Category O for Takiff sl_2

We investigate various ways to define an analogue of BGG category $\mathcal{O}$ for the non-semi-simple Takiff extension of the Lie algebra $\mathfrak{sl}_2$. We describe Gabriel quivers for blocks of these analogues of category $\mathcal{O}$ and prove extension fullness of one of them in the category of all modules.


Introduction and description of the results
The celebrated BGG category O, introduced in [BGG], is originally associated to a triangular decomposition of a semi-simple finite dimensional complex Lie algebra. The definition of O is naturally generalized to all Lie algebras admitting some analogue of a triangular decomposition, see [MP]. These include, in particular, Kac-Moody algebras and Virasoro algebra. Category O has a number of spectacular properties and applications to various areas of mathematics, see for example [Hu] and references therein.
The paper [DLMZ] took some first steps in trying to understand structure and properties of an analogue of category O in the case of a non-reductive finite dimensional Lie algebra. The investigation in [DLMZ] focuses on category O for the so-called Schrödinger algebra, which is a central extension of the semi-direct product of sl 2 with its natural 2-dimensional module. It turned out that, for the Schrödinger algebra, the behavior of blocks of category O with non-zero central charge is exactly the same as the behavior of blocks of category O for the algebra sl 2 . At the same, block with zero central charge turned out to be significantly more difficult. For example, it was shown in [DLMZ] that some blocks of O for the Schrödinger algebra have wild representation type, while all blocks of O for sl 2 have finite representation type.
In the present paper we look at a different non-reductive extension of the algebra sl 2 , namely, the corresponding Takiff Lie algebra g defined as the semi-direct product of sl 2 with the adjoint representation. Such Lie algebras were defined and studied by Takiff in [Ta] with the primary interest coming from invariant theory. Alternatively, the Takiff Lie algebra g can be described as the tensor product sl 2 ⊗ C C[x]/(x 2 ) . The latter suggests an obvious generalization of the notion of a triangular decomposition for g by tensoring the components of a triangular decomposition for sl 2 with C[x]/(x 2 ).
Having defined a triangular decomposition for g, we can define Verma modules and try to guess a definition for category O. The latter turned out to be a subtle task as the most obvious definition of category O does not really work as expected, in particular, it does not contain Verma modules. This forced us to investigate two alternative definitions of category O: • the first one analogous to the definition of the so-called thick category O, see, for example, [Sor], in which the action of the Cartan subalgebra is only expected to be locally finite and not necessarily semi-simple as in the classical definition; • and the second one given by the full subcategory of the thick category O from the first definition with the additional requirement that the Cartan subalgebra of sl 2 acts diagonalizably.
The results of this paper fall into the following three categories: • We describe the linkage between simple object in both our versions of category O and in this way explicitly determine all (indecomposable) blocks, see Theorem 11.
• We prove that thick category O is extension full in the category of all g-modules, see Theorem 6.
For some of the blocks, we also obtain not only a Gabriel quiver, but also a fairly explicit description of the whole block, see Theorem 17. Some of the results are unexpected and look rather surprizing. For example, the trivial g-module exhibits behavior different from the behaviour of all other simple finite dimensional g-modules, compare Lemma 25 and Proposition 26.
The paper is organized as follows: All preliminaries are collected in Section 2, in particular, in this section we define all main protagonists of the paper and describe their basic properties. In Section 3 we prove extension fullness of thick category O in the category of all g-modules. Section 4 is devoted to the study of the decomposition of both O and its thick version O into indecomposable blocks. As usual, generic Verma modules over g are simple. In Section 5 we describe the structure of those Verma modules that are not simple. Finally, in Section 6, we compute first extensions between simple highest weight modules and in this way determine the Gabriel quivers of all block in O and O.
This paper is a revision, correction and extension of the master thesis [Sod] of the second author written under the supervision of the first author.
2. Takiff sl 2 and its modules 2.1. Takiff sl 2 . In this paper we work over the field C of complex numbers. Consider the Lie algebra sl 2 with the standard basis {e, h, f } and the Lie bracket Let D := C[x]/(x 2 ) be the algebra of dual numbers. Consider the associated Takiff Lie algebra g = sl 2 ⊗ C D with the Lie bracket where a, b ∈ sl 2 and i, j ∈ {0, 1} with the Lie bracket on the right hand side being the sl 2 -Lie bracket. Set Consider the standard triangular decomposition where n − is generated by f , h is generated by h and n + is generated by e. Let n − be the subalgebra of g generated by e and e, h the subalgebra of g generated by h and h, and n + be the subalgebra of g generated by f and f . The we have the following triangular decomposition of g: We set b := h ⊕ n + and b := h ⊕ n + .
