Cotorsion dimensions and FP-projective dimensions of weak Hopf actions

Abstract

Let H be a finite dimensional weak Hopf algebra and A be an H-module algebra. In this paper, we mainly discuss the relations of cotorsion dimension and FP-projective dimension between A\mathrm{♯}H and A. As applications, sufficient conditions are given for \mathrm{L}\mathrm{C}\mathrm{D}\left(A\mathrm{♯}H\right)=\mathrm{L}\mathrm{C}\mathrm{D}\left(A\right) and \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\mathrm{♯}H\right)=\mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\right).

Keywords

Cotorsion dimension / FP-projective dimension / weak Hopf algebra

In this paper, k is a field on which all algebras and coalgebras are discussed. H is a k-weak Hopf algebra. Hopf algebra turns out an important structure obtained by weakening the unitality of Hopf algebra comultiplication and multiplicativity of counit in [1]. As a result, many Hopf algebra conclusions can be generalized in weak Hopf algebra theories.

Let R be a ring and M be a left R-module. According to [2, 7], the cotorsion dimension {\mathrm{C}\mathrm{D}}_R\left(M\right)=n of the module M refers to the smallest positive integer such that {\mathrm{E}\mathrm{x}\mathrm{t}}_R^{n+1}(F,M)=0 for any flat R-module F. The left cotorsion dimension \mathrm{L}\mathrm{C}\mathrm{D}\left(R\right) of R is the supremum of the cotorsion dimensions of all left R-modules. In particular, if \mathrm{C}\mathrm{D}\left(M\right)=0, then M is called a cotorsion module. The left FP-projective dimension \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(M\right)=n of the module M means the smallest positive integer such that {\mathrm{E}\mathrm{x}\mathrm{t}}_R^{n+1}(M,N)=0 for any FP-injective left R-module N. This index implies how far the ring R is from being Noetherian. If H is a Hopf algebra, then \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left({}_{\mathrm{A}}M\right)=\mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left({}_{A\mathrm{♯}H}\mathrm{M}\right) (see [2, 7]). This paper mainly discusses whether this equation holds under the condition of weak Hopf algebra as well as the relations between \mathrm{C}\mathrm{D}\left(A\right) and \mathrm{C}\mathrm{D}\left(A\mathrm{♯}H\right) or not. Please see references [3, 6, 7, 9] for details of group ring and cotorsion dimensions of modules. Then, we will review some basic concepts of weak Hopf algebra. Please see references [1, 5, 8] for details.

Definition 1　A weak bialgebra (WBA) is a quintet (A,\mu ,\eta ,\mathrm{\Delta },\epsilon ) satisfying the following axioms 1‒3. If (A,\mu ,\eta ,\mathrm{\Delta },\epsilon ,S) further satisfies Axiom 4, then it is called a weak Hopf algebra (WHA).

Axiom 1　 A is a finite dimensional associative algebra over the field k with multiplication \mu and unit \eta.

Axiom 2　 A is a finite dimensional coalgebra over the field k with comultiplication \mathrm{\Delta } and counit \epsilon.

Axiom 3　The comultiplication satisfies multiplicativity:

⟨1⟩ \mathrm{\Delta }\left(xy\right)=\mathrm{\Delta }\left(x\right)\mathrm{\Delta }\left(y\right);

the weak multiplicativity of counit:

⟨2⟩ \epsilon \left(xyz\right)=\sum \epsilon \left(x{y}_1\right)\epsilon \left(y_{2}z\right),\epsilon \left(xyz\right)=\sum \epsilon \left(x{y}_2\right)\epsilon \left(y_{1}z\right);

⟨3⟩ {\mathrm{\Delta }}^2(1)=(\mathrm{\Delta }(1)\otimes 1)(1\otimes \mathrm{\Delta }(1)),{\mathrm{\Delta }}^2(1)=(1\otimes \mathrm{\Delta }(1))(\mathrm{\Delta }(1)\otimes 1).

Axiom 4 S:A\to A is a k-linear map, called an antipode, satisfying

⟨4⟩\sum x_{1}S\left(x_2\right)=\sum \epsilon \left(1_{1}x\right)1_2,

　 \sum S\left(x_1\right)\left(x_2\right)=\sum \epsilon \left(x{1}_2\right)1_1,

　 \sum S\left(x_1\right)x_{2}S\left(x_3\right)=S\left(x\right).

