# Engineering high Chern number insulators

Sungjong Woo<EMAIL_ADDRESS>Department of Physics, Pukyong National
University, Pusan, Republic of Korea  Seungbum Woo<EMAIL_ADDRESS>Department of Physics and Astronomy, Seoul National University, Seoul,
Republic of Korea  Jung-Wan Ryu<EMAIL_ADDRESS>Center for Theoretical
Physics of Complex Systems, Institute for Basic Science, Daejeon, Republic of
Korea  Hee Chul Park<EMAIL_ADDRESS>Department of Physics, Pukyong National
University, Pusan, Republic of Korea

###### Abstract

The concept of Chern insulators is one of the most important buliding block of
topological physics, enabling the quantum Hall effect without external
magnetic fields. The construction of Chern insulators has been typically
through an guess-and-confirm approach, which can be inefficient and
unpredictable. In this paper, we introduce a systematic method to directly
construct two-dimensional Chern insulators that can provide any nontrivial
Chern number. Our method is built upon the one-dimensional Rice-Mele model,
which is well known for its adjustable polarization properties, providing a
reliable framework for manipulation. By extending this model into two
dimensions, we are able to engineer lattice structures that demonstrate
predetermined topological quantities effectively. This research not only
contributes the development of Chern insulators but also paves the way for
designing a variety of lattice structures with significant topological
implications, potentially impacting quantum computing and materials science.
With this approach, we are to shed light on the pathways for designing more
complex and functional topological phases in synthetic materials.

##  I Introduction

Topological physics, which has been one of the most hot topics in condensed
matter physics in the last decade, historically began with quantum Hall effect
based on the electronic Landau levels v. Klitzing et al. (1980). Haldane
predicted quantum Hall effect without external magnetic field on a graphene,
so called a Chern insulator Haldane (1988), which attracted huge attention
after topological insulators were found Kane and Mele (2005); Bernevig and
Zhang (2006); Hasan and Kane (2010). Although Haldane’s quantum Hall effect on
a graphene is much weaker than was predicted, it has its significance in
understanding topological materials. It is the very basic building block for
the theory of band topology. Differently from the quantum Hall effect under
external magnetic field, the Haldane’s original model supports Chern number
$N_{\rm C}$ only $\pm 1$ or 0. Chern insulators with Chern number larger than
one will be very useful for quantum technology such as quantum computing based
on electronics, photonics, and so on. Graphene-like hexagonal lattice with
long range hoppings was theoretically proposed for a high Chern number
insulator Sticlet and Piéchon (2013). Recently, multilayer quantum anomalous
Hall system was also proposed and experimentally realized for high Chern
number insulators with various Chern numbers et. al. (2020a, b); Wang and Li
(2021); Wenxuan Zhu and Pan (2022).

Typically, to make a new Chern insulator in a tight-binding model, one has to
guess the bulk structure of a system and the complex hopping strengths to find
out the Chern number only after system-dependent Berry flux integration. In
this work, we propose a systematic procedure to build Chern insulators with
any Chern number. The procedure is based on the Rice-Mele(RM) model Rice and
Mele (1982), a generalized version of the Su-Schrieffer-Heeger(SSH) model W.
P. Su and Heeger (1979).

##  II Theoretical concept

Figure 1(a) represents RM model with unit cell size $a$, whose Hamiltonian is

$\displaystyle\begin{aligned}
H=\sum_{R}&\left[{\varepsilon}\left(c_{R,A}^{\dagger}c_{R,A}-c_{R,B}^{\dagger}c_{R,B}\right)\right.\\\
&+\left.(t+\delta t)\left(c_{R,A}^{\dagger}c_{R,B}+h.c.\right)+(t-\delta
t)\left(c_{R+a,A}^{\dagger}c_{R,B}+h.c.\right)\right].\end{aligned}$ (1)

Here, $R=na$ represents the cell vectors with $n=\cdots,-2,-1,0,1,2,\cdots$
and $a$ being the cell size. The value of $2\varepsilon$ represents the on-
site energy difference between $A$ and $B$ sites, $t+\delta t$ and $t-\delta
t$ are inter- and intra-cell hopping constants, $c_{R,A(B)}^{\dagger}$ is the
creation operator on the site $A$($B$) in the unit cell at $R$, and $h.c.$
represents hermitian conjugate. With Fourier transformation,
$c_{R,A(B)}^{\dagger}=\sqrt{\frac{1}{G}}\int_{0}^{G}dk~{}c_{k,A(B)}^{\dagger}e^{-ik(R+d_{A(B)})}$,
together with positions of two sites within a unit cell, $d_{A}=\frac{a}{4}$
and $d_{B}=\frac{3a}{4}$, the Hamiltonian can be rewritten as a Bloch form of
$H=\int_{0}^{G}dk~{}\phi_{k}^{\dagger}\mathcal{H}(k)\phi_{k}$ with

$\displaystyle\mathcal{H}(k)=2t\cos\left(\frac{ka}{2}\right)\sigma_{1}-2\delta
t\sin\left(\frac{ka}{2}\right)\sigma_{2}+{\varepsilon}\sigma_{3},$ (2)

where $G=\frac{2\pi}{a}$ is the reciprocal lattice vector,
$\phi^{\dagger}\equiv(c_{k,A}^{\dagger},c_{k,B}^{\dagger})$ and $\sigma_{i}$s
are the Pauli matrices (see Appendix for details). Differently from the SSH
model, the Zak phase,
$\gamma=i\int_{0}^{G}dk~{}\bra{u(k)}\ket{{\partial_{k}}u(k)}$ Zak (1989), can
have all the values from 0 to $2\pi$ up to multiple of $2\pi$ phase ambiguity,
owing to the energy difference between $A$ and $B$ sites. Here, $\ket{u(k)}$
is the eigenvector of $\mathcal{H}(k)$. We calculate the Zak phase numerically
by obtaining $\ket{u(k)}$ and using

$\displaystyle\gamma=-\imaginary\log\prod_{j=0}^{N-1}\bra{u(k_{j})}\ket{u(k_{j+1})},$
(3)

instead of its differential form, where, $k_{j}=\frac{G}{N}j$ Resta (2000).
Eq. (3) does not need the differentiability of $\ket{u(k)}$ in terms of $k$.
One has to keep in mind that $\ket{u\left(G\right)}=e^{-iG\hat{x}}\ket{u(0)}$
with $\hat{x}$ being the position operator that has two values, $d_{A}$ or
$d_{B}$.

Figure 1: (a) The unit cell of Rice-Mele model (b) A colored contour plot of
Zak phase of Rice-Mele model as a function of two parameters, on-site energy
difference between two subsites, $\varepsilon$, and half of the difference
between neigbouring hopping constants, $\delta t$.

Figure 1(b) shows $\gamma$ calculated numerically. It demonstrates that
$\gamma\approx\pi+\arg{\left({2\delta t+i2\varepsilon}\right)}$ for small
$\delta t$ and $\varepsilon$ compared to $t$, where $\arg(z)$ means the
argument of a complex number $z$. A pair of parameters can always be found for
any given $\gamma$ as

$\displaystyle\delta t\approx-2t^{\prime}\cos(\gamma),\hskip
14.22636pt\varepsilon\approx-2t^{\prime}\sin(\gamma),$ (4)

with an arbitrary parameter $t^{\prime}(\ll t)$. The condition $t^{\prime}\ll
t$, that is required for the approximation of Eq. (4), is in fact not
necessary for the topological implication since what only matters is the
$2\pi$ phase winding around the sigular point in Fig. 1(b). The extra
multiples of two are for the simplicity of the formulas in the following part.
Rice-Mele model is a one-dimensional (1D) system that can modulate $\gamma$
with two control parameters, $\delta t$ and $\varepsilon$. For example,
positive $\delta t$ with zero $\varepsilon$ yields $\gamma=\pi$, so that the
expectation value of position operator becomes
$\expectationvalue{x}=\frac{\gamma}{G}=\frac{a}{2}$; it represents the middle
of the strong bond in Fig. 1(a). This case corresponds to the trivial
topological phase with no edge states. In many literatures, trivial phase
corresponds to $\gamma=0$. It, however, is only a matter of convention of
setting the positions of two subsites $A$ and $B$ while $\gamma=0$ for trivial
topological phase can be achieved from our formalism by setting
$d_{A}=-\frac{a}{4}$ and $d_{B}=\frac{a}{4}$. The reason of choice of our
convention is to confine two sites within a unit cell.

On the other hand, a Chern insulator is a two-dimensional (2D) system with
$\gamma$ along one axis winding the two-torus of the first Brillouin zone with
integral number of times. For a 2D system in a rectangular coordinate system,
the matrix representation of a Bloch Hamiltonian can be written as
$\mathcal{H}(k_{x},k_{y})$. Let us assume a rectangular lattice system with
two orthogonal unit vectors of ${\bf u}_{x}=(a_{x},0)$ and ${\bf
u}_{y}=(0,a_{y})$, so that $0\leq k_{x}<G_{x}(=\frac{2\pi}{a_{x}})$ and $0\leq
k_{y}<G_{y}(=\frac{2\pi}{a_{y}}$). For a fixed $k_{x}$, one can calculate
$\gamma$ along the $y$ axis,
$\gamma_{y}(k_{x})=i\int_{0}^{G_{y}}dk_{y}~{}\bra{u}{{\partial
k_{y}}}\ket{u}$. If this value varies by $2\pi N_{\rm C}$ as $k_{x}$ changes
by $G_{x}$, the system is topologically classified to have Chern number of
$N_{\rm C}$ Hasan and Kane (2010).

##  III results and discussion

We propose a procedure to construct a 2D Chern insulator using 1D RM model by
building up $2\pi N_{\rm C}$ change of $\gamma$ along an additional dimension
of Bloch momentum space; A 2D system satisfying $\gamma_{y}(k_{x})=N_{\rm
C}k_{x}a_{x}$ makes $\gamma_{y}$ change by $2\pi N_{\rm C}$ as $k_{x}$ changes
by $G_{x}$. Using Eq. (4), $\delta t$ and $\varepsilon$ for a given $k_{x}$
can be set to be

$\displaystyle\delta t=-2t^{\prime}\cos\left(N_{\rm C}k_{x}a_{x}\right),\hskip
14.22636pt\varepsilon=-2t^{\prime}\sin\left(N_{\rm C}k_{x}a_{x}\right),$

and hence Eq. (2), with $k\rightarrow k_{y}$, can be extended to 2D
Hamiltonian with Chern number $N_{\rm C}$ such that $H_{N_{\rm C}}=\int_{\rm
BZ}d^{2}k~{}\phi_{\bf k}^{\dagger}\mathcal{H}_{N_{\rm C}}({\bf k})\phi_{\bf
k}$, where $\phi^{\dagger}\equiv(c_{{\bf k},A}^{\dagger},c_{{\bf
k},B}^{\dagger})$, ${\bf k}=(k_{x},k_{y})$ and

$\displaystyle\begin{aligned} \mathcal{H}_{N_{\rm C}}&({\bf
k})=2t\cos\left(\frac{k_{y}a_{y}}{2}\right)\sigma_{1}+4t^{\prime}\cos\left(N_{\rm
C}k_{x}a_{x}\right)\sin\left(\frac{k_{y}a_{y}}{2}\right)\sigma_{2}-2t^{\prime}\sin(N_{\rm
C}k_{x}a_{x})\sigma_{3}\\\ &=2t\cos\left(\frac{{\bf k}\cdot{\bf
u}_{y}}{2}\right)\sigma_{1}+4t^{\prime}\cos\left(N_{\rm C}{\bf k}\cdot{\bf
u}_{x}\right)\sin\left(\frac{{\bf k}\cdot{\bf
u}_{y}}{2}\right)\sigma_{2}-2t^{\prime}\sin(N_{\rm C}{\bf k}\cdot{\bf
u}_{x})\sigma_{3}.\end{aligned}$ (5)

So is built, this Hamiltonian has $\gamma_{y}$ varying by the amount of
$2\pi{N_{\rm C}}$ as $k_{x}$ changes by $G_{x}$; a Chern insulator with a
Chern number of $N_{\rm C}$.

In order to find out the corresponding real space representation of this
$H_{N_{\rm C}}$, one can use 2D inverse Fourier transformation of the creation
operator,

$\displaystyle c_{{\bf k},A(B)}^{\dagger}={1\over 2\pi}\sum_{\bf R}c_{{\bf
R},A(B)}^{\dagger}e^{i{\bf k}\cdot\left({\bf R}+{\bf d}_{A(B)}\right)},$ (6)

where ${\bf R}=(na_{x},ma_{y})$ and ${\bf d}_{A(B)}=(0,d_{A(B)})$ with $a_{x}$
and $a_{y}$ being the sizes of the unit cells along the $x$ and $y$ axis and
$n,m=\cdots,-2,-1,0,1,2,\cdots$. With $d_{A}=\frac{a}{4}$ and
$d_{B}=\frac{3a}{4}$, $H_{N_{\rm C}}$ can be transformed back into real space
(see Appendix B for details) as

$\displaystyle\begin{aligned} &H_{N_{\rm C}}=t\sum_{{\bf R}}\left(c_{{\bf
R},A}^{\dagger}c_{{\bf R},B}+c_{{\bf R}+{\bf u}_{y},A}^{\dagger}c_{{\bf
R},B}\right)+h.c.\\\ &+t^{\prime}\sum_{{\bf R}}\left(c_{{\bf R}-N_{\rm C}{\bf
u}_{x},A}^{\dagger}c_{{\bf R},B}-c_{{\bf R}-N_{\rm C}{\bf u}_{x}+{\bf
u}_{y},A}^{\dagger}c_{{\bf R},B}+c_{{\bf R}+N_{\rm C}{\bf
u}_{x},A}^{\dagger}c_{{\bf R},B}-c_{{\bf R}+N_{\rm C}{\bf u}_{x}+{\bf
u}_{y},A}^{\dagger}c_{{\bf R},B}\right)+h.c.\\\ &+it^{\prime}\sum_{{\bf
R}}\left(c_{{\bf R}-N_{\rm C}{\bf u}_{x},A}^{\dagger}c_{{\bf R},A}-c_{{\bf
R}-N_{\rm C}{\bf u}_{x},B}^{\dagger}c_{{\bf R},B}\right)+h.c.\end{aligned}$
(7)

Figure 2(a) shows the hopping configuration corresponding to such a system
with $N_{\rm C}=1$.

Figure 2: The hopping configurations of constructed lattice structures for (a)
$N_{\rm C}=1$ and (b) $N_{\rm C}=2$. In terms of the hopping configuration,
(b) is two copies of (a) that are mutually shifted in space. Find the details
in the main text.

