Lecture 7 Notes: Categories and Morita Equivalence

Lecture 7: Categories and Morita Equivalence 7 March 2 - Categories and Morita equivalence  Remark 7.1: We can also discuss projective covers of graded modules over graded rings. Let 𝑅 = 𝑛⩾0 𝑅𝑛 with 𝑅0 Artinian and let 𝐿 be an irreducible graded module over 𝑅 that is concentrated in one degree. WLOG we can assume 𝐿 is concentrated in degree 0. Then 𝑃 = 𝑅𝑒𝐿 is a graded projective cover of 𝐿; 𝑒𝐿 ∈ 𝑅0 is the idempotent corresponding to the projective cover of 𝐿 as an 𝑅0 -module. 7.1 Morita equivalence Definition 7.2: We say that two rings are Morita equivalent if their categories of modules are equivalent. (Below, we will recall some facts about categories.) Theorem 7.3: A ring 𝑆 is Morita equivalent to a ring 𝑅 iff 𝑆 = End𝑅 (𝑃 ) op where 𝑃 is a finitely generated projective generator of the category of 𝑅-modules. Definition 7.4: A projective module 𝑃 over a ring 𝑅 is a projective generator if Hom(𝑃, 𝑀 ) ≠ 0 for every nonzero 𝑅-module 𝑀 . Lemma 7.5: 𝑀 is a generator iff 𝑅 is a direct summand in 𝑀 𝑛 for some 𝑛. Proof. If 𝑀 is a generator, then for every module 𝑁 , the images of all possible homomorphisms 𝑀 → 𝑁 generate 𝑁 . This is because if 𝑆 is the sum of all the images of such maps, then Hom(𝑀, 𝑆 ) → Hom(𝑀, 𝑁 ) is an isomorphism, and since 𝑀 is a generator, this implies that 𝑆  𝑁 .  Now if 𝑁 is finitely generated, say with generators 𝑛𝑖 , and 𝑛𝑖 = 𝑓𝑖 𝑗 (𝑚 𝑗 ) where 𝑓𝑖 𝑗 ∈ Hom(𝑀, 𝑁 ), then only images for those finitely many 𝑓𝑖 𝑗 are needed to generate 𝑁 . Hence there is a surjection 𝑀 𝑛 ↠ 𝑁 . In particular, if we take 𝑁 = 𝑅, 𝑅 is projective, so the surjection splits and 𝑅 is a summand of 𝑀 𝑛 . In the other direction, if 𝑅 is a summand of 𝑀 𝑛 , this implies 𝑀 𝑛 is a generator, and hence 𝑀 is a generator also. □ Example 7.6: 𝑅 is Morita equivalent to itself. In this case, take 𝑃 = 𝑅 (the rank 1 free module), and 𝑅 = End𝑅 (𝑅) op . More generally, if we take 𝑃 = 𝑅𝑛 , then 𝑆 = End𝑅 (𝑅𝑛 ) op = Mat𝑛 (𝑅) is Morita equivalent to 𝑅 also. 𝑃𝑖𝑚𝑖 is a Using the lemma, we see that if 𝑅 is Artinian with indecomposable projectives 𝑃1, . . . , 𝑃𝑛 , 𝑃 = op projective generator iff 𝑚𝑖 ⩾ 1 for all 𝑖 . In particular, if we take 𝑚𝑖 = 1 for all 𝑖 , then 𝑆 = End𝑅 (𝑃 ) is what’s known as a based ring, meaning that each irreducible 𝐿𝑖 is a one-dimensional vector space over 𝐷𝑖 = End𝑅 (𝐿𝑖 ). Proposition 7.7: Let 𝑃 = 𝑅𝑒 for an idempotent 𝑒 ∈ 𝑅. Then 𝑃 is a generator iff 𝑅 = 𝑅𝑒 𝑅.  