2. Preliminaries

Throughout this article, k is a field, and $S=k[x_1,\ldots ,x_N]$ is a graded polynomial ring, with $\deg (x_i)=1$ for every $i=1,\ldots ,N$ . We will refer to this as the standard grading on S. We let $\mathfrak {m}=(x_1,\ldots ,x_N)$ be the ideal of S generated by elements of positive degree. If $M = \bigoplus _{j\in \mathbb {Z}} M_j$ is a finitely generated $\mathbb {Z}$ -graded S-module, we let $M_{\leqslant j}$ be the graded S-module generated by $\bigoplus _{p \leqslant j} M_p$ . We let $\beta _{i,j}(M) = \dim _k(\operatorname {\mathrm {Tor}}_i^S(M,k)_j)$ be the $(i,j)$ th graded Betti number of M as an S-module, and $t_i(M) = \text {sup}\{j \mid \beta _{i,j}(M) \ne 0\}$ . We also let $\beta _{i,\leqslant j}(M) = \sum _{p \leqslant j}\beta _{i,p}(M)$ and $\beta _i(M) = \sum _{j \in \mathbb {Z}} \beta _{i,j}(M)$ . The socle of M is $\operatorname {soc}(M) = 0:_M \mathfrak {m}$ . Recall that $\operatorname {soc}(M)$ is a finitely generated k-vector subspace of $H^0_{\mathfrak {m}}(M)$ , which is non-zero if and only if $\operatorname {depth}(M)=0$ . The projective dimension of M is defined as

$$\begin{align*}\operatorname{pd}(M) = \text{sup}\{i \in \mathbb{Z}_{\geqslant 0} \mid \beta_{i,j}(M) \ne 0 \text{ for some } j\in \mathbb{Z}\} \end{align*}$$

and the Castelnuovo–Mumford regularity is

$$\begin{align*}\operatorname{reg}(M) = \text{sup} \{t_i(M)-i \mid i \in \mathbb{Z}_{\geqslant 0} \}. \end{align*}$$

Now let $\preccurlyeq $ be a monomial order on S. Given a set of polynomials $f_1,\ldots ,f_s \in S$ as input, which we may assume being monic, we will call an iteration of Buchberger’s algorithm the following procedure:

1. 1. Compute the S-polynomials $S_{ij}$ between $f_i$ and $f_j$ for all $1 \leqslant i < j \leqslant s$ :

   $$\begin{align*}S_{ij} = \frac{\operatorname{in}(f_j)}{\gcd\left(\operatorname{in}(f_i),\operatorname{in}(f_j)\right)} f_i - \frac{\operatorname{in}(f_i)}{\gcd\left(\operatorname{in}(f_i),\operatorname{in}(f_j)\right)} f_j. \end{align*}$$

2. 2. Perform a division algorithm to get a standard expression of each $S_{ij}$ in terms of $f_1,\ldots ,f_s$ . That is, write

   $$\begin{align*}S_{ij} = \sum_{t=1}^s g_tf_t + r_{ij}, \end{align*}$$
   with $\operatorname {in}(g_tf_t) \preccurlyeq \operatorname {in}(S_{ij})$ for all t and either $r_{ij}=0$ or $\operatorname {Supp}(r_{ij}) \cap (\operatorname {in}(f_1),\ldots ,\operatorname {in}(f_s)) = \emptyset $ .

The outputs are the original polynomials $f_1,\ldots ,f_s$ , together with the remainders $r_{ij}$ that are not zero, rescaled so that they are themselves monic.

It is well-known that, given an ideal $I=(f_1,\ldots ,f_s)$ , one can obtain a Gröbner basis of I for the monomial order $\preccurlyeq $ after a finite number of iterations of this algorithm, starting from $f_1,\ldots ,f_s$ and taking as the next input the output of the previous iteration.

The following remark, even if rather straightforward, will be used several times in the rest of the article. For a more general statement, which holds for orders induced by weights and also controls the first syzygies of the initial ideal of I (see [Reference Caviglia and De Stefani4, Proposition 2.4]).

In characteristic zero, Borel-fixed monomial ideals coincide with strongly stable ideals (see, e.g., [Reference Herzog and Hibi11, Section 4.2.2]). In positive characteristic, however, the class of Borel-fixed monomial ideals is strictly larger; for instance, $I=(x_1^p,x_2^p)$ in $S=\mathbb {F}_p[x_1,x_2]$ is Borel-fixed but not strongly stable. If $\preccurlyeq $ is a monomial order, and $I \subseteq S$ is homogeneous, then its generic initial ideal $\operatorname {gin}_{\preccurlyeq }(I)$ is Borel-fixed [Reference Bayer and Stillman2], [Reference Galligo10].