The solid, dashed and dotted lines without arrows represent bondings with real
hopping constants of $t$, $t^{\prime}$ and $-t^{\prime}$, respectively. Solid
and empty circles repesent $A$ and $B$ sites. The yellow rectangle is a unit
cell with ${\bf u}_{x}$ and ${\bf u}_{y}$ as the unit vectors. The lines with
arrows represent hopping constants $it^{\prime}$ along the direction of the
arrow and $-it^{\prime}$ along the opposite direction. The convention of
hopping direction is that the coefficient of $c_{\beta}^{\dagger}c_{\alpha}$
corresponds to the hopping constant from the $\alpha$ to the $\beta$ site. It
implies staggered magnetic flux perpendicular to the plane which cancels out
within each unit cell similar to the Haldane’s Chern insulator. It is
noteworthy that the on-site energy difference between two subsites is zero
even though it is built from the RM model that has non-zero on-site energy
difference. Instead, the imaginary hopping constants along the $A$ and $B$
sites have opposite signs. The inter-site next nearest hopping constants also
have two opposite signs within a unit cell.

Figure 3: Band structure with an open boundary condition with two edges. For
the system with $N_{C}=1$ with edges (a) along the $x$ axis and (c) along the
$y$ axis. For the system with $N_{C}=3$ with edges (b) along the $x$ axis and
(d) along the $y$ axis.

We have calculated the chiral edge states with open boundary conditions along
the $x$ and $y$ axis, respectively. Figure 3(a) and (c) show the band
structures for the case with a pair of edges along the $x$ and $y$ axis
respectively for $N_{\rm C}=1$. The value of $t^{\prime}$ that decides the gap
size is set to be $t^{\prime}=0.6t$. The number of unit cells between edges
are chosen to be 60. The number of gapless chiral edge band lines is 2 since
there are two edges.

Figure 2(b) shows the configuration for $N_{\rm C}=2$. For a general case of
$N_{\rm C}>1$, the $n$-th site is connected to the $(n\pm N_{\rm C})$-th site
by hopping along the $x$ axis so that only $(n\pm\eta N_{\rm C})$-th sites are
connected to $n$-th sites with $\eta=1,2,3,\cdots$. In such a manner, the
system is partitioned into $N_{\rm C}$ decoupled subsystems. In Fig. 2(b), two
subsystems are colored distinctly as red and blue. Furthermore, each decoupled
system is identical to the $N_{\rm C}=1$ system in terms of the hopping graph.
With an open boundary condition, since the structure consists of $N_{\rm C}$
copies of a Chern number one system, it should show $N_{\rm C}$ gapless
boundary band lines for each edge. This means that its Chern number is $N_{\rm
C}$ as supposed. Figure 3(b) and (d) show the band structures for open
boundary condition with a pair of edges along the $x$ and $y$ axis
respectively, for $N_{\rm C}=3$. Each of the two band lines corresponding to
the edge states on two edges in Fig. 3(d) has three-fold degeneracy.

Figure 4: (a) Bulk DOS of a mixed system of $N_{\rm C}=1$ and $N_{\rm C}=2$
for various mixing ratio $r$. The horizontal and vertical axes are mixing
ratio and energy, respectively. We confirm that it is identical to that for a
mixed system of $N_{\rm C}=2$ and $N_{\rm C}=3$. (b) Band structures of an
open system with varying mixing ratio for a mixed system of $N_{\rm C}=1$ and
$N_{\rm C}=2$. (c) Those for a mixed system of $N_{\rm C}=2$ and $N_{\rm
C}=3$. (d) 2D band structure for the unmixed Chern insulator with $N_{\rm
C}=2$ (left), for a mixed one between $N_{\rm C}=1$ and $N_{\rm C}=2$ at the
transition point making a Dirac cone (right), for a general mixed one with
Chern number two (middle).

One might say that this result is trivial since the constructed high-$N_{\rm
C}$ Chern insulator is simply an overlap of $N_{\rm C}$ copies of partially
translated systems with $N_{\rm C}=1$. A nontrivial one, however, can further
be constructed by mixing Hamiltonians with distinct Chern numbers. For
example, the Hamiltonian that mixes $N_{\rm C}=1$ and $N_{\rm C}=2$ is

$\displaystyle H_{1,2}(r)=(1-r)H_{N_{\rm C}=1}+rH_{N_{\rm C}=2}$

where $r$ is the mixing ratio. If $r=0$ the Hamiltonian becomes one with
$N_{\rm C}=1$ while if $r=1$ it becomes one with $N_{\rm C}=2$. Therefore, as
$r$ varies from 0 to 1, the Chern number of $H_{1,2}(r)$ must undergo at least
one topological phase transition. At the topological transition, the bulk band
gap must be closed. Figure 4(a) shows the bulk density of state (DOS) as a
function of $r$ demonstrating gap closing at $r=0.5$; $N_{\rm C}=1$ for
$r<0.5$ and $N_{\rm C}=2$ for $r>0.5$. Figure 4(b) shows how the number of
gapless band lines increases. Further numerical calculations confirm that the
change of DOS as a funtion of the mixing ratio from $N_{\rm C}=2$ to $3$ with
$H_{2,3}(r)$ is identical to the case from $N_{\rm C}=1$ to $2$. Figure 4(c)
shows the discrete change of number of gap closing band lines from 4 to 6 as
$r$ changes; $N_{\rm C}$ makes a transition from 2 to 3 at $r=0.5$. Figure
4(d) shows the 2D band structure for the unmixed Chern insulator with $N_{\rm
C}=2$ (left), for a mixed one between $N_{\rm C}=1$ and $2$ at the transition
point which closes gap making a Dirac cone (right), and for a general mixed
one with Chern number two that shows quadratic dispersion at the zone boundary
(middle). This demonstrates that by mixing different unmixed Hamiltonians,
$H_{N_{\rm C}}$, with different $N_{\rm C}$ values in a various way,
nontrivial high Chern number insulators can be constructed on demand.

Figure 5: $H_{\alpha,\beta}(r)=(1-r)H_{N_{\rm C}=\alpha}+rH_{N_{\rm C}=\beta}$
where $H_{N_{\rm C}}$ represents the Hamiltonian of a pure system with the
Chern number of $N_{\rm C}$. The ldos at $E=0$ within the bulk energy gap with
energy width of $0.02t$ for isolated systems with $30\times 30$ unit cells.
(a) Since Chern number is zero there is no gap states. The ldos within the gap
is zero. (b) and (c) demonstrate the chrial edge states for Chern number one.
(d) corresponds to Chern number two.

We further calculate local density of states (ldos) at $E=0$ with energy width
of $0.02t$ for isolated systems of size $30\times 30$ unit cells with an open
boundary condition by varying the value of $r$ for $H_{0,1}(r)$ and
$H_{1,2}(r)$ to demonstrate the emergence of chiral edge states. Figure 5
shows zero-energy ldos showing gap state density for (a) $H_{0,1}(r=0.2)$, (b)
$H_{0,1}(r=0.8)$, (c) $H_{1,2}(r=0.2)$, and (d) $H_{1,2}(r=0.8)$. It
demonstrates that the chiral edge states emerge once $r$ becomes larger than
0.5 in the case with $H_{0,1}(r)$ [Figs. 5(a, b)]. Figures 5(b, c) have Chern
number of one. Figure 5(d), that corresponds to Chern number two, shows that
the edge state density covers two rows on each edge along the $y$ axis,
differently from (b) and (c) where it covers only one row. Although not being
a general manefestation of Chern number, it can be used for a signature of
topological phase transition for this specific system if tested experimentally
in the future.

Figure 6: Berry flux in the first Brillouin zone. (a) Mixed hamiltonian of
$N_{\rm C}=0$ and $N_{\rm C}=3$ with mixing ratio $r=0,0.47,0.53,1$. The
topological phase transition occurs at $r=0.5$ from Chern number 0 to 3. (b)
Mixed hamiltonian of $N_{\rm C}=1$ and $N_{\rm C}=2$ with the same mixing
ratios. The topological phase transition also occurs at $r=0.5$ from Chern
number 1 to 2.

In order to further confirm the Chern number, we calculate Berry flux within
the first Brillouin zone across the phase transition. Figure 6(a) and (b)
shows the Berry flux for systems with $H_{0,3}(r)$ and $H_{1,2}(r)$ for
different values of $r$. Both cases undergo phase transition at $r=0.5$. For
$H_{0,3}(r)$, Chern number changes directly from $0$ to $3$ at $r=0.5$. The
second and third plots are Berry flux right before and after phase transtion
with Chern numbers, $0$ and $3$. The integration of Berry flux over the
Brillouin zone divided by $2\pi$ is the Chern number. For the case before the
phase transition with $r=0.47$, integration of the dispersed positive Berry
flux (yellow) exactly cancels out with that of the three localized spots of
negative Berry flux (blue) to make the Chern number zero. On the other hand,
for the case after the phase transition with $r=0.53$, the localized spots of
Berry flux becomes singular when $r=0.5$, changing their sign of flux to plus
making the Chern number three. For $H_{1,2}(r)$, the story is similar with
phase transition from Chern number one to two through one singular spot of
Berry flux. We confirm that singular points of Berry flux correspond to gap
closing points.

Lastly, we would like to mention briefly about experimental realization and
future plans. As for the experimental realization, due to its relatively
complicated nature of hopping structures it might be challenging in solid
state systems. However, complicated hopping structures have been widely
achieved recently in the fields such as optics and electric circuits. The next
nearest neighbor and long-range hopping can be implemented in photonic systems
such as an array of coupled optical-ring resonators and coupled-resonator
optical waveguides, which consist of site resonators and link resonators M.
Hafezi and Taylor (2013); Sunil Mittal and Hafezi (2016); Daniel Leykam and
Chong (2018); JungYun Han and Leykam (2019). In a synthetic dimension, long-
range hopping can also be implemented by utilizing frequency modes in two
coupled ring resonators Kai Wang and Fan (2021). Asymmetric long-range hopping
between resonant elements in a photonic arrangement have been achieved in
active optical platforms Yuzhou G. N. Liu and Khajavikhan (2021) and complex
hopping between neighboring lattice sites have been realized in electronic
circuit layouts Jia Ningyuan and Simon (2015); Xiang Ni and Alù (2020);
Lingling Song and Yan (2022). Our future work will include investigation of
dynamical responses of topological edges states to varying Chern number by
tuning the hopping parameters of these systems. It is also important to note
that the experimental realization of high Chern number insulators are mostly
multilayered systems et. al. (2020a, b); Wang and Li (2021); Wenxuan Zhu and
Pan (2022). They consist of layers each of which has Chern number one while
multilayers make copies of each layer introducing high Chern number. Such
mechanism is analogous to our structure; our systems consist of laterally
shifted copies of Chern number one systems.

##  IV Conclusion

In conclusion, we propose a systematic method to construct a two-dimensional
lattice structure with proper hopping parameters that can give an arbitrary
Chern number, based on extended 1D Rice-Mele model. With constructed
structures, we successfully demonstrate topological phase transitions between
different Chern numbers by modulating hopping parameters. We also demonstrate
the emergence of topological chiral edge states across the phase transition.
Generalization of our method to different lattice structures such as hexagonal
lattice as well as extension to higher dimensions for higher-order topological
insulators shall be of interest for future projects.

##  V acknowledgements

This work was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT) (No. RS-2023-00278511), by the
Pukyong National University Research Fund in (No.202303660001), by Korea
Institute for Advanced Study (KIAS), and by the Institute for Basic Science in
Korea (No. IBS-R024-D1).

##  VI references

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##  Appendix A Fourier Transformation

As in Eq. (1), the Hamiltonian of a RM model is

$\displaystyle\begin{aligned}
H=\sum_{R}&\left[{\varepsilon}\left(c_{R,A}^{\dagger}c_{R,A}-c_{R,B}^{\dagger}c_{R,B}\right)\right.\\\
&+\left.(t+\delta t)\left(c_{R,A}^{\dagger}c_{R,B}+h.c.\right)+(t-\delta
t)\left(c_{R+a,A}^{\dagger}c_{R,B}+h.c.\right)\right].\end{aligned}$ (8)

With Fourier transformation,
$c_{R,A(B)}^{\dagger}=\sqrt{\frac{1}{G}}\int_{0}^{G}dk~{}c_{k,A(B)}^{\dagger}e^{-ik(R+d_{A(B)})}$,
each term can be transformed into $k$-space. The term of $\sum
c_{R,A}^{\dagger}c_{R,A}$ is transformed as

$\displaystyle\begin{aligned} \sum_{R}&c_{R,A}^{\dagger}c_{R,A}\\\
&=\frac{1}{G}\sum_{R}\int^{G}_{0}dk~{}c_{k,A}^{\dagger}e^{-ik(R+d_{A})}~{}\int^{G}_{0}dk^{\prime}~{}c_{k^{\prime},A}e^{ik^{\prime}(R+d_{A})}\\\
&=\iint^{G}_{0}dk~{}dk^{\prime}~{}c_{k,A}^{\dagger}c_{k^{\prime},A}~{}\frac{1}{G}\sum_{R}e^{-i(k-k^{\prime})R}\\\
&=\iint^{G}_{0}dk~{}dk^{\prime}~{}c_{k,A}^{\dagger}c_{k^{\prime},A}~{}\delta(k-k^{\prime})\\\
&=\iint^{G}_{0}dk~{}c_{k,A}^{\dagger}c_{k,A}.\end{aligned}$ (9)

The term, $\sum c_{R,B}^{\dagger}c_{R,B}$ is transformed similarly as

$\displaystyle\begin{aligned}
\sum_{R}&c_{R,B}^{\dagger}c_{R,B}=\iint^{G}_{0}dk~{}c_{k,B}^{\dagger}c_{k,B}.\end{aligned}$
(10)

The term, $\sum c_{R,A}^{\dagger}c_{R,B}$ is transformed as

$\displaystyle\begin{aligned} \sum_{R}&c_{R,A}^{\dagger}c_{R,B}\\\
&=\frac{1}{G}\sum_{R}\int^{G}_{0}dk~{}c_{k,A}^{\dagger}e^{-ik(R+d_{A})}\int^{G}_{0}dk^{\prime}~{}c_{k^{\prime},B}e^{ik^{\prime}(R+d_{B})}\\\
&=\iint^{G}_{0}dk~{}dk^{\prime}~{}c_{k,A}^{\dagger}c_{k^{\prime},B}~{}e^{-ikd_{A}+ik^{\prime}d_{B}}~{}\frac{1}{G}\sum_{R}e^{-i(k-k^{\prime})R}\\\
&=\iint^{G}_{0}dk~{}dk^{\prime}~{}c_{k,A}^{\dagger}c_{k^{\prime},B}~{}e^{-ikd_{A}+ik^{\prime}d_{B}}~{}\delta(k-k^{\prime})\\\
&=\int^{G}_{0}dk~{}c_{k,A}^{\dagger}c_{k,B}~{}e^{-ik(d_{A}-d_{B})}.\end{aligned}$