Proof. Suppose 𝑅 = 𝑅𝑒𝑅. Then we can write 1 = 𝑎𝑖 𝑒𝑏𝑖 for 𝑎𝑖 , 𝑏𝑖 ∈ 𝑅, so the map 𝑃 𝑛 → 𝑅 given by (𝑥 1, . . . , 𝑥𝑛 ) → ↦  𝑥𝑖 𝑏𝑖 is onto. So by the lemma 7.5, 𝑃 is a generator. In the other direction, 𝑀 = 𝑅/𝑅𝑒𝑅 satisfies Hom(𝑃 , 𝑀 ) = 𝑒 𝑀 = 0, so if 𝑀 ≠ 0, 𝑃 can’t be a generator. □ 7.2 Categories and the Yoneda Lemma Quick review: a (small) category C consists of a set of objects Ob(C), a set of morphisms Hom C (𝑋 , 𝑌 ) for all 𝑋 , 𝑌 ∈ Ob(C), an identity morphism id𝑋 ∈ Hom(𝑋 , 𝑋 ), and an associative composition operation. Remark 7.8: Small categories are those where Ob(C) is actually a set. Since there is no such thing as the “set of all sets”, categories like Set or 𝑅-Mod are not small. We could get around this by fixing a universe and only considering sets from this universe. We could also consider “large” categories, whose objects form a collection more general than a set, called a class. We will ignore all these set-theoretic issues. Given two categories C1, C2 , we can talk about the category of functors Fun(C1, C2 ) whose objects are functors and whose morphisms are natural transformations. Definition 7.9: A functor 𝐹 is faithful if the map Hom(𝑋 , 𝑌 ) → Hom(𝐹 (𝑋 ), 𝐹 (𝑌 )) is injective for all 𝑋 , 𝑌 . Definition 7.10: A functor 𝐹 is fully faithful if the map Hom(𝑋 , 𝑌 ) → Hom(𝐹 (𝑋 ), 𝐹 (𝑌 )) is an isomorphism. Definition 7.11: A functor 𝐹 is essentially surjective if it is surjective on isomorphism classes of objects. Definition 7.12: A functor 𝐹 : C1 → C2 is an equivalence of categories if there exists 𝐺 : C2 → C1 such that 𝐹 ◦ 𝐺 , 𝐺 ◦ 𝐹 are isomorphic to the respective identity functors (that is, they are naturally equivalent to the identity functors). Lemma 7.13: A functor 𝐹 is an equivalence of categories iff it is fully faithful and essentially surjective. Proof. Since we are ignoring set-theoretic considerations, we get to use the axiom of choice. It’s clear that if 𝐹 is an equivalence, then it’s fully faithful and essentially surjective. In the other direction, if 𝐹 is essentially surjective, the axiom of choice allows us to choose 𝑋 ∈ Ob(C2 ) and 𝐺 (𝑋 ) ∈ Ob(C1 ) such that 𝑖𝑋 : 𝑋  𝐹 (𝐺 (𝑋 )). Then we can define 𝐺 (𝑓 : 𝑋 → 𝑌 ) as follows: first 𝑖𝑌−1 ◦ 𝑓 ◦ 𝑖𝑋 gives a map 𝐹 (𝐺 (𝑋 )) → 𝐹 (𝐺 (𝑌 )), and because 𝐹 is fully faithful, this corresponds to a unique 𝐺 (𝑓 ) : 𝐺 (𝑋 ) → 𝐺 (𝑌 ). Then one can verify that 𝐺 is indeed a functor and □ that 𝐹 ◦ 𝐺 and 𝐺 ◦ 𝐹 are equivalent to id C𝑖 . Lemma 7.14 (Yoneda Lemma): For a category C, consider the functors 𝑅 : C op → Fun(C, Set) and 𝐶 : C → Fun(C op, Set) where 𝑅 (𝑋 ) : 𝑇 ↦→ Hom(𝑋 , 𝑇 ) and 𝐶 (𝑋 ) : 𝑇 → Hom(𝑇 , 𝑋 ). Then 𝑅, 𝐶 are fully faithful. Here 𝑅 is for “represent” and 𝐶 for “corepresent”. Proof (Sketch). For 𝑋 , 𝑌 ∈ Ob(C), there’s a natural map Hom(𝑋 , 𝑌 ) → Hom(𝑅 (𝑋 ), 𝑅 (𝑌 )) given by composing with the map 𝑋 → 𝑌 . In the other direction, given 𝜑 : 𝑅 (𝑋 ) → 𝑅 (𝑌 ), send it to the element 𝜑 (id𝑋 ) ∈ Hom(𝑋 , 𝑌 ). It’s easy to see these are inverse bijections. The argument for 𝐶 is similar. □ That is, an object in C is uniquely defined up to unique isomorphism up to the functor it (co)represents. Example 7.15: The initial (resp. final) object of a category C is an object 𝐼 (resp. 