Let A be a weak bialgebra. Then the maps {\Pi }^l and {\Pi }^r:A\to A, {\Pi }^l\left(x\right)=x^l=\sum \epsilon \left(1_{1}x\right)1_2,{\Pi }^r\left(x\right)=x^r=\sum \epsilon \left(x{1}_2\right)1_1. Let H^l=\left\{h^l|h\in H\right\},H^r=\{h^r|h\in H\}.

Definition 2　Let H be a weak bialgebra over the field k. If an algebra A is a left H-module and satisfies the following conditions, then A is called a left H-module algebra:

⟨5⟩ h\cdot ab=\sum (h_1\cdot a)(h_2\cdot b),

⟨6⟩ h\cdot 1=h^l\cdot 1.

Lemma 1　 Let H be a weak Hopf algebra over the field k and A be a left H-module algebra. For any h\in H and a\in A, we have

⟨7⟩ h^l\cdot a=(h^l\cdot 1)a,h^r\cdot a=a(h^r\cdot 1).

Let H be a weak Hopf algebra over the field k and A be a left H-module algebra. The Smash product A\mathrm{♯}H is A{\otimes }_{H^l}H as a space, and its multiplication is defined as \left(a\mathrm{♯}h\right)\left(b\mathrm{♯}g\right)=\sum a(h_1\cdot b)\mathrm{♯}{h}_{2}g. It has been proved that A\mathrm{♯}H is an associative algebra with unit A\mathrm{♯}H, and the maps i_A:A\to A\mathrm{♯}H,a\mapsto a\mathrm{♯}1 and i_H:H\to A\mathrm{♯}H,h\mapsto 1\mathrm{♯}h are all algebraic maps. For more details, see reference [5].

Let A\mathrm{♯}H be defined as mentioned above, M and N be two A\mathrm{♯}H-modules, then {\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N) is a rightH-module, and its module action can be defined as:

\left(f\leftharpoonup h\right)\left(m\right)=\sum S\left(h_1\right)f\left(h_{2}m\right),f\in {\mathrm{H}\mathrm{o}\mathrm{m}}_A\left(M,N\right),h\in H,m\in M.

In fact, for any h,g\in H,f\in {\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N) and m\in M, there are

\left(\right(f\leftharpoonup h)\leftharpoonup g)\left(m\right)=\sum S\left(g_1\right)S\left(h_1\right)f\left(h_2{g}_{2}m\right)=(f\leftharpoonup (hg\left)\right)\left(m\right);

\begin{array}{rl}(f\leftharpoonup 1)\left(m\right)& =\sum S\left(1_1\right)f\left(1_{2}m\right)=\sum (1\mathrm{♯}S(1_1))(S^{-1}(1_2)\cdot 1\mathrm{♯}1)f\left(m\right)\\ & =\sum \left(\right(S{\left(1_1\right)}_1{S}^{-1}\left(1_2\right))\cdot 1\mathrm{♯}S{\left(1_1\right)}_2)f\left(m\right)\\ & =\sum \left(1_{1}S\right(1_2)1_3\cdot 1\mathrm{♯}1)f\left(m\right)=f\left(m\right).\end{array}

Lemma 2　 Let A\mathrm{♯}H be defined as mentioned above, M and N be A\mathrm{♯}H-modules. Then there exists a natural isomorphism from functor {\mathrm{H}\mathrm{o}\mathrm{m}}_{A\mathrm{♯}H}(-,-) to functor {\mathrm{H}\mathrm{o}\mathrm{m}}_H(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(-,-\left)\right).

Proof　For {\iota }_{M,N}:{\mathrm{H}\mathrm{o}\mathrm{m}}_H(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N\left)\right)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N{)}^H,f\mapsto f(1), and{\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N{)}^H=\{f\in {\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N)\mid f\leftharpoonup h=f\leftharpoonup h^r\} is isomorphic. Therefore, we only need to prove

{\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N{)}^H={\mathrm{H}\mathrm{o}\mathrm{m}}_{A\mathrm{♯}H}(M,N).