For a Lie algebra a, we denote by U (a) the corresponding universal enveloping algebra.
The natural projection g ։ sl 2 induced an inclusion of sl 2 -Mod to g-Mod, which we denote by ι.
By a direct calculation, it is easy to check that the Casimir element (1) c := hh + 2h + 2f e + 2f e belongs to the center of U (g), see Example 1.2 in [Mo].
2.2. (Generalized) weight modules. A g-module M is called a generalized weight module provided that the action of U (h) on M is locally finite. As U (h) is just the polynomial algebra in h and h, every generalized weight module M admits a decom- where M λ denotes the set of all vectors in M killed by some power of the maximal ideal m λ of U (h) corresponding to λ. We will say that a generalized weight module M is a weight module provided that the action of h on M is diagonalizable. We will say that a weight module M is a strong weight module provided that the action of h on M is diagonalizable.
Note that submodules, quotients and extensions of generalized weight modules are generalized weight modules. Also submodules and quotients of (strong) weight modules are (strong) weight modules. From the commutation relations in g, for any generalized weight modules M and any λ ∈ h * , we have where α ∈ h * is given by α(h) = 2, α(h) = 0.
If M is a generalized weight module, then the set of all λ for which M λ = 0 is called the support of M and denoted supp(M ).
The following simplicity criterion for ∆(λ) can be deduced from the main result of [Wi], however, we include a short proof for the sake of completeness. Proof. Let v λ be the canonical generator of ∆(λ). Assume first that λ(h) = 0 and consider the element w = f v λ . Then we have ew = ew = 0 and hence, from the PBW Theorem and (2), it follows that the submodule N in ∆(λ) generated by w satisfies N λ = 0 and thus is a non-zero proper submodule. Therefore ∆(λ) is reducible in this case.
Now assume that λ(h) = 0. We need to show that any non-zero submodule N of ∆(λ) contains v λ . If N λ = 0, then the fact that v λ ∈ N is clear. Assume now that N λ = 0 and let n ∈ Z >0 be minimal such that N λ−nα = 0. Let w ∈ N λ−nα be a non-zero element. Using the PBW Theorem, we may write Denote the maximal value of i such that c i = 0 by k. Then it is easy to check that (h − λ(h)) k w equals f n v λ up to a non-zero scalar. In particular, N contains f n v λ .
But then it is easy to check that ef n v λ equals f n−1 v λ , up to a non-zero scalar. In particular, N λ−nα+α = 0. The obtained contradiction proves that N λ = 0 and the claim of the proposition follows.
Corollary 2. For λ ∈ h * , we have Proof. If λ(h) = 0, then the claim is just a part of Proposition 1. If λ(h) = 0, then the unique up to scalar non-zero vector in ι(L sl2 (λ| h )) λ generates a b-submodule of ι(L sl2 (λ| h )) isomorphic to C λ . By adjunction, we obtain a non-zero homomorphism from ∆(λ) to ι(L sl2 (λ| h )) which must be surjective as the latter module is simple. Consequently, ι(L sl2 (λ| h )) must be isomorphic to L(λ) by the definition of L(λ). As U (g) is noetherian, the categories O, O and O are abelian categories closed under taking submodules, quotients and finite direct sums. Directly from the definition, it also follows that O is closed under taking extensions, in particular, O is a Serre subcategory of g-mod.
Proposition 3. For each λ ∈ h * , the module ∆(λ) belongs to both O and O. However, Proof. That ∆(λ) ∈ O follows from (4). That ∆(λ) ∈ O follows by combining the fact that ∆(λ) ∈ O and that the adjoint action of h on U (g) is diagonalizable (implying that the action of h on ∆(λ) is diagonalizable).
That ∆(λ) ∈ O follows from the fact that the matrix of the action of h in the basis and hence is not diagonalizable. Proof. Let L be a simple module in O and v a non-zero element in L. Since the vector space By adjunction, we obtain a non-zero homomorphism from ∆(λ) to L. This implies L ∼ = L(λ) and proves claim (a). Claim (b) is proved similarly.
