ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS H. FAUSK, P. HU, AND J.P. MAY Abstract.There are many contexts in algebraic geometry, algebraic topol- ogy, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a s* *hift of the left adjoint specified by an appropriate üd alizing object". Typical* *ly the left adjoint is well understood while the right adjoint is more mysterio* *us, and the result identifies the right adjoint in familiar terms. We give a cat* *egorical discussion of such results. One essential point is to differentiate betw* *een the classical framework that arises in algebraic geometry and a deceptively * *similar, but genuinely different, framework that arises in algebraic topology. An* *other is to make clear which parts of the proofs of such results are formal. * *The analysis significantly simplifies the proofs of particular cases, as we * *illustrate in a sequel discussing applications to equivariant stable homotopy theor* *y. Contents 1. The starting point: an adjoint pair (f*, f*) 2 2. The general context: adjoint pairs (f*, f*) and (f!, f!) 4 3. Isomorphisms in the Verdier-Grothendieck context 7 4. The Wirthmüller isomorphism 10 5. Preliminaries on triangulated categories 13 6. The formal isomorphism theorems 15 References 17 We give a categorical discussion of Verdier and Grothendieck isomorphisms on the one hand and formally analogous results whose proofs involve different issu* *es on the other. Our point is to explain and compare the two contexts and to differen* *tiate the formal issues from the substantive issues in each. The philosophy goes back* * to Grothendieck's "six operations" formalism. We fix our general framework, explain what the naive versions of our theorems say, and describe which parts of their proofs are formal in xx1-4. This discussion does not require triangulated categ* *ories. Its hypotheses and conclusions make sense in general closed symmetric monoidal categories, whether or not triangulated. In practice, that means that the argum* *ents apply equally well before or after passage to derived categories. After giving some preliminary results about triangulated categories in x5, we explain the formal theorems comparing left and right adjoints in x6. Our öf rmal Grothendieck isomorphism theorem" is an abstraction of results of Amnon Neeman, and our öf rmal Wirthmüller isomorphism theorem" borrows from his ideas. His ____________ Date: April 10, 2002. The second and third authors were partially supported by the NSF. 1 2 H. FAUSK, P. HU, AND J.P. MAY paper [22] has been influential, and he must be thanked for catching a mistake * *in a preliminary version by the third author. We thank Gaunce Lewis for discussions of the topological context, and we thank Sasha Beilinson and Madhav Nori for making clear that, contrary to our original expectations, the context encounter* *ed in algebraic topology is not part of the classical context familiar to algebraic g* *eometers. We also thank Johann Sigurdsson for corrections and emendations. 1. The starting point: an adjoint pair (f*, f*) We fix closed symmetric monoidal categories C and D with respective unit ob- jects S and T . We write and Hom for the product and internal hom functor in either category, and we write X (sometimes also W ) and Y (sometimes also Z) generically for objects of C and objects of D, respectively. We write C (W, X) * *and D(Y, Z) for the categorical hom sets. We let DX = Hom (X, S) denote the dual of X. We let ev: Hom (X, W ) X -! W denote the evaluation map, that is, the counit of the ( , Hom ) adjunction C (X X0, W ) ~=C (X, Hom (X0, W )). We also fix a strong symmetric monoidal functor f* :D -! C . This means that we are given isomorphisms (1.1) f*T ~=S and f*(Y Z) ~=f*Y f*Z, the second natural, that commute with the associativity, commutativity, and unit isomorphisms for in C and D. We assume throughout that f* has a right adjoint f*, and we write ": f*f*X -! X and j :Y -! f*f*Y for the counit and unit of the adjunction. This general context is fixed throug* *hout. The notation (f*, f*) meshes with standard notation in algebraic geometry, where one starts with a map f :A -! B of spaces or schemes and f* and f* are pullback and pushforward functors on sheaves. In our generality there need * *be no underlying map "f" in sight. Some simple illustrative examples are given in * *x3. The assumption that f* is strong symmetric monoidal has several basic impli- cations. To begin with, the adjoints of the isomorphism f*T ~=S and the map f*(f*W f*X) ~=f*f*W f*f*X _"_"_//W X are maps (1.2) T -! f*S and f*W f*X -! f*(W X). These are not usually isomorphisms. This means that f* is lax symmetric monoida* *l. The adjoint of the map f*(ev) * f* Hom (Y, Z) f*Y ~=f*(Hom (Y, Z) _Y_)//_f Z is a natural map (1.3) ff: f* Hom (Y, Z) -! Hom (f*Y, f*Z). It may or may not be an isomorphism in general, and we say that f* is closed symmetric monoidal if it is. However, the adjoint of the composite map Hom(id,") * f* Hom (Y, f*X)ff_//Hom(f*Y, f*f*X)_________//Hom(f Y, X) ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS 3 is a natural isomorphism (1.4) Hom (Y, f*X) ~=f*Hom (f*Y, X). In particular, Hom (Y, f*S) ~=f*Df*Y . Indeed, we have the following two chains of isomorphisms of functors. D(Z, Hom (Y, f*X)) ~=D(Z Y, f*X) ~=C (f*(Z Y ), X) D(Z, f*Hom (f*Y, X)) ~=C (f*Z, Hom (f*Y, X)) ~=C (f*Z f*Y, X) By the Yoneda lemma and a check of maps, these show immediately that the as- sumed isomorphism of functors in (1.1) is equivalent to the claimed isomorphism of functors (1.4). That is, the isomorphism of left adjoints in (1.1) is adjoin* *t to the isomorphism of right adjoints in (1.4). Systematic recognition of such öc njuga* *te" pairs of isomorphisms can substitute for quite a bit of excess verbiage in the * *liter- ature. We call this a öc mparison of adjointsä nd henceforward leave the detai* *ls of such arguments to the reader. Using the isomorphism (1.4), we obtain the following map fi, which is analogo* *us to both ff and the map of (1.2). Like the latter, it is not usually an isomorph* *ism. f*Hom(",id) * ~= (1.5) fi :f*Hom (X, W_)________//f*Hom (f f*X, W_)__//Hom(f*X, f*W ). Using (1.2), we also obtain a natural composite (1.6) ß :Y f*X_j_id//_f*f*Y f*X__//_f*(f*Y X). Like ff, it may or may not be an isomorphism in general. When it is, we say that the projection formula holds. As noted by Lipman [1, p. 119], there is already a non-trivial öc herence pro* *blem" in this general context, the question of determining which compatibility diagra* *ms relating the given data necessarily commute. An early reference for coherence in closed symmetric monoidal categories is [8], and the volume [14] contains sever* *al papers on the subject and many references. In particular, a paper of G. Lewis in [14] gives a partial coherence theorem for closed monoidal functors. The catego* *rical literature of coherence is relevant to the study of öc mpatibilities" that focu* *ses on base change maps and plays an important role in the literature in algebraic geometry (e.g. [1, 4, 7, 6, 11]). A study of that is beyond the scope of this n* *ote. A full categorical coherence theorem is not known and would be highly desirable. We illustrate by recording a particular commutative coherence diagram, namely ~= f*(ev) (1.7) f*DY f*Y ____//_f*(DY Y_)___//f*T ff id|| |~=| fflffl| fflffl| Df*Y f*Y __________ev__________//S. We shall need a consequence of this diagram. There is a natural map :DX W -! Hom (X, W ), namely the adjoint of DX W X ~=DX X W -ev-id--!S W ~=W. 4 H. FAUSK, P. HU, AND J.P. MAY The commutativity of the diagram (1.7) implies the commutativity of the diagram ~= f* (1.8) f*DY f*Z ____//_f*(DY Z)_____//f* Hom (Y, Z) ff id|| |ff| fflffl| fflffl| Df*Y f*Z _____________________//Hom(f*Y, f*Z). We assume familiarity with the theory of üd alizable" (alias "strongly dualiz- ableö r "finite") objects; see [18] for a recent exposition. The defining prop* *erty is that :DX X -! Hom (X, X) is an isomorphism. It follows that is an isomorphism if either X or W is dualizable. It also follows that the natural map æ : X -! DDX is an isomorphism, but the converse fails in general. When X0 is dualizable, we have the duality adjunction (1.9) C (X X0, X00) ~=C (X, DX0 X00). As observed in [16, III.1.9], (1.1) and the definitions imply the following r* *esult. Proposition 1.10. If Y 2 D is dualizable, then DY , f*Y , and Df*Y are dualiz- able and the map ff of (1.3) restricts to an isomorphism (1.11) f*DY ~=Df*Y. This implies that ff and ß are often isomorphisms for formal reasons. Proposition 1.12. If Y 2 D is dualizable, then ff: f* Hom (Y, Z) -! Hom (f*Y, f*Z) and ß :Y f*X -! f*(f*Y X) are isomorphisms for all objects X 2 C and Z 2 D. Thus, if all objects of D are dualizable, then f* is closed symmetric monoidal and the projection formula hol* *ds. Proof.For the first statement, ff coincides with the composite f* Hom (Y, Z) ~=f*(DY Z) ~=f*DY f*Z ~=Df*Y f*Z ~=Hom (f*Y, f*Z). For the second statement, ß induces the isomorphism of represented functors D(Z, Y f*X) ~=D(Z DY, f*X) ~=C (f*(Z DY ), X) ~=C (f*Z f*DY, X) ~=C (f*Z Df*Y, X) ~=C (f*Z, f*Y X) ~=D(Z, f*(f*Y X)). 2.The general context: adjoint pairs (f*, f*) and (f!, f!) In addition to the adjoint pair (f*, f*) of the previous section, we here ass* *ume given a second adjoint pair (f!, f!) relating C and D, with f!:C -! D being the left adjoint. We write oe :f!f!Y -! Y and i :X -! f!f!X for the counit and unit of the second adjunction. The adjunction D(Y, f*X) ~=C (f*Y, X) can be recovered from the more general "internal Hom adjunction" Hom (Y, f*X) ~=f*Hom (f*Y, X) of (1.4) by applying the functor D(T, -) and using the assumption that f*T ~= S. Analogously, it is natural to hope that the adjunction D(f!X, Y ) ~=C (X, f!Y ) can be recovered by applying the functor D(T, -) to a similar internal Hom adjunction Hom (f!X, Y ) ~=f*Hom (X, f!Y ). ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS 5 However, unlike (1.4), such an adjunction does not follow formally from our h* *y- potheses. Motivated by different specializations of the general context, we con* *sider two triads of basic natural maps that we might ask for relating our four functo* *rs. For the first triad, we might ask for either of the following two duality maps,* * the first of which is a comparison map for the desired internal Hom adjunction. (2.1) fl :f*Hom (X, f!Y ) -! Hom (f!X, Y ). (2.2) ffi :Hom (f*Y, f!Z) -! f!Hom (Y, Z). We might also ask for a projection formula map (2.3) ^ß:Y f!X -! f!