For any f\in {\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N{)}^H,h\in H,m\in M, there is

\begin{array}{rl}(f\leftharpoonup h)\left(m\right)& =(f\leftharpoonup h^r)\left(m\right)\iff \sum S\left(h_1\right)f\left(h_{2}m\right)\\ & =\sum S\left(1_1\right)f\left(1_2{h}^{r}m\right)=f\left(h^{r}m\right).\end{array}

Hence,

\begin{array}{rl}hf(m)& =\sum h_{2}f(S^{-1}{(h_1)}^{l}m)=\sum h_{3}f(S^{-1}(h_2)h_{1}m)\\ & =\sum f(S^{-1}{(h_2)}^r{h}_{1}m)=f(hm).\end{array}

a\mathrm{♯}h=\left(a\mathrm{♯}1\right)\left(1\mathrm{♯}h\right), so f\in {\mathrm{H}\mathrm{o}\mathrm{m}}_{A\mathrm{♯}H}(M,N).

Conversely, if f\in {\mathrm{H}\mathrm{o}\mathrm{m}}_{A\mathrm{♯}H}(M,N), then

(f\leftharpoonup h)\left(m\right)=\sum S\left(h_1\right)f\left(h_{2}m\right)=f\left(h^{r}m\right)=\sum S\left(1_1\right)f\left(1_2{h}^{r}m\right)=(f\leftharpoonup h^r)\left(m\right).

Next, we need to prove that the naturality establishes. For any f\in {\mathrm{H}\mathrm{o}\mathrm{m}}_{A\mathrm{♯}H}(M,M^{\prime }), assume

\overline{f}={\mathrm{H}\mathrm{o}\mathrm{m}}_H(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(f,N\left)\right),

i.e., \overline{f}:{\mathrm{H}\mathrm{o}\mathrm{m}}_H(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(M^{\prime },N\left)\right)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_H(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N\left)\right),\overline{f}\left(\alpha \right)\left(z\right)\left(m\right)=\alpha \left(z\right)f\left(m\right), then on one hand ({\mathrm{H}\mathrm{o}\mathrm{m}}_{A\mathrm{♯}H}(f,N)\circ {\iota }_{M^{\prime },N})\left(\alpha \right)= {\mathrm{H}\mathrm{o}\mathrm{m}}_{A\mathrm{♯}H}(f,N)({\iota }_{M^{\prime },N}(\alpha ))=\alpha \left(1\right)f, and on the other ({\iota }_{M,N}\circ \overline{f})\left(\alpha \right)= {\iota }_{M,N}(\overline{f}(\alpha ))=\alpha (1)f. In the same way, the naturality on N is established. Therefore, the proof is completed. □

In the following part, we assume that H is a semisimple weak Hopf algebra, that is, every H-module is completely reducible.

Lemma 3　 Let H be a semisimple weak Hopf algebra, A be a left H-module algebra, M and N be A\mathrm{♯}H-modules satisfying {\mathrm{E}\mathrm{x}\mathrm{t}}_A^p(M,N)= 0,p>0. Then {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^n(M,N)=0,n>0.

Proof　Let 0\to C\to P\to M\to 0 be an A\mathrm{♯}H-module exact sequence, whereP is a projective A\mathrm{♯}H-module. Since {\mathrm{E}\mathrm{x}\mathrm{t}}_A^p(M,N)=0,p>0, there are exact sequences

0\to {\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_A(P,N)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_A(C,N)\to 0,

and {\mathrm{E}\mathrm{x}\mathrm{t}}_A^p(C,N)\cong {\mathrm{E}\mathrm{x}\mathrm{t}}_A^{p+1}(M,N)=0,p>0. Therefore, the commutative diagram is obtained:

\begin{array}{rl}{\mathrm{H}\mathrm{o}\mathrm{m}}_H(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(P,N\left)\right)& \to {\mathrm{H}\mathrm{o}\mathrm{m}}_H(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(C,N\left)\right)\\ & \to {\mathrm{E}\mathrm{x}\mathrm{t}}_H^1(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N\left)\right)\to 0,\end{array}

{\mathrm{H}\mathrm{o}\mathrm{m}}_{A\mathrm{♯}H}(P,N)\to {\mathrm{H}\mathrm{o}\mathrm{m}}_{A\mathrm{♯}H}(C,N)\to {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^1(M,N)\to 0.