Proposition 4 has the following consequence.
Corollary 5. The functor ι induces an equivalence between the category O for sl 2 and the category O.
Proof. By construction, the functor ι is full and faithful and maps the category O for sl 2 to the category O. Hence, what we need to prove is that this restriction of ι is dense. By Proposition 4(c), ι hits all isomorphism classes of simple objects in O. In particular, h annihilates all simple objects in O. Since, by the definition of O, the action of h on any object in O is semi-simple, it follows that h annihilates all objects in O.
Since the ideal of g generated by h contains both e and f , it follows that the latter two elements annihilate all object in O. This yields that every object in O is, in fact, isomorphic to an object in the image of ι. The claim follows.
Due to Corollary 5, the category O is fairly well-understood, see e.g. [Ma] for a very detailed description. Therefore, in what follows, we focus on studying the categories O and O.

Extension fullness of O in g-Mod
The inclusion functor Φ : O ֒→ g-Mod is exact and hence induces, for each M, N ∈ O and i ≥ 0, homomorphisms The main result of this section is the following statement.
Theorem 6 is a generalization of Theorem 16 in [CM2] to our setup. We refer the reader to [CM1] and [CM2] for more details on extension full subcategories.
Proof. We follow the proof of Theorem 2 in [CM1]. Denote byÔ the full subcategory of g-Mod consisting of all modules, the action of U (b) on which is locally finite. The difference betweenÔ andÕ is that, in the case ofÔ, we drop the condition on modules to be finitely generated.
First we note thatÕ is extension full inÔ. Indeed, if M ∈Ô, N ∈Õ and α : M → N is a surjective homomorphism, we can use that N is finitely generated to claim that N is in the image of a finitely generated submodule of M . Therefore the fact thatÕ is extension full inÔ follows from Proposition 3 in [CM1] (applied in the situation B =Õ and A =Ô).
To complete the proof of the theorem, it remains to prove thatÔ is extension full in g-Mod. For a locally finite dimensional U Note that, by Theorem 6 in [BM], the action of U (b) on M (V ) is locally finite. The same computation as in the proof of Lemma 3 in [CM1] shows that, for any V as above, any N ∈Ô and any i ≥ 0, the natural map is an isomorphism. Therefore the extension fullness ofÔ in g-Mod follows from Proposition 1 in [CM1] (applied in the situation A = g-Mod, B =Ô and B 0 consisting of modules of the form M (V ), for V as above). This completes the proof.

Description of blocks
4.1. Characters and composition multiplicities.
Lemma 7. Let X ∈ {O, O} and M ∈ X . There exist k ∈ Z >0 and λ 1 , λ 2 , . . . , λ k ∈ h * such that Proof. If two modules M 1 and M 2 have the properties described in the formulation of the lemma, then any extension of M 1 and M 2 also has similar properties. By definition, M is finitely generated, and hence, taking the first sentence into account, without loss of generality we may assume that M is generated by one element Moreover, since, considered as an adjoint h-module, all generalized weight spaces of U (n − ) are finite dimensional, it follows that all M µ are finite dimensional. This completes the proof.
Consider the set F of all functions χ : h * → Z ≥0 having the property that the support {λ ∈ h * : χ(λ) = 0} of χ belongs to µ, for some µ as above. The set F has the natural structure of an additive monoid with respect to the pointwise addition of functions. The neutral element of this monoid is the zero function.
Let X ∈ {O, O}. Given M ∈ X , we define the character ch(M ) as the function from h * to Z ≥0 sending λ to dim(M λ ). By Lemma 7, we have ch(M ) ∈ F. Clearly, characters are additive on short exact sequences, that is, for any short exact sequence (b) For every λ ∈ h * , the function k λ : Ob(X ) → Z ≥0 has the following properties: (i) k λ (L(λ)) = 1; (M ), and hence the sum in (a) can be taken over supp (M ) instead of the whole h * .
