A CONVENIENT SOURCE OF HOMOLOGICAL EXAMPLES OVER ARTINIAN RINGS

B. ZIMMERMANN HUISGEN Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

ABSTRACT

We introduce a simplified quiver for artinian rings R with J(R)^{2} = 0 which allows finding various homological dimensions without computational effort. For these rings, we prove that the categories of finitely generated modules of finite projective dimension are contravariantly finite in R-mod, resp., mod-R, and we show how to determine the projective dimensions of the corresponding minimal approximations of the simple modules. For arbitrary artinian rings, we present a description of those projective modules which appear as direct summands of higher syzygies.

1. Introduction and notation

At the 1991 summer workshop on homological aspects of finite dimensional algebras, M. Auslander posed the question whether, given an Artin algebra R, the left and right finitistic dimensions of R coincide provided that both the categories P^{∞}(_{R}R) and P^{∞}(R_{R}) of finitely generated left and right R-modules of finite projective dimension are contravariantly finite in R -mod and mod- R, respectively. Recall that P^{∞}(_{R}R) is contravariantly finite in R -mod, in the sense of Auslander-Smalø [2] and Auslander-Reiten [1], in case each finitely generated left R-module X has a right P^{∞}(_{R}R)-approximation, i.e., a homomorphism f : A → X with A in P^{∞}(_{R}R) such that each map from an object of P^{∞}(_{R}R) to X factors through f. One of the purposes of this note is to describe a graphically intuitive road to a source of homological examples that can be evaluated without any computational effort. In particular, this class contains specimens answering the above question in the negative.

We start by exhibiting a speedy optical method for determining various homological dimensions, including the finitistic dimensions, for left artinian rings with zero radical square. This is done by replacing the quiver Γ(R) of such a ring R by a simplified quiver Γ̃(R) which, e.g., reduces the computation of the projective dimension of a minimal right P^{∞}(_{R}R)-approximation of a simple left R-module to a count of the lengths of paths leaving the corresponding vertex in Γ̃(R). This last application

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16E10,16P10,16P20.

This research was partially supported by a National Science Foundation grant.

of Γ̃(R) is relevant inasmuch as we prove that right P^{∞}(_{R}R)-approximations always exist for this class of rings. In other words, the P^{∞}-categories are contravariantly finite whenever R is artinian with vanishing radical square.

In the third section we upgrade a criterion of Bass [3] and Auslander-Reiten [1] for the vanishing of the left finitistic dimension of a right artinian ring R (here no restriction is placed on the radical). More specifically, we characterize the projective summands of higher syzygies as follows: an indecomposable projective left module Re is isomorphic to a direct summand of some syzygy Ω^{d}(M), where d ≥ 1 and M is an R-module, if and only if the simple right module eR/eJ does not occur in the right socle of R; here e is a primitive idempotent and J the Jacobson radical of R. For the case of a path algebra modulo relations, we then give a graphtheoretic interpretation of this criterion which will permit us to recognize projective summands of syzygies at a glance.

Throughout R will be an (at least one-sided) artinian ring with Jacobson radical J, and e_{1}, …, e_{n} will be a complete set of primitive idempotents. Recall that the left quiver Γ(R) of R has vertices e_{1}, …, e_{n} and m arrows from e_{i} to e_{j} if m is the multiplicity of the simple left R-module S_{j} = Re_{j}/Je_{j} in Je_{i}/J^{2}e_{i}. Whenever we refer to a path in a quiver, we mean a directed path, and given paths p, q such that the endpoint of p coincides with the starting point of q, the composite path is denoted qp.

The full subcategory of R -mod having as objects the modules of finite projective dimension will be abbreviated by P^{∞}(R) = P^{∞}(_{R}R); the definition of P^{∞}(R_{R}) is analogous. The left (little) finitistic dimension of R, finally, is

l fin dim R = sup{p dim M | M is an object of P^{∞}(_{R}R)},

where p dim M denotes the projective dimension of M. Observe that, when J^{2} = 0, we have

l fin dim R = 1 + sup{p dim S | S ⊆ J is a simple module in P^{∞}(_{R}R)}.

2. The homology of left artinian rings with J^{2} = 0

The following lemma will make it a cinch to find l fin dim R on the basis of Γ(R).

Lemma 1. Let R be left artinian with J^{2} = 0, and let i ∈ {1, …, n}. Then the projective dimension of S_{i} is finite if and only if the quiver Γ(R) contains no (directed) path of length ≥ 0 starting in the vertex e_{i} and ending on an oriented cycle.

In case this condition is satisfied, p dim S_{i} equals the length of the longest path of Γ(R) starting in e_{i}.