From Lemma 2, there exists the commutative diagram:

0\to {\mathrm{E}\mathrm{x}\mathrm{t}}_H^p(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(C,N\left)\right)\to {\mathrm{E}\mathrm{x}\mathrm{t}}_H^{p+1}(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N\left)\right)\to 0,

0\to {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^p(C,N)\to {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^{p+1}(M,N)\to 0.

It is obvious that {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^n(M,N)\cong {\mathrm{E}\mathrm{x}\mathrm{t}}_H^n(H^l,{\mathrm{H}\mathrm{o}\mathrm{m}}_A(M,N\left)\right)=0,n>0. □

Lemma 4　 Let A\mathrm{♯}H be defined as mentioned above, and A\mathrm{♯}H is a flat right A-module. Then for any A\mathrm{♯}H-module M, {\mathrm{C}\mathrm{D}}_A\left(M\right)\le {\mathrm{C}\mathrm{D}}_{A\mathrm{♯}H}\left(M\right).

Proof　Assume . Let F be a flat A-module, then A\mathrm{♯}H{\otimes }_{A}F is a flat A\mathrm{♯}H-module. Then {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^{n+1}(A\mathrm{♯}H{\otimes }_{A}F,M)=0 and {\mathrm{E}\mathrm{x}\mathrm{t}}_A^{n+1}(F,M)\cong {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^{n+1}(A\mathrm{♯}H{\otimes }_{A}F,M) (see reference [9]). Hence, {\mathrm{E}\mathrm{x}\mathrm{t}}_A^{n+1}(F,M)=0, which indicates {\mathrm{C}\mathrm{D}}_A\left(M\right)\le n.

Theorem 1　 Let A\mathrm{♯}H be defined as mentioned above and A\mathrm{♯}H is a flat right A-module, then for any A\mathrm{♯}H-module M, {\mathrm{C}\mathrm{D}}_A\left(M\right)= {\mathrm{C}\mathrm{D}}_{A\mathrm{♯}H}\left(M\right).

Proof　According to Lemma 4, we need to prove {\mathrm{C}\mathrm{D}}_A\left(M\right)\ge {\mathrm{C}\mathrm{D}}_{A\mathrm{♯}H}\left(M\right). Assume . According to reference [3], there exists a A\mathrm{♯}H-module exact sequence 0\to M\to X_0\to \cdots \to X_{n-1}\to X\to 0, where X_0,\dots ,X_{n-1} are all cotorsion A\mathrm{♯}H-modules. According to reference [6] and Lemma 6, X_0,\dots ,X_{n-1},X are all cotorsion A-modules. Then for any flat A-module F and p>0, {\mathrm{E}\mathrm{x}\mathrm{t}}_A^p(F,X)=0. Based on Lemma 3, for any flat A\mathrm{♯}H-module F^{\prime }, {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^1(F^{\prime },X)=0, so {\mathrm{C}\mathrm{D}}_{A\mathrm{♯}H}\left(X\right)=0. According to reference [6], there exists an exact sequence

0\to M\to X_0\to \cdots \to X_{n-1}\to X_n\to 0,

where X_0,\dots ,X_{n-1},X_n are all cotorsion modules, so {\mathrm{C}\mathrm{D}}_{A\mathrm{♯}H}\left(M\right)\le n. □

Corollary 1　 Let A\mathrm{♯}H be defined as mentioned above and A\mathrm{♯}H be a flat right A-module, then \mathrm{L}\mathrm{C}\mathrm{D}\left(A\mathrm{♯}H\right)\le \mathrm{L}\mathrm{C}\mathrm{D}\left(A\right).

It is known that the ring R is left perfect if and only if \mathrm{L}\mathrm{C}\mathrm{D}\left(R\right)=0. As an application, we will give several conditions such that \mathrm{L}\mathrm{C}\mathrm{D}\left(A\mathrm{♯}H\right) =\mathrm{L}\mathrm{C}\mathrm{D}\left(A\right).

Theorem 2　 Let H be a semisimple weak Hopf algebra which is free on H^l, and A be a left H-module algebra. If one of the following conditions is met, then \mathrm{L}\mathrm{C}\mathrm{D}\left(A\mathrm{♯}H\right)=\mathrm{L}\mathrm{C}\mathrm{D}\left(A\right):

(1) H^{\ast } is semisimple;

(2) A is commutative.

Proof　According to Corollary 1, we need to prove \mathrm{L}\mathrm{C}\mathrm{D}\left(A\mathrm{♯}H\right)\ge \mathrm{L}\mathrm{C}\mathrm{D}\left(A\right).