Assume first that, for M ∈ X , we have where all a λ and b λ are in Z ≥0 . Assume that there is some λ such that a λ = b λ . Let X := {λ : a λ > b λ } and Y := supp(M ) \ X. Then we have By our assumptions, χ ∈ F is non-zero. Then there exists ν ∈ h * such that χ(ν) = 0 but χ(ν + mα) = 0, for all m ∈ Z >0 . If ν ∈ X, then ν ∈ Y and from the property χ(ν + mα) = 0, for all m ∈ Z >0 , we see that χ(ν) = 0 is not possible if we compute χ using the second expression. Similarly, if ν ∈ Y , then ν ∈ X and we get that χ(ν) = 0 is not possible if we compute χ using the first expression. The obtained contradiction shows that, if a decomposition of the form as in (a) exists, then it is unique.
Let us now prove existence of (a). If M = 0, we set k λ (M ) = 0, for all λ. Let M ∈ X be non-zero and λ ∈ supp(M ) be such that λ + mα ∈ supp (M ), for all m ∈ Z >0 . Let v ∈ M λ be a non-zero element which is an eigenvector for both h and h and set K := U (g)v ⊂ M and N := M/K. By adjunction, there is a non-zero epimorphism from ∆(λ) to K sending the canonical generator of ∆(λ) to v. Let K ′ denote the image, under this epimorphism, of the unique maximal submodule of ∆(λ). By construction, we have two short exact sequences: Note that supp(N ) ⊂ supp (M ) and dim(N λ ) < dim(M λ ), moreover, we also have supp(K ′ ) ⊂ supp (M ) and λ ∈ supp(K ′ ). Therefore, thanks to Lemma 7, Formula (6) gives an iterative procedure which, after a finite number of iterations, completely determines k µ (M ) such that (a) holds by construction.
It remains to check that k µ (M ) defined above have all the properties listed in (b). Properties (bi)-(biii) follow directly from the definition in the previous paragraph. Property (bi) follows from the equality in (a) and the fact that characters are additive on short exact sequences.
The number k µ (M ) will be called the composition multiplicity of L(µ) in M .
This means that L(λ) and L(µ) have different central characters and hence (7) splits. The claim of the proposition follows.
Following the proof of Proposition 9, we also obtain the following claim.

Easy blocks. Let X ∈ {O, O}. Set
, denote by X (λ) the full subcategory of X consisting of all modules with support {λ − Z ≥0 α}.

Difficult blocks.
For ξ ∈ h * 0 /Zα, denote by X (ξ) the full subcategory of X consisting of all modules whose support is contained in ξ.

Block decomposition.
Theorem 11. For X ∈ {O, O}, we have a decomposition of X into a direct sum of indecomposable abelian subcategories (blocks).
It remains to prove that all summands in the right hand side of (8) are indecomposable.
That each X (λ), where λ ∈ h * 1 , is indecomposable, is clear as X (λ) contains, by construction, only one simple module, up to isomorphism.
Up to a positive integer, e i f i v λ is a multiple of v λ with the coefficient As λ(h) / ∈ Z ≥0 , we obtain that the quotient ∆(λ)/∆(λ − α) is a simple module and hence is isomorphic to L(λ). The claim follows.
(a) Then there is a filtration Moreover, all subquotients in this filtration are simple modules and we also have i∈Z ≥0 ∆(λ − iα) = 0.
(b) The filtration given by (a) is the unique composition series of ∆(λ), in other words, ∆(λ) is a uniserial module.
Note that, under the assumptions of Corollary 13, the module ∆(λ) has infinite length. This emphasizes the difference with the classical sl 2 -situation, see Subsection 3.2 in [Ma] for the latter.
Proof. Existence of such filtration and the claim that all subquotients in this filtration are simple follows directly from Proposition 12. The claim that i∈Z ≥0 ∆(λ − iα) = 0 follows from the fact that i∈Z ≥0 supp(∆(λ − iα)) = ∅, which, in turn, is a consequence of (3). This proves claim (a).
To prove claim (b) we only need to show that any non-zero submodule M of ∆(λ) has the form ∆(λ − iα), for some i. Choose i such that λ − iα is the highest weight of M . From (a) it follows that any simple subquotient of ∆(λ)/∆(λ − iα) has a weight of the form λ − jα, where j < i. Therefore M ⊂ ∆(λ − iα). That M = ∆(λ − iα) follows from the fact that ∆(λ − iα) is generated by its highest weight vector. This proves claim (b) and completes the proof of the corollary.
