arXiv:1802.10139v2 [math.AC] 7 Jan 2019
STILLMAN’S CONJECTURE VIA GENERIC INITIAL IDEALS
JAN DRAISMA AND MICHAŁ LASO
´
N AND ANTON LEYKIN
Abstract. Using recent work by Erman-Sam-Snowden, we show that finitely gen-
erated ideals in t he ring of bounded-degree formal power series in infinitely many
variables have finitely gene rate d Gr ¨obner bases relative to the graded reverse lex-
icographic order. We then combine this result with the first author’s work on
topological Noetherianity of polynomial functors to give an algorithmic proof of
the following stateme nt: ideals in polynomial rings generated by a fixed number of
homogeneous polynomials of fixed degrees only have a finite number of possible
generic initial ideals, inde pendently of the number of variables that they involve
and independe ntly of the characteristic of the ground field. Our algorithm outputs
not only a finite list of possible generic initial ideals, but also finite descriptions of
the corresponding strata in the space of coefficients.
1. Introduction
Grevlex series and Gr¨obner bases. Let A be a ring and let R
A
be the A-algebra
of formal power series over A of bounded degree in the infinitely many variables
x
1
, x
2
, . . .. In other words, each element of R
A
is a formal infinite sum
X
α∈N
Z
≥0
,|α|≤d
c
α
x
α
where d is some nonnegative integer and c
α
∈ A for ea ch sequence α = (α
1
, α
2
, . . .)
of nonnegative integers whose sum |α| is (finite and) at most d. Addition and
multiplication are as usual.
We equip the polynomial ring R
A
with the graded reverse lexicographic order
grevlex, in which x
α
> x
β
if either |α| > |β| or |α| = |β| and the last non-zero entry
of α − β is negative. So, for instance, the monomials of degree 3 a re ordered as
follows:
x
3
1
> x
2
1
x
2
> x
1
x
2
2
> x
3
2
> x
2
1
x
3
> x
1
x
2
x
3
> x
2
2
x
3
> x
1
x
2
3
> x
2
x
2
3
> x
3
3
> x
2
1
x
4
> . . .
To remind the reader that this is the only monomial order considered in this paper,
we call the elements of R
A
grevlex series over A. If f is a nonzero element of R,
then lm( f ) denotes the largest monomial that has a nonzero coefficient in f , lc( f )
denotes that coefficient, and lt( f ) = lc( f )lm( f ) is the leading term. The ring R
A
carries a unique topology in which a basis of open neighborhoods of f ∈ R
A
is
given b y all sets {g ∈ R
A
| lm( f − g) < x
α
} as α varies.
Let L be a field. A Gr¨obner basis of an ideal I ⊆ R
L
is a subset B ⊆ I such that for
each h ∈ L there exists an f ∈ B with lm( f )|lm(h). We do not require that B be finite.
As in the classical setting, a Gr¨obner basis B of I generates I as an ideal (Lemma 7).
Our first main result is the following.
JD is partially supported by a Vici grant from the Netherlands Organisation for Scientific Research
(NWO).
Research of AL is supported in part by DMS-1719968 award from NSF.
1
2 JAN DRAISMA AND MICHAŁ LASO
´
N AND ANTON LEYKIN
Theorem 1. For every field L, every finitely generated homogeneous ideal in the ring R
L
has a finite Gr¨obner basis with respect to grevlex.
The ana logous statement certainly does not hold for all monomial orders: in
[Sne98a, Appendix A.2] it is shown that the ideal generated by a generic quadric
and a generic cubic has a non-finitely generated initial ideal relative to the lexi-
cographic order. Theorem 1 implies a positive answer to [Sne98b, Question 7.1];
in that paper a positive answer is given in the case where the ide al is generated
by series
P
|α|=d
i
c
i,α
x
α
, i = 1, . . . , k whose coefficients (c
i,α
)
i∈[k],|α|=d
i
are algebraically
independent over the prime field of L.
The natural question arises whether a Gr¨obner ba sis as in the theorem can be
computed in finite time. A straightforward variant SeriesBuchberger of Buch-
berger’s algorithm shows that this would, indeed, be the case—if only we could
work effectively with infinite series.