With $d_{A}=\frac{a}{4}$ and $d_{B}=\frac{3a}{4}$, it becomes

$\displaystyle\sum_{R}$ $\displaystyle
c_{R,A}^{\dagger}c_{R,B}=\int^{G}_{0}dk~{}c_{k,A}^{\dagger}c_{k,B}~{}e^{i\frac{ka}{2}}.$
(11)

The term, $\sum c_{R+a,A}^{\dagger}c_{R,B}$ is transformed as

$\displaystyle\begin{aligned} \sum_{R}&c_{R+a,A}^{\dagger}c_{R,B}\\\
&=\frac{1}{G}\sum_{R}\int^{G}_{0}dk~{}c_{k,A}^{\dagger}e^{-ik(R+a+d_{A})}\int^{G}_{0}dk^{\prime}~{}c_{k^{\prime},B}e^{ik^{\prime}(R+d_{B})}\\\
&=\iint^{G}_{0}dk~{}dk^{\prime}~{}c_{k,A}^{\dagger}c_{k^{\prime},B}~{}e^{-ik(a+d_{A})+ik^{\prime}d_{B}}~{}\frac{1}{G}\sum_{R}e^{-i(k-k^{\prime})R}\\\
&=\iint^{G}_{0}dk~{}dk^{\prime}~{}c_{k,A}^{\dagger}c_{k^{\prime},B}~{}e^{-ik(a+d_{A})+ik^{\prime}d_{B}}~{}\delta(k-k^{\prime})\\\
&=\int^{G}_{0}dk~{}c_{k,A}^{\dagger}c_{k,B}e^{-ik(a+d_{A}-d_{B})}.\end{aligned}$

With $d_{A}=\frac{a}{4}$ and $d_{B}=\frac{3a}{4}$, it becomes

$\displaystyle\sum_{R}$ $\displaystyle
c_{R+a,A}^{\dagger}c_{R,B}=\int^{G}_{0}dk~{}c_{k,A}^{\dagger}c_{k,B}~{}e^{-i\frac{ka}{2}}.$
(12)

Summing up terms from Eqs. (9, 10, 11, 12), Eq. (8) can be represented in
$k$-space as $H=\int^{G}_{0}dk~{}\mathcal{H}(k)$ with

$\displaystyle\begin{aligned}
\mathcal{H}=&\varepsilon\left(c_{k,A}^{\dagger}c_{k,A}-c_{k,B}^{\dagger}c_{k,B}\right)\\\
&+(t+\delta
t)\left(c_{k,A}^{\dagger}c_{k,B}~{}e^{i\frac{ka}{2}}+h.c.\right)+(t-\delta
t)\left(c_{k,A}^{\dagger}c_{k,B}~{}e^{-i\frac{ka}{2}}+h.c.\right)\\\
=&\varepsilon\left(c_{k,A}^{\dagger}c_{k,A}-c_{k,B}^{\dagger}c_{k,B}\right)+2t\left(c_{k,A}^{\dagger}c_{k,B}+c_{k,B}^{\dagger}c_{k,A}\right)\cos{\left(\frac{ka}{2}\right)}+\\\
&+2i\delta
t\left(c_{k,A}^{\dagger}c_{k,B}-c_{k,B}^{\dagger}c_{k,A}\right)\sin{\left(\frac{ka}{2}\right)}\\\
=&\begin{pmatrix}c^{\dagger}_{k,A}&c^{\dagger}_{k,B}\end{pmatrix}\begin{pmatrix}\varepsilon&2t\cos\left(\frac{ka}{2}\right)+2i\delta
t\sin\left(\frac{ka}{2}\right)\\\ 2t\cos\left(\frac{ka}{2}\right)-2i\delta
t\sin\left(\frac{ka}{2}\right)&-\varepsilon\end{pmatrix}\begin{pmatrix}c_{k,A}\\\
c_{k,B}\end{pmatrix}\\\
=&\phi^{\dagger}_{k}\left(2t\cos\left(\frac{ka}{2}\right)\sigma_{1}-2\delta
t\sin\left(\frac{ka}{2}\right)\sigma_{2}+\varepsilon\sigma_{3}\right)\phi_{k},\end{aligned}$
(13)

where $\phi^{\dagger}\equiv(c_{k,A}^{\dagger},c_{k,B}^{\dagger})$ and
$\sigma_{i}$s are Pauli matrices.

##  Appendix B Inverse Fourier Transformation

From Eq. (5), the hamiltonian with Chern number $N_{\rm C}$ in the $\bf
k$-space representation consists of three terms as

$\displaystyle H_{N_{\rm C}}=$ $\displaystyle\int_{\rm BZ}d^{2}k~{}\phi_{\bf
k}^{\dagger}\mathcal{H}_{N_{\rm C}}({\bf k})\phi_{\bf k}$ $\displaystyle=$
$\displaystyle+2t\int_{\rm BZ}d^{2}k\cos\left({{\bf k}\cdot{\bf a}\over
2}\right)\phi_{\bf k}^{\dagger}\sigma_{1}\phi_{\bf k}$ (14a)
$\displaystyle+4t^{\prime}\int_{\rm BZ}d^{2}k\cos\left(N_{\rm C}{\bf
k}\cdot{\bf b}\right)\sin\left({{\bf k}\cdot{\bf a}\over 2}\right)\phi_{\bf
k}^{\dagger}\sigma_{2}\phi_{\bf k}$ (14b) $\displaystyle-2t^{\prime}\int_{\rm
BZ}d^{2}k\sin(N_{\rm C}{\bf k}\cdot{\bf b})\phi_{\bf
k}^{\dagger}\sigma_{3}\phi_{\bf k}.$ (14c)

Each term can be transformed into the representation of real space using Eq.
(6). The first term, Eq. (14a), transforms as

$\displaystyle\begin{aligned} 2t\int_{\rm BZ}&d^{2}k~{}\cos\left({{\bf
k}\cdot{\bf a}\over 2}\right)\phi_{\bf k}^{\dagger}\sigma_{1}\phi_{\bf k}\\\
&=2t\int_{\rm BZ}d^{2}k~{}c_{{\bf k},A}^{\dagger}c_{{\bf k},B}+h.c.\\\
&=2t\int_{\rm BZ}d^{2}k~{}\cos\left({{\bf k}\cdot{\bf a}\over
2}\right){1\over(2\pi)^{2}}\sum_{{\bf R},{\bf R^{\prime}}}c_{{\bf
R},A}^{\dagger}e^{i{\bf k}\cdot({\bf R}+{\bf d}_{A})}c_{{\bf
R}^{\prime},B}e^{-i{\bf k}\cdot({\bf R}^{\prime}+{\bf d}_{B})}+h.c.\\\
&=2t\sum_{{\bf R},{\bf R^{\prime}}}c_{{\bf R},A}^{\dagger}c_{{\bf
R}^{\prime},B}{1\over(2\pi)^{2}}\int_{\rm BZ}d^{2}k~{}\cos\left({{\bf
k}\cdot{\bf a}\over 2}\right)e^{i{\bf k}\cdot({\bf R}-{\bf
R}^{\prime}-\frac{\bf b}{2})}+h.c.\\\ &=t\sum_{{\bf R},{\bf
R^{\prime}}}c_{{\bf R},A}^{\dagger}c_{{\bf
R}^{\prime},B}{1\over(2\pi)^{2}}\int_{\rm BZ}d^{2}k~{}\left(e^{i\frac{{\bf
k}\cdot{\bf a}}{2}}+e^{-i\frac{{\bf k}\cdot{\bf a}}{2}}\right)e^{i{\bf
k}\cdot({\bf R}-{\bf R}^{\prime}-{\frac{\bf a}{2}})}+h.c.\\\ &=t\sum_{{\bf
R},{\bf R^{\prime}}}c_{{\bf R},A}^{\dagger}c_{{\bf
R}^{\prime},B}{1\over(2\pi)^{2}}\int_{\rm BZ}d^{2}k~{}\left(e^{i{\bf
k}\cdot\left({\bf R}-{\bf R}^{\prime}\right)}+e^{i{\bf k}\cdot({\bf R}-{\bf
R}^{\prime}-{\bf a})}\right)+h.c.\\\ &=t\sum_{{\bf R},{\bf R^{\prime}}}c_{{\bf
R},A}^{\dagger}c_{{\bf R}^{\prime},B}\left(\delta_{{\bf R},{\bf
R}^{\prime}}+\delta_{{\bf R},{\bf R}^{\prime}+{\bf a}}\right)+h.c.\\\
&=t\sum_{{\bf R}}\left(c_{{\bf R},A}^{\dagger}c_{{\bf R},B}+c_{{\bf R}+{\bf
a},A}^{\dagger}c_{{\bf R},B}\right)+h.c.\end{aligned}$ (15)

Here, ${\bf d}_{B}-{\bf d}_{A}={\bf{a}\over 2}$ is used. The sencond term, Eq.
(14b), transforms as

$\displaystyle\begin{aligned} 4&t^{\prime}\int_{\rm
BZ}d^{2}k~{}\cos\left(N_{\rm C}{\bf k}\cdot{\bf b}\right)\sin\left({{\bf
k}\cdot{\bf a}\over 2}\right)\phi_{\bf k}^{\dagger}\sigma_{2}\phi_{\bf k}\\\
&=4t^{\prime}\int_{\rm BZ}d^{2}k~{}\cos\left(N_{\rm C}{\bf k}\cdot{\bf
b}\right)\sin\left({{\bf k}\cdot{\bf a}\over 2}\right)\left(-ic_{{\bf
k},A}^{\dagger}c_{{\bf k},B}\right)+h.c.\\\ &=4t^{\prime}\int_{\rm
BZ}d^{2}k~{}\cos\left(N_{\rm C}{\bf k}\cdot{\bf b}\right)\sin\left({{\bf
k}\cdot{\bf a}\over 2}\right){-i\over(2\pi)^{2}}\sum_{{\bf R},{\bf
R^{\prime}}}c_{{\bf R},A}^{\dagger}e^{i{\bf k}\cdot({\bf R}+{\bf
d}_{A})}c_{{\bf R}^{\prime},B}e^{-i{\bf k}\cdot({\bf R}^{\prime}+{\bf
d}_{B})}+h.c.\\\ &=4t^{\prime}\sum_{{\bf R},{\bf R^{\prime}}}c_{{\bf
R},A}^{\dagger}c_{{\bf R}^{\prime},B}{-i\over(2\pi)^{2}}\int_{\rm
BZ}d^{2}k~{}\cos\left(N_{\rm C}{\bf k}\cdot{\bf b}\right)\sin\left({{\bf
k}\cdot{\bf a}\over 2}\right)e^{i{\bf k}\cdot({\bf R}-{\bf
R}^{\prime}-\frac{\bf a}{2})}+h.c.\\\ &=-t^{\prime}\sum_{{\bf R},{\bf
R^{\prime}}}c_{{\bf R},A}^{\dagger}c_{{\bf
R}^{\prime},B}{1\over(2\pi)^{2}}\int_{\rm BZ}d^{2}k~{}\left(e^{iN_{\rm C}{{\bf
k}\cdot{\bf b}}}+e^{-iN_{\rm C}{{\bf k}\cdot{\bf
b}}}\right)\left(e^{i\frac{{\bf k}\cdot{\bf a}}{2}}-e^{-i\frac{{\bf
k}\cdot{\bf a}}{2}}\right)e^{i{\bf k}\cdot({\bf R}-{\bf R}^{\prime}-{\frac{\bf
a}{2}})}+h.c.\\\ &=-t^{\prime}\sum_{{\bf R},{\bf R^{\prime}}}c_{{\bf
R},A}^{\dagger}c_{{\bf R}^{\prime},B}\\\ &\hskip
14.22636pt\times{1\over(2\pi)^{2}}\int_{\rm BZ}d^{2}k~{}\left(e^{i{\bf
k}\cdot\left(N_{\rm C}{\bf b}\right)}-e^{i{\bf k}\cdot\left(N_{\rm C}{\bf
b}-{\bf a}\right)}+e^{i{\bf k}\cdot\left(-N_{\rm C}{\bf b}\right)}-e^{i{\bf
k}\cdot\left(-N_{\rm C}{\bf b}-{\bf a}\right)}\right)e^{i{\bf k}\cdot({\bf
R}-{\bf R}^{\prime})}+h.c.\\\ &=-t^{\prime}\sum_{{\bf R},{\bf
R^{\prime}}}c_{{\bf R},A}^{\dagger}c_{{\bf R}^{\prime},B}\left(\delta_{{\bf
R},{\bf R}^{\prime}-N_{\rm C}{\bf b}}-\delta_{{\bf R},{\bf R}^{\prime}-N_{\rm
C}{\bf b}+{\bf a}}+\delta_{{\bf R},{\bf R}^{\prime}+N_{\rm C}{\bf
b}}-\delta_{{\bf R},{\bf R}^{\prime}+N_{\rm C}{\bf b}+{\bf a}}\right)+h.c.\\\
&=-t^{\prime}\sum_{{\bf R}}\left(c_{{\bf R}-N_{\rm C}{\bf
b},A}^{\dagger}c_{{\bf R},B}-c_{{\bf R}-N_{\rm C}{\bf b}+{\bf
a},A}^{\dagger}c_{{\bf R},B}+c_{{\bf R}+N_{\rm C}{\bf b},A}^{\dagger}c_{{\bf
R},B}-c_{{\bf R}+N_{\rm C}{\bf b}+{\bf a},A}^{\dagger}c_{{\bf
R},B}\right)+h.c.\end{aligned}$ (16)