Then the element f n+1 v λ generates a submodule K(λ) of ∆(λ) such that the module M n := ∆(λ)/K(λ) is uniserial and has a filtration Proof. We prove this statement by induction on n. The induction step moves λ − α to λ and hence changes n − 2 to n. Therefore we have two different cases for the basis of the induction.
As e · f n+1 v λ = 0 by the sl 2 -theory, we may assume q = 0. If x = a = p = 0, then the elements f y · f n+1 v λ are linearly independent and do not belong to ∆(λ − α). Therefore, if a linear combination of elements of the form , then at least one of y, a or q must be non-zero. Commuting the corresponding overlined basis element to the right and using ev λ = hv λ = 0, one shows that our linear combination ends up in K(λ − α).
Consider now the following diagram: The solid part of this diagram consists of natural inclusions. By the previous paragraph, This solid part is, in fact, a commutative pullback diagram. The vertical dashed arrows are natural projections. The horizontal dashed arrow is induced by the solid part such that the dashed box commutes. The horizontal dashed arrow is injective since the solid part is a pullback. The dotted part of the diagram is given by the Snake Lemma and the whole diagram (12) commutes. By the Snake Lemma, all rows and all columns of (12) are short exact sequences.
From the Second Isomorphism Theorem and definitions, we have Therefore, the upper row of (12) gives a short exact sequence As L(λ) is a unique simple top of ∆(λ), the module L(λ) also must be a unique simple top of M n . Now all necessary claims follow by induction.
As an immediate consequence of the above, we obtain: Corollary 16. Assume that λ ∈ h * is such that λ(h) = 0 and λ(h) = n ∈ Z ≥0 . The Hasse diagram of the partially ordered, by inclusion, set of submodules of ∆(λ) of the form ∆(λ − iα) and K i is as follows (here k = ⌈ n−1 2 ⌉): . . .
Theorem 17. For λ ∈ h * 1 , we have: Recall that the Gabriel quiver of a block is a directed graph whose • vertices are isomorphism classes of simple objects in the block; • the number of arrows from a vertex L to a vertex S equals the dimension of Ext 1 (L, S).
As an immediate consequence of Theorem 17, we obtain: In particular, as simple modules in X are uniquely determined by their characters, it follows that L(λ) ⋆ ∼ = L(λ), for all λ ∈ h * . In other words, the duality ⋆ is simple preserving.
Proof. The left hand side of the equality is obtained from the right hand side by applying the simple preserving duality ⋆.
We note that ⋆ does not extend to the whole of X as ⋆ messes up the property of being finitely generated. For example, for an infinite length Verma module ∆(λ) ∈ X as in Subsection 5.1, the module ∆(λ) ⋆ is not finitely generated and hence does not belong to X .
Assume first that k > 0 and that (13) does not split. In this case M must be generated by M λ and hence, by adjunction, is a quotient of the Verma module ∆(λ). Under the assumptions λ(h) = 0 and λ(h) / ∈ Z, all submodules of ∆(λ) are described in Corollary 13. Out of all possible quotients of ∆(λ), only the quotient ∆(λ)/∆(λ − 2α) has length two. This quotient has composition subquotients L(λ) and L(λ − α). This implies that It remains to compute Ext 1 O (L(λ), L(λ)). Consider a non-split short exact sequence (13) in O, with λ = µ. The vector space M λ is, naturally, a U (h)-module. If this module were semi-simple, by adjunction there would exist two linearly independent homomorphisms from ∆(λ) to M and hence (13) would be split. Therefore M λ must be an indecomposable U (h)-module. As h is supposed to act diagonalizably, such module M λ is unique, up to isomorphism. In particular, there is a basis {v, w} of M λ such that the matrix of the action of h in this basis is 0 0 1 0 .
Consider now the module ∆ (M λ Hence, we just need to check how many submodules K of ∆(M λ ) have the property that ∆(M λ )/K has length two with both composition subquotients isomorphic to L(λ). We claim that such submodule is unique, which implies that Ext 1 O (L(λ), L(λ)) ∼ = C. In fact, since k λ (∆(M λ )) = 2 by construction, the uniqueness of K, provided that K exists, is clear.