Next we focus on the following setting where we can indee d work with such
series. Let S
∞
=
S
n
S
n
be the union of all symmetric groups, and le t S
>n
be the
subgroup of all permutations fixing 1, . . . , n elementwise. Suppose that we are
given a n action of S
>n
0
on A by means of ring automorphisms, and let S
>n
0
act on
the variab les x
1
, x
2
, . . . via πx
i
= x
π(i)
. This action extends to an action of S
>n
0
by
(continuous) ring automorphisms on R
A
via
π
X
α
c
α
x
α
= π
X
α
c
α
Y
i
x
α
i
i
=
X
α
π(c
α
)
Y
i
x
α
i
π(i)
=
X
α
π(c
α
)x
α◦π
−1
.
We call an f ∈ R
A
eventually invariant if there exists an n ≥ n
0
such that π( f ) = f for
all π ∈ S
>n
. To specify an eventually invariant grevlex series we need only a finite
number of coefficients: if f is invariant under S
>n
and has degree d, then S
>n
has only
finitely many orbits on monomials in x
1
, x
2
, . . . of degree at most d—the grevlex-
largest element in ea ch orbit is of the form x
α
where α(n + 1) ≥ α(n+ 2) ≥ . . .. Then f
is uniquely determined by its coefficients on these grevlex-largest representatives
x
α
1
, . . . , x
α
s
. We call
ˆ
f :=
P
s
i=1
c
α
i
x
α
i
the n-representation of f =
P
α
c
α
x
α
. Often we
will suppress n from this notation.
Theorem 2. Suppose that A = L is a field. There exist a finite algorithm that on input
a finite list
ˆ
f
1
, . . . ,
ˆ
f
k
of representations of eventually invariant grevlex series f
1
, . . . , f
k
outputs a finite list
ˆ
g
1
, . . . ,
ˆ
g
l
representing an eventually invariant Gr¨obner basis g
1
, . . . , g
l
of h f
1
, . . . , f
k
i
R
L
.
Stillman’s conjecture. The condition of eventual invariance seems rather restric-
tive, but it is tailored to a proof of the following theorem.
Theorem 3. There exists a finite algorithm that on input k ∈ Z a nd d
1
, . . . , d
k
∈ Z
≥0
outputs a finite sequence S
1
, . . . , S
t
, each S
i
a finite set of monomials in the x
j
, such that
the following holds: For every infinite field K, all n ∈ N, and all homogeneous polynomials
f
1
, . . . , f
k
∈ K[x
1
, . . . , x
n
] of degrees d
1
, . . . , d
k
, respectively, th e generic grevlex initial ideal
of h f
1
, . . . , f
k
i
K[x
1
,...,x
n
]
equals hS
i
i
K[x
1
,...,x
n
]
for some i.
In short: ideals in polynomial rings generated by homogeneous polynomials
of degrees d
1
, . . . , d
k
have only finitely many possible ge ner ic grevlex initial ideals,
independently of the number of va riables. Via [Eis95, Corollary 19.11], which is
based on [BS87], this implies that the projective dimension of an ideal generated
by homogeneous forms of fixed degrees but in an arbitrary number of variables
STILLMAN’S CONJECTURE VIA GENERI C INIT IAL IDEALS 3
and in arbitra ry characteristic is uniformly bounded. This is Stillman’s conjecture
from the title; see [PS09].
This is the fourth proof of Stillman’s conjecture, after the first proof by Ananyan-
Hochster [AH16] and two recent proofs by Erman-Sam-Snowden [ESS18]. Our
proof is the same in spirit as the second proof in the latter paper in that it uses
Draisma’s theorem on topological Noetherianity of polynomial functors [Dra17].
However, unlike the second proof in [ESS18] (but like the first proof there, and
like A nanyan-Hochster’s proof), our theorem yields S
1
, . . . , S
t
that are valid in all
characteristics. Also, our theorem is constructive in the sense that we give an
algorithm for computing the possible initial ideals and the corresponding strata
given by equations and disequations for field characteristics and coefficients of the
input serie s. All these are represented finitely.
In [ESS18] the authors raise the question whether a version over Z of Draisma’s
theorem holds, as this would also make their second proof characteristic-indepen-
dent. We do not settle this question. Instead, the algorithm of Theorem 3 simulates
a generic ideal computation in all characteristics, branching along c onstructible
subsets of Spec Z whenever necessary. We a rgue that, if there were an infinite
branch in this computation, then this bra nch would also be infinite over some
field; and that this would contradict Draisma’s theorem over that field.
In [ESS17] (see also [DES17, Theorem 1 .9]), using Stillman’s c onjecture and
Draisma’s theorem, the same authors estab lish a ge ner alization of Stillman’s con-
jecture to ideal invariants that a re upper semicontinuous in flat families and p re-
served under adding a variable to the polynomial ring. We hav e not pursued the
question to what ex tent (an algorithmic ve rsion of) this generalisation also follows
from our Theorem 3.
