Gröbner-coherent rings and modules
Let be a graded ring. We introduce a class of graded -modules called Gröbner-coherent modules. Roughly, these are graded -modules that are coherent as ungraded modules because they admit an adequate theory of Gröbner bases. The class of Gröbner-coherent modules is formally similar to the class of coherent modules: for instance, it is an abelian category closed under extension. However, Gröbner-coherent modules come with tools for effective computation that are not present for coherent modules.
Let be a graded ring. The purpose of this paper is to isolate a particular class of graded -modules: the Gröbner-coherent modules. Roughly speaking, these are graded -modules that are coherent as ungraded -modules but not simply by happenstance: they have a “reason” for their coherence related to the grading (that reason basically being an adequate theory of Gröbner bases). Formally, the class of Gröbner-coherent modules over behaves similarly to the class of coherent modules: for instance, it is an abelian category and closed under extensions. However, Gröbner-coherent modules enjoy an advantage over coherent modules in that they come with effective computation procedures. For instance, if is a Gröbner-coherent ring (meaning Gröbner-coherent as a module over itself) and one can compute with finitely presented graded -modules then one can also compute with finitely presented (ungraded) -modules.
Our original motivation for developing the theory of Gröbner-coherence was in relation to our study of divided power algebras [NS]. Let be the divided power algebra in one variable over a noetherian ring . Typically, is not noetherian. We show that is a Gröbner-coherent ring. As a consequence, if one can compute with finitely generated -modules then one can also compute with finitely presented -modules.
2. Background on coherence
Let be a ring. An -module is coherent if it is finitely generated and every finitely generated submodule is finitely presented; clearly, a coherent module is finitely presented. The category of coherent modules is abelian and closed under extensions. The ring is coherent if it is coherent as a module over itself. In this case, every finitely presented module is coherent. Whenever we refer to coherence of a graded ring or module we ignore the grading.
Suppose now that is a graded ring. Then a graded -module is graded-coherent if it is finitely generated and every finitely generated homogeneous submodule is finitely presented. Once again, a graded-coherent module is finitely presented, and the category of graded-coherent modules is abelian and closed under extensions. The ring is graded-coherent if it is graded-coherent as a module, and then every finitely presented graded module is graded-coherent.
3. Gröbner bases
Gröbner bases are typically employed to study ideals in a polynomial ring, or, more generally, submodules of free modules over polynomial rings. However, the ideas apply equally well to study inhomogeneous submodules of graded modules over an arbitrary graded ring. In this section, we develop the theory of Gröbner bases in this greater generality.
Let be a graded ring, and let be a graded -module. We assume is supported in non-negative degrees and that for . Let be a nonzero element of with . We define the degree of , denoted , to be the maximal such that . We define the initial term of , denoted , to be where . Given an (inhomogeneous) submodule of , we define the initial submodule, denoted , to be the homogeneous submodule generated by over all nonzero .
Let be a submodule of a graded module . A collection of elements in is a Gröbner basis for if the generate as an -module. ∎
Let be a submodule of a graded module , let be a collection of elements of , and let be another element of . An expression with is reduced if for all . ∎
Let be a graded -module, let be a submodule, and let be a collection of elements of . Let , for in an index set , be a homogeneous generating set for the module of syzygies of the , and put . Then the following are equivalent:
The form a Gröbner basis for .
Every element of has a reduced expression in terms of the .
The generate and each has a reduced expression in terms of the .
(a) (b). Let be the statement “every element of of degree has a reduced expression in terms of the .” For , the statement is obviously true. Now suppose that is true, and let us prove . Thus let be an element of degree . We then have an expression where . Let . Then and , so by we have an expression with . Thus is reduced expression for , and follows. We conclude that holds for all by induction, and so (b) holds.
(b) (a). Let be a nonzero element of , and let us show that can be generated by the . Using (b), we have an expression where . Let be the set of indices such that has degree equal to . Then . Thus the form a Gröbner basis, and so (a) holds.
(b) (c) is immediate, (b) implies that the generate , and explicitly states that the admit reduced expressions.
(c) (b). Let be given. Let be an expression for in terms of the ’s with minimal. We claim , and so the expression is reduced. Assume not. Write where is the degree piece of (and has smaller degree). Then . This is a homogeneous syzygy of the , and so we have an expression in terms of the ’s: there exist homogeneous elements satisfying such that . We have
Using the reduced expression for the ’s, this gives an expression with , a contradiction. Thus as claimed, and (b) holds. ∎
The above proposition leads to Buchberger’s algorithm for finding a Gröbner basis. Let be a generating set for . The algorithm proceeds as follows:
Compute the set as in the proposition.
If each has a reduced expression, output and terminate.
Otherwise, replace with and return to step (a).