𝐹 ) such that Hom(𝐼 , 𝑋 ) (resp. Hom(𝑋 , 𝐹 )) is a singleton. By the Yoneda lemma, initial and final objects are unique up to unique isomorphism (if they exist). For example, in the category 𝑅-Mod, the zero module is both initial and final. Definition 7.16:The coproduct representing (resp. corepresenting) the product of  (resp. product) is the object   Hom sets: Hom ( 𝑋𝑖 , 𝑇 ) = Hom(𝑋𝑖 , 𝑇 ) and Hom (𝑇 , 𝑋𝑖 ) = Hom(𝑇 , 𝑋𝑖 ). These are unique up to unique isomorphism if they exist. Example 7.17: In 𝑅-Mod, these both exist; coproduct is the direct sum and product is the usual product. Remark 7.18: We can characterize the statement that a finite  product  in  direct sum is the same as a finite categorical terms. Using the final object 0, there is a morphism 𝑋𝑖 → 𝑋𝑖 . Hence, there is a map 𝑋𝑖 → 𝑋𝑖 , and this is an isomorphism when the 𝑋𝑖 form a finite collection. Remark 7.19: This can also be used to show that Hom(𝑀, 𝑁 ) has an abelian group structure. You can define the sum of two maps 𝑓 , 𝑔 : 𝑀 → 𝑁 as the composition 𝑓 ×𝑔 id𝑁 ⨿ id𝑁 𝑀 −−−→ 𝑁 × 𝑁  𝑁 ⨿ 𝑁 −−−−−−−→ 𝑁 . 7.3 Proof of Morita equivalence theorem Proof (of Theorem 7.3). Suppose 𝐹 : 𝑆 -Mod → 𝑅-Mod is an equivalence. We will show that 𝑃 := 𝐹 (𝑆 ) is a finitely generated projective generator in 𝑅-Mod and that 𝑆 = End𝑆 (𝑆 ) op = End𝑅 (𝑃 ) op . This follows from the following observations: • 𝐹 sends projective 𝑆 -modules to projective 𝑅-modules. 𝑀 is projective iff Hom(𝑀, −) is exact, i.e. sends a surjective map of modules to a surjective map of sets. A map of modules 𝑇1 → 𝑇2 is surjective iff Hom(𝑇2, 𝑋 ) ↩→ Hom(𝑇1, 𝑋 ) is injective for all 𝑋 . Using essential surjectivity of 𝐹 , we find 𝑁 1, 𝑁 2, 𝑌 ∈ 𝑆 -Mod such that 𝐹 (𝑁𝑖 )  𝑇𝑖 and 𝐹 (𝑋 )  𝑌 ; then the full faithfulness of 𝐹 implies that 𝑁 1 ↠ 𝑁 2 . Then Hom(𝑀, 𝑁 1 ) ↠ Hom(𝑀, 𝑁 2 ) combined with full faithfulness of 𝐹 translates this into Hom(𝐹 (𝑀 ), 𝑇1 ) ↠ Hom(𝐹 (𝑀 ), 𝑇2 ). • 𝐹 sends a projective generator to a projective generator, since Hom(𝑀, 𝑁 ) = 0 ⇔ Hom(𝐹 (𝑀 ), 𝐹 (𝑁 )) = 0 by full faithfulness of 𝐹 . • 𝐹 sends finitely generated projective 𝑆 -modules to finitely generated projective 𝑅-modules. Use the following characterization of finitely generated projectives: a projective  𝑃 is finitely generated