(f*Y X), which should be thought of as a generalized analogue of the map ß of (1.6). The* *se three maps are not formal consequences of the given adjunctions, but rather must be constructed by hand. However, it suffices to construct any one of them. Proposition 2.4. Suppose given any one of the natural maps fl, ffi, and ^ß. Then it determines the other two by conjugation. The map ffi is an isomorphism for a* *ll dualizable Y if and only if its conjugate ^ßis an isomorphism for all dualizabl* *e Y . If any one of the three conjugately related maps is a natural isomorphism, then* * so are the other two. The second triad results from the first simply by changing the direction of t* *he arrows. That is, we can ask for natural maps in the following directions. (2.5) ~fl:Hom(f!X, Y ) -! f*Hom (X, f!Y ). (2.6) ~ffi:f!Hom (Y, Z) -! Hom (f*Y, f!Z). (2.7) ~ß:f!(f*Y X) -! Y f!X. Here ~ffiis to be viewed as a generalized analogue of the map ff of (1.3). Proposition 2.8. Suppose given any one of the natural maps ~fl, ~ffi, and ~ß. T* *hen it determines the other two by conjugation. The map ~ffiis an isomorphism for a* *ll dualizable Y if and only if its conjugate ~ßis an isomorphism for all dualizabl* *e Y . If any one of the three conjugately related maps is a natural isomorphism, then* * so are the other two. Of course, when the three maps are isomorphisms, the triads are inverse to each other and there is no real difference. However, there are two very differ* *ent interesting specializations: we might have f! = f*, or we might have f! = f*. The first occurs frequently in algebraic geometry, and is familiar. The second occurs in algebraic topology and elsewhere, but seems less familiar. With the f* *irst specialization, the first triad of maps arises formally since we can take ^ßto * *be the map ß of (1.6). With the second specialization, the second triad arises formal* *ly since we can take ~ffito be the map ff of (1.3). Recall the isomorphism (1.4), * *the map fi of (1.5), and Proposition 1.12. Proposition 2.9. Suppose f!= f*. Taking ^ßto be the projection map ß of (1.6), the conjugate map fl is the composite fi Hom(id,oe) f*Hom (X, f!Y_)___//Hom(f*X, f*f!Y_)_______//_Hom(f*X, Y ) 6 H. FAUSK, P. HU, AND J.P. MAY and the conjugate map ffi is the adjoint of the map Hom(id,oe) f*Hom (f*Y, f!Z) ~=Hom (Y, f*f!Z)______//_Hom(Y, Z). Moreover, ß and ffi are isomorphisms if Y is dualizable. When f! = f*, passage to adjoints from S ~=f*T and the natural map W X _i_i_//f*f!W f*f!X ~=f*(f!W f!X) gives maps, not usually isomorphisms, (2.10) f!S -! T and f!(W X) -! f!W f!X. This means that f!is an op-lax symmetric monoidal functor. Proposition 2.11. Suppose f! = f*. Taking ~ffito be the map ff of (1.3), the conjugate map ~ßis the composite f!(f*Y X)_____//f!f*Y f!Xoe/id/_Y f!X and the conjugate map ~flis the adjoint of the map Hom(i,id) * f* Hom (f!X, Y_)ff_//Hom(f*f!X, f*Y_)_______//Hom(X, f Y ). Moreover ff and ~ßare isomorphisms if Y is dualizable. Definition 2.12. We introduce names for the different contexts in sight. (i) The Verdier-Grothendieck context: There is a natural isomorphism ^ßas in (2* *.3) (projection formula); taking ~ß= ^ß-1, there are conjugately determined natural isomorphisms fl = ~fl-1, and ffi = ~ffi-1. (ii) The Grothendieck context: f!= f* and the projection formula holds. (iii) The Wirthmüller context: f! = f* and f* is closed symmetric monoidal. Thus, in the Grothendieck context, the strong symmetric monoidal functor f* is the left adjoint of a left adjoint. In the Wirthmüller context, it is a left* * and a right adjoint. The Verdier-Grothendieck context encapsulates the properties that hold for su* *it- able derived categories C and D of sheaves over locally compact spaces A and B and maps f : A -! B; see [2, 13, 25]. Here f!is given by pushforward with compa* *ct supports. The same abstract context applies to suitable derived categories C and D of complexes of OA -modules and of OB -modules for schemes A and B and maps f : A -! B. In either context, we have f!= f* when the map f is proper. For the scheme theoretic context, see [5, 6, 11] and, for more recent reworkings and ge* *n- eralizations, [1, 4, 15, 22]. There is a highly non-trivial categorical, more p* *recisely 2-categorical, coherence problem concerning composites of base change functors * *in the Verdier-Grothendieck context. A start on this has been made by Voevodsky [7]. Since his discussion focuses on base change relating quadruples (f*, f*, f* *!, f!), ignoring and Hom , it is essentially disjoint from our discussion. The relev* *ant coherence problem simplifies greatly in either the Grothendieck or the Wirthmül* *ler context, due to the canonicity of the maps in Propositions 2.9 and 2.11. We repeat that our categorical results deduce formal conclusions from formal hypotheses and therefore work equally well before or after passage to derived c* *ate- gories. Much of the work in passing from categories of sheaves to derived categ* *ories can be viewed as the verification that formal properties in the category of she* *aves ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS 7 carry over to the same formal properties in derived categories, although other * *prop- erties only hold after passage to derived categories. A paper by Lipman in [1] * *takes a similarly categorical point of view. While the proofs of Propositions 2.4 and 2.9 are formal, in the applications * *to algebraic geometry they require use of unbounded derived categories, since