Organization. This paper is organized as follows. In Section 2 we prove Theorem 1
using work from [ ESS18]. In Section 3 we use this existence result to prove that a
version of Buchberger’s algorithm for eventually invariant ser ie s terminates; this
yields Theorem 2. In Se c tion 4 we review topological Noetherianity of a specific
polynomial functor, which follows from [Dra17]. Finally, in Section 5 we d erive
Theorem 3 from Theorem 2 and Draisma’s theorem.
2. The existence of finite Gr
¨
obner bases
We will use two results from [ESS18], the first of which is the following.
Theorem 4 (T heorem 1.2 from [ESS18]). If L is perfect, then R
L
contains an (un-
countable) set of homogeneous elements {g
j
: j ∈ J} such t hat the unique L- algebra
homomorphism L[(x
j
)
j∈J
] → R
L
sending x
j
to g
j
is a n L-a lgebra isomorphism.
For each n ∈ Z
≥0
we write R
(n)
L
:= L[x
1
, . . . , x
n
]. There is a natural L-algebra
homomorphism R
L
→ R
(n)
L
, f 7→ f
(n)
that retains only the terms involving only the
variables x
1
, . . . , x
n
. We may think of a degree-at-most-d element of R
L
as a sequence
( f
(0)
, f
(1)
, . . .) in which each f
(n)
is a polynomial in R
(n)
L
of degree at most d such that
f
(n)
is the image of f
(n+1)
under discarding all terms divisible by x
n+1
. Conversely,
R
(n)
L
is an L-subalgebra of R
L
. Observe that, for any f ∈ R
L
and n ∈ Z
≥0
, the image
(lm( f ))
(n)
is either zero or equal to lm( f
(n)
) in the grevlex order on L[x
1
, . . . , x
n
].
4 JAN DRAISMA AND MICHAŁ LASO
´
N AND ANTON LEYKIN
Theorem 5 (Theorem 5.4 from [ESS18]). A sequence g
1
, . . . , g
l
∈ R
L
of homogeneous
elements is a regular sequence in R
L
if and only if g
(n)
1
, . . . , g
(n)
l
is a regular sequence in R
(n)
L
for all n ≫ 0.
The following lemma is straightforward from [ESS18, Section 5], but we include
its proof using the two results ab ove.
Lemma 6. Let f
1
, . . . , f
k
∈ R
L
. Then the natural map between first syzygies
Syz
R
(n+1)
L
( f
(n+1)
1
, . . . , f
(n+1)
k
) → Syz
R
(n)
L
( f
(n)
1
, . . . , f
(n)
k
)
is surjective for all n ≫ 0.
Proof. This surjectivity is not affected by enlarging the field, so we may assume
that L is perfect. By Theorem 4, there ex ist homogeneous g
1
, . . . , g
l
∈ R
L
such that
f
1
, . . . , f
k
∈ L[g
1
, . . . , g
l
] and such that g
1
, . . . , g
l
are par t of a system of variables
for the polynomial ring R
L
. In particular, they are a regular sequence in R
L
, and
hence by Theorem 5 the polynomials g
(n)
1
, . . . , g
(n)
l
are a regular sequence in R
(n)
L
for n ≫ 0. We draw two conclusions from this. First, for n ≫ 0, g
(n)
1
, . . . , g
(n)
l
are
algebraically independent over L, f
(n)
1
, . . . , f
(n)
k
are elements of the polynomial ring
A
(n)
:= L[g
(n)
1
, . . . , g
(n)
l
], and
Syz
A
(n+1)
( f
(n+1)
1
, . . . , f
(n+1)
k
) → Syz
A
(n)
( f
(n)
1
, . . . , f
(n)
k
)
is a bijection. Second, still f or n ≫ 0, R
(n)
is a free module over A
(n)
. Therefore,
Syz
A
(n)
( f
(n)
1
, . . . , f
(n)
k
) ⊆ (A
(n)
)
k
generates Syz
R
(n)
( f
(n)
1
, . . . , f
(n)
k
) ⊆ (R
(n)
)
k
as an R
(n)
-
module. Combining these two statements we find the surjectivity claimed in the
lemma.
Proof of T heorem 1. Let f
1
, . . . , f
k
∈ R
L
be nonzero, homogeneous, and let n ∈ Z
≥0
.