The third term, Eq. (14c) transforms as

$\displaystyle\begin{aligned} -2t^{\prime}&\int_{\rm
BZ}d^{2}k~{}\sin\left(N_{\rm C}{\bf k}\cdot{\bf b}\right)\phi_{\bf
k}^{\dagger}\sigma_{3}\phi_{\bf k}\\\ &=-2t^{\prime}\int_{\rm
BZ}d^{2}k~{}\sin\left(N_{\rm C}{\bf k}\cdot{\bf b}\right)\left(c_{{\bf
k},A}^{\dagger}c_{{\bf k},A}-c_{{\bf k},B}^{\dagger}c_{{\bf k},B}\right)\\\
&=-2t^{\prime}\int_{\rm BZ}d^{2}k~{}\sin\left(N_{\rm C}{\bf k}\cdot{\bf
b}\right){1\over(2\pi)^{2}}\sum_{{\bf R},{\bf R^{\prime}}}\left(c_{{\bf
R},A}^{\dagger}c_{{\bf R}^{\prime},A}e^{i{\bf k}\cdot({\bf R}-{\bf
R}^{\prime})}-c_{{\bf R},B}^{\dagger}c_{{\bf R}^{\prime},B}e^{i{\bf
k}\cdot({\bf R}-{\bf R}^{\prime})}\right)\\\ &=-2t^{\prime}\sum_{{\bf R},{\bf
R^{\prime}}}\left(c_{{\bf R},A}^{\dagger}c_{{\bf R}^{\prime},A}-c_{{\bf
R},B}^{\dagger}c_{{\bf R}^{\prime},B}\right){1\over(2\pi)^{2}}\int_{\rm
BZ}d^{2}k~{}\sin\left(N_{\rm C}{\bf k}\cdot{\bf b}\right)e^{i{\bf k}\cdot({\bf
R}-{\bf R}^{\prime})}\\\ &=it^{\prime}\sum_{{\bf R},{\bf
R^{\prime}}}\left(c_{{\bf R},A}^{\dagger}c_{{\bf R}^{\prime},A}-c_{{\bf
R},B}^{\dagger}c_{{\bf R}^{\prime},B}\right){1\over(2\pi)^{2}}\int_{\rm
BZ}d^{2}k~{}\left(e^{i{\bf k}\cdot N_{\rm C}{\bf b}}-e^{-i{\bf k}\cdot N_{\rm
C}{\bf b}}\right)e^{i{\bf k}\cdot({\bf R}-{\bf R}^{\prime})}\\\
&=it^{\prime}\sum_{{\bf R},{\bf R^{\prime}}}\left(c_{{\bf
R},A}^{\dagger}c_{{\bf R}^{\prime},A}-c_{{\bf R},B}^{\dagger}c_{{\bf
R}^{\prime},B}\right)\left(\delta_{{\bf R},{\bf R}^{\prime}-N_{\rm C}{\bf
b}}-\delta_{{\bf R},{\bf R}^{\prime}+N_{\rm C}{\bf b}}\right)\\\
&=it^{\prime}\sum_{{\bf R}}\left(c_{{\bf R}-N_{\rm C}{\bf
b},A}^{\dagger}c_{{\bf R},A}-c_{{\bf R}+N_{\rm C}{\bf b},A}^{\dagger}c_{{\bf
R},A}-c_{{\bf R}-N_{\rm C}{\bf b},B}^{\dagger}c_{{\bf R},B}+c_{{\bf R}+N_{\rm
C}{\bf b},B}^{\dagger}c_{{\bf R},B}\right)\\\ &=it^{\prime}\sum_{{\bf
R}}\left(c_{{\bf R}-N_{\rm C}{\bf b},A}^{\dagger}c_{{\bf R},A}-c_{{\bf
R}-N_{\rm C}{\bf b},B}^{\dagger}c_{{\bf R},B}\right)+h.c.\end{aligned}$ (17)

Summing up terms from Eqs. (15, 16, 17), the Hamiltonian can be written in the
real space as

$\displaystyle\begin{aligned} H_{N_{\rm C}}&=t\sum_{{\bf R}}\left(c_{{\bf
R},A}^{\dagger}c_{{\bf R},B}+c_{{\bf R}+{\bf a},A}^{\dagger}c_{{\bf
R},B}\right)+h.c.\\\ &+\frac{t^{\prime}}{2}\sum_{{\bf R}}\left(c_{{\bf
R}-N_{\rm C}{\bf b},A}^{\dagger}c_{{\bf R},B}-c_{{\bf R}-N_{\rm C}{\bf b}+{\bf
a},A}^{\dagger}c_{{\bf R},B}+c_{{\bf R}+N_{\rm C}{\bf b},A}^{\dagger}c_{{\bf
R},B}-c_{{\bf R}+N_{\rm C}{\bf b}+{\bf a},A}^{\dagger}c_{{\bf
R},B}\right)+h.c.\\\ &+it^{\prime}\sum_{{\bf R}}\left(c_{{\bf R}-N_{\rm C}{\bf
b},A}^{\dagger}c_{{\bf R},A}-c_{{\bf R}-N_{\rm C}{\bf b},B}^{\dagger}c_{{\bf
R},B}\right)+h.c.\end{aligned}$ (18)


# A comparison of dg algebra resolutions with prime residual characteristic

Michael DeBellevue  Mathematics Department, Syracuse University, Syracuse, NY
13244 U.S.A<EMAIL_ADDRESS>and  Josh Pollitz  Department of Mathematics,
University of Utah, Salt Lake City, UT 84112, U.S.A<EMAIL_ADDRESS>
###### Abstract.

In this article we fix a prime integer $p$ and compare certain dg algebra
resolutions over a local ring whose residue field has characteristic $p$.
Namely, we show that given a closed surjective map between such algebras there
is a precise description for the minimal model in terms of the acyclic
closure, and that the latter is a quotient of the former. A first application
is that the homotopy Lie algebra of a closed surjective map is abelian. We
also use these calculations to show deviations enjoy rigidity properties which
detect the (quasi-)complete intersection property.

###### Key words and phrases:

prime characteristic, dg algebra, acyclic closure, minimal model, homotopy Lie
algebra, deviations, complete intersection, closed homomorphisms

###### 2020 Mathematics Subject Classification:

13D02, 13A35 (primary); 13D07, 16E45 (secondary)

## Introduction

This work concerns homological properties of a surjective map of commutative
noetherian local rings $\varphi\colon R\rightarrow S$. These properties are
studied through certain differential graded (from now on abbreviated to dg)
algebra resolutions of $S$ over $R$, which provide a relationship between
homological and ring-theoretic data of $\varphi$. A dg algebra resolution
consists of a free resolution of $S$ over $R$ further equipped with a graded-
commutative multiplication compatible with the differential; see Section 1 and
the references therein for more information.

The prototypical example of a dg algebra resolution is the Koszul complex
resolving the residue field of a regular local ring. Its multiplicative
structure is that of the exterior algebra, and its differential maps the
exterior variables bijectively to a set of minimal generators of the maximal
ideal of $R$. When $R$ is singular the Koszul complex is not acyclic, but Tate
[31] realized that a dg algebra resolution may still be obtained by adjoining
additional variables to the Koszul complex.

Tate’s construction consists of minimally adjoining exterior and divided power
variables in odd and even degrees, respectively. Its importance was
demonstrated by Schoeller [30] and Gulliksen [17], who proved that this always
yields a minimal resolution of the residue field. More generally, Tate’s
construction always produces a (not necessarily minimal) dg algebra resolution
of $S$ over $R$ called the _acyclic closure_ of $\varphi$, denoted $R\langle
Y\rangle$ with $Y$ the set of exterior and divided power variables.

If polynomial variables are used in lieu of divided power variables one
obtains the minimal models which were imported to local algebra from rational
homotopy theory by Avramov [2]; cf. 1.3. We write $R[X]$ for a minimal model
for $\varphi$ where $X$ is the set of exterior and polynomial variables
adjoined. Their use has been instrumental in the detection of the complete
intersection property, and showing that the gap between complete intersection
rings and other singularities is vast. In particular, they were applied to
show the Betti numbers of the residue field have polynomial growth over
complete intersection rings and exponential growth in all other cases [2, 4].
More recently, minimal models were also used by Briggs [16] to settle
Vasconcelos conjecture on the conormal module [32].

When the common residue field $k$ of $R$ and $S$ has characteristic zero, it
is well known that the acyclic closure and minimal model of $\varphi\colon
R\rightarrow S$ coincide (see Remark 3.2); in general, they can differ
drastically. The main result in this article gives a precise relationship
between these two resolutions when $k$ has characteristic $p>0$ and the
acyclic closure of $\varphi$ is minimal as a complex of $R$-modules; such maps
are called closed and were introduced in [10].

######  Theorem A.

Let $\varphi\colon R\rightarrow S$ be a surjective map of local rings whose
residue field $k$ has characteristic $p>0$. If $\varphi$ is closed, then its
acyclic closure $R\langle Y\rangle$ is a quotient of its minimal model $R[X]$,
and there is an exact sequence of graded $k$-spaces

$0\to\bigoplus_{i=1}^{\infty}(kY_{\rm{even}}^{(p^{i})}\oplus{\scriptstyle\mathsf{\Sigma}}kY_{\rm{even}}^{(p^{i})})\to
kX\to kY\to 0\,,$

where ${\scriptstyle\mathsf{\Sigma}}$ is the usual suspension functor of a
graded object and $Y_{\rm{even}}^{(p^{i})}$ consists of all divided power
monomials $y^{(p^{i})}$ for $y\in Y_{\rm{even}}$.

Closed maps arise frequently. For example, large and quasi-complete
intersection homomorphisms are well-studied classes of ring maps, and each
belong to the class of closed maps; see 1.13 for more details. Understanding
the structure of quasi-complete intersection maps is of particular importance
due to their role in a long-standing conjecture of Quillen [27]. The
conclusion of A holds under the more technical assumption that $\varphi$ is
_weakly-closed_ , as defined in Definition 1.10; cf. Remark 5.7. A refinement
of A can be found in Theorem 3.6.

The main application of Theorem 3.6 is a computation of the homotopy Lie
algebra $\pi(F^{\varphi})$ of $\varphi$ whenever $\varphi$ is closed; this
calculation is purely in terms of the acyclic closure of $\varphi$. The
homotopy Lie algebra—also adopted from rational homotopy theory to local
algebra by Avramov [2]—is a graded Lie algebra naturally associated to a local
homomorphism. Properties of $\pi(F^{\varphi})$ relate to those of $\varphi$,
such as whether $\varphi$ is complete intersection [8] or Golod [3], making
$\pi(F^{\varphi})$ a useful computational tool. For example, it was recently
used by Briggs to settle a long-standing conjecture of Vasconcelos [16]. See
4.2 for more details on the homotopy Lie algebra.

######  Theorem B.

If $\varphi$ is closed, then $\pi(F^{\varphi})$ is an abelian Lie algebra with
$k$-basis dual to

$\begin{cases}{\scriptstyle\mathsf{\Sigma}}Y&\operatorname{char}k=0\\\
{\scriptstyle\mathsf{\Sigma}}Y\cup\bigcup_{i=1}^{\infty}\left({\scriptstyle\mathsf{\Sigma}}Y_{\rm{even}}^{(p^{i})}\cup{\scriptstyle\mathsf{\Sigma}}^{2}Y_{\rm{even}}^{(p^{i})}\right)&\operatorname{char}k>0\,.\end{cases}$

Furthermore if $\operatorname{char}k\neq 2$, then $\pi(F^{\varphi})$ also has
trivial reduced square.

By an abelian Lie algebra we mean a Lie algebra whose bracket is trivial. In
characteristic zero, B follows directly from the equality of acyclic closure
and minimal models as discussed in Remark 3.2. In positive characteristic, B
follows a fortiori from Theorem 4.4.

B should be contrasted with [8, Theorem C] which shows, regardless of the
residual characteristic, that when $\varphi$ has finite projective dimension
the only instance $\pi(F^{\varphi})$ is abelian is when $\varphi$ is complete
intersection. Hence, without the assumption $\varphi$ has finite projective
dimension, B provides many examples for which $\pi(F^{\varphi})$ is abelian
and $\varphi$ is not complete intersection; cf. Remark 4.6.

An initial motivation for this project was to better understand certain
homological invariants of quasi-complete intersection maps. Theorem 4.4
calculates the homotopy Lie algebra of a quasi-complete intersection map,
showing that it exhibits the following dichotomy:

######  Theorem C.

Let $\varphi$ be a quasi-complete intersection map with residue field $k$, and
set $U=\pi^{2}(F^{\varphi})$ and $V=\pi^{3}(F^{\varphi})$. There is an
isomorphism of abelian Lie algebras

$\pi(F^{\varphi})\cong\begin{cases}U\oplus V&\operatorname{char}k=0\\\ U\oplus
V\oplus\bigoplus_{t=1}^{\infty}{\scriptstyle\mathsf{\Sigma}}^{-2p^{t}+2}V\oplus\bigoplus_{t=1}^{\infty}{\scriptstyle\mathsf{\Sigma}}^{-2p^{t}+1}V&\operatorname{char}k>0\end{cases}$

and the reduced square operation is given by:

  1. (1)

If $p\neq 2$, the reduced square is trivial on $\pi(F^{\varphi})$.

  2. (2)

If $p=2$, then there is a basis $\mathcal{B}$ of $V$ for which the reduced
square on $\bigoplus_{t=1}^{\infty}{\scriptstyle\mathsf{\Sigma}}^{-2p^{t}+2}V$
is determined by

${\scriptstyle\mathsf{\Sigma}}^{-2^{t}+2}b\mapsto{\scriptstyle\mathsf{\Sigma}}^{-2^{t+1}+1}b$

for $b\in\mathcal{B}$ and extended according to the axioms of a reduced
square. The reduced square is trivial otherwise.

Suitably interpreted, a converse to C holds when $\varphi$ is assumed to be
weakly-closed; see Corollary 5.4. This provides a homological invariant
distinguishing quasi-complete intersection maps.

A final application of the results above, given in Corollary 5.3, establishes
a characterization of the complete intersection property in terms of the
sequence of deviations $\\{\varepsilon_{i}(\varphi)\\}$ of $\varphi$. These
record the $k$-vector space dimensions of the graded pieces of
$\pi(F^{\varphi})$. When $\varphi$ is a map of finite projective dimension,
much work has been done to relate vanishing $\varepsilon_{i}(\varphi)$ for
large $i$ to the complete intersection property [4, 8, 10, 18, 20]; a central
motivation for these works was to resolve Quillen’s conjecture, which claims
all such maps must be complete intersection homomorphisms. The following
theorem notably lacks the requirement that $\varphi$ has finite projective
dimension, a necessary property of prior rigidity results.

######  Theorem D.

Let $\varphi\colon R\rightarrow S$ be a closed local homomorphism with
residual characteristic $p>0$. The following are equivalent:

  1. (1)

$\varphi$ is complete intersection;

  2. (2)

$\varepsilon_{i}(\varphi)=0$ for $i\gg 0$;

  3. (3)

$\varepsilon_{i}(\varphi)=0$ for $i=2p^{t}+1$ or $i=2p^{t}+2$ for some
$t\geqslant 1$.

######  Acknowledgements.

The authors thank Benjamin Briggs, Srikanth Iyengar, and Alexandra Seceleanu
for helpful comments. They also thank an anonymous referee for a careful
reading and their comments. The first author also thanks Mark Walker for
funding through NSF grant DMS 1901848. The second author was partially
supported by NSF grants DMS 1840190 and DMS 2002173.

