On the Semi-Abelianness of Affine Group Schemes

On the Semi-Ab elianness of Affine Group Sc hemes Da vid F orsman Univ ersité catholique de Louv ain david.forsman@uclouvain.be 24 F ebruary , 2026 Abstract W e pro ve that the category of comm utative Hopf algebras ov er a field k is co- semi-ab elian. Consequen tly , the category of affine group k -schemes is semi-ab elian. W e establish coregularit y b y iden tifying the orthogonal factorization system of sur- jections and faithfully flat injections, and w e deduce co exactness from T ak euchi’s corresp ondence b et ween normal Hopf ideals and Hopf subalgebras of commutativ e Hopf k -algebras. 1 In tro duction It is a w ell-established result that the category of co comm utativ e Hopf k -algebras o ver an arbitrary field k forms a semi-ab elian category [ 5 , 18 ], first prov en for zero character- istic fields [ 4 ]. These results hav e motiv ated a broader in vestigation into the categorical prop erties of different Hopf structures; for instance, the semi-ab elian condition has also b een successfully analyzed for certain categories of co comm utative color Hopf algebras [ 15 ]. The pro of that co comm utativ e Hopf k -algebras form a semi-ab elian category heav- ily relies on Newman’s bijective correspondence b et w een the Hopf subalgebras and the bi-ideals of co comm utativ e Hopf k -algebras [ 12 ]. Our approac h for the dual problem relies on T ak euchi’s results ab out injectiv e morphisms of comm utative Hopf k -algebras being faithfully flat and T akeuc hi’s corresp ondence, whic h characterizes normal Hopf-ideals syn- tactically among Hopf-ideals [ 16 ]. In tro duced by Janelidze, Márki, and Tholen, semi-ab elian categories provide a unified framew ork for homological algebra in non-ab elian settings, p ossessing enough structure to supp ort k ey lemmas and constructions in homological algebra [ 6 ]. The established semi-ab elian nature of co commutativ e Hopf k -algebras naturally raises the dual question: what is the categorical nature of comm utativ e Hopf algebras? Coprotomo dularit y of comm utative Hopf algebras w as noted in the talk [ 17 ] related to the pap er by García- Martínez and V an der Linden [ 3 ]. Homological self-duality of commutativ e Hopf algebras w as sho wn in [ 13 ]. W e answ er this question by proving that the category Hopf com k of comm utativ e Hopf algebras o ver a field k is co-semi-ab elian , meaning its opposite category of affine group sc hemes ov er k (see [ 10 ]) is a semi-ab elian category . F urthermore, since T akeuc hi’s results extend to the context of super Hopf algebras [ 9 ], the same pro of demonstrates that the category of comm utative sup er Hopf algebras o ver a field k ( char( k )  = 2 ) also forms a lo cally presen table co-semi-ab elian category . The pap er is structured as follo ws. Section 2 cov ers the necessary categorical and algebraic preliminaries. In Section 3, we pro ve the main theorem for commutativ e Hopf algebras ov er a field. 1 2 Preliminaries W e remind the reader of the definition of semi-ab elian categories and note ho w exactness can b e seen as a stabilit y prop ert y of normal sub objects (kernels) in a homological setting. Definition 2.1. W e say that a category C is: 1. P ointed if it has a zero-ob ject, an ob ject that is b oth initial and terminal. 2. Regular if it has finite limits and every morphism f factors f = me , where m is a monomorphism and e is a stably extremal epimorphism. 1 2 3. Exact (in the sense of Barr) if it is regular and every in ternal equiv alence relation is a k ernel pair of some morphism. 4. Protomo dular if pullbacks along retractions exist and for every retraction r : x → y in C with a section s and a pullback k : a → x along r , the pair ( s, k ) is jointly extremal epimorphic. a b x y k ⌟ r 5. Semi-ab elian if it is p ointed, protomodular, exact, and has binary copro ducts. The category C is co-semi-ab elian if its dual C op is semi-ab elian. 6. Lo cally presen table , if C is equiv alent to a reflexive sub category of a pre-sheaf category [ K , Set ] closed under large enough directed colimits for some small category K [ 1 ]. As ab elian categories are the same as exact additiv e ones [ 2 ], semi-ab elian categories can b e viewed as ab elian categories where additivity is weak ened to p ointed protomodu- larit y with binary copro ducts. Protomo dularit y can be reflected b et ween categories via a jointly conserv ativ e family ( F i : C → D i ) i of pullback preserving functors. This allows an easy w a y to v erify that the category Grp( C ) of in ternal groups of a finitely com- plete category C is protomo dular via the functors Grp( C ) → Grp( Set ) induced by the represen tables. Lemma 2.2. L et C b e a p ointe d, r e gular, and pr otomo dular c ate gory. Then the fol lowing ar e e quivalent: 1. The c ate gory C is exact. 