Proof. Suppose first that Γ(R) contains a path starting in e_{i} and ending in a vertex that belongs to an oriented cycle. Let p be a path with this property having minimal length ℓ ≥ 0. We show p dim S_{i} = ∞ by induction on ℓ. If ℓ = 0, then the vertex e_{i} sits on an oriented cycle, of length r ≥ 1 say. Since clearly S_{j} is isomorphic to a direct summand of Ω^{1}(S_{i}) if and only if there exists an arrow e_{i} → e_{j}, we see that S_{i} is isomorphic to a direct summand of Ω^{r}(S_{i}), and as r ≥ 1 we conclude that indeed p dim S_{i} = ∞. Now let ℓ ≥ 1 and write p = p′α, where α is an arrow and length(p′) ≥ 0. If the endpoint of α is e_{k}, we know that S_{k} is a direct summand of Ω^{1}(S_{i}). Since length(p′) = ℓ − 1, the induction hypothesis yields p dim S_{k} = ∞ and thus also p dim S_{i} = ∞.

Now suppose that Γ(R) contains no path starting in e_{i} and ending on an oriented cycle. We prove p dim S_{i} = d by induction on the number

d = max{length(q) | q is a path in Γ(R) starting in e_{i}}.

If d = 0, our claim is trivial, so suppose that d ≥ 1, and let i(1), …, i(m) be the endpoints of arrows starting in e_{i}. Then Ω^{1}(S_{i}) is a direct sum of copies of the simple modules S_{i(1)}, …, S_{i(m)} and, by the induction hypothesis, p dim(⨁_{1≤j≤m} S_{i(j)}) = d − 1. This completes the proof. □

For less confusing optics, we consider the following simplified (left) quiver Γ̃(R) of R. To obtain Γ̃(R) from Γ(R), delete all those arrows of Γ(R) which can be extended to a path terminating on an oriented cycle; in particular, this requires deletion of all arrows that themselves belong to an oriented cycle.

Proposition 2. If R is left artinian with J^{2} = 0, then l fin dim R equals the length of the longest path of the simplified quiver Γ̃(R).

Proof. In view of the obvious fact that

l fin dim R = 1+sup{p dim S | S ⊆ J is a simple left R-module with p dim S < ∞},

the claim follows from Lemma 1. □

Our first example illustrates the merits of considering the simplified quiver.

Example 3. Let k be a field and R the path algebra kΓ/I, where Γ is the quiver and I the ideal of kΓ generated by all paths of length 2. Then the simplified quiver Γ̃(R) is

and l fin dim R = 6.

If we reverse the arrow 4 → 6 in Γ and let R′ be the corresponding algebra with vanishing radical square, then the simplified quiver Γ̃(R′) is

and l fin dim R′ = 2. □

Suppose that R is left artinian and X a finitely generated left R-module. Recall the definition of a right P^{∞}(R)-approximation of X from the introduction. A P^{∞}(R)-approximation f : A → X is minimal provided A has minimal composition length with this property; in case of existence, minimal P^{∞}(R)-approximations are unique up to isomorphism [1]. If right P^{∞}(R)-approximations exist globally, i.e. for all objects in R -mod, one says that P^{∞}(R) is contravariantly finite in R -mod.

In view of an example of Igusa, Smalø, Todorov [6], contravariant finiteness of P^{∞}(R) in R -mod may fail, even for finite dimensional algebras with J^{3} = 0. We include this example to illustrate its simplicity.

Example 4. (See [6]) Let Γ be the quiver

k any field and R = kΓ/⟨γβ, αγ, βγ⟩, i.e., the graphs of the indecomposable projective left R-modules (in the sense of [4,5]) are:

Then Re_{1}/Je_{1} does not have a right P^{∞}(R)-approximation by [6], and hence P^{∞}(R) is not contravariantly finite in R -mod. Moreover, using [4, Section 5] we immediately compute that l fin dim R = 1. On the other hand, it is clear from the graphs above that

Soc(_{R}R) = (Re_{1}/Je_{1})^{2} ⨁(Re_{2}/Je_{2}),

whence r fin dim R = 0 (see Section 3). In particular, P^{∞}(R_{R}) is contravariantly finite in mod- R. □

By contrast, contravariant finiteness of P^{∞}(R) is guaranteed if J^{2} = 0. In fact, assuming J^{2} = 0, we can explicitly describe the minimal right P^{∞}(R)-approxima-tion of an arbitrary finitely generated left R-module X.

Start with the obvious fact that any semisimple left R-module Y has a unique decomposition Y = Y_{fin} ⨁ Y_{inf}, where Y_{fin} is the sum of the simple submodules of Y having finite projective dimension and Y_{inf} is the sum of the simple submodules of infinite projective dimension.

Theorem 5. If R is left artinian with J^{2} = 0, then P^{∞}(R) is contravariantly finite in R -mod. The minimal right P^{∞}(R)-approximation of a finitely generated left R-module X is the canonical map

\overline{f} : P/(Ω^{1}(X))_{fin} → X,

where 0 ⟶ Ω^{1}(X) ⟶ P ⟶^{f} X ⟶ 0 is a projective cover of X.