othe* *r- wise we would not have closed symmetric monoidal categories to begin with. These were not available until Spaltenstein's paper [24], and he noticed one of our f* *ormal implications [24, x6]. Unfortunately, as he makes clear, in the classical sheaf* * con- text his methods fail to give the (f!, f!) adjunction for all maps f between lo* *cally compact spaces. It seems possible that a model theoretic approach to unbounded derived categories would allow one to resolve this problem. In any case, a comp* *lete reworking of the theory in model theoretical terms would be of considerable val* *ue. In the algebraic geometry setting, smooth maps lead to a context close to the Wirthmüller context, but that is not our motivation. In that context, we think * *of f* as a forgetful functor which does not alter underlying structure, f!as a kin* *d of extension of scalars functor, and f* as a kind of öc extension of scalars" func* *tor. For example, let f : H -! G be an inclusion of a subgroup in a group G and let C and D be the categories of H-objects and G-objects in some Cartesian clos* *ed category, such as topological spaces. Let f* : D -! C be the evident forgetful functor. Certainly f*(Y x Z) ~=f*Y x f*Z. The left and right adjoints of f* send an H-object X to GxH X and to MapH (G, X* *). Clearly GxH (X xX0) is not isomorphic to (GxH X)x(GxH X0). Our motivating example is a spectrum level analogue of this for which there is a Wirthmüller isomorphism theorem [16, 27]. Our formal Wirthmüller isomorphism theorem below substantially simplifies its proof [20]. One can hope for such a result in any * *context where group actions and triangulated categories mix. For another example, let f : A -! B be an inclusion of cocommutative Hopf algebras over a field k and let C and D be the categories of A-modules and of B-modules. These are closed symmetric monoidal categories under the functors k and Hom k. Indeed, using the coproduct on A, we see that if M and N are A-modules, then so are M k N and Hom k(M, N). The commutativity of k requires the cocommutativity of A. The unit object in both C and D is k. Again, if f* : D -! C is the evident forgetful functor, then f*(Y k Z) = f*Y k f*Z. The left and right adjoints of f* send an A-module X to B A X and to Hom A(B, X* *), and again B A (X k X0) is not isomorphic to (B A X) k (B A X0). This example deserves investigation on the level of derived categories. 3.Isomorphisms in the Verdier-Grothendieck context We place ourselves in the Verdier-Grothendieck context in this section. Definition 3.1. For an object W 2 C , define DW X = Hom (X, W ), the W -twisted dual of X. Of course, if X or W is dualizable, then DW X ~= DX W . Let æW : X -! DW DW X be the adjoint of the evaluation map DW X X -! W . We say that X is W -reflexive if æW is an isomorphism. 8 H. FAUSK, P. HU, AND J.P. MAY Replacing Y by Z in (2.1) and letting W = f!Z, the isomorphisms fl and ffi ta* *ke the following form: (3.2) f*DW X ~=DZf!X and DW f*Y ~=f!DZY. This change of notation and comparison with the classical context of algebraic geometry explains why we think of fl and ffi as duality maps. If f!X is Z-refle* *xive, the first isomorphism implies that (3.3) f!X ~=DZf*DW X. If Y is isomorphic to DZY 0for some Z-reflexive object Y 0, the second isomorph* *ism implies that (3.4) f!Y ~=DW f*DZY. These observations and the classical context suggest the following definition. Definition 3.5. A dualizing object for a full subcategory C0 of C is an object W of C such that if X 2 C0, then DW X is in C0 and X is W -reflexive. Thus DW specifies an auto-duality of the category C0. Remark 3.6. In algebraic geometry, we often encounter canonical subcategories C0 C and D0 D such that f!C0 D0 and f!D0 C0 together with a dualizing object Z for D0 such that W = f!Z is a dualizing object for C0. In such context* *s, (3.3) and (3.4) express f!on C0 and f! on D0 in terms of f* and f*. For any objects Y and Z of D, the adjoint of the map f!(f*Y f!Z) ~=Y f!f!Zid_oe//Y Z is a natural map (3.7) OE: f*Y f!Z -! f!(Y Z). It specializes to (3.8) OE: f*Y f!T -! f!Y, which of course compares a right adjoint to a shift of a left adjoint. A Verdi* *er- Grothendieck isomorphism theorem asserts that the map OE is an isomorphism; in the context of sheaves over spaces, such a result was announced by Verdier in [* *25, x5]. The following observation, abstracts a result of Neeman [22, 5.4]. In it* *, we only assume the projection formula for dualizable Y . Proposition 3.9. The map OE: f*Y f!Z -! f!(Y Z) is an isomorphism for all objects Z and all dualizable objects Y . Proof.Using Proposition 1.10, the projection formula, duality adjunctions (1.9), and the (f!, f!) adjunction, we obtain isomorphisms C (X, f*Y f!Z) ~=C (f*DY X, f!Z) ~=D(f!(f*DY X), Z) ~=D(DY f!X, Z) ~=D(f!X, Y Z) ~=C (X, f!(Y Z)). Diagram chasing shows that the composite isomorphism is induced by OE. It is natural to ask when OE is an isomorphism in general, and we shall retur* *n to that question in the context of triangulated categories. Of course, this discus* *sion specializes and remains interesting in the Grothendieck context f!= f*. We give some elementary examples of the Verdier-Grothendieck context. ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS 9 Example 3.10. An example of the Verdier-Grothendieck context is already avail- able with C = D and f* = f* = Id. Fix an object C of C and set f!X = X C and f!