2. R e gular epis map kernels to kernels: F or every c ommutative squar e a b x y f ′ k m f wher e f and f ′ ar e r e gular epis, k a kernel and m a monomorphism, m is a kernel. 1 A family ( f i : c i → d ) i ∈ I of morphisms is join tly extremal epimorphic, if a factorization f i = me i , where m is a fixed monomorphism for all i ∈ I , guaran tees that m is an isomorphism. A morphism is an extremal epimorphism if its singleton family is jointly extremal epimorphic. A stably extremal epimorphism is a morphism whose all pullbacks are extremal epimorphisms. 2 Extremal epimorphisms in regular categories are regular by [ 7 , Prop osition 2.2 (b)]. 2 3. F or every r eflexive r elation r 1 R ⇒ x r 2 in C , the morphism r 1 k 2 is a kernel, wher e k 2 is the kernel of r 2 . Pr o of. Pro of is essentially giv en in [ 6 , pp. 381–382]. 3 Prop erties of Comm utativ e Hopf Algebras Consider a symmetric monoidal category A and consider the category Mon com A of commu- tativ e monoids in A with resp ect to the tensor pro duct. Notice how the tensor pro duct de- fines the copro duct in Mon com A . The category Hopf com A of comm utative Hopf A -algebras is defined as the category Grp(( Mon com A ) op ) op of internal cogroup ob jects in Mon com A . As a shorthand, w e write Hopf com R for the category Hopf com Mo d R of Hopf R -algebras for a comm utative ring R . It is known that ( Hopf com R ) op is equiv alen t to the category of affine group schemes o ver a commutativ e ring R [ 10 ]. Prop osition 3.1. L et A b e a lo c al ly pr esentable symmetric monoidal c ate gory. A ssume that the tensor pr o duct pr eserves dir e cte d c olimits c omp onentwise. Then the c ate gory of Hopf com A of c ommutative Hopf A -algebr as is a lo c al ly pr esentable, p ointe d, and pr otomo d- ular c ate gory. Pr o of. Lo cal presentabilit y of Mon com A is sho wn in [ 14 ]. As coalgebraic structures within a lo cally presen table category form a lo cally presentable category b y [ 1 , Remark 2.63], Hopf com A is lo cally presen table. Since the representable functors induce a jointly conserv a- tiv e family of con tinuous functors Grp( C ) → Grp( Set ) for any finitely complete category C , it follows that Hopf com A is p oin ted coprotomo dular. 3.1 Coregularit y Coregularit y is pro ven in t wo steps. First, we lift the orthogonal factorization system of surjections and injections from Mon com k to Hopf com k , using the fact that injectiv e mor- phisms in Mon com k are closed under tensoring. The lifting of suc h factorization systems along monadic functors resp ecting the left class is established in [ 19 , Proposition 3.7], where it is sho wn that the prop erness assumption from Lin ton’s original result [ 8 ] is not necessary . With a sligh t mo dification, this pro of extends to algebras in any finitely complete category , provided the left class is closed under products. Second, w e use the result by T ak euc hi, which states that injective morphisms in Hopf com k are faithfully flat, ensuring that injections are pushout-stable extremal monomorphisms. Definition 3.2. Let f : R → S b e a morphism of commutativ e rings. W e sa y that f is faithfully flat if the induced functor S ⊗ R ( − ) : Mo d R → Mo d S b et w een the mo dule categories preserves finite limits and is faithful. Theorem 3.3 (F aithful Flatness) . L et k b e a field. Then any inje ctive morphism H 1 → H 2 of c ommutative Hopf k -algebr as is faithful ly flat as a morphism of c ommutative rings. Pr o of. A pro of can b e found in [ 16 , Theorem 3.1]. Theorem 3.4 (Coregularity) . The c ate gory Hopf com k is c or e gular, with surje ctive and faithful ly flat morphisms b eing the classes of epimorphisms and extr emal monomorphisms, r esp e ctively. 