##  1\. Minimal models and acyclic closures

Throughout $\varphi\colon R\to S$ is a surjective map of commutative
noetherian local rings with common residue field $k$, and
$A=\\{A_{i}\\}_{i\geqslant 0}$ will be a dg $R$-algebra.

In this article, an arbitrary element $x$ of a graded object
$X=\\{X_{i}\\}_{i\in\mathbb{Z}}$ will be implicitly taken to be homogeneous
and $|x|$ will denote that integer $i$ so that $x\in X_{i}$. All dg algebras
will be strictly graded-commutative in the sense that $ab=(-1)^{|a||b|}ba$ for
all $a,b$ in $A$ and $a^{2}=0$ when $|a|$ is odd. The defining property making
$A$ a _dg_ algebra is that it is equipped with a degree $-1$ differential
$\partial$ satisfying that Leibniz rule:

$\partial(ab)=\partial(a)b+(-1)^{|a|}a\partial(b)\,.$

Said differently, $(A,\partial)$ is a complex of $R$-modules where the
multiplication map $A\otimes_{R}A\to A$ and algebra unit $R\to A$ are
morphisms of complexes.

######  1.1.

Suppose $A_{0}$ is the local ring $(R,\mathfrak{m})$. If
$\partial(A_{1})\subseteq\mathfrak{m}$, then the maximal dg ideal
$\mathfrak{m}_{A}$ of $A$ is given by

$(\mathfrak{m}_{A})_{i}\coloneqq\begin{cases}\mathfrak{m}&i=0\\\ A_{i}&i>0\\\
0&i<0\end{cases}\,.$

This gives rise to the complex of $k$-spaces

${\operatorname{ind}}A\coloneqq\dfrac{\mathfrak{m}_{A}}{\mathfrak{m}+\mathfrak{m}_{A}^{2}}\,$

which is called the indecomposable complex of $A$.

######  1.2.

We write $A[X]$ to denote a semifree extension of $A$ on the graded set of
variables $X=X_{1},X_{2},X_{3},\ldots$ as defined in [5, Section 2.1]. That
is, each variable of $X_{i}$ has homological degree $i$: When $i$ is even
$X_{i}$ consists of polynomial variables and when $i$ is odd $X_{i}$ consists
of exterior variables. In particular, $A[X]^{\natural}$ is the free strictly
graded-commutative $A$-algebra on $X$ where $(-)^{\natural}$ forgets the
differential of a differential graded object. Note that when $A=R$, the
underlying graded $k$-space of ${\operatorname{ind}}(R[X])$ is $kX;$ cf. 1.1.

######  1.3.

A minimal model for $\varphi$ is a factorization of $\varphi$ as $R\to
R[X]\xrightarrow{\simeq}S$ where ${\operatorname{ind}}(R[X])$ has zero
differential. Note that ${\operatorname{ind}}(R[X])$ has zero differential if
and only if the differential of $R[X]$ is decomposable in the sense that
$\partial(R[X])$ is contained in $\mathfrak{m}_{R}R[X]+(X)^{2}.$ By [5,
Section 7.2], minimal models always exist and are unique up to an isomorphism
of dg algebras. Their construction begins with the Koszul complex $R[X_{1}]$
with the set of exterior variables $X_{1}$ mapping bijectively to a minimal
set of generators for $\ker\varphi$. The construction proceeds inductively by
adjoining to the dg algebra $R[X_{\leqslant i-1}]$ a set of variables $X_{i}$
mapping bijectively to a minimal generating set for the homology module ${\rm
H}_{i-1}(R[X_{\leqslant i-1}])$; to guarantee that the direct limit of this
procedure $R[X]$ is a semifree extension, the sets $X_{i}$ alternate between
polynomial and exterior variables as described in 1.2. We will slightly abuse
terminology and call $R[X]$ the minimal model for $\varphi$.

######  1.4.

Fix a semifree extension $A[X]$ of $A$. It is well known that $A[X]$ enjoys
the following lifting property: given a dg $A$-algebra map $f\colon A[X]\to C$
and a surjective quasi-isomorphism of dg $A$-algebras $g\colon
B\xrightarrow{\simeq}C$, then there exists a dg $A$-algebra map
$\tilde{f}\colon A[X]\to B$, extending $f$, with $g\tilde{f}=f$ that is unique
up to $A$-linear homotopy; see, for example, [5, Proposition 2.1.9]. As a
consequence, whenever $R[X]$ is a minimal model of $\varphi$ and
$B\xrightarrow{\simeq}S$ is a dg $R$-algebra resolution, there exists a quasi-
isomorphism $R[X]\xrightarrow{\simeq}B$ compatible with the augmentation maps
to $S$.

Our analysis in Section 3 relies on the following general lifting property for
semifree extensions; its proof is implicitly contained in the proof of [5,
Proposition 2.1.9]. We sketch the argument below.

######  Lemma 1.5.

Let $\alpha\colon A\rightarrow B$ be a morphism of dg algebras, and $Z$ a set
of cycles in $A$ whose image $\alpha(Z)$ is a set of boundaries in $B$. If
$\\{b_{z}:z\in Z\\}$ is a collection of elements of $B$ with
$\alpha(z)=\partial(b_{z})$, then there exists a unique map of dg algebras,
extending $\alpha$,

$\tilde{\alpha}:A[X\,|\,\partial(x_{z})=z]\to B\ \text{ with }\
\tilde{\alpha}(x_{z})=b_{z}$

where $X=\\{x_{z}:|x_{z}|=|z|+1\\}_{z\in Z}$.

###### Proof.

By induction it suffices to check when $Z=\\{z\\}$, in which case we write $b$
for $b_{z}$ and $x$ for $x_{z}.$ As $A[x\mid\partial x=z]^{\natural}$ is free
in the category of graded-commutative algebras there is a uniquely defined
graded algebra map $\tilde{f}\colon A[x\mid\partial x=z]\to B$ with
$\tilde{f}|_{A}=f$ and $x\mapsto b$. It remains to show $\tilde{f}$ commutes
with the differentials:

$\displaystyle\partial^{B}\tilde{f}\left(\sum a_{i}x^{i}\right)$
$\displaystyle=\sum\partial^{B}f(a_{i})b^{i}+(-1)^{|a_{i}|}f(a_{i})\partial^{B}(b^{i})$
$\displaystyle=\sum\partial^{B}f(a_{i})b^{i}+(-1)^{|a_{i}|}if(a_{i})f(z)b^{i-1}$
$\displaystyle=\sum
f\partial^{A}(a_{i})b^{i}+(-1)^{|a_{i}|}if(a_{i})f(z)\tilde{f}(x)^{i-1}$
$\displaystyle=\tilde{f}\left(\sum\partial^{A}(a_{i})b^{i}+(-1)^{|a_{i}|}ia_{i}zx^{i-1}\right)$
$\displaystyle=\tilde{f}\partial^{A}\left(\sum a_{i}x^{i}\right)\,;$

the equalities are all evident, and so $\tilde{f}$ is the desired (unique) dg
algebra extension of $f$ mapping $x$ to $z$. ∎

######  Discussion 1.6.

Let $A[x]$ be a semifree extension of $A$ on a single variable of even degree
$x$, and suppose $n$ is a positive integer. The element $\partial(x)x^{n-1}$
is always a cycle in $A[x]$. It is a boundary if and only if $n$ is invertible
in $A_{0}$, in which case the image of $\frac{1}{n}x^{n}$ under the
differential is $\partial(x)x^{n-1}$. Tate [31] realized that the dependence
on the invertiblility of $n$ can be eliminated by adjoining $x$ as a _divided
power variable_ , as recalled in the sequel.

######  1.7.

We write $A\langle Y\rangle$ for a semifree extension obtained by adjoining a
graded set of divided power variables $Y=Y_{1},Y_{2},Y_{3},\ldots$ to $A$; see
[5, Section 6.1] as well as [19, Chapter 1]. We refer to the variables of $Y$
as $\Gamma$-variables, and to $A\langle Y\rangle$ as a semifree
$\Gamma$-extension of $A$. Analogous to 1.2, each $\Gamma$-variable of $Y_{i}$
has homological degree $i$ with $Y_{i}$ consisting of divided power variables
when $i$ is even, and exterior variables when $i$ is odd. Recall that when
$y\in Y_{\rm{even}}$, its set of divided powers
$\\{y^{(i)}:|y^{(i)}|=|y|i\\}_{i\geqslant 0}$ with $y^{(0)}=1$ and $y^{(1)}=y$
satisfy the equalities

$y^{(n)}y^{(m)}=\binom{n+m}{n}y^{(n+m)}\ \text{ and }\
\partial(y^{(n)})=\partial(y)y^{(n-1)}\,.$

These fundamental equations are crucial to the analysis in the next section.

######  1.8.

Consider a semifree $\Gamma$-extension $A\langle Y\rangle$. We may well-order
$Y$ first by homological degree and then by ordering each set $Y_{n}$.
Associated to $A\langle Y\rangle$, with this choice of ordering, is a
canonical $A$-linear basis called the _normal $\Gamma$-monomials_; this basis
consists of $1$ together with the set of terms of the form

(1) $y_{\lambda_{1}}^{(i_{1})}\cdots y_{\lambda_{n}}^{(i_{n})}$

with $y_{\lambda_{1}}<\ldots<y_{\lambda_{n}}$ and each $i_{j}$ is a positive
integer, along with the additional constraint that $i_{j}=1$ when
$y_{\lambda_{j}}$ is of odd degree; see [5, Section 6] for further details.

Suppose $A$ is the local ring $R$ with maximal ideal $\mathfrak{m}$ and
residue field $k$. Let $\mathfrak{m}_{R\langle Y\rangle}^{(2)}$ denote the
ideal of $R\langle Y\rangle$ generated by all normal $\Gamma$-monomials Eq. 1
with $i_{1}+\ldots+i_{n}\geqslant 2.$ The complex of $\Gamma$-indecomposables
of $A\langle Y\rangle$ is the complex of $k$-spaces

${\operatorname{ind^{\gamma}}}R\langle
Y\rangle\coloneqq\dfrac{\mathfrak{m}_{R\langle Y\rangle}}{\mathfrak{m}R\langle
Y\rangle+\mathfrak{m}_{R\langle Y\rangle}^{(2)}}\,;$

it is clear that as a graded $k$-vector space
${\operatorname{ind^{\gamma}}}R\langle Y\rangle$ is simply $kY.$

######  1.9.

An acyclic closure for $\varphi$ is a factorization of $\varphi$ as $R\to
R\langle Y\rangle\xrightarrow{\simeq}S$ where
${\operatorname{ind^{\gamma}}}(R\langle Y\rangle)$ has zero differential.
Acyclic closures may be constructed inductively by adjoining
$\Gamma$-variables to minimally kill cycles generating the homology modules
${\rm H}_{n}(R\langle Y_{\leqslant n}))$, as originally described by Tate [31]
(see also [5, 19]). The procedure is identical to the construction process for
minimal models described in 1.3, except that divided power variables are used
in place of polynomial variables. Uniqueness, up to an isomorphism of dg
$\Gamma$-algebras, is contained in [19, Theorem 1.9.5]. As with minimal
models, by a slight abuse of terminology $R\langle Y\rangle$ is referred to as
an acyclic closure for $\varphi$.

We end this section with a discussion of a property of $\varphi$ which allows
for the comparison of dg algebra resolutions performed in Section 3.

######  Definition 1.10.

We say $\varphi$ is _weakly-closed_ provided that an acyclic closure $R\langle
Y\rangle$ of $\varphi$ has decomposable differential; that is,

$\partial(R\langle Y\rangle)\subseteq\mathfrak{m}_{R}R\langle
Y\rangle+(Y)^{2}\,.$

######  Remark 1.11.

Let $R\langle Y\rangle$ be a semifree $\Gamma$-extension of $R$. In analogy to
Definition 1.10, the containment

$\partial(R\langle Y\rangle)\subseteq\mathfrak{m}_{R}R\langle
Y\rangle+\mathfrak{m}_{R\langle Y\rangle}^{(2)}\,$

is called $\Gamma$-decomposability of the differential. It implies that the
differential of ${\operatorname{ind^{\gamma}}}R\langle Y\rangle$ is zero, and
if $R\langle Y\rangle$ is a resolution of $S$, it is equivalent to $R\langle
Y\rangle$ being an acyclic closure; see, for example, [5, Lemma 6.3.2].

This classical notion is similar to the one introduced in Definition 1.10, but
they are distinct. A precise description of their difference is provided by
Lemma 2.6 and Remark 2.7.

######  Remark 1.12.

As any two acyclic closures of $\varphi$ are isomorphic as local dg algebras
(in fact, as local dg $\Gamma$-algebras) the property of $\varphi$ being
weakly-closed is independent of choice of acyclic closure. If
$\mathbb{Q}\subseteq R$, then $\varphi$ is trivially weakly-closed. Indeed, in
this case $\mathfrak{m}_{R\langle Y\rangle}^{(2)}=(Y)^{2}$; see for example
Remark 3.2. The previous remark therefore guarantees $\varphi$ is weakly-
closed.

Nontrivial examples of such maps—regardless of characteristic
considerations—are closed maps, defined in the sequel. See also the discussion
in Remark 5.7.

######  1.13.

Following [11, 1.3], we say $\varphi$ is closed if some (equivalently, every)
acyclic closure $R\langle Y\rangle$ of $\varphi$ is minimal as a complex of
$R$-modules. That is, $\partial(R\langle
Y\rangle)\subseteq\mathfrak{m}_{R}R\langle Y\rangle$. Clearly any closed
homomorphism is weakly-closed. Examples of closed morphisms are plentiful:

  1. (1)

The augmentation map to the residue field is closed. This is the content of a
celebrated theorem of Gulliksen and Schoeller [17, 30]

  2. (2)

More generally, any large homomorphism—that is, $\varphi$ satisfying that the
induced map on Tor algebras
${\operatorname{Tor}}^{R}(k,k)\to{\operatorname{Tor}}^{S}(k,k)$ is
surjective—is closed by a theorem of Avramov and Rahbar-Rochandel; see [23,
Theorem 2.5]. A prominent example of a family of large homomorphisms is
provided by algebra retracts, in the sense that there exists a ring map from
$S$ to $R$ which is right inverse to $\varphi$.