We now recall the Taylor resolution of a monomial ideal (see, for instance, [Reference Eisenbud8, Exercise 17.11]). Let $u_1,\ldots ,u_r$ be monomials in S. For $\Lambda \subseteq \{1,\ldots ,r\},$ we let $u_\Lambda = \operatorname {lcm}(u_i \mid i \in \Lambda )$ . If $a_\Lambda $ is the exponent vector of $u_\Lambda $ , and $S[-a_\Lambda ]$ denotes a free cyclic $\mathbb {Z}^N$ -graded S-module with generator in multi-degree $a_\Lambda $ , then for $0 \leqslant i \leqslant r,$ we let

$$\begin{align*}F_i = \bigoplus_{\substack{\Lambda \subseteq \{1,\ldots,r\} \\|\Lambda|=i}} S[-a_\Lambda]. \end{align*}$$

If $\{e_{\Lambda }\}_{\Lambda \subseteq \{1,\ldots ,r\}}$ denotes a graded free basis of $\bigoplus _{i=0}^r F_i$ , we define differentials $d_i:F_i \to F_{i-1}$ as

$$\begin{align*}d_i(e_{\Lambda}) = \sum_{j \in \Lambda} \mathrm{sign}(j,\Lambda) \frac{u_{\Lambda}}{u_{\Lambda \smallsetminus \{j\}}} e_{\Lambda \smallsetminus \{j\}}, \end{align*}$$

where $\mathrm {sign}(j,\Lambda )$ denotes $(-1)^{s+1}$ if j is the sth element of $\Lambda $ in the natural order. This construction provides a free resolution of $S/I$ , where $I=(u_1,\ldots ,u_r)$ .

Equivalently, we have that $\ell $ is filter regular for M if $\ell \notin \mathfrak {p}$ for all $\mathfrak {p} \in \left (\operatorname {Ass}_S(M) \smallsetminus \{\mathfrak {m}\}\right )$ . Clearly, any M-regular element is filter regular; moreover, we note that if k is infinite, then any sufficiently general linear form is filter regular for M. We note that if $\ell $ is filter regular for M but not M-regular, then necessarily $\operatorname {depth}(M)=0$ .

3. Uniform upper bounds

Let $S=k[x_1,\ldots ,x_N]$ , and M be a finitely generated $\mathbb {Z}$ -graded S-module. We recall that $\operatorname {depth}(M) = \inf \{r \mid H^r_{\mathfrak {m}}(M) \ne 0\}$ , where $\mathfrak {m}=(x_1,\ldots ,x_N)$ . If $\operatorname {depth}(M) = r$ , we let $\alpha (M) = \min \{i+r \mid [\operatorname {soc}(H^r_{\mathfrak {m}}(M))]_i \ne 0\}$ .

Proof. Applying local cohomology to the short exact sequence

gives a long exact sequence

First note that $H^{r-1}_{\mathfrak {m}}(M)=0$ . Applying the functor $\operatorname {Hom}_S(k,-)$ gives an exact sequence on socles:

Since the multiplication by $\ell $ is the zero map on socles, we get graded isomorphisms

$$\begin{align*}\operatorname{soc}\left(H^{r-1}_{\mathfrak{m}}\left(\frac{M}{\ell M}\right)\right)_n \cong \operatorname{soc}\left(H^r_{\mathfrak{m}}\left(M\right)\right)_{n-1} \end{align*}$$

for all $n \in \mathbb {Z}$ . In particular, this yields the desired equality.

Theorem 3.3. Let $I \subseteq S$ be an ideal generated by $\mu $ homogeneous elements, and let $\alpha =\alpha (S/I)$ . Then

$$\begin{align*}\operatorname{pd}(S/I) \leqslant \mu^{2^{\alpha}}. \end{align*}$$

As already noted in Remark 2.1, in place of $\mu ,$ we could actually use the minimal number of generators of I of degree at most $\alpha +1$ . As a consequence of this observation, and using [Reference Caviglia and De Stefani4], we obtain the following estimate for unmixed radical ideals.

Theorem 3.4. Let $I \subseteq S$ be a homogeneous unmixed radical ideal of height h, and let $\alpha = \alpha (S/I)$ . Then

$$\begin{align*}\operatorname{pd}(S/I) \leqslant \left(h^{2^{\alpha+3}-3} \right)^{2^{\alpha}}. \end{align*}$$

Any bound on the projective dimension of an ideal, together with the knowledge of the number and degrees of its generators, allows to obtain a double exponential bound on the Castelnuovo–Mumford regularity [Reference Peeva14, Theorem 29.5].