If the algorithm terminates, then its output is a Gröbner basis. If the input set is finite and is graded-coherent, then will remain finite after each step, and so if the algorithm terminates it will produce a finite Gröbner basis. Note that in the usual description of Buchberger’s algorithm, one does not add the ’s to , but their remainders after applying the generalized division algorithm. This is more efficient, but makes no theoretical difference.
In general, analysis of Buchberger’s algorithm can be quite difficult. However, there is one situation that we can analyze easily:
Let be a graded module, a submodule, and a generating set for . Suppose that there exists an integer such that every element admits an expression of the form with . Then Buchberger’s algorithm terminates after at most steps.
This is like (c) (b) from the previous proposition: each step in the algorithm lets us reduce by at least one, so we eventually get down to . ∎
The following proposition shows how Gröbner bases can be used to compute syzygies.
Let be a graded module, let be a submodule, and let be a Gröbner basis for . Let , for , be a homogeneous generating set for the module of syzygies of the , let , and let be a reduced expression. Let be the map defined by . We regard as homogeneous of degree . Let . Then is a Gröbner basis for .
By definition, is independent of . We have , and so as well. Thus . Now, suppose that is an element of . Thus . Let , and write where is the homogeneous degree piece of . Then is a homogeneous syzygy, and so there are homogeneous element with such that . We have
Note that and . Thus the equation above shows that and so is generated by the . This implies that is a Gröbner basis for . ∎
4. Gröbner coherence
The primary definitions are:
Let be a graded ring. We say that a graded -module is Gröbner-coherent if it is graded-coherent and every finitely generated inhomogeneous submodule admits a finite Gröbner basis. ∎
We say that a graded ring is Gröbner-coherent if it is so as a module over itself. ∎
The two conditions in Definition 4.1 (namely, graded-coherent and every submodule admits a finite Gröbner basis) play off of each other nicely, as computations with Gröbner bases often reduce to computations with leading terms, and the graded-coherence ensures that such computations behave well. ∎
Let be a Gröbner-coherent graded module. Then is coherent.
Let be a finitely generated inhomogeneous submodule of . Let be a finite Gröbner basis for , and let be the surjection defined by . It follows from Proposition 3.5, that has a finite Gröbner basis (note that the graded-coherence of implies that the set in Proposition 3.5 is finite), and so is finitely generated. This shows that is finitely presented, completing the proof. ∎
Let be a Gröbner-coherent -module and let be a finitely generated homogeneous submodule. Then and are both Gröbner-coherent.
It follows from basic properties of graded-coherence that and are graded-coherent. If is a finitely generated submodule of , then it is also one of , and thus admits a finite Gröbner basis. Thus is Gröbner-coherent.
Now let be a finitely generated submodule of , and let be its inverse image in , which is finitely generated. Let be a finite Gröbner basis for , and let be the image of in . Let and let be a lift of to with . Let be a reduced expression. Then is also a reduced expression, and so is a finite Gröbner basis for . This shows that is Gröbner-coherent. ∎
Let be graded -modules such that and are Gröbner-coherent. Then is Gröbner-coherent.
It follows from basic properties of graded-coherence that is graded-coherent. Let be a finitely generated submodule, and let be its image in . Let be a finite Gröbner basis for , and let be a lift of . Note that we cannot necessarily pick to have the same degree as . Let . Since is coherent, the kernel of the map is a finitely generated submodule of . Let be a Gröbner basis for .
Now let be an element of , and let be its image in . Let be a reduced expression, so that . We have . Put . This is an element of satisfying . Let be a reduced expression, so that . We thus have an expression where and . This expression shows that the and generate , and it follows from Proposition 3.4 that Buchberger’s algorithm applied to this generating set stops after at most steps, producing a finite Gröbner basis for . Thus is Gröbner-coherent. ∎
A finite direct sum of Gröbner-coherent modules is Gröbner-coherent.
The category of Gröbner-coherent -modules is an abelian subcategory of the category of all graded -modules, and is closed under extension.
Suppose is a map of Gröbner-coherent modules. Since the category of graded-coherent modules is abelian, we know that kernel, cokernel, and image of are graded-coherent. In particular, these objects are finitely generated. Hence , and are Gröbner-coherent by Proposition 4.5. The statement about extensions follows from Proposition 4.6. ∎
Let be Gröbner-coherent and let be a graded -module. Then is Gröbner-coherent if and only if is finitely presented.
Suppose that is finitely presented, and write where is a finite rank free module and is a finitely generated submodule. Then , being a sum of shifts of , is Gröbner-coherent by Corollary 4.7, and so is Gröbner-coherent by Proposition 4.5. Conversely, a Gröbner-coherent module (over any ring) is coherent, and thus finitely presented. ∎
The following proposition shows that one can effectively compute a Gröbner basis for an inhomogeneous submodule of a Gröbner-coherent module using Buchberger’s algorithm, starting from any set of generators.