  3. (3)

Quasi-complete intersection maps, defined and studied by Avramov, Henriques
and Şega, are closed; cf. [9, 1.6]. Recall $\varphi$ is quasi-complete
intersection if in the acyclic closure $R\langle Y\rangle$ of $\varphi$ we
have $Y_{i}=\emptyset$ for all $i\geqslant 3.$ An elementary example is when
$\varphi$ is a complete intersection homomorphism—in the usual sense that the
kernel of $\varphi$ is generated by an $R$-regular sequence—in which case
$Y_{2}=\emptyset$. The augmentation $R\rightarrow k$, when $R$ is a complete
intersection ring, is an additional example of a quasi-complete intersection;
when $R$ is singular the augmentation is not a complete intersection [31]. An
additional concrete example is the quotient $R\rightarrow R/(r)$ when $r$ is
an exact zero divisor as introduced in [14].

######  Remark 1.14.

Quasi-complete intersection maps are central to a conjecture due to Quillen on
the cotangent complex [27]. The latter has prompted numerous investigations
into the former; see, for example, [9, 13, 14, 22, 25, 26, 29, 33]. C from the
introduction, as well as Corollary 5.4, are two of the main results of this
article as they reveal further structural restrictions on quasi-complete
intersection maps. We hope these insights can lead to progress towards
settling Quillen’s conjecture, as analogous restrictions for complete
intersection maps were fundamental for resolving an analogous conjecture of
Quillen and understanding these maps more generally.

##  2\. Divided powers in prime residual characteristic

This section contains various facts involving divided power algebras and
binomial coefficients that are needed for our analysis in Section 3.
Foundational material on divided power algebras is contained in [1, 28], which
includes some of the formulas in this section; we provide a self-contained
treatment below. Throughout this section $p$ will be a fixed prime number.

######  2.1.

Recall that every integer $n$ may be written uniquely in its base $p$
expansion as

$n_{0}p^{0}+n_{1}p^{1}+\dots+n_{l}p^{l}$

where each $n_{i}$ is a nonnegative integer strictly less than $p$; in this
case we write $n=[n_{0}n_{1}\dots n_{l}]_{p}$. We use $\equiv_{p}$ to denote
equivalence of integers modulo $p$.

In the notation above, we recall the following classical theorem of Lucas
[24]: _If $n=[n_{0}\dots n_{l}]_{p}$ and $m=[m_{0}\dots m_{l}]_{p}$, then_

$\binom{n}{m}\equiv_{p}\binom{n_{0}}{m_{0}}\cdots\binom{n_{l}}{m_{l}}.$

In applying this theorem, note the convention that $\binom{n}{m}=0$ whenever
$m>n$.

######  Notation 2.2.

Let $y$ be a divided power variable over a ring $R$. Using 1.7 it follows
easily that for any positive integer partition of
$n=n_{0}+n_{1}+\ldots+n_{l}$, there is the equality

$y^{(n_{0})}\cdots y^{(n_{l})}=\binom{n}{n_{0},n_{1},\dots,n_{l}}y^{(n)}\
\text{ where }\
\binom{n}{n_{0},n_{1},\dots,n_{l}}=\dfrac{n!}{n_{0}!n_{1}!\cdots n_{l}!}\,.$

Define the following nonnegative integers

$b_{t,m}=\binom{tp^{m}}{\,\underbrace{p^{m},p^{m},\dots,p^{m}}_{t\text{-times}}}\
\text{ and }\ c_{n}\coloneqq\binom{n}{n_{0},n_{1}p,\dots,n_{l}p^{l}}$

for each $m,n,t\in\mathbb{N},$ where $n=[n_{0}\dots n_{l}]_{p}.$ These arise
as the coefficients in the following multiplications:

$\displaystyle(y^{(p^{m})})^{t}$ $\displaystyle=b_{t,m}y^{(tp^{m})}$
$\displaystyle y^{(n_{0})}y^{(n_{1}p)}\cdots y^{(n_{l}p^{l})}$
$\displaystyle=c_{n}y^{(n)}\,\text{ where }\,n=[n_{0}\dots n_{l}]_{p}\,.$

The coefficient $b_{p-1,m}$ arises sufficiently often that we will denote it
$b_{m}$ for simplicity.

######  Lemma 2.3.

Adopting 2.2, there are equivalences

$b_{t,m}\equiv_{p}t!$ |  |  | $b_{m}\equiv_{p}-1$ |  |  | $c_{n}\equiv_{p}1$
---|---|---|---|---|---|---

for any nonnegative integers $m$ and $n$ and for $t<p$. In particular, if $R$
has residual charactersitic $p$, then $b_{t,m}$, $b_{m}$, and $c_{n}$ are
units in $R$.

###### Proof.

For the first equivalence

$b_{t,m}=\binom{p^{m}}{p^{m}}\binom{2p^{m}}{p^{m}}\cdots\binom{tp^{m}}{p^{m}}\equiv_{p}t!$

where the equivalence follows from Lucas’ theorem, recalled in 2.1. Wilson’s
theorem [21] says $(p-1)!\equiv_{p}-1$, giving the second equivalence.

For the third equivalence, we induct on $l$; the base case, when $l=0$, is
trivial. Next assume $l>0$ and set $n^{\prime}=[n_{0}\dots n_{l-1}]_{p}$.
Observe that

$c_{n}=c_{n^{\prime}}\binom{n}{n_{l}p^{l}}\equiv_{p}1$

where the first equality uses $n=n^{\prime}+n_{l}p^{l}$ and the equivalence
holds by Lucas’ theorem and induction. ∎

######  Remark 2.4.

The precise formulas for the coefficients specified in 2.2 are primarily
needed in Section 3 to handle the case when $R$ has mixed characteristic. When
$R$ itself has characteristic $p$, the arguments in the proof of Lemma 2.3
show that $b_{m}=-1$ and $c_{n}=1$. That is,

$(y^{(p^{m})})^{p-1}=-y^{((p-1)p^{m})}\ \text{ and }\
y^{(n)}=y^{(n_{0})}y^{(n_{1}p)}\cdots y^{(n_{l}p^{l})}\,$

where $n=[n_{0}\dots n_{l}]_{p}$.

######  Notation 2.5.

Suppose $(R,\mathfrak{m},k)$ is local and $k$ has characteristic $p>0.$ For
any semifree $\Gamma$-extension $R\langle Y\rangle$ of $R$, we let
$kY^{(p^{\infty})}$ denote the subcomplex of ${\operatorname{ind}}R\langle
Y\rangle$ consisting of cycles of the form $y^{(p^{i})}$ with $y\in
Y_{\rm{even}}$ and $i>0.$

######  Lemma 2.6.

Suppose $(R,\mathfrak{m},k)$ is local and $k$ has characteristic $p>0.$ For
any semifree $\Gamma$-extension $R\langle Y\rangle$ of $R$, there is an exact
sequence of complexes of $k$-spaces

$0\rightarrow kY^{(p^{\infty})}\to{\operatorname{ind}}R\langle
Y\rangle\rightarrow{\operatorname{ind^{\gamma}}}R\langle Y\rangle\rightarrow
0\,.$

###### Proof.

Set $A=R\langle Y\rangle$. The containment of subcomplexes

(2)
$\mathfrak{m}+\mathfrak{m}_{A}^{2}=\mathfrak{m}A+\mathfrak{m}_{A}^{2}\subseteq\mathfrak{m}A+\mathfrak{m}_{A}^{(2)}$

induces a surjection
${\operatorname{ind}}A\rightarrow{\operatorname{ind^{\gamma}}}A.$ It suffices
to examine when $y^{(n)}$ is zero in ${\operatorname{ind}}A$ with $y\in
Y_{\textrm{even}}$, and $n>1$ with $n=[n_{0}\dots n_{l}]_{p}$.

When $n$ is not a power of $p$, there are two cases. First if at least two
$n_{i}$ are nonzero, then using 2.2 and Lemma 2.3, we obtain the equality

$y^{(n)}=c_{n}^{-1}y^{(n_{0})}\ldots y^{(n_{l}p^{l})}\,.$

The second case is that $n_{i}=0$ for $l>i$ and $n_{l}>1$. Here

$y^{(n)}=y^{(n_{l}p^{l})}=b_{n_{l},l}^{-1}(y^{(p^{l})})^{n_{l}}$

where the second equality again uses 2.2 and Lemma 2.3. Hence, in either case,
we have shown $y^{(n)}$ is zero in ${\operatorname{ind}}A$.

Now we show that when $n$ is a power of $p$, say $p^{i}$, then $y^{(n)}$ is
not zero in ${\operatorname{ind}}A.$ Assuming to contrary, in $A\otimes_{R}k$
one can write $y^{(n)}$ as a $k$-linear combination of $\Gamma$-monomials of
the form

$y^{(i_{1})}\cdots y^{(i_{t})}$

with $i_{1}+\ldots+i_{t}=n$ and $t>1.$ Set $m=i_{1}+\dots+i_{t-1}$. Using
Lemma 2.3, we have

$y^{(i_{1})}\cdots
y^{(i_{t})}=\binom{m}{n_{1},\dots,n_{t-1}}\binom{n}{i_{t}}y^{(p^{i})}$

and since $t\geqslant 1$, it follows that $i_{t}<n$. Observe that
$\binom{n}{i_{t}}\equiv_{p}0$ by Lucas’ theorem. As a consequence
$y^{(i_{1})}\cdots y^{(i_{t})}$ equals zero in $A\otimes k$. ∎

######  Remark 2.7.

A simple consequence of Lemma 2.6, that is easy to see independently, is that
decomposability of the differential of $R\langle Y\rangle$ implies
$\Gamma$-decomposability of its differential. The main content of Lemma 2.6 is
that the converse holds if no summand of the form $y^{(p^{i})}$, for $y$ in
$Y_{\textrm{even}}$ and positive integer $i$, appears as a summand in the
differential of any element of $R\langle Y\rangle.$

##  3\. Comparison map

Throughout, we fix a surjective homomorphism of commutative local noetherian
rings $\varphi\colon R\to S$ with common residue field $k$. Let
$R[X]\xrightarrow{\simeq}S$ be a minimal model for $\varphi$ and let $R\langle
Y\rangle\xrightarrow{\simeq}S$ denote an acyclic closure for $\varphi$.

######  3.1.

By 1.4, there exists a quasi-isomorphism of dg $R$-algebras
$R[X]\xrightarrow{\simeq}R\langle Y\rangle$ which is unique up to homotopy and
compatible with the augmentations to $S$. In this section we construct an
explicit representative of this map, which will denoted by
$\operatorname{\gamma}^{\varphi}$ and referred to as the _comparison map of_
$\varphi$. As $\operatorname{\gamma}^{\varphi}$ is a morphism of local dg
algebras, along with an analogous containment to Eq. 2, the comparison map
induces a map on complexes of $k$-spaces

${\operatorname{ind}}(\operatorname{\gamma}^{\varphi})\colon{\operatorname{ind}}(R[X])\to{\operatorname{ind^{\gamma}}}(R\langle
Y\rangle)\,.$

######  Remark 3.2.

When the characteristic of $k$ is further assumed to be zero, it is then
standard that $R[X]$ and $R\langle Y\rangle$ are isomorphic, and easy to see
that $\operatorname{\gamma}^{\varphi}$ is itself an isomorphism of dg algebras
which induces the isomorphism
${\operatorname{ind}}(\operatorname{\gamma}^{\varphi})$.

Indeed, if $R$ is local and $\operatorname{char}k=0$, then $R$ contains a copy
of $\mathbb{Q}$. In this setting, for any cycle $z$ in a dg $R$-algebra $A$,
the extensions $A[x\,|\,\partial(x)=z]$ and $A\langle
x\,|\,\partial(x)=z\rangle$ coincide when $|x|$ is odd. When $|x|$ is even,
the map $A[x\,|\,\partial(x)=z]\to A\langle x\,|\,\partial(x)=z\rangle$ given
by

$\sum_{i\geq 0}a_{i}x^{i}\mapsto\sum_{i\geq 0}i!a_{i}x^{(i)}$

establishes an isomorphism of dg $R$-algebras; the fact this map is invertible
is the only place that $\mathbb{Q}\subseteq R$ is used. Thus $R[X]$ and
$R\langle Y\rangle$ are each a direct limit of isomorphic extensions and
$\operatorname{\gamma}^{\gamma}$ is the direct limit of these isomorphisms.

The goal of this section is to provide insight on the situation when $k$ has
prime characteristic $p>0$, which we assume for the remainder of the section.

In this section, we prove A which is recast, in the notation set above, in
Corollary 3.3. It is an immediate consequence of Theorem 3.6 presented at the
end of the section. Recall that for a graded object $V$, the $i$-fold
suspension of $V$ is ${\scriptstyle\mathsf{\Sigma}}^{i}V$ where
$({\scriptstyle\mathsf{\Sigma}}^{i}V)_{j}=V_{j-i}.$

######  Corollary 3.3.

Suppose $\varphi\colon R\to S$ is a surjective map of local rings whose common
residue field $k$ has characteristic $p>0$. If $\varphi$ is weakly-closed,
then the comparison map $\operatorname{\gamma}^{\varphi}$ is surjective and
induces the following exact sequence of graded $k$-spaces

$0\to
kY^{(p^{\infty})}\oplus{\scriptstyle\mathsf{\Sigma}}kY^{(p^{\infty})}\to{\operatorname{ind}}(R[X])\xrightarrow{{\operatorname{ind}}(\operatorname{\gamma}^{\varphi})}{\operatorname{ind^{\gamma}}}(R\langle
Y\rangle)\to 0.\qed$

######  Construction 3.4.

Let $A$ be a dg $R$-algebra and $z\in A_{\rm{odd}}$ be a cycle. By Lemma 1.5,
there is a canonical map of dg $A$-algebras

$\operatorname{\gamma}\colon A[x\,|\,\partial x=z]\to A\langle y\,|\,\partial
y=z\rangle$

completely determined by $x\mapsto y$. We define a semifree extension

$A[x]\hookrightarrow A[\\{x_{i},x_{i+1}^{\prime}\\}_{i\geqslant 0}]$

and a map
$\operatorname{\gamma}_{z}(A):A[\\{x_{i},x_{i+1}^{\prime}\\}_{i\geqslant
0}]\to A\langle y\rangle$ of dg $A$-algebras extending
$\operatorname{\gamma}$. Recall the coefficients $b_{t,m}$ and $c_{n}$ defined
in 2.2. For notational convenience, set

(3)
$d_{i}=c_{p^{i}-1}\prod_{j=0}^{i-1}(b_{j}c_{p^{j}-1}^{p-1})^{p^{i-1-j}}\,.$

Set $x_{0}\coloneqq x$ and by induction assume we have constructed
$A[\\{x_{i},x_{i+1}^{\prime}\\}_{0\leqslant i\leqslant n}]$, denoted $A(n)$,
with differential determined by