In particular, the canonical maps A_{i} = Re_{i}/(Je_{i})_{fin} → Re_{i}/Je_{i} = S_{i} are the minimal right P^{∞}(R)-approximations of the simple modules S_{i}. An alternate description of A_{i} is as follows:

A_{i} = Re_{i}/( ⨁_{1≤j≤m(i)} Rα_{ij}),

where α_{i1}, …, α_{i,m(i)} are precisely the arrows departing from e_{i} in Γ̃(R), and p dim A_{i} is the length of the longest path with starting point e_{i} in Γ̃(R).

Proof. To back up the first assertion, let g ∈ Hom(B, X), where B is any finitely generated module of finite projective dimension. For a projective cover

0 ⟶ Ω^{1}(B) ⟶ Q ⟶^{h} B ⟶ 0

this means that Ω^{1}(B) = Ker(h) ⊆ (JQ)_{fin}. In view of the projectivity of Q we moreover obtain a commutative diagram

Q ⟶^{ℓ}  P  ⟶^{f} X
∥  ^{canon}↓  ∥
Q ⟶^{\overline{ℓ}} P/(Ω^{1}(X))_{fin} ⟶^{\overline{f}} X
∥  ∥
Q ⟶^{h}  B  ⟶^{g} X

Now ℓ(Ω^{1}(B)) ⊆ ℓ((JQ)_{fin}) ⊆ (JP)_{fin}, and ℓ(Ω^{1}(B)) ⊆ Ker(f) = Ω^{1}(X), since fℓ(Ω^{1}(B)) = gh(Ω^{1}(B)) = 0. We conclude that ℓ(Ω^{1}(B)) ⊆ (JP)_{fin} ∩ Ω^{1}(X) = (Ω^{1}(X))_{fin}. Consequently, \overline{ℓ}(Ω^{1}(B)) = 0, where \overline{ℓ} is as indicated in the above commutative diagram, and so \overline{ℓ} = g′h for a suitable homomorphism g′ : B → P/(Ω^{1}(X))_{fin}. Since gh = \overline{f}\overline{ℓ} = \overline{f}g′h and h is onto, it follows that g = \overline{f}g′. Thus the canonical map \overline{f} : P/(Ω^{1}(X))_{fin} → X is indeed a right P^{∞}(R)-approximation of X.

To see that the composition length of P/(Ω^{1}(X))_{fin} is minimal among the lengths of the right P^{∞}(R)-approximations of X, let ϕ : A → X be any right P^{∞}(R)- approximation. On one hand, the projective cover f : P → X then factors through ϕ, and hence ϕ is onto. On the other hand, ϕ factors through \overline{f} : P/(Ω^{1}(X))_{fin} → X, i.e., there exists g ∈ Hom_{R}(A, P/(Ω^{1}(X))_{fin}) such that \overline{f}g = ϕ. But \overline{f} having superfluous kernel, this shows us that g is in turn onto, and hence the length of P/(Ω^{1}(X))_{fin} cannot exceed that of A. This proves our first claim.

The supplementary statements about A_{i} are now immediate consequences of Lemma 1. □

Let us compute the projective dimensions of the minimal right P^{∞}(R)- approximations A_{i} → S_{i} of the simple left modules S_{i} in Example 3. In view of the simplified quiver exhibited there, Theorem 5 immediately yields that p dim A_{i} = 0 for i ∈ {1, 2, 3, 6, 13, 14, 17}, while p dim A_{4} = 1, p dim A_{5} = 2, p dim A_{7} = 1, p dim A_{8} = 2, p dim A_{9} = 3, p dim A_{10} = 4, p dim A_{11} = 4, p dim A_{12} = 5, p dim A_{15} = 6, p dim A_{16} = 1.

Example 6. For any positive integer n, we now present a finite dimensional algebra R such that P^{∞}(R) ⊆ R -mod and P^{∞}(R_{R}) ⊆ mod- R are contravariantly finite, while l fin dim R − r fin dim R = n. Let k be a field and R = kΓ/I, where Γ is the quiver and I the ideal of the path algebra kΓ generated by all paths of length 2. Then both contravariant finiteness conditions are satisfied by Theorem 5. Moreover, the simplified left quiver Γ̃(R) is

1 ⟶ 2 ⟶ ⋯ ⟶ n ⟶ n + 1,

whence l fin dim R = n by Lemma 3, and the simplified right quiver of R is

1 2 ⋯ n n + 1,

whence r fin dim R = 0. □

This graphical method can be regarded as a supplement to the algorithms presented in [4] and [5].