(Y ) = Hom (C, Y ). The projection formula f!(f*Y Z) ~=Y f!Z is the associativity isomorphism (Y Z) C ~=Y (Z C). The map OE: f*Y f!Z -! f!(Y Z) is the canonical map :Y Hom (C, Z) -! Hom (C, Y Z). It is an isomorphism if Y is dualizable, and it is an isomorphism for all Y if * *and only if C is dualizable. The shift of an adjunction by an object of C used in the previous example generalizes to give a shift of any Verdier-Grothendieck context by an object of* * C . Definition 3.11. For an adjoint pair (f!, f!) and an object C 2 C , define the twisted adjoint pair (fC!, f!C) by (3.12) fC!(X) = f!(X C) and f!CY = Hom (C, f!Y ). Proposition 3.13. If (f*, f*) and (f!, f!) are in the Verdier-Grothendieck cont* *ext, then so are (f*, f*) and (fC!, f!C ). Proof.The isomorphism ^ßof (2.3) shifts to a corresponding isomorphism ^ßC. We also give a simple example of the context of Definition 3.5. Recall that dualizable objects are S-reflexive, but not conversely in general. The followi* *ng observation parallels part of a standard characterization of üd alizing complex* *es" [11, V.2.1]. Let dC denote the full subcategory of dualizable objects of C . Proposition 3.14. S is W -reflexive if and only if all X 2 dC are W -reflexive. Proof.Since S is dualizable, the backwards implication is trivial. Assume that S is W -reflexive. Since W ~=DW S, Hom (W, W ) = DW W ~=DW DW S. In any closed symmetric monoidal category, such as C , we have a natural isomorphism Hom (X X0, X00) ~=Hom (X, Hom (X0, X00)), where X, X0, and X00are arbitrary objects. When X is dualizable, :DX X0- ! Hom (X, X0) is an isomorphism for any object X0. Therefore DW DW X ~=Hom (DX W, W ) ~=Hom (DX, Hom (W, W )) ~=DDX DW DW S. Identifying X with X S, is easy to check that æW corresponds under this isom* *or- phism to æS æW . The conclusion follows. Corollary 3.15. Let W be dualizable. Then the following are equivalent. (i)W is a dualizing object for dC . (ii)S is W -reflexive. (iii)W is invertible. (iv)DW :dC op-! dC is an auto-duality of dC . 10 H. FAUSK, P. HU, AND J.P. MAY Proof.If X is dualizable, then DW X ~=DX W is dualizable. The proposition shows that (i) and (ii) are equivalent, and it is clear that (iii) and (iv) are* * equivalent. Since W is dualizable, DW DW S ~=Hom (W, W ) ~=W DW , with æW corresponding to the coevaluation map coev : S -! W DW . By [18, 2.9], W is invertible if a* *nd only if coev is an isomorphism. Therefore (ii) and (iii) are equivalent. Finally, we have a shift comparison of Grothendieck and Wirthmüller contexts. Remark 3.16. Start in the Grothendieck context, so that f!= f*, and assume that the map OE: f*Y f!T -! f!Y of (3.8) is an isomorphism. Assume further that f!T is invertible and let C = Df!T . Define a new functor f!by f!X = f*(X DC). Then f!is left adjoint to f*. Replacing X by X C, we see that f*X ~=f!(X C). In the next section, we shall consider isomorphisms of this general form in the Wirthmüller context. Conversely, start in the Wirthmüller context, so that f! =* * f*, and assume given a C such that f*S ~=f!C and the map ! :f*X -! f!(X C) of (4.7) below is an isomorphism. Define a new functor f! by f!Y = Hom (C, f*Y ) and note that f!T ~= DC. Then f! is right adjoint to f*. If either C or Y is dualizable, then Hom (C, f*Y ) ~=f*Y DC and thus f*Y f!T ~=f!Y , which is an isomorphism of the same form as in the Grothendieck context. 4. The Wirthmüller isomorphism We place ourselves in the Wirthmüller context in this section, with f! = f*. Here the specialization of the Verdier-Grothendieck isomorphism is of no intere* *st. In fact, OE reduces to the originally assumed isomorphism (1.1). However, there* * is now a candidate for an isomorphism between the right adjoint f* of f* and a shi* *ft of the left adjoint f!. This is not motivated by duality questions, and it can * *already fail on dualizable objects. We assume in addition to the isomorphisms ff = ~ffi* *, hence ~ßand ~fl, that we are given an object C 2 C together with an isomorphism (4.1) f*S ~=f!C. Observe that the isomorphism ~flspecializes to an isomorphism (4.2) Df!X ~=f*DX. Taking X = S in (4.2) and using that DS ~=S, we see that (4.1) is equivalent to (4.3) Df!S ~=f!C. This version is the one most naturally encountered in applications, since it ma* *kes no reference to the right adjoint f* that we seek to understand. In practice, f* *!S is dualizable and C is dualizable or even invertible. It is a curious feature o* *f our discussion that it does not require such hypotheses. Replacing C by S C in (4.1), it is reasonable to hope that it continues to * *hold with S replaced by a general X. That is, we can hope for a natural isomorphism (4.4) f*X ~=f]X, where f]X f!