Let be a Gröbner-coherence -module, let be a finitely generated inhomogeneous submodule of , and let be a set of generators for . Then Buchburger’s algorithm applied to terminates after finitely many steps and yields a Gröbner basis for .
Since is Gröbner-coherent, admits a finite Gröbner basis, say . Since the generate , we can write for scalars . Let be the maximum value of over all and . Now, let be an arbitrary element. Since forms a Gröbner basis, we have a reduced expression . This gives with . We have
Thus Buchberger’s algorithm applied to terminates after at most steps by Proposition 3.4. ∎
5. Further results
The following result gives a potentially useful way of establishing Gröbner-coherence: once graded-coherence is known, Gröbner-coherence is, in a sense, local.
Let be a graded ring and let . Let be a graded-coherent -module. Then is Gröbner-coherent if and only if is Gröbner-coherent for each maximal ideal of .
First, suppose is Gröbner-coherent and let be a maximal ideal of . Let be a finitely generated submodule of . Then there is a finitely generated module of such that . Let be a finite Gröbner basis of and let be the image of under the natural map . Let . Then there is an element of the same degree as such that . Let be a reduced expression. Then is a reduced expression as well. This shows that is a Gröbner basis of .
Conversely, suppose is Gröbner-coherent for each maximal ideal of . Let be a finitely generated inhomogeneous submodule. Suppose that is a finite Gröbner basis for for some maximal ideal . Multiplying by an element if necessary, we may assume that each are of the form for some where is the localization map , and by adding in finitely many elements we may assume that generate . Let be as in Proposition 3.3. Since is graded-coherent can be assumed to be finite. Each admits a reduced expression in terms of the after localizing at . Since there are only finitely many scalars involved, it follows that admits such an expression after inverting a single element . Thus by Proposition 3.3, the image of under the localization define a Gröbner basis for .
Since is Gröbner-coherent for each , the above argument shows that there are generating the unit ideal and sets for such that for each the set is a finite Gröbner basis of and generates . We claim that is a Gröbner basis of . To see this let . Multiplying a reduced expression of in terms of by a large enough power we can obtain a reduced expression for . Since generate the unit ideal there are satisfying . Now is a reduced expression for . This shows that is a Gröbner basis of , completing the proof. ∎
The next results show that Gröbner properties behave well along flat maps.
Let be a flat map of graded rings, let be a graded -module, let be an inhomogeneous submodule, and let be a Gröbner basis for . Then is a Gröbner basis for .
Let , for in an index set , be a homogeneous generating set for the module of syzygies of the . This syzygy module is simply the kernel of the map sending the th basis vector to . Since is flat, it follows that the map has kernel . Thus the relations , for , form a homogeneous generating set for the module of syzygies of the . Let be as in Proposition 3.3. Since the form a Gröbner basis, each admits a reduced expression in terms of the . It follows that also admits a reduced expression in terms of the , and thus the form a Gröbner basis by Proposition 3.3. ∎
Let be a directed system of Gröbner-coherent graded rings such that for all the transition map is flat. Then the direct limit is Gröbner-coherent.
It is a standard fact that is graded-coherent ([So, Proposition 20]). Now let be a finitely generated ideal of . Since is finitely generated, there is some such that is the extension of an ideal along the map . Since this map is flat, we have , and so a Gröbner basis of gives one of by Proposition 5.2. Thus has a finite Gröbner basis, and so is Gröbner-coherent. ∎
A direct limit of noetherian graded rings with flat transition maps is Gröbner-coherent. In particular, a polynomial ring (with any cardinality of variables) over a noetherian coefficient ring is Gröbner-coherent. ∎
6. Relation between different notions of coherence
As we have seen, the following implications hold:
We now show that both implications are strict.
First, let be a coherent ring such that is not coherent. By [So, Proposition 18], such a ring exists: in fact, can be taken to be a countable direct product of ’s. It is easy to show that is graded-coherent (this also follows from [NS, §4.5]). Thus is an example of a ring that is graded-coherent but not coherent.
Next, let be a valuation ring with non-archimedean valuation group. Thus, letting be the valuation on , there exist non-zero with and for all . We claim that is coherent but not Gröbner-coherent. The coherence of follows from [RG, pg. 25] (or see [Gl, Theorem 7.3.3]). Now consider the ideal in . It is easy to see that the degree zero piece of is the -ideal generated by elements of the form for , and is clearly not finitely generated. Hence is not finitely generated, completing the proof of the claim. (This example comes from [Ye, pg. 10].)
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