3. Projective syzygies

Definition. Call a left R-module P a projective syzygy if P is projective and isomorphic to a direct summand of a syzygy Ω^{k}(M) for some k ≥ 1 and R-module M.

Note that, given any non-projective module M of finite projective dimension k, Ω^{k}(M) is a projective syzygy. The following characterization of projective syzygies was inspired by Bass [3, Theorem 6.3].

Proposition 7. For a \underline{right} artinian ring R and a primitive idempotent e ∈ R the following conditions are equivalent:

(1) Re is a projective syzygy.

(2) Re embeds into a finite direct sum J ⨁ ⋯ ⨁ J.

(3) The simple module eR/eJ does not occur in the right socle, Soc(R_{R}), of R.

Proof. Without loss of generality, R is basic and e = e_{1}.

‘(1)⟺(2)’ is obvious.

‘(2)⟹(3)’. By (2), there exist a positive integer t and an element a = (a_{1}, …, a_{t}) in J^{(t)} with e_{1}a = a and l. ann_{R}(a) = l. ann_{R}(e_{1}) = ⨁_{2≤j≤n} Re_{j}. Observe that the right ideal A = ∑_{2≤j≤n} e_{j}R + ∑_{1≤i≤t} a_{i}R of R is contained in the maximal right ideal B = ⨁_{2≤j≤n} e_{j}R ⨁ e_{1}J because a_{i} ∈ e_{1}J for 1 ≤ i ≤ t. Next note that occurrence of a copy of e_{1}R/e_{1}J in Soc(R_{R}) is tantamount to l. ann_{R} B ≠ 0; so, to prove non-occurrence of e_{1}R/e_{1}J, it suffices to show that l. ann_{R} A = 0. But for any r ∈ l. ann_{R} A we have r(a_{1}, …, a_{t}) = 0 and hence re_{1} = 0, which, in view of re_{2} = ⋯ = re_{n} = 0, implies r = 0.

‘(3)⟹(2)’. If B is as in the preceding paragraph, then (3) says l. ann_{R} B = 0. This clearly implies that l. ann_{Re_{1}}(e_{1}J) = 0. Since R is right artinian, e_{1}J = ∑_{j=1}^{s} c_{j}R for suitable elements c_{j} ∈ e_{1}J. Then c = (c_{1}, …, c_{s}) ∈ J^{(s)} satisfies e_{1}c = c and l. ann_{Re_{1}} c = 0. But this means that the map Re_{1} → Rc ⊆ J^{(s)} which sends r to rc is an embedding. □

Since, clearly, l fin dim R = 0 is equivalent to the non-existence of nonzero projective left syzygies, the following result of Bass [3] and Auslander-Reiten [1] is an immediate consequence.

Corollary 8. If R is right artinian, the following statements are equivalent: (a) l fin dim R = 0.

(b) Each simple right R-module occurs in Soc(R_{R}). □

For quick detection of projective syzygies over path algebras modulo relations, the following reformulation of Proposition 7 is helpful.

Observation 9. Let R = kΓ/I be a finite dimensional path algebra modulo relations over a field k and e_{i} ∈ R a primitive idempotent which we identify with a vertex of Γ. Then the following are equivalent:

(1) Re_{i} is a projective syzygy.

(2) Whenever x is a linear combination in kΓ of paths starting in e_{i} such that xα ∈ I for all arrows α, then x ∈ I. □

We use this observation to compute the projective syzygies in a number of examples.

Examples 10. Let Γ be the quiver

and R_{j} = kΓ/I_{j} for j = 1, …, 5, where

I_{1} = ⟨γβα_{1} − ϵδα_{1}⟩
I_{2} = ⟨γβα_{i} − ϵδα_{i} | i = 1, 2⟩
I_{3} = ⟨γβ − ϵδ⟩
I_{4} = ⟨γβα_{1}, γβα_{2}⟩
I_{5} = ⟨γβα_{1}⟩.

For all of the rings R = R_{j}, j = 1, …, 5, the left modules Re_{3}, Re_{4}, Re_{5} are projective syzygies by Observation 9, while Re_{1} is never a projective syzygy. Using that same observation, we see that the module Re_{2} is a projective syzygy if R is R_{1}, R_{3}, or R_{5}, but not if R is R_{2} or R_{4}. □

6. References

1. M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Advances in Math. 86 (1991) 111-152.

2. M. Auslander and S. O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980) 61-122.

3. H. Bass, Finitistic dimensions and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960) 466-488.

4. B. Zimmermann Huisgen, Predicting syzygies over monomial relation algebras, manuscripta math. 70 (1992) 157-182.

5. , Homological domino effects and the first finitistic dimension conjecture, Invent. Math. 108 (1992) 369-383.

6. K. Igusa, S. O. Smalø, and G. Todorov, Finite projectivity and contravariant finiteness, Proc. Amer. Math. Soc. 109 (1990) 937-941.