Corollary 3.5. Let $I \subseteq S$ be an ideal generated by $\mu \geqslant 2$ homogeneous elements of degree at least $2$ and at most D. If we let $\alpha = \alpha (S/I)$ , then

$$\begin{align*}\operatorname{reg}(I) \leqslant D^{2^{\left(\mu^{2^{\alpha}}-2\right)}}. \end{align*}$$

Theorem 3.3 allows us to recover the other implication of [Reference Peeva14, Theorem 29.5] as well: a bound on the Castelnuovo–Mumford regularity in terms of the number and the degrees of generators of an ideal provides one for its projective dimension, as proved by the first named author.

Corollary 3.6. Let $I \subseteq S$ be a homogeneous ideal generated by $\mu $ elements. Then

$$\begin{align*}\operatorname{pd}(S/I) \leqslant \mu^{2^{\operatorname{reg}(S/I)}}. \end{align*}$$

Proof. Let $r=\operatorname {depth}(S/I)$ . It suffices to observe that

$$\begin{align*}\alpha(S/I) \leqslant \max\{i+r \mid [H^r_{\mathfrak{m}}(S/I)]_i \ne 0\} \leqslant \max\{i+j \mid [H^j_{\mathfrak{m}}(S/I)]_i \ne 0 \} = \operatorname{reg}(S/I), \end{align*}$$

and combine this with the first bound in Theorem 3.3.

We now turn our attention to our second main result. In addition to information on the minimal number of generators of an ideal I and their degrees, we require knowledge of the degrees of the generators of the first syzygy modules of sufficiently general hyperplane sections. This allows once again to obtain a double exponential bound for the projective dimension of I in terms of the given data.

Theorem 3.7. Assume that k is infinite. Let $I \subseteq S$ be a homogeneous ideal generated by $\mu $ forms of degree at most D. Let $r=\operatorname {depth}(S/I)$ , and assume that $t_2(S/(I+(\ell _1,\ldots ,\ell _{r+1}))) \leqslant C$ for a sufficiently general choice of linear forms $\ell _1,\ldots ,\ell _{r+1}$ . If we let $\gamma = \max \{C,D\}$ , then

$$\begin{align*}\operatorname{pd}(S/I) \leqslant \mu^{2^{\gamma-1}}. \end{align*}$$

Proof. After going modulo general linear forms $\ell _1,\ldots ,\ell _r$ , we can assume that $\operatorname {pd}(S/I) = N$ . Moreover, after performing a sufficiently general change of coordinates, we may assume that $J:=\operatorname {in}_{\mathrm {revlex}}(I)$ is Borel-fixed and that $\ell _{r+1} = x_N$ . By assumption, we then have that $t_2(S/(I+(x_N))) \leqslant C$ . The graded short exact sequence

induces a long exact sequence of $\operatorname {\mathrm {Tor}}$ modules

which, in turn, gives that

$$\begin{align*}\displaystyle t_1\left(\frac{S}{I:x_N}\right)+1 \leqslant \max\left\{t_2\left(\frac{S}{I+(x_N)}\right),t_1\left(\frac{S}{I}\right)\right\} \leqslant \gamma. \end{align*}$$

Since $I:x_N \ne I$ , if we let $j_0 = \min \{j \mid \left [(I:x_N)/I\right ]_j \ne 0\},$ then $I:x_N$ must contain a minimal generator of degree $j_0$ . As a consequence of the above inequalities, we must have $j_0 \leqslant \gamma -1$ . Recall that $\operatorname {in}_{revlex}(I:x_N) = J:x_N$ (see [Reference Eisenbud8, Section 15.7]). Since passing to initial ideals does not change Hilbert functions, we still have $\left [(J:x_N)/J)\right ]_{j_0} \ne 0$ , and this implies that there is a monomial $u \in (J:x_N) \smallsetminus J$ of degree $j_0$ .

Now let $J':=J_{\leqslant \gamma }$ . Since the Borel group acts on S preserving degrees, and J is Borel-fixed, we have that $J'$ is Borel-fixed as well. If we let $\ell $ be a filter regular linear element for $J'$ , we can perform an upper triangular linear change of coordinates and assume that $\ell =x_N$ . Since $J'$ is Borel-fixed, we conclude that $x_N$ is filter regular for $J'$ . However, $x_N$ is not $J'$ -regular since $ux_N \in J'$ but $u \notin J'$ . It follows that $\operatorname {depth}(S/J')=0$ , that is, $\operatorname {pd}(S/J')=N$ . Now, by Remark 2.1, we have that $\beta _0(J') = \beta _{0,\leqslant \gamma }(J') \leqslant \mu ^{2^{\gamma -1}}$ , and by Remark 2.3, we conclude that $N \leqslant \mu ^{2^{\gamma -1}}$ , as desired.