Set I := h f
1
, . . . , f
k
i ⊆ R
L
. Consider a monomial u ∈ lm(I) ∩ R
(n+1)
divisible by
x
n+1
. There exist homogeneous a
1
, . . . , a
k
∈ R
(n)
with deg(a
i
) = deg(u) − deg( f
i
) and
homogeneous b
1
, . . . , b
k
∈ R
(n+1)
with deg(b
i
) = deg(u) − deg( f
i
) − 1 such that
u = lm((a
1
+ b
1
x
n+1
) f
(n+1)
1
+ · · · + (a
k
+ b
k
x
n+1
) f
(n+1)
k
).
Now (a
1
, . . . , a
k
) ∈ Syz
R
(n)
( f
(n)
1
, . . . , f
(n)
k
)—otherwise, the right-hand side would equal
lm(
P
i
a
i
f
(n)
i
), which is not divisible by x
n+1
. By Lemma 6, if n ≫ 0, the syzygy
(a
1
, . . . , a
k
) can be lifted to a syzygy (c
1
, . . . , c
k
) ∈ Syz
R
(n+1)
L
( f
(n+1)
1
, . . . , f
(n+1)
k
). Write
c
i
= a
i
+ x
n+1
b
′
i
for each i. Then
u = lm((b
1
− b
′
1
)x
n+1
f
(n+1)
1
+ · · · + (b
k
− b
′
k
)x
n+1
f
(n+1)
),
but then we see that u/x
n+1
∈ lm(I). Hence for n ≫ 0, lm(I) does not contain minimal
generators divisible by x
n+1
. It follows that for such an n, lm(I) is generated by any
finite genera ting list m
1
, . . . , m
t
of lm(I
(n)
). Now h
1
, . . . , h
t
∈ I such that lm(h
i
) = m
i
form a Gr ¨obner basis of I.
Buchberger’s algorithm for grevlex series
To turn Theorem 1 into an algorithm, we derive a version of Buchberger’s
algorithm.
STILLMAN’S CONJECTURE VIA GENERI C INIT IAL IDEALS 5
Lemma 7 (Division with remainder). Let f
1
, . . . , f
k
∈ R
L
be monic and h ∈ R
L
. Then
there exist q
1
, . . . , q
k
∈ R
L
such that lm(q
i
f
i
) ≤ lm(h) for all i and such that no term of the
remainder h −
P
i
q
i
f
i
is divisible by any lm( f
i
).
In particular, if f
1
, . . . , f
k
is a Gr¨obner basis of the ideal that they generate, then
the remainder must be zero.
Proof. Initialize r := h and q
i
:= 0 for all i. While some term of r is divisible by
some lm( f
i
), pick the grevlex-largest such term cx
α
in r, subtract c (x
α
/lm( f
i
)) f
i
from
r and add cx
α
/lm( f
i
) to q
i
. This does not change terms in q
i
larger than the term just
added, and hence in the product topology on R
k
L
the vector q converges a solution
vector q as desired.
function SeriesBuchberger( f
1
, . . . , f
k
)
assume f
1
, . . . , f
k
∈ R
L
homogeneous grevlex series.
m := 1; B := ∅ (the basis); Q := { f
1
, . . . , f
k
} (the queue);
while Q , ∅ do
while Q contains an f with f
(m)
, 0 do
Q := Q \ { f };
f := f /lc( f );
B := B ∪ { f };
for h ∈ B \ { f } do
γ := lcm(lm(h), lm( f ));
s := (γ/lm(h))h − (γ/lm( f )) f ;
compute a remainder r of s after division by B;
if r , 0 then
Q := Q ∪ {r};
end if;
end for;
end while;
m := m + 1;
end while;
return B;
end function
Proposition 8. Assuming an implementation for addition, multiplication, and division
with remainder of grevlex series, SeriesBuchberger on page 5 terminates after a finite
number of steps and outputs a Gr¨obner basis of the ideal g enerated by the series in the
input.
Proof. Fix any natur al number n. The loops with m ranging from 1 to n really
compute a Gr¨obner basis for I := h f
(n)
1
, . . . , f
(n)
k
i while dragging the tails of the
series along. In particular, these n loops terminate. If an element is added to the
queue Q in the (n + 1)st run of the loop, then this implies that lm(I) ∩ R
(n)
L
does not
generate lm(I). By Theorem 1, this cannot happen infinitely often, so the algorithm
terminates. That the output is, indeed, a Gr¨obner ba sis, follows from the ordinary
Buchberger criterion.