(X C). Note that we twist by C before applying f!. We shall shortly define a particular natural map ! :f*X -! f]X. A Wirthmüller isomorphism theorem asserts that ! is an isomorphism. We shall show that if f!S is dualizable and X is a retract* * of some f*Y , then ! is an isomorphism. However, even for dualizable X, ! need not be an isomorphism in general. A counterexample is given in the sequel [20]. We ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS 11 shall also give a categorical criterion for ! to be an isomorphism for a partic* *ular object X. An application is also given in [20]. Using the map T -! f*S of (1.2), the assumed isomorphism f*S ~=f!C gives rise to maps ø :T -! f*S ~=f!C and , :f*f!C ~=f*f*S_"__//S such that , O f*ø = id:S -! S. Using the alternative defining property (4.3) of C, we can obtain alternative d* *escrip- tions of these maps that avoid reference to the functor f* we seek to understan* *d. Lemma 4.5. The maps ø and , coincide with the maps T ~=DT _Doe_//Df!f*T ~=Df!S ~=f!C and f*f!C ~=f*Df!S ~=Df*f!S_Di__//DS ~=S. Proof.The isomorphism Df!S ~=f!C used in the displays is the composite of the given isomorphism (4.1) and the special case (4.2) of the isomorphism ~fl. The * *proofs are diagram chases that use the naturality of j and ", the triangular identitie* *s for the (f!, f*) adjunction, and the description of ~flin Proposition 2.11. Using the isomorphism (2.7), we extend ø to the natural map (4.6) ø :Y ~=Y T_id_ø//_Y f!C ~=f!(f*Y C) = f]f*Y. Specializing to Y = f*X, we obtain the desired comparison map ! as the composite f]" (4.7) ! :f*X__ø__//f]f*f*X___//_f]X. An easy diagram chase using the triangular identity Ö f*j = idshows that (4.8) ! O j = ø :Y -! f]f*Y. If ! is an isomorphism, then ø must be the unit of the resulting (f*, f]) adjun* *ction. Similarly, using (1.1) and (2.7), we extend , to the natural map (4.9) , :f*f]f*Y = f*f!(f*Y C) ~=f*Y f*f!Cid_,//_f*Y S ~=f*Y. We view , as a partial counit, defined not for all X but only for X = f*Y . Sin* *ce , O f*ø = id:S -! S, it is immediate that (4.10) , O f*ø = id:f*Y -! f*Y, which is one of the triangular identities for the desired (f*, f]) adjunction. * *Define (4.11) _ :f]f*Y -! f*f*Y to be the adjoint of ,. The adjoint of the relation (4.10) is the analogue of (* *4.8): (4.12) _ O ø = j :Y -! f*f*Y. Proposition 4.13. If Y or f!S is dualizable, then ! : f*f*Y -! f]f*Y is an iso- morphism with inverse _. If _ is an isomorphism for all Y , then f!S is dualiza* *ble. If X is a retract of some f*Y , where Y or f!S is dualizable, then ! : f*X -! f* *]X is an isomorphism. 12 H. FAUSK, P. HU, AND J.P. MAY Proof.With X = f*Y , the first part of the proof of the following result gives * *that _ O ! = id, so that ! = _-1 when _ is an isomorphism. We claim that _ coincides with the following composite: f]f*Y = f!(f*Y C) ~=Y D(f!S) -!Hom (f!S, Y ) ~=f*Hom (S, f*Y ) = f*Y. Here the isomorphisms are given by (2.7) and (4.3) and by (2.5). Since is an isomorphism if Y or f!S is dualizable, the claim implies the first statement. N* *ote that _ = f*, O j and that the isomorphism ~flof (2.5) is f*Hom (i, id) O f*ff O* * j. Using the naturality of j and the description of , in Lemma 4.5, an easy, if le* *ngthy, diagram chase shows that the diagram (1.8) gives just what is needed to check the claim. The second statement is now clear by the definition of dualizabilit* *y: it suffices to consider Y = f!S. The last statement follows from the first sinc* *e a retract of an isomorphism is an isomorphism. We extract a criterion for ! to be an isomorphism for a general object X from the usual proof of the uniqueness of adjoint functors [17, p. 85]. Proposition 4.14. If there is a map , :f*f]X = f*f!(X C) -! X such that (4.15) f], O ø = id:f]X -! f]X and the following (partial naturality) diagram commutes, then ! :f*X -! f]X is an isomorphism with inverse the adjoint _ of ,. (4.16) f*f]f*f*X __,__//f*f*X f*f]"|| |"| fflffl| fflffl| f*f]X ___,_____//X Moreover, (4.15) holds if and only if the following diagram commutes. (4.17) X C ______i_____//f*f!(XOO C) i|| f*f!(,|id)| fflffl| | f*f!(X C)_f*ø//_f*f!(f*f!(X C) C) Proof.In the diagram (4.16), the top map , is given by (4.9). The diagram and the relation , O f*ø = id of (4.10) easily imply the relation , O f*! = ", which is complementary to the defining relation Ö f*_ = , for the adjoint _. Passage to adjoints gives that _ O ! = id. The following diagram commutes by (4.8), the triangular identity f*Ö j = id, the naturality of j and !, and the fact that * *_ is adjoint to ,. It gives that ! O _ = f], O ø = id. ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS 13 f]X|_______________ø_______________//_If]f*f]X| | IIIjI rrrrrr|rrr | II rrrrrr | || II$$ ! rrrrr || | f*f*f]X_____//f]f*f]X | | | _ || f*f*_|| |f]f*_| |f],| | fflffl|! fflffl| | | f*f*f*X ____//_f]f*f*X | | u::u LLL | | j uu | LLf]"L | | uuu |f*" LLL | fflffl|uu fflffl| L&&fflffl| f*X ________f*X __________!__________//_f]X. The last statement is clear by adjunction. Remark 4.18. The map ! can be generalized to the Verdier-Grothendieck context. For that, we assume given an object W of C such that f!C ~=Df!f!T ; compare (4.3). As in Lemma 4.5, we then have the map ø :T ~=DT _Doe//_Df!f!T ~=f!C. This allows us to define the comparison map f!(" id) ! :f*X ~=f*X T_id_ø//_f*X f!W ~=f!(f*f*X _C)__//f!(X C. A study of when this map ! is an isomorphism might be of interest, but we have no applications in mind. We illustrate the idea in the context of Example 3.10. Example 4.19. Returning to Example 3.10, we seek an object C0 of C such that f!C0~=D(f!f!S), which is C0 C ~=D(DC C). If C is dualizable, then the right side is isomorphic to C DC ~=DC C and we can take C0= DC. Here the map ! :X = f*X -! f!(X DC) = X DC C turns out to be id (fl O coev), where coev :S -! C DC is the coevaluation map of the duality adjunction (1.9) and fl is the commutativity isomorphism for . * *We conclude (e.g., by [18, 2.9]) that ! is an isomorphism if and only if C is inve* *rtible. 