Following the same strategy as in Corollary 3.5, we obtain the following upper bound on the regularity.

Corollary 3.8. Assume that k is infinite. Let $I \subseteq S$ be a homogeneous ideal generated by $\mu \geqslant 2$ forms of degree at least $2$ and at most D. Let $r=\operatorname {depth}(S/I)$ , and assume that $t_2(S/(I+(\ell _1,\ldots ,\ell _{r+1}))) \leqslant C$ for a sufficiently general choice of linear forms $\ell _1,\ldots ,\ell _{r+1}$ . If we let $\gamma = \max \{C,D\}$ , then

$$\begin{align*}\operatorname{reg}(I) \leqslant D^{2^{\left(\mu^{2^{\gamma-1}}-2\right)}}. \end{align*}$$

In order to state our last main result, we recall some notation from the introduction. For $r \in \mathbb {Z}_{>0}$ , we let

$$\begin{align*}\operatorname{reg}_{\frac{1}{r}}(S/I) = \text{sup}\left\{\operatorname{reg}\left(\operatorname{\mathrm{Tor}}_i^S(S/I,k)\right) - i \ \bigg| \ 0 \leqslant i \leqslant \left\lceil \frac{\operatorname{pd}(S/I)}{r} \right\rceil\right\}. \end{align*}$$

Note that $\operatorname {reg}_1(S/I)$ coincides with $\operatorname {reg}(S/I)$ , while $\operatorname {reg}_{\frac {1}{2}}(S/I)$ only takes into account the first half of the resolution.

Theorem 3.9. Let $I \subseteq S$ be an ideal generated by $\mu $ homogeneous elements. Suppose that $\operatorname {reg}_{\frac {1}{r}}(S/I) \leqslant \delta $ for some $r \in \mathbb {Z}_{>0}$ . Then

$$\begin{align*}\operatorname{pd}(S/I) \leqslant r \cdot \mu^{2^{\delta}}. \end{align*}$$

Proof. Let $s=\lceil \frac {\operatorname {pd}(S/I)}{r} \rceil -1$ . Because $\operatorname {reg}_{\frac {1}{r}}(S/I) \leqslant \delta $ , we have that

$$\begin{align*}\beta_{s+1}(S/I) = \sum_{j \leqslant \delta} \beta_{s+1,j+s+1}(S/I) = \sum_{j \leqslant \delta+1} \beta_{s,j+s}(I). \end{align*}$$

Now fix a monomial order $\preccurlyeq $ , and let $J=\operatorname {in}_{\preccurlyeq }(I)$ . By upper semi-continuity, we have that $\beta _{s,j}(I) \leqslant \beta _{s,j}(J)$ for all $j \in \mathbb {Z}$ , and it follows that

$$\begin{align*}\beta_s(I) = \sum_{j \leqslant \delta+1} \beta_{s,j+s}(I) \leqslant \sum_{j \leqslant \delta+1} \beta_{s,j+s}(J) \leqslant \sum_{j \leqslant \delta+1} \beta_{s,j+s}(J_{\leqslant \delta+1}) \leqslant \beta_{s}(J_{\leqslant \delta+1}). \end{align*}$$

The second inequality above is a consequence of the fact that, to compute $\beta _{s,j+s}(J) = \dim _{k}(\operatorname {\mathrm {Tor}}_s^S(J,k)_{j+s})$ , one only needs to know J in degrees $j-1$ and j. In particular, we have that $\operatorname {pd}(S/J_{\leqslant \delta +1}) \geqslant \operatorname {pd}(S/I) \geqslant s+1$ . Since I is generated by $\mu $ elements, by Remark 2.1, we obtain that $\beta _{0,\leqslant \delta +1}(J) =\beta _{0,\leqslant \delta +1}(J_{\leqslant \delta +1}) \leqslant \mu ^{2^{\delta }}$ . We conclude by Remark 2.3 that $s+1 \leqslant \mu ^{2^{\delta }}$ , and hence $\operatorname {pd}(S/I) \leqslant r \cdot (s+1) \leqslant r \cdot \mu ^{2^{\delta }}$ .

Corollary 3.10. Let $I \subseteq S$ be an ideal generated by $\mu \geqslant 2$ homogeneous elements. Suppose that $\operatorname {reg}_{\frac {1}{r}}(S/I) \leqslant \delta $ for some $r \in \mathbb {Z}_{>0}$ . Then

$$\begin{align*}\operatorname{reg}(I) \leqslant (\delta+1)^{2^{\left(r\mu^{2^{\delta}}-2\right)}}. \end{align*}$$