iff Hom(𝑃 , −) commutes  with arbitrary coproducts (i.e. Hom(𝑃 , 𝑋𝑖 ) = Hom (𝑃, 𝑋𝑖 ). If 𝑃 is projective and finitely generated, it’s a direct summand of 𝑆 𝑛 , which has this property, so 𝑃 also has this property. In the other direction, suppose  Hom(𝑃, −) commutes with coproducts. We is the direct summand of some free module, say 𝐼 𝑆,  know 𝑃 Hom(𝑃 , 𝑆 ), so the image of 𝑃 ↩→ 𝑆 must land in which then splits as 𝑃 ⊕  𝑄 . Then Hom(𝑃, 𝐼 𝑆 ) = 𝐼 𝐼 𝑛 will also split as 𝑃 ⊕ (𝑄 ∩ 𝑆 𝑛 ), so 𝑃 is in fact finitely generated. 𝑆 , |𝐽 | < ∞. 𝑆 a finite direct sum 𝑆 𝑛 = 𝐽 Since 𝐹 is an equivalence of categories, it preserves the property that Hom(𝐹 (𝑃 ), −) commutes with arbitrary coproducts, so 𝐹 (𝑃 ) is also finitely generated projective. Combining these three, we get that 𝐹 (𝑆 ) is a finitely generated projective generator. Because 𝐹 is fully faithful, Hom𝑆 (𝑆 , 𝑆 )  Hom𝑅 (𝐹 (𝑆 ), 𝐹 (𝑆 )) = End𝑅 (𝑃 ), so 𝑆 = End𝑅 (𝑃 ) op . In the other direction, we want to show that if 𝑆 = End𝑅 (𝑃 ) op for 𝑃 a finitely generated projective generator 𝑃 of 𝑅-Mod, the functor 𝐹𝑃 : 𝑀 ↦→ Hom𝑅 (𝑃 , 𝑀 ) is the desired equivalence of categories. Here 𝑀 ∈ 𝑅-Mod and Hom𝑅 (𝑃 , 𝑀 ) has an 𝑆 -action via composition. 𝐹𝑃 induces an isomorphism Hom𝑅 (𝑃 , 𝑁 )  Hom𝑆 (𝐹𝑃 (𝑃 ), 𝐹𝑃 (𝑁 )) for all 𝑁 : the RHS will be Hom𝑆 (𝑆, 𝐹𝑃 (𝑁 ))  𝐹𝑃 (𝑁 ) = Hom(𝑃, 𝑁 ). This isomorphism coincides with the 𝐹𝑃 -action on morphisms. Since 𝑃 is finitely generated and projective, 𝐹𝑃 commutes with coproducts. Moreover, 𝑃 is a projective generator, we claim we can find an exact sequence 𝑃 ⊕𝐽 → 𝑃 ⊕𝐼 → 𝑀 → 0. Lemma 7.20: A projective module 𝑃 is a generator iff the free module 𝑅 is a direct summand in 𝑃 𝑛 for some 𝑛 iff every module is a quotient of 𝑃 ⊕𝐼 . Now we want to show that Hom(𝑀, 𝑁 ) → Hom(𝐹𝑃 (𝑀 ), 𝐹𝑃 (𝑁 )) is an isomorphism. Notice that if this is true for 𝑀1, 𝑀2 , it’s also true for coker(𝑓 ), 𝑓 : 𝑀1 → 𝑀2 because exactness of 𝐹𝑃 implies that both Hom-spaces are the kernel of the map Hom(𝑀2, 𝑁 ) → Hom(𝑀1, 𝑁 ). So by the above, it suffices to show that this is true for 𝑀 = 𝑃 ⊕𝐼 , but that is what we proved above. So 𝐹𝑃 is fully faithful. 𝑓 To see that 𝐹𝑃 is essentially surjective, take 𝑁 ∈ 𝑆 -Mod, which fits in an exact sequence 𝑆 ⊕𝐽 → − 𝑆 ⊕𝐼 → 𝑁 → 0. ⊕𝐽 ⊕𝐼 Because 𝐹𝑃 is fully faithful, 𝑓 = 𝐹𝑃 (𝑔) for 𝑔 : 𝑃 → 𝑃 . Hence 𝑁  𝐹𝑃 (coker(𝑔)). Thus, 𝐹𝑃 is an equivalence of categories. □ Example 7.21: Now it’s interesting to consider notions that are invariant under Morita equivalence. We will see that the center 𝑍 (𝑅) and cocenter 𝐶 (𝑅) of a ring are such notions, i.e. if 𝑅, 𝑆 are Morita equivalent, they have the same center and the same cocenter.