Proof of (1).

From reference [8], A\mathrm{♯}H\mathrm{♯}{H}^{\ast }\cong \mathrm{E}\mathrm{n}\mathrm{d}(A\mathrm{♯}H{)}_A\cong M_n(A). If H^{\ast } is semisimple, we can obtain \mathrm{L}\mathrm{C}\mathrm{D}\left(A\mathrm{♯}H\right)\le \mathrm{L}\mathrm{C}\mathrm{D}\left(A\right) directly from Corollary 1.

Proof of (2).

First, we assert that A\mathrm{♯}H is an (A,A)-bimodule, with actions defined as b\rightharpoonup \left(a\mathrm{♯}h\right)=ab\mathrm{♯}h and \left(a\mathrm{♯}h\right)\leftharpoonup b=ab\mathrm{♯}h,a,b\in A,h\in H. In fact, for any a,b,c\in A,z\in H^l,h\in H, there are

\begin{array}{rl}b\rightharpoonup (a\mathrm{♯}zh-a\cdot z\mathrm{♯}h)& =ab\mathrm{♯}zh-(S^{-1}(z)\cdot a)b\mathrm{♯}h\stackrel{⟨7⟩}{=}ab\mathrm{♯}zh-(S^{-1}(z)\cdot 1)ab\mathrm{♯}h\\ & \stackrel{⟨7⟩}{=}ab\mathrm{♯}zh-ab\mathrm{♯}zh=0;\end{array}

(a\mathrm{♯}zh-a\cdot z\mathrm{♯}h)\leftharpoonup b=ab\mathrm{♯}zh-(S^{-1}(z)\cdot a)b\mathrm{♯}h\stackrel{⟨7⟩}{=}ab\mathrm{♯}zh-ab\mathrm{♯}zh=0;

b\rightharpoonup \left(c\rightharpoonup \left(a\mathrm{♯}h\right)\right)=acb\mathrm{♯}h=abc\mathrm{♯}h=\left(bc\right)\rightharpoonup \left(a\mathrm{♯}h\right),1\rightharpoonup \left(a\mathrm{♯}h\right)=a\mathrm{♯}h,

\left(\left(a\mathrm{♯}h\right)\leftharpoonup c\right)\leftharpoonup b=acb\mathrm{♯}h=\left(a\mathrm{♯}h\right)\leftharpoonup \left(cb\right),\left(a\mathrm{♯}h\right)\leftharpoonup 1=a\mathrm{♯}h.

Next, we prove \alpha :A\mathrm{♯}H\to A,a\mathrm{♯}h\mapsto h^l\cdot a is an (A,A)-bimodule epimorphism. In reality, for any a,b,c\in A,z\in H^l,h\in H, there are

\begin{array}{rl}\alpha (a\mathrm{♯}zh-a\cdot z\mathrm{♯}h)& =(zh{)}^l\cdot a-h^l{S}^{-1}(z)\cdot a\stackrel{⟨7⟩}{=}(zh{)}^l\cdot a-(h^l\cdot a)(S^{-1}(z)\cdot 1)\\ & =(zh{)}^l\cdot a-(h^l\cdot a)(z\cdot 1)\stackrel{⟨7⟩}{=}\sum \epsilon (1_{1}zh)1_2\cdot a-z^l{h}^l\cdot a\\ & =\sum \epsilon \left(z_{1}h\right)z_2\cdot a-z^l{h}^l\cdot a=z^l{h}^l\cdot a-z^l{h}^l\cdot a=0;\end{array}

b\alpha \left(a\mathrm{♯}h\right)=b(h^l\cdot a)\stackrel{⟨7⟩}{=}(h^l\cdot 1)ab=h^l\cdot \left(ab\right)=\alpha (b\rightharpoonup (a\mathrm{♯}h\left)\right);

\alpha \left(a\mathrm{♯}h\right)b=(h^l\cdot a)b\stackrel{⟨7⟩}{=}(h^l\cdot 1)ab=h^l\cdot \left(ab\right)=\alpha \left(\right(a\mathrm{♯}h)\leftharpoonup b).