$\partial(x_{i})=zx_{0}^{p-1}x_{1}^{p-1}\ldots x_{i-1}^{p-1}\ \text{ and }\
\partial(x_{i}^{\prime})=x_{i-1}^{p}-px_{i},$

and a dg algebra map $\operatorname{\gamma}_{z}(n):A(n)\to A\langle y\rangle$
determined by $\operatorname{\gamma}_{z}(n)(x_{0})=y$ and for all $i\geqslant
1$:

$\operatorname{\gamma}_{z}(n)(x_{i})=d_{i}y^{(p^{i})}\ \text{ and }\
\operatorname{\gamma}_{z}(n)(x_{i}^{\prime})=0\,.$

By Lemma 2.3, each $d_{i}$ is a product of units and hence is a unit of $R$. A
tedious direct calculation, using Lemma 2.3, yields

$\operatorname{\gamma}_{z}(n)(zx_{0}^{p-1}\cdots
x_{n}^{p-1})=d_{n+1}\partial(y^{(p^{n+1})})\,.$

Also, as $z$ has odd degree, it follows that $zx_{0}^{p-1}x_{1}^{p-1}\ldots
x_{n}^{p-1}$ is a cycle of degree $(|x|p^{n+1}-1)$. Therefore by Lemma 1.5,
there is a unique dg $A$-algebra map

(4) $A(n)[x_{n+1}\,|\,\partial x_{n+1}=zx_{0}^{p-1}x_{1}^{p-1}\ldots
x_{n}^{p-1}]\to A\langle y\rangle$

extending $\operatorname{\gamma}_{z}(n)$ with $x_{n+1}\mapsto
d_{n+1}y^{(p^{n+1})}.$

Under this extension the image of $x_{n}^{p}-px_{n+1}$ is a cycle of degree
$|x|p^{n+1}$. Using the definitions of $d_{n}$ and $c_{i}$, the image of this
cycle under Eq. 4 is

$\displaystyle d_{n}^{p}(y^{(p^{n})})^{p}-pd_{n+1}y^{(p^{n+1})}$
$\displaystyle=\left(b_{n}d_{n}^{p}\binom{p^{n+1}}{p^{n}}-pd_{n+1}\right)y^{(p^{n+1})}$
$\displaystyle=\left(\binom{p^{n+1}}{p^{n}}\frac{c_{p^{n}-1}}{c_{p^{n+1}-1}}-p\right)d_{n+1}y^{(p^{n+1})}$
$\displaystyle=0\,.$

Hence $\operatorname{\gamma}_{z}(n)$ can be further extended, again applying
Lemma 1.5, to a dg $A$-algebra map $\operatorname{\gamma}_{z}(n+1):A(n+1)\to
A\langle y\rangle$ where

$A(n+1)\coloneqq A[\\{x_{i},x_{i+1}^{\prime}\\}_{0\leqslant i\leqslant
n+1}\,|\,\partial x_{i+1}=zx_{0}^{p-1}\ldots x_{i}^{p-1},\ \partial
x_{i+1}^{\prime}=x_{i}^{p}-px_{i+1}]$

with $\operatorname{\gamma}_{z}(n+1)(x_{i})=d_{i}y^{(p^{i})}$ and
$\operatorname{\gamma}_{z}(n+1)(x_{i}^{\prime})=0$ for each $i$, which
completes the induction.

Now taking the colimit of the maps $\operatorname{\gamma}_{z}(n)$, we obtain
the dg $A$-algebra map
$\operatorname{\gamma}_{z}(A):A[\\{x_{i},x_{i+1}^{\prime}\\}_{i\geqslant
0}]\to A\langle y\rangle$ extending $\operatorname{\gamma}$ with

$\operatorname{\gamma}_{z}(A)(x_{i})=d_{i}y^{(p^{i})}\ \text{ and
}\operatorname{\gamma}_{z}(A)(x_{i}^{\prime})=0\,.$

######  Lemma 3.5.

In the notation of 3.4, the dg $A$-algebra map

$\operatorname{\gamma}_{z}(A):A[\\{x_{i},x_{i+1}^{\prime}\\}_{i\geqslant
0}]\to A\langle y\rangle$

is a quasi-isomorphism.

###### Proof.

As the regular element $x_{i}^{p}-px_{i+1}$ is the boundary of the
corresponding $x_{i+1}^{\prime}$, by iteratively applying [31, Theorem 3] it
follows that the canonical map

$A[\\{x_{i},x_{i+1}^{\prime}\\}_{i\geqslant
0}]\xrightarrow{\simeq}A[\\{x_{i}\\}_{i\geqslant
0}]/(\\{x_{i}^{p}-px_{i+1}\\}_{i\geqslant 0})$

is a quasi-isomorphism. Next observe that $\operatorname{\gamma}_{z}(A)$
factors through the canonically induced map

$\overline{\operatorname{\gamma}}:A[\\{x_{i}\\}_{i\geqslant
0}]/(\\{x_{i}^{p}-px_{i+1}\\}_{i\geqslant 0})\to A\langle y\rangle$

with $x_{i}\mapsto d_{i}y^{(p^{i})}$ for each $i\geqslant 0$; cf. 3.4Eq. 3 for
the definition of $d_{i}$ which is defined in terms of the coefficients
introduced in 2.2.

Let $n$ be a nonnegative integer and write $n=[n_{0}n_{1}\ldots n_{l}]_{p}$.
Observe that

$\displaystyle\overline{\operatorname{\gamma}}(x_{0}^{n_{0}}x_{1}^{n_{1}}\cdots
x_{l}^{n_{l}})$ $\displaystyle=\prod_{i=1}^{l}(d_{i}y^{(p^{i})})^{n_{i}}$
$\displaystyle=\prod_{i=1}^{l}d_{i}^{n_{i}}b_{n_{i},i}y^{(n_{i}p^{i})}$
$\displaystyle=\left(\prod_{i=1}^{l}d_{i}^{n_{i}}b_{n_{i},i}\right)c_{n}y^{(n)}$

the first equality holds as $\overline{\operatorname{\gamma}}$ is an algebra
map and the other equalities use 2.2. Also, an $A$-linear basis for
$A[\\{x_{i}\\}_{i\geqslant 0}]/(\\{x_{i}^{p}-px_{i+1}\\}_{i\geqslant 0})$ is

$\\{x_{0}^{n_{0}}x_{1}^{n_{1}}\cdots x_{l}^{n_{l}}:0\leqslant n_{i}<p\\}$

and it is standard that an $A$-linear basis for $A\langle y\rangle$ is
$\\{y^{(n)}\\}_{n\geqslant 0}$, the divided powers of $y$. By Lemma 2.3, the
nonnegative integers

$\left(\prod_{i=1}^{l}d_{i}^{n_{i}}b_{n_{i},i}\right)c_{n}$

are units in $R$ and so $\overline{\operatorname{\gamma}}$ is an isomorphism
of dg $A$-algebras, which finishes the proof of the lemma once recalling
$\operatorname{\gamma}_{z}(A)$ factors as

$A[\\{x_{i},x_{i+1}^{\prime}\\}_{i\geqslant
0}]\xrightarrow{\simeq}A[\\{x_{i}\\}_{i\geqslant
0}]/(\\{x_{i}^{p}-px_{i+1}\\}_{i\geqslant
0})\xrightarrow{\overline{\operatorname{\gamma}}}A\langle y\rangle.\qed$

######  Theorem 3.6.

Suppose $\varphi\colon R\to S$ is a surjective morphism of local rings of
residual characteristic $p>0$. If $\varphi$ is weakly-closed, then the minimal
model $R[X]$ for $\varphi$ has the form

$R[X(0),X(1),X^{\prime}(1),X(2),X^{\prime}(2),\ldots]$

where

  1. (1)

$X(0)=\\{x_{0}(y):y\in Y\\}$ with $|x_{0}(y)|=|y|$,

  2. (2)

for $i\geqslant 1$, $X(i)=\\{x_{i}(y):y\in Y_{\rm{even}}\\}$ with
$|x_{i}(y)|=|y|p^{i}$,

  3. (3)

for $i\geqslant 1$, $X^{\prime}(i)=\\{x_{i}^{\prime}(y):y\in Y_{\rm{even}}\\}$
with $|x_{i}^{\prime}(y)|=|y|p^{i}+1$,

and there exists a surjective quasi-isomorphism
$\operatorname{\gamma}^{\varphi}\colon R[X]\rightarrow R\langle Y\rangle$
determined by the formulas

$\operatorname{\gamma}^{\varphi}(x_{i}(y))=d_{i}y^{(p^{i})}\ \text{ and
}\operatorname{\gamma}^{\varphi}(x_{i}^{\prime}(y))=0$

where $d_{i}$ is the unit defined in 3.4Eq. 3. Furthermore, the differential
of $R[X]$ is given by

$\partial(x_{i}(y))=\tilde{z}\prod_{j=0}^{i-1}x_{j}(y)^{p-1}\ \text{ and }\
\partial(x_{i}^{\prime}(y))=x_{i-1}(y)^{p}-px_{i}(y)$

where $\tilde{z}$ is a cycle lifting the cycle $\partial(y)$ along
$\operatorname{\gamma}^{\varphi}$.

###### Proof.

Assume, by induction, we have constructed a surjective quasi-isomorphism of dg
$R$-algebras

$\operatorname{\gamma}(n):R[X(0)_{\leqslant n},X(1)_{\leqslant
n},X^{\prime}(1)_{\leqslant n},\ldots]\to R\langle Y_{\leqslant n}\rangle$

with the desired properties for some $n\geqslant 0$, and let $A(n)$ denote the
source of this map.

Let $z$ be a cycle in $R\langle Y_{\leqslant n}\rangle$ of homological degree
$n$. Since $\operatorname{\gamma}(n)$ is a surjective quasi-isomorphism, there
exists a cycle $\tilde{z}$ in $A(n)$ with
$\operatorname{\gamma}(n)(\tilde{z})=z.$ Furthermore, by Remark 2.7 the
assumption that $\varphi$ is weakly-closed guarantees no summand of $z$ is of
the form $y^{(p^{i})}$ for $y\in Y_{\text{even}}$, which implies that
$\tilde{z}$ can be chosen to be in
$\mathfrak{m}_{R}A(n)+\mathfrak{m}_{A(n)}^{2}$.

When $n$ is even, $\operatorname{\gamma}(n)$ extends to the quasi-isomorphism
of dg $R$-algebras

$A(n)\langle x\,|\,\partial x=\tilde{z}\rangle\xrightarrow{\simeq}R\langle
Y_{\leqslant n}\rangle\langle y\,|\,\partial y=z\rangle;$

cf. [5, Lemma 7.2.10]. As $x$ is an exterior variable, $A(n)[x\,|\,\partial
x=\tilde{z}]$ and $A(n)\langle x\rangle$ coincide.

Now assume $n$ is odd. In this case we obtain a surjective quasi-isomorphism

(5) $A(n)[\\{x_{i},x^{\prime}_{i+1}\\}_{i\geqslant
0}]\xrightarrow{\simeq}A(n)\langle y\,|\,\partial y=\tilde{z}\rangle,$

with $\partial x_{i}=\tilde{z}x_{0}^{p-1}\ldots x_{i-1}^{p-1}$ and $\partial
x_{i+1}^{\prime}=x_{i}^{p-1}-px_{i+1}$, using Lemma 3.5. Furthermore, another
application of [5, Lemma 7.2.10], extends $\operatorname{\gamma}(n)$ to a
surjective quasi-isomorphism

(6) $A(n)\langle y\,|\,\partial y=\tilde{z}\rangle\xrightarrow{\simeq}R\langle
Y_{\leqslant n}\rangle\langle y\,|\,\partial y=z\rangle.$

Composing the surjective quasi-isomorphisms from Eq. 5 and Eq. 6 yield another
one

$A(n)[\\{x_{i},x^{\prime}_{i+1}\\}_{i\geqslant 0}]\xrightarrow{\simeq}R\langle
Y_{\leqslant n}\rangle\langle y\,|\,\partial y=z\rangle.$

Repeating this for each cycle of degree $n$, extends
$\operatorname{\gamma}(n)$ to a surjective quasi-isomorphism

$\operatorname{\gamma}(n+1):R[X(0)_{\leqslant n+1},X(1)_{\leqslant
n+1},X^{\prime}(1)_{\leqslant n+1},\ldots]\to R\langle
Y_{\leqslant{n+1}}\rangle.$

Taking the colimit of these maps yields the desired surjective quasi-
isomorphism

$\operatorname{\gamma}^{\varphi}\colon
R[X(0),X(1),X^{\prime}(1),X(2),X^{\prime}(2),\ldots]\xrightarrow{\simeq}R\langle
Y\rangle$

satisfying all of the desired properties. ∎

##  4\. The homotopy Lie algebra

Fix a surjective map $\varphi\colon R\to S$ of local rings with common residue
field $k$. First we recall the homotopy Lie algebra $\pi(F^{\varphi})$
introduced by Avramov. Its structure reflects interesting ring-theoretic
properties of $\varphi$; suitable references include [2, 5, 7, 8].

######  Remark 4.1.

Homotopy Lie algebras in local algebra play an analogous role to those arising
in rational homotopy theory. Inspired by their utility in the study of finite-
type simply-connected CW complexes, Avramov, Halperin, Roos, and others
initiated a study of the corresponding Lie algebras in local algebra. A
thorough dictionary between the Lie algebras in rational homotopy theory and
those in local algebra may be found in [7]. Notable applications in
commutative algebra include resolving conjectures of Quillen regarding
cotangent cohomology [8, 12], as well as settling a conjecture of Vasoncelos
on the conormal module [16]. Their construction is described in the sequel.

