have already constructed the minimal projective resolution and applied the Nakayama unctor to it. It only remains to compute the kernel o ; thus D ker D 3 : Remark.3. The Auslander Reiten translation has been introduced by Auslander and Reiten in []..4 Extensions and Ext In this section, Q always denotes a quiver without oriented cycles. We give here a short account on the Ext -groups; or urther inormation we reer to [53, Sect. 7.]. Let M rep Q and take a projective resolution P P g M o M in rep Q. Thus P and P are projective representations and the above sequence is exact. Let N be any representation in rep Q. Then we can apply the unctor Hom. ;N/ to this projective resolution, and as a result we get the exact sequence Hom(M,N) g Hom(P,N) Hom(P,N) Ext (M,N), where Ext.M; N / D coker is called the irst group o extensions o M and N.

31 .4 Extensions and Ext 63 Remark.3. In arbitrary categories, projective resolutions do not necessarily stop ater two steps; in act, they might not even stop at all. Thus a projective resolution in a general category is o the orm P n n P n P P M and applying Hom. ;N/yields a so-called cochain complex Hom(M, N) Hom(P,N) n Hom(P n,n), which means that i i D, or all i. One then deines the ith extension group Ext i.m; N / or i to be the ith cohomology group o this complex, that is, Ext i.m; N / D ker ic ı im i : One can show that this deinition does not depend on the choice o the projective resolution; see, or example, [53, Proposition 6.4]. In the category rep Q, all the Ext i -groups, with i, vanish, because the minimal projective resolutions are o the orm P P g M. On the other hand, the Ext -groups provide very interesting inormation. Our next goal is to show that the vector space Ext.M; N / is isomorphic to the vector space o extensions o M by N. Deinition.. An extension o M by N is a short exact sequence! N! E! g M!. Two extensions and are called equivalent i there is a commutative diagram: ζ : N E g M = = = ζ : N E g M Example.. Let Q be the quiver α β,

32 64 Projective and Injective Representations let N D S./, M D S./ be the two simple modules and let E = k k and E = k k. Then the short exact sequences ζ : ζ : S() S() E E g g S() S() are not equivalent, because E and E are not isomorphic. An extension is split i the short exact sequence is split, that is, i the extension is equivalent to the short exact sequence: N N M M. Given two extensions and o M by N, we deine their sum C as ollows: Let E D.x; x / E E j g.x/ D g.x /g be the so-called pull back o g and g, and deine F to be the quotient o E by the subspace..n/;.n// E E j n N g, compare with Exercises.8 and.9 in Chap.. Then C is N F M. The set o equivalence classes E.M; N / o extensions o M by N together with the sum o extensions is an abelian group, and the class o the split extension is the zero element o that group. There is an isomorphism o groups E.M; N /! Ext.M; N / which is deined as ollows. Let W! N! u E! v M! be a representative o a class in g E.M; N /, and let! P! P! M! be a projective resolution. Then since P is projective, it ollows that there exists a morphism Hom.P ;E/ such that g D v. Now since is exact, the universal property o the kernel implies that there exists also a morphism u Hom.P ;N/such that the ollowing diagram commutes: P P g M u = ζ : N u E v M.

33 .4 Extensions and Ext 65 Recall that Ext.M; N / is the cokernel o, which is the quotient Hom.P ;N/=im. The isomorphism E.M; N /! Ext.M; N / is sending the class o to the class o u. Example.3. Let us compute the sum o the two short exact sequences and in Example.. Thus Q is the quiver α β, N D S./, M D S./ and E = k k and E = k k. Using our notation E D.E i ;' / iq ; Q and E D.E i ;' /iq ; Q,wehave E Š E Š k ' D 'ˇ D E Š E Š k ' D 'ˇ D Let us denote the elements o E as pairs.e ;e / E E and those o E as.e ;e / E E : To compute the sum C, we irst need to compute the pull back E. By deinition E D.e ;e /;.e ;e / E E j g.e ;e / D g.e ;e / : Since g and g are both projections on the irst component, we have E D.e ;e /;.e ;e / E E : We want to write E as a representation E D.E i ;' above shows that E This shows that Š k and E Š k. Let s compute ' /iq ; Q. Our computation and ' ˇ. Wehave '..e ;e /;.e ;e // D.'.e ;e /; '.e ;e // D..; e /;.; // ' ˇ..e ;e /;.e ;e // D.'ˇ.e ;e /; ' ˇ.e ;e // D..; /;.; e // E = k k.

34 66 Projective and Injective Representations Now we compute F. By deinition and using the act that both and are inclusions in the second component, we have F D E =..; n/;.; n// j n kg : Writing F D.F i ; /, we see that F Š F Š k. Moreover, i.e ;e /;.e ;e / E and.e ;e /;.e ;e / denotes its class in F, then and ˇ.e ;e /;.e ;e / D '.e ;e /;.e ;e / D..; e /;.; //.e ;e /;.e ;e / D ' ˇ.e ;e /;.e ;e / D..; /;.; e //: In particular,.e ;e /;.e ;e / D ˇ.e ;e /;.e ;e /. This shows that F = k k. Finally, we see that the sum C is the short exact sequence S() F g S() with the inclusion in the second component and g the projection on the irst component. Problems Exercises or Chap... Let Q be the quiver 3 4. Prove that x ab P() = k x k cd k, z k y y

35 .4 Extensions and Ext 67 i and only i. all our maps have maximal rank, that is, ad bc,.x ;x /.; /;.y ;y /.; /; z and ab x ab y. the vectors and are linearly independent. cd cd x.. Compute the indecomposable projective representations P.i/ and the indecomposable injective representations I.i/ or the ollowing quivers: y Compute a projective resolution or each simple representation S.i/ or each o the quivers in (.).. Compute the dimension vector o S.i/or each simple representation S.i/ o the quivers in (.)..4. Prove Proposition.3.5. Prove Proposition.5.6. Show that or each i Q, the sequence radp(i) P(i) S(i) is a minimal projective resolution..7. Let M D.M i ; / be a representation o Q. Prove that or any vertex i in Q, there is an isomorphism o vector spaces: Hom.M; I.i// Š M i :.8. Let i and j be vertices in the quiver Q. Prove that the vector space Hom.I.i/; I.j // has a basis consisting o all paths rom j to i in Q. In particular, End.I.i// Š k:.9. Prove that the representation I.j/ is a simple representation i and only i Hom.I.j /; I.i// D or all i j.

36 68 Projective and Injective Representations.. Prove Proposition.7... Prove that P is projective i and only i Ext.P; N / D or all representations N... Let Q be the quiver α β and let M be the representation deined in Exercise.6 o Chap... Show that the short exact sequences are not equivalent i.. Show that M D M, or all. M λ, M μ,,

37