3 Pr o of. Consider the orthogonal factorization system ( E , M ) of Mon com k of surjective and injectiv e morphisms. As k is a field, the class M is closed under tensoring and it follows that ( E ′ , M ′ ) is an orthogonal factorization system on Hopf com k , where E ′ = U − 1 ( E ) , M ′ = U − 1 ( M ) and U : Hopf com k → Mon com k is the forgetful functor [ 19 , Prop osition 3.7]. By T ak euc hi’s theorem, every morphism in M ′ is faithfully flat. Next, we demonstrate that the class M ′ is pushout-stable. Consider a morphism f ∈ M ′ , its pushout ¯ f and the cok ernel pair of ¯ f in Hopf com k . These pushouts are computed in the underlying category of Mon com k : H X G ⊗ H X G G ⊗ H X G ⊗ H G ⊗ H X f g ⌜ ¯ f ¯ f ⌜ s 2 ¯ g s 1 Note that G ⊗ H ¯ f = s 1 is a section and b y faithfulness of G ⊗ H ( − ) we witness the injectivit y of ¯ f : X → G ⊗ H X . Thus M ′ is pushout stable. F or any epimorphism f ∈ M ′ , the morphism G ⊗ H f is an isomorphism as one of the morphisms of the cokernel pair of f . The faithfulness of G ⊗ H ( − ) implies that f is b oth a monic and epic morphism of mo dules and thus is an isomorphism. Hence, every morphism in M ′ is a pushout stable extremal monomorphism, proving coregularity . Conv ersely , as every surjection is an epimorphism, w e hav e b y orthogonality that every extremal monomorphism is in M ′ . This shows that E ′ and M ′ are the classes of epimorphisms and extremal monomorphisms, resp ectively . 3.2 Co exactness W e prov e the co exactness of Hopf com k . Again, b y T akeuc hi’s work, we hav e a crucial c haracterization of normal Hopf ideals, and this is the key to sho wing the rest of the co exactness of Hopf com k . These observ ations assemble the necessary comp onen ts to prov e the co-semi-ab elianness of Hopf com k . Definition 3.5 (Hopf ideal) . Let X be a comm utative Hopf k -algebra and let I b e a subset of X . W e call I a Hopf ideal of X , if there is a morphism f : X → Y of Hopf com k , where I = f − 1 (0 Y ) . If f ma y b e c hosen as a cok ernel in Hopf com k , then I is called a normal Hopf ideal of X . This notion of a Hopf ideal is equiv alen t to the standard one [ 11 ]. Theorem 3.6. L et X b e a c ommutative Hopf k -algebr a with a Hopf ide al I of X . Then the fol lowing ar e e quivalent. 1. I is a normal Hopf ide al. 2. I = X A + for some Hopf sub algebr a A of X , wher e A + : = ε − 1 A (0) is the augmentation ide al of A and ε A : A → k is the augmentation of A . 3. F or e ach x ∈ I , we have x 1 S ( x 3 ) ⊗ x 2 ∈ X ⊗ I . 3 Pr o of. The equiv alence of the first t wo conditions is a straigh tforward verification, and the third condition is equiv alen t to the second b y Theorem 4.3 in [ 16 ]. Theorem 3.7. The c ate gory Hopf com k is c o exact. 3 The symbol S denotes the an tip ode. W e employ Sweedler’s notation, where for the comultiplication w e write ∆( x ) as x 1 ⊗ x 2 and (∆ ⊗ id ) ◦ ∆ = ( id ⊗ ∆) ◦ ∆( x ) as x 1 ⊗ x 2 ⊗ x 3 . 4 Pr o of. Theorem 3.4 shows that Hopf com k is coregular. By Lemma 2.2 , it suffices to show that the image along a regular epimorphism in ( Hopf com k ) op of a normal subob ject is normal. Consider the commutativ e diagram X Z Y W f ′ k k ′ f in Hopf k , where f is a cokernel, f ′ is an epimorphism, and k , k ′ are inclusions. W e sho w that f ′ is a cok ernel. Since f ′ is surjectiv e, it suffices to show that the Hopf ideal I = f ′− 1 (0) is normal. Let x ∈ I . By Theorem 3.6 it suffices to sho w that x ′ : = x 1 S ( x 3 ) ⊗ x 2 ∈ X ⊗ I . Since x ∈ J : = f − 1 (0) , we ha ve that x ′ ∈ ( Y ⊗ J ) ∩ ( X ⊗ X ) = ( Y ∩ X ) ⊗ ( J ∩ X ) = X ⊗ I b y the fact that intersections and tensor pro ducts satisfy interc hange b y the comp onen t- wise left-exactness of the tensor ⊗ k . Th us I is a normal Hopf ideal, which shows that f ′ is a cok ernel. 3.3 Conclusion Theorem 3.8. The c ate gory Hopf com k of c ommutative Hopf algebr as over a field k is a c o-semi-ab elian c ate gory. Pr o of. The category is p oin ted, locally presen table, coprotomo dular (Prop osition 3.1 ), and co exact (Theorem 3.7 ). Remark 3.9. The coexactness of Hopf com A , where A is a symmetric monoidal ab elian category with a comp onen twise exact tensor pro duct, is pro ven the same wa y as Theo- rem 3.7 , assuming similar characterizations of faithfully flat morphisms and Hopf-ideals. Consider the symmetric monoidal category A of super k -v ector spaces o ver a field k of c haracteristic other than 2 . As these c haracterizations of faithfully flat morphisms and normal Hopf-ideals hav e b een work ed out in [ 9 , Corollary 5.5, Theorem 5.9] for the category Hopf com A , w e attain that the category Hopf com A of comm utativ e super Hopf k -algebras is co-semi-ab elian. In a future pap er, w e aim to extend the semi-ab elianness result to affine group sc hemes o ver comm utative v on Neumann regular rings by applying a lo cal-to-global principle. A c kno wledgemen ts This research was funded by the F onds de la Rec herche Scientifique (Belgium) through an Aspiran t fellowship. 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