Finally, we prove that \alpha is a split (A,A)-bimodule epimorphism. Define \beta :A\to A\mathrm{♯}H,\beta \left(a\right)=a\mathrm{♯}1. It is proved that \beta is an (A,A)-bimodule map and \alpha \beta ={id}_A. Therefore, as an (A,A)-bimodule, A, is the direct summand of A\mathrm{♯}H, and for anyA-module M, there is

A\mathrm{♯}H{\otimes }_{A}M\cong M\oplus \left(X{\otimes }_{A}M\right).

Hence, {\mathrm{C}\mathrm{D}}_A\left(M\right)\le {\mathrm{C}\mathrm{D}}_A\left(A\mathrm{♯}H{\otimes }_{A}M\right). From Lemma 4,{\mathrm{C}\mathrm{D}}_A\left(A\mathrm{♯}H{\otimes }_{A}M\right)\le {\mathrm{C}\mathrm{D}}_{A\mathrm{♯}H}\left(A\mathrm{♯}H{\otimes }_{A}M\right), so we can get \mathrm{L}\mathrm{C}\mathrm{D}\left(A\mathrm{♯}H\right)\ge \mathrm{L}\mathrm{C}\mathrm{D}\left(A\right). □

Corollary 2　 Let H be a semisimple weak Hopf algebra which is free on H^l, and A be a left H-module algebra. If one of the following conditions is met, then A is left perfect if and only if A\mathrm{♯}H is left perfect:

(1) H^{\ast } is semisimple;

(2) A is commutative.

Theorem 3　Let H be a semisimple weak Hopf algebra which is free on H^l. Assume A is a left H-module algebra and M is a A\mathrm{♯}H-module, then M is a finitely presented A\mathrm{♯}H-module if and only if M is a finitely presented A-module.

Proof　If M is a finitely presented left A\mathrm{♯}H-module, then there exists an exact sequence 0\to C\to P\to M\to 0, where C is a finitely generated A\mathrm{♯}H-module, and P is a finitely generated projective A\mathrm{♯}H-module. Based on reference [7], C is a finitely generated A-module, and P is a finitely generated projective A-module, so M is a finitely presented A-module.

On the contrary, if M is a finitely presented A-module, then there exists an exact sequence 0\to C\to P\to M\to 0, where C is a finitely generated A-module and Pis finitely generated projective A-module. Since H is free on H^l, there exists an A\mathrm{♯}H-module exact sequence 0\to A\mathrm{♯}H{\otimes }_{A}C\to A\mathrm{♯}H{\otimes }_{A}P\to A\mathrm{♯}H{\otimes }_{A}M\to 0. Obviously, A\mathrm{♯}H{\otimes }_{A}C is finitely generated, and A\mathrm{♯}H{\otimes }_{A}P is finitely generated projective. This means that A\mathrm{♯}H{\otimes }_{A}M is finitely presented, and M is the direct summand of A\mathrm{♯}H{\otimes }_{A}M as an A\mathrm{♯}H-module, so M is a finitely presented A\mathrm{♯}H-module.

□

Theorem 4　 Let H be a semisimple weak Hopf algebra which is free on H^l. Assume A is a left H-module algebra, and M is a A\mathrm{♯}H-module. Then M is a FP-projective left A\mathrm{♯}H-module if and only if M is a FP-projective left A-module.

Proof　If M is an FP-projective left A\mathrm{♯}H-module, then M is the direct summand of N as an A\mathrm{♯}H-module and N is the union of based on reference [4] , where for is a finitely presented A\mathrm{♯}H-module. From Theorem 3, is a finitely presented A-module, so M is an FP-projective A-module.

Conversely, if M is an FP-projective left A-module, we first assert that for any FP-injective A\mathrm{♯}H-module N, it is also an FP-injective A-module. LetPbe a finitely presented A-module, based on reference [9], {\mathrm{E}\mathrm{x}\mathrm{t}}_A^1(P,N)\cong {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^1(A\mathrm{♯}H{\otimes }_{A}P,N). According to Theorem 3, A\mathrm{♯}H{\otimes }_{A}P is a finitely presented A\mathrm{♯}H-module, and

{\mathrm{E}\mathrm{x}\mathrm{t}}_A^1(P,N)\cong {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^1(A\mathrm{♯}H{\otimes }_{A}P,N)=0,

M is an FP-injective A-module. For any FP-injective A\mathrm{♯}H-module N, there are {Ext}_A^1(M,N)=0 and