######  4.2.

Let $R[X]\xrightarrow{\simeq}S$ be a minimal model for $\varphi$, and let
$k[X]$ denote $k\otimes_{R}R[X].$ Also let $kX^{n}$ be the $k$-linear space
generated by all monomials in $X$ of degree $n$; the space $kX^{1}=kX$ is
canonically isomorphic to ${\operatorname{ind}}(R[X])$ (as defined in 1.1),
and so these will be naturally identified. The assumption that $R[X]$ has
decomposable differential yields the decomposition

$\partial^{k[X]}=\partial^{[2]}+\partial^{[3]}+\ldots$

with $\partial^{[i]}|_{kX}:kX\to kX^{i}$. The equality
$\partial^{[2]}\partial^{[2]}=0$ defines a graded Lie algebra structure on
$({\scriptstyle\mathsf{\Sigma}}kX)^{*}$ as recalled below. In what follows,
$(-)^{*}$ denotes $k$-linear duality, and we equip
$({\scriptstyle\mathsf{\Sigma}}kX)^{*}$ with the dual basis of functionals
${\scriptstyle\mathsf{\Sigma}}x^{*}$ where

${\scriptstyle\mathsf{\Sigma}}x^{*}({\scriptstyle\mathsf{\Sigma}}x^{\prime})=\begin{cases}1&x=x^{\prime}\\\
0&x\neq x^{\prime}\end{cases}$

for $x,x^{\prime}\in X.$

As a graded $k$-space the homotopy Lie algebra of $\varphi$, denoted
$\pi(F^{\varphi})$, is $({\scriptstyle\mathsf{\Sigma}}kX)^{*}$. The Lie
structure on $\pi(F^{\varphi})$ is defined using $\partial^{[2]}$ along with a
fixed well-ordering of $X$, as usual first ordered by homological degree.
Namely, writing

$\partial^{[2]}(x_{l})=\sum_{i<j}q_{ij}^{l}x_{i}x_{j}+\sum_{i}q_{i}^{l}x_{i}^{2}\,,$

the Lie bracket and reduced square on $({\scriptstyle\mathsf{\Sigma}}kX)^{*}$
are defined, with $j>i$, as

$[{\scriptstyle\mathsf{\Sigma}}x_{j}^{*},{\scriptstyle\mathsf{\Sigma}}x_{i}^{*}]=\sum_{l}q^{l}_{ij}{\scriptstyle\mathsf{\Sigma}}x_{l}^{*}\
\text{ and }\
({\scriptstyle\mathsf{\Sigma}}x_{i}^{*})^{[2]}=-\sum_{l}q_{i}^{l}{\scriptstyle\mathsf{\Sigma}}x_{l}^{*}\,.$

######  Discussion 4.3.

There are two functors involved in the formation of the homotopy Lie algebra
of a surjective local homomorphism, which explains our choice of notation
above. Namely, the first functor $F$ associates to the surjective local
homomorphism $\varphi\colon R\to S$ the (derived) fiber

$F^{\varphi}\coloneqq k\otimes^{\sf L}_{R}S\simeq k[X]\,$

in the notation of 4.2. The functor $\pi$ is naturally defined on the category
of semifree dg $k$-algebras with decomposable differential as discussed in
_loc. cit._ ; the target category is that of graded Lie algebras over $k$ of
finite type.

One can also regard $\pi$ as the functor which associates to a semifree dg
$k$-algebra with decomposable differential the homology of the dg Lie algebra
of $\Gamma$-derivations of an acyclic closure of $k$ over $A$; this is the
perspective taken in [5, Chapter 10], though this will be less relevant for
what follows.

We now arrive at one of the main applications of Theorem 3.6 which shows the
homotopy Lie algebra of a non-complete intersection weakly-closed map always
has an infinitely generated abelian Lie subalgebra.

######  Theorem 4.4.

Let $\varphi\colon R\rightarrow S$ be a surjective local map with prime
residual characteristic $p>0$ and let $R\langle Y\rangle$ be an acyclic
closure of $\varphi$. If $\varphi$ is weakly-closed, then there is an
isomorphism of graded Lie algebras

$\pi(F^{\varphi})\cong L\times L^{\infty}\ \text{ with }\
[\pi(F^{\varphi}),L^{\infty}]=0$

where $L\cong({\scriptstyle\mathsf{\Sigma}}kY)^{*}$ and
$L^{\infty}\cong({\scriptstyle\mathsf{\Sigma}}kY^{(p^{\infty})}\oplus{\scriptstyle\mathsf{\Sigma}}^{2}kY^{(p^{\infty})})^{*}$
as $k$-spaces. Furthermore, when $p>2$ the reduced square on $L^{\infty}$ is
trivial. When $p=2$, the reduced square of
$({\scriptstyle\mathsf{\Sigma}}y^{(2^{i})})^{*}\in L^{\infty}$ is given by
$({\scriptstyle\mathsf{\Sigma}}^{2}y^{(2^{i+1})})^{*}$ and trivial otherwise.

###### Proof.

Take $R[X]=R[X(0),X(1),X^{\prime}(1),\dots]$ as described in Theorem 3.6 to be
a minimal model of $\varphi$. Define the $k$-subspaces of $\pi(F^{\varphi})$:

$L\coloneqq({\scriptstyle\mathsf{\Sigma}}kX(0))^{*}\ \text{ and }\
L^{\infty}\coloneqq\left(\bigoplus_{i=1}^{\infty}{\scriptstyle\mathsf{\Sigma}}kX(i)\oplus{\scriptstyle\mathsf{\Sigma}}^{2}kX^{\prime}(i)\right)^{*}\,.$

The asserted isomorphisms on $k$-spaces are induced by the obvious
isomorphisms

$kY\cong kX(0)\ \text{ and }\
kY^{(p^{\infty})}\cong\bigoplus_{i=1}^{\infty}kX(i)$

given by $y\mapsto x_{0}(y)$ for each $y\in Y$ and $y^{(p^{j})}\mapsto
x_{j}(y)$ for each $y\in Y_{\textrm{even}}$ and any $j\geqslant 1$,
respectively.

When $p\neq 2$, a direct calculation using Theorem 3.6 shows that for $i>0$ no
element of $X(i)$ or $X^{\prime}(i)$ appears as a summand of any element in
the image of $\partial^{[2]}$ and that $\partial^{[2]}(X(i)\cup
X^{\prime}(i))=0$. Using these facts and 4.2 to compute the Lie bracket and
reduced square on $\pi(F^{\varphi})$ it follows that $L$ and $L^{\infty}$ are
Lie subalgebras of $\pi(F^{\varphi})$ with the property
$[\pi(F^{\varphi}),L^{\infty}]=0$ with trivial reduced square on $L^{\infty}$.

Finally, when $p=2$, a similar calculation together with the equality

$\partial^{[2]}(x_{i+1}^{\prime}(y))=x_{i}(y)^{2}$

yields the last conclusion. ∎

The next corollary, along with Theorem 4.4, establishes B from the
Introduction.

######  Corollary 4.5.

If $\varphi$ is closed, then $\pi(F^{\varphi})$ is an abelian Lie algebra.
Furthermore, $\pi(F^{\varphi})$ is abelian as a restricted Lie algebra
whenever the residue field has characteristic different from 2.

###### Proof.

When $R$ has characteristic zero, Remark 3.2 and the definition of a closed
map ensure that $\partial^{[2]}=0$ on all of $X$, so the formula for the
bracket in 4.2 yields that $\pi(F^{\varphi})$ is abelian as a restricted Lie
algebra.

Next, consider the case when $\operatorname{char}k=p>0$. The assumption that
$\varphi$ is closed guarantees $\partial^{[2]}$ vanishes on any variable
belonging to $X(0)$, in the notation of Corollary 3.3. Hence the Lie algebra
$L$ in Theorem 4.4 is an abelian restricted Lie subalgebra of
$\pi(F^{\varphi})$. ∎

######  Remark 4.6.

When $\varphi$ is a map of finite projective dimension—in the sense that $S$
has finite projective dimension over $R$—the only case that $\pi(F^{\varphi})$
is abelian is when $\varphi$ is complete intersection; cf. [8, Theorem C].
Corollary 4.5 provides a wide class of maps (not necessarily of finite
projective dimension) for which $\pi(F^{\varphi})$ is abelian; see the
examples in 1.13. A consequence is that there cannot be a direct analog to [8,
Theorem C] for detecting the quasi-complete intersection property in terms of
an abelian Lie algebra structure of $\pi(F^{\varphi})$.

##  5\. Applications involving deviations and closing remarks

Let $\varphi\colon R\rightarrow S$ be a surjective local map, and let $k$
denote the common residue field of $R$ and $S$. In this section we present
applications involving the deviations of $\varphi$; see 5.1, below, for the
definition.

The rigidity and vanishing of the deviations of a local homomorphism have been
well-studied; see for example, [4, 8, 10, 15]. However, a crucial assumption
for many of these works is that $\varphi$ is a map of finite projective
dimension. The corollaries in this section add to the rigidity results above,
without the restriction on maps of finite projective dimension. Instead, our
results apply when $\varphi$ is a weakly-closed surjective local map with
residual characteristic $p>0$; the generic examples in 1.13 are maps of
infinite projective dimension.

######  5.1.

The numbers $\varepsilon_{i}(\varphi):=\dim_{k}\pi^{i}(F^{\varphi})$ are
called the _deviations of $\varphi$_. They are encoded in the Poincaré series

$P(t)=\sum_{i\geqslant 0}\dim_{k}({\operatorname{Tor}}^{k\otimes^{\sf
L}_{S}R}_{i}(k,k))t^{i}$

of $k$ over the derived fiber $k\otimes^{\sf L}_{R}S$ according to the
equality

$P(t)=\frac{\prod_{i=1}^{\infty}(1+t^{2i-1})^{\varepsilon_{2i-1}(\varphi)}}{\prod_{i=1}^{\infty}(1-t^{2i})^{\varepsilon_{2i}(\varphi)}}\,.$

Furthermore, upon fixing a minimal model $R[X]$ for $\varphi$, we see that the
number of variables adjoined in $X_{i}$ is exactly
$\varepsilon_{i+1}(\varphi).$ These naturally generalize the deviations of a
local ring; a standard reference for the latter is [5, Section 7], and see the
references contained in _loc. cit_.

######  5.2.

When $\varphi$ is a weakly-closed surjective map and $R$ has residual
characteristic $p>0$, with acyclic closure $R\langle Y\rangle$, Corollary 3.3
shows that the deviations of $\varphi$ are completely determined by the
numbers
$\varepsilon^{\gamma}_{i}(\varphi):=\dim_{k}({\operatorname{ind^{\gamma}}}_{i-1}(R\langle
Y\rangle))$; the latter value is exactly the number of $\Gamma$-variables
adjoined in $Y_{i-1}$ and referred to as the $i^{\text{th}}$
$\Gamma$-deviation of $\varphi$. In particular, we have

$\varepsilon_{i}(\varphi)=\begin{dcases}\sum_{s=0}^{t}\varepsilon^{\gamma}_{2jp^{s}+1}(\varphi)&i=2jp^{t}+1\\\
\varepsilon^{\gamma}_{2jp^{t}+2}(\varphi)+\sum_{s=0}^{t-1}\varepsilon^{\gamma}_{2jp^{s}+1}(\varphi)&i=2jp^{t}+2\\\
\varepsilon^{\gamma}_{i}(\varphi)&\text{otherwise}\end{dcases}$

Furthermore, since the first steps in the inductive constructions of an
acyclic closure and a minimal model coincide, the equalities below always hold

$\varepsilon_{2}(\varphi)=\varepsilon^{\gamma}_{2}(\varphi)\ \text{ and }\
\varepsilon_{3}(\varphi)=\varepsilon^{\gamma}_{3}(\varphi)\,.$

It is known that the eventual vanishing of the deviations of a surjective
local map is equivalent to the map being complete intersection when it is a
map of finite projective dimension, or more generally, when the map has finite
weak category; see [8, Theorem C] for the former, and [4, Section 3] for the
latter (as well as [10, Theorem 5.4]). The numeric relations listed in 5.2
establishes the equivalence for a completely different class of surjective
homomorphisms:

######  Corollary 5.3.

If $\varphi\colon R\rightarrow S$ is a weakly-closed surjective local map with
residual characteristic $p>0$, then the following are equivalent:

  1. (1)

$\varphi$ is complete intersection;

  2. (2)

$\varepsilon_{i}(\varphi)=0$ for all $i\gg 0$;

  3. (3)

$\varepsilon_{i}(\varphi)=0$ for $i=2p^{t}+1$ or $i=2p^{t}+2$ for some
$t\geqslant 1$.

Our last main result, another immediate consequence of 5.2, shows that
rigidity of certain deviations detects the quasi-complete intersection
property in positive characteristic among weakly-closed maps; this should be
compared with [9, Theorem 5.3] and [15, Theorem 33] which characterizes the
quasi-complete intersection property in terms of the functorial map
$\pi(F^{\rho})\to\pi(F^{\varphi\rho})$ where $\rho\colon Q\to R$ is a Cohen
presentation of $R$.

######  Corollary 5.4.

Let $\varphi\colon R\rightarrow S$ be a weakly-closed surjective local map
with residual characteristic $p>0$. Then $\varphi$ is a quasi-complete
intersection homomorphism if and only if there are equalities

$\varepsilon_{i}(\varphi)=\begin{cases}\varepsilon_{3}(\varphi)&\text{ if
}i=2p^{t}+1\text{ or }i=2p^{t}+2\\\ 0&\text{ otherwise}\end{cases}$

for all $i\geqslant 4$ and $t\geqslant 1$.

######  Remark 5.5.

Corollaries 5.3 and 5.4 fail when the residual characteristic is zero. In
particular, if $\varphi$ is a quasi-complete intersection that is not a
complete intersection, then $\varphi$ is a closed map satisfying
$\varepsilon_{3}(\varphi)\neq 0$ and $\varepsilon_{i}(\varphi)=0$ for $i\geq
4$. Hence $\varphi$ satisfies conditions (2) and (3) of Corollary 5.3, while
(1) fails. Furthermore, $\varphi$ is a quasi-complete intersection with
$\varepsilon_{i}(\varphi)=0\neq\varepsilon_{3}(\varphi)$ whenever $i=2p^{t}+1$
or $i=2p^{t}+2$, so the forward implication of Corollary 5.4 fails.

We end this article with some closing remarks.

######  Remark 5.6.

The results in the present article were stated for surjective homormophisms;
they naturally extend—and can be deduced from the surjective case—to the
setting of (arbitrary) local homomorphisms using the theory of Cohen
factorizations introduced in [6]. We leave these deductions to the interested
reader.

######  Remark 5.7.

Recall by Remark 2.7, $\varphi$ is weakly-closed provided for an acyclic
closure $R\langle Y\rangle$ for $\varphi$, the boundary $\partial y$ does not
contain a nonzero summand from $kY^{(p^{\infty})}$ for each $y\in Y$; this is
needed in the core result Theorem 3.6. The examples of such maps listed in
1.13 are closed, which is a priori stronger than requiring $\varphi$ be
weakly-closed. An example of a weakly-closed map that is _not_ closed, when
$k$ has characteristic $p$, is unknown to the authors.

######  Remark 5.8.

The Lie algebra $L$ in Theorem 4.4 is intrinsic to the map $\varphi$. Namely,
as a $k$-space $L$ is
$({\scriptstyle\mathsf{\Sigma}}{\operatorname{ind^{\gamma}}}R\langle
Y\rangle)^{*}$ where $R\langle Y\rangle$ is an acyclic closure of $\varphi$.
If the characteristic of $k$ is different from 2 or 3, without the assumption
$\varphi$ is weakly-closed, then $L$ acquires a Lie algebra structure by
adjusting the formulas for the bracket and reduced square in 4.2. An
investigation of this Lie algebra is reserved for another time as the
development of this Lie algebra is not entirely relevant to the content in
this work.

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