5. Preliminaries on triangulated categories We now go beyond the hypotheses of xx1-4 to the triangulated category situa- tions that arise in practice. We assume that C and D are triangulated and that the functors (-) X and f* are exact (or triangulated). This means that they a* *re additive, commute up to isomorphism with , and preserve distinguished triangle* *s. For (-) X, this is a small part of the appropriate compatibility conditions t* *hat relate distinguished triangles to and Hom in well-behaved triangulated closed symmetric monoidal categories; see [19] for a discussion of this, as well as fo* *r basic observations about what triangulated categories really are: the standard axiom * *sys- tem is redundant and unnecessarily obscure. We record the following easily prov* *en observation relating adjoints to exactness (see for example [21, 3.9]). 14 H. FAUSK, P. HU, AND J.P. MAY Lemma 5.1. Let F :A -! B and G : B -! A be left and right adjoint functors between triangulated categories. Then F is exact if and only if G is exact. We also record the following definitions (see for example [12, 22]). Definition 5.2. A full subcategory B of a triangulated category C is thick if a* *ny retract of an object of B is in B and if the third object of a distinguished tr* *iangle with two objects in B is also in B. The category B is localizing if it is thick* * and closed under coproducts. The smallest thick (respectively, localizing) subcateg* *ory of C that contains a set of objects G is called the thick (respectively, locali* *zing) subcategory generated by G . Definition 5.3. An object X of an additive category A is compact, or small, if * *the functor A (X, -) converts coproducts to direct sums. The category A is compactly generated if it has arbitrary coproducts and has a set G of compact objects that detects isomorphisms, in the sense that a map f in A is an isomorphism if and only if A (X, f) is an isomorphism for all X 2 G . When A is symmetric monoidal, we require its unit object to be compact; thus it can be included in the set G . In the triangulated case, this is equivalent to Neeman's definition [22, 1.7]* *. With our version, we have the following generalization of a result of his [22, 5.1]. Lemma 5.4. Let A be a compactly generated additive category with generating set G and let B be any additive category. Let F :A -! B be an additive functor with right adjoint G. If G preserves coproducts, then F preserves compact objec* *ts. Conversely, if F (X) is compact for X 2 G , then G preserves coproducts. Proof.Let X 2 A and let {Yi} be a set of objects of B. Then the evident map f : qG(Yi) -! G(qYi) induces a map f*: A (X, qG(Yi)) -! A (X, G(qYi)). If X is compact and f* is an isomorphism, then, by adjunction and compactness, it induces an isomorphism qB(F (X), Yi) -! B(F (X), qYi), which shows that F (X) is compact. Conversely, if X and F (X) are both compact, then f* corresponds under adjunction to the identity map of qB(F (X), Yi) and is therefore an isomorphism. Restricting to X 2 G , it follows from Definition 5.3* * that f is an isomorphism. While this result is elementary, it is fundamental to the applications. We ge* *n- erally have much better understanding of left adjoints, so that the compactness criterion is verifiable, but it is the preservation of coproducts by right adjo* *ints that is required in all of the formal proofs. Returning to triangulated categories, we justify the term "generating set" by* * the following result. Its first part is [22, 3.2], and its second part is [12, 2.1.* *3(d)]. Proposition 5.5. Let A be a compactly generated triangulated category with gen- erating set G . Then the localizing subcategory generated by G is A itself. * *If the objects of G are dualizable, then the thick subcategory generated by G is the f* *ull subcategory of dualizable objects in A , and an object is dualizable if and onl* *y if it is compact. The following standard observation works in tandem with the previous result. ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS 15 Proposition 5.6. Let F, F 0:A -! B be exact functors between triangulated cate- gories and let OE: F -! F 0be a natural transformation that commutes with . Th* *en the full subcategory of A whose objects are those X for which OE is an isomorph* *ism is thick, and it is localizing if F and F 0preserve coproducts. Proof.Since a retract of an isomorphism is an isomorphism, closure under retrac* *ts is clear. Closure under triangles is immediate from the five lemma. A coproduct of isomorphisms is an isomorphism, so closure under coproducts holds when F and F 0preserve coproducts. 6. The formal isomorphism theorems We assume throughout this section that C and D are closed symmetric monoidal categories with compatible triangulations and that (f*, f*) is an adjoint pair * *of functors with f* strong symmetric monoidal and exact. For the Wirthmüller context, we assume in addition that f* has a left adjoint f!. The maps (2.7)-(2.6) are then given by (1.3) and Proposition 2.11. When ~ß:f!(f*Y X) -! Y f!X is an isomorphism, the map ! : f*X -! f!