{\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^1(A\mathrm{♯}H{\otimes }_{A}M,N)\cong {\mathrm{E}\mathrm{x}\mathrm{t}}_A^1(M,N)=0,

M is the direct summand of A\mathrm{♯}H{\otimes }_{A}M, so {\mathrm{E}\mathrm{x}\mathrm{t}}_{A\mathrm{♯}H}^1(M,N)=0, that is, M is an FP-projectie A\mathrm{♯}H-module. □

Theorem 5　 Let H be a semisimple weak Hopf algebra which is free on H^l. Assume A is a coherent H-module algebra, and M is an A\mathrm{♯}H-module, then {\mathrm{F}\mathrm{P}\mathrm{D}}_A\left(M\right)={\mathrm{F}\mathrm{P}\mathrm{D}}_{A\mathrm{♯}H}\left(M\right)={\mathrm{F}\mathrm{P}\mathrm{D}}_{A\mathrm{♯}H}\left(A\mathrm{♯}H{\otimes }_{A}M\right).

Proof　Since M is the direct summand of A\mathrm{♯}H{\otimes }_{A}M as an A\mathrm{♯}H-module, there is

{\mathrm{F}\mathrm{P}\mathrm{D}}_{A\mathrm{♯}H}\left(M\right)\le {\mathrm{F}\mathrm{P}\mathrm{D}}_{A\mathrm{♯}H}\left(A\mathrm{♯}H{\otimes }_{A}M\right).

Next, we prove {\mathrm{F}\mathrm{P}\mathrm{D}}_A\left(M\right)\le {\mathrm{F}\mathrm{P}\mathrm{D}}_{A\mathrm{♯}H}\left(M\right). If , there exists an A\mathrm{♯}H-module exact sequence 0\to P_n\to P_{n-1}\to \cdots \to P_0\to M\to 0, where P_n,P_{n-1},\dots ,P_0 are all FP-projective A\mathrm{♯}H-modules. From Theorem 4, they are all FP-projective A-modules, so {\mathrm{F}\mathrm{P}\mathrm{D}}_A\left(M\right)\le n.

At last, we must prove {\mathrm{F}\mathrm{P}\mathrm{D}}_{A\mathrm{♯}H}\left(A\mathrm{♯}H{\otimes }_{A}M\right)\le {\mathrm{F}\mathrm{P}\mathrm{D}}_A\left(M\right). Assuming , there exists an A-module exact sequence 0\to P_n\to P_{n-1}\to \cdots \to P_0\to M\to 0, where P_n,P_{n-1},\dots ,P_0 are all FP-projective A-modules. From reference [7], we can get the exact sequence

0\to A\mathrm{♯}H{\otimes }_A{P}_n\to A\mathrm{♯}H{\otimes }_A{P}_{n-1}\to \cdots \to A\mathrm{♯}H{\otimes }_A{P}_0\to A\mathrm{♯}H{\otimes }_{A}M\to 0,

where

A\mathrm{♯}H{\otimes }_A{P}_n,A\mathrm{♯}H{\otimes }_A{P}_{n-1},\dots ,A\mathrm{♯}H{\otimes }_A{P}_0

are all FP-projective A\mathrm{♯}H-modules, so {\mathrm{F}\mathrm{P}\mathrm{D}}_{A\mathrm{♯}H}\left(A\mathrm{♯}H{\otimes }_{A}M\right)\le n. □

Corollary 3　 Let H be a semisimple weak Hopf algebra which is free on H^l. AssumeA is a left coherent H-module algebra, then \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\mathrm{♯}H\right)\le \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\right).

Corollary 4　 Let H be a semisimple weak Hopf algebra which is free on H^l. Assume A is a left coherent H-module algebra and H^{\ast } is semisimple, then \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\right)=\mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\mathrm{♯}H\right).

Proof　From Corollary 3, we only need to prove \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\right)\le \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\mathrm{♯}H\right). According to reference [8], A\mathrm{♯}H\mathrm{♯}{H}^{\ast } is isomorphic to M_n\left(A\right) and M_n\left(A\right) is isomorphic to A. Based on Corollary 3, we can obtain \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\right)\le \mathrm{L}\mathrm{F}\mathrm{P}\mathrm{D}\left(A\mathrm{♯}H\right). □

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