To prove existence, we consider the submodule K of ∆(M λ ) generated by f w and λ(h)f v − f w (note that λ(h) = 0 by our assumptions). It is easy to check that both these vectors are annihilated by e and e. The vector f w generates a submodule of ∆(M λ ) isomorphic to ∆(λ − α). The image of λ(h)f v − f w in the quotient ∆(M λ )/∆(λ) generates in this quotient a submodule isomorphic to ∆(λ − α). Therefore, from Proposition 12 it follows that ∆(M λ )/K indeed has length two with both simple subquotients isomorphic to L(λ). The claim follows.
As an immediate corollary from Proposition 20, we have: Corollary 21. Assume that λ ∈ h * is such that λ(h) = 0 and λ(h) / ∈ Z. Then, for ξ := λ + Zα, the Gabriel quiver of O(ξ) has the form: Now we can proceed to O.
Proposition 22. Assume that λ ∈ h * is such that λ(h) = 0 and λ(h) / ∈ Z. Then, for µ ∈ λ + Zα, we have Proof. The case µ = λ is proved by exactly the same arguments as in Proposition 20. The case µ = λ is also similar, but requires some small adjustments which we describe below.
Consider a non-split short exact sequence (13) in O, with λ = µ. The vector space M λ is an indecomposable U (h)-module of length two, namely, a self-extension of the simple U (h)-module C λ corresponding to λ. The space of such self-extensions is twodimensional (as h is two-dimensional). In fact, using the arguments as in the proof of Proposition 20, we can show that parabolic induction, followed by taking a canonical quotient, defines a surjective map from Ext 1 U(h) (C λ , C λ ) to Ext 1 O (L(λ), L(λ)) which sends isomorphic module to isomorphic and non-isomorphic modules to non-isomorphic (the latter claim is obvious by restricting the action to the generalized λ-weight space). This, clearly, implies the necessary claim. Here are the details.
There is a basis {v, w} of M λ such that the matrices of the action of h and h in this basis are λ(h) 0 p λ(h) and 0 0 q 0 , respectively, where p and q are complex numbers at least one of which is non-zero.
Similarly to the proof of Proposition 20, one shows that the submodule K of ∆(M λ ) generated by f w and λ(h) q f v − f w, in case q = 0, or f v, in case q = 0, is the unique submodule of ∆(M λ ) such that ∆(M λ )/K is isomorphic to M . The claim follows.
As an immediate corollary from Proposition 22, we have: Corollary 23. Assume that λ ∈ h * is such that λ(h) = 0 and λ(h) / ∈ Z. Then, for ξ := λ + Zα, the Gabriel quiver of O(ξ) has the form: Proof. By Weyl's complete reducibility theorem, h acts diagonalizably on any finite dimensional g-module. Hence any self-extension of L(λ) lives in O. Therefore it is enough to prove that Ext 1 O (L(λ), L(λ)) ∼ = C. Let M be a self-extension of L(λ). Then M λ is an C[h]-module and, similarly to Proposition 20, M is indecomposable if and only if M λ is. As C[h] is a polynomial algebra in one variable, this implies that Ext 1 O (L(λ), L(λ)) is at most one dimensional. To prove that Ext 1 O (L(λ), L(λ)) is exactly one-dimensional, it is enough to construct one non-split self-extension of L(λ), which we do below.
As the action of h on v 0 is non-zero (here the condition n > 0 is crucial!), the module M is a non-split self-extension of L(λ). This completes the proof.
Proof. We start with claim (a). Assume that is a non-split short exact sequence. Then, similarly to Proposition 20, M must be a quotient of ∆(λ). If λ(h) / ∈ Z ≥0 , then from Corollary 13 it follows that ∆(λ) has a unique quotient with correct composition subquotients. If λ(h) ∈ Z ≥0 , then from Lemma 15 it follows that ∆(λ) has a unique quotient with correct composition subquotients. This completes the proof of claim (a) We proceed with claim (b). Assume that λ(h) = n ∈ Z ≥0 and 0 → L(λ − (n + 1)α) → M → L(λ) → 0 is a non-split short exact sequence. Then, from Lemma 14 it follows that ∆(λ) has a unique quotient with correct composition subquotients. This completes the proof of claim (b).
Proposition 12, Lemma 14 and Lemma 15 imply that the only socle components possible in length two quotients of ∆(λ) are ∆(λ − α) and ∆(λ − (n + 1)α), and the latter one is only possible under the additional assumption that λ(h) = n ∈ Z ≥0 . This implies claim (c) and completes the proof.