(X C) is defined. Observe that ~ßis a map between exact left adjoints and that ~ßand ! commute with . The results of the previous section give the following conclusi* *on. Theorem 6.1 (Formal Wirthmüller isomorphism). Let C be compactly generated with a generating set G such that ~ßand ! are isomorphisms for X 2 G . Then ~ßis an isomorphism for all X 2 C . If the objects of G are dualizable, then ! * *is an isomorphism for all dualizable X. If f*X is compact for X 2 G , then ! is an isomorphism for all X 2 C . The force of the theorem is that no construction of an inverse to ! is requir* *ed: we need only check that ! is an isomorphism one generating object at a time. Proposition 4.14 explains what is needed for that verification. For the Grothendieck context, we can use the following basic results of Neeman [22, 3.1, 4.1] to construct the required right adjoint f! to f* in favorable ca* *ses. A main point of Neeman's later monograph [23] and of Franke's paper [9] is to rep* *lace compact generation by a weaker notion that makes use of cardinality considerati* *ons familiar from the theory of Bousfield localization in algebraic topology. Theorem 6.2 (Triangulated Brown representability theorem). Let A be a com- pactly generated triangulated category. A functor H :A op-! A b that takes dis- tinguished triangles to long exact sequences and converts coproducts to product* *s is representable. Theorem 6.3 (Triangulated adjoint functor theorem). Let A be a compactly gen- erated triangulated category and B be any triangulated category. An exact funct* *or F :A -! B that preserves coproducts has a right adjoint G. Proof.Take G(Y ) to be the object that represents the functor B(F (-), Y ). The map ß :Y f*X -! f*(f*Y X) 16 H. FAUSK, P. HU, AND J.P. MAY of (1.6) commutes with . When ß is an isomorphism, OE: f*Y f!Z -! f!(Y Z) is defined and commutes with . We obtain the following conclusion. Theorem 6.4 (Formal Grothendieck isomorphism). Let D be compactly generated with a generating set G such that f*Y is compact and ß is an isomorphism for Y 2 G . Then f* has a right adjoint f!, ß is an isomorphism for all Y 2 D, and * *OE is an isomorphism for all dualizable Y . If the functor f! preserves coproducts* *, then OE is an isomorphism for all Y 2 D. Proof.As a right adjoint of an exact functor, f* is exact by Lemma 5.1, and it preserves coproducts by Lemma 5.4. Thus f! exists by Theorem 6.3. Now ß is an isomorphism for all Y by Proposition 5.6, OE is an isomorphism for dualizable Y* * by Proposition 3.9, and the last statement holds by Propositions 5.5 and 5.6. When f! is obtained abstractly from Brown representability, the only sensible way to check that it preserves coproducts is to appeal to Lemma 5.4, requiring C to be compactly generated and f*X to be compact when X is in the generating set. For the Verdier-Grothendieck context, we assume that we have a second adjunc- tion (f!, f!), with f!exact. We also assume given a map ^ß:Y f!X ~=f!(f*Y X) that commutes with . When ^ßis an isomorphism, the map OE: f*Y f!Z -! f!(Y Z) is defined and commutes with . Using Proposition 3.9 and the results of the previous section, we obtain the following conclusion. Theorem 6.5 (Formal Verdier isomorphism). Let D be compactly generated with a generating set G such that f*Y is compact and ^ßis an isomorphism for Y 2 G . Then ^ßis an isomorphism for all Y 2 D, and OE is an isomorphism for all dualiz* *able Y . If the functor f! preserves coproducts, then OE is an isomorphism for all Y* * 2 D. Remark 6.6. In many cases, one can construct a more explicit right adjoint f!0f* *rom some subcategory D0 of D to some subcategory C0 of C , as in Remark 3.6. In such cases we can combine approaches. Indeed, assume that we have an adjoint pair (f!, f!0) on full subcategories C0 and D0 such that objects isomorphic to * *objects in C0 (or D0) are in C0 (or D0). Then, by the uniqueness of adjoints, the right adjoint f! to f!given by Brown representability restricts on D0 to a functor wi* *th values in C0 that is isomorphic to the explicitly constructed functor f!0. That* * is, the right adjoint given by Brown representability can be viewed as an extension* * of the functor f!0to all of D. This allows quotation of Proposition 2.4 or 2.9 for* * the construction and comparison of the natural maps (2.3)-(2.2). We give an elementary example and then some remarks on the proofs of the results that we have quoted from the literature, none of which are difficult. Example 6.7. Return to Example 3.10, but assume further that C is a compactly generated triangulated category. Here the formal Verdier duality theorem says t* *hat OE = : Y Hom (C, Z) -! Hom (C, Y Z) is an isomorphism if and only if the functor Hom (C, -) preserves coproducts. That is, an object C is dualizable if * *and only if Hom (C, -) preserves coproducts. ISOMORPHISMS BETWEEN LEFT AND RIGHT ADJOINTS 17 Remark 6.8. Clearly Theorem 6.3 is a direct consequence of Theorem 6.2. In turn, Theorem 6.2 is essentially a special case of Brown's original categorical repre* *senta- tion theorem [3]. Neeman's self-contained proof closely parallels Brown's argum* *ent. The first statement of Proposition 5.5 is used as a lemma in the proof, but it * *is also a special case. To see this, let B be the localizing subcategory of A generated* * by G . Then, applied to the functor A (-, X) on B for X 2 A , the representability theorem gives an object Y 2 B and an isomorphism Y ~= X in A . 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