[b]S. Bacchio

# Disconnected contribution to the LO HVP term of muon g-2 from ETMC

C. Alexandrou  P. Dimopoulos  J. Finkenrath  R. Frezzotti  G. Gagliardi  M.
Garofalo  K. Hadjiyiannakou  B. Kostrzewa  K. Jansen  V. Lubicz  M. Petschlies
F. Sanfilippo  S. Simula  C. Urbach  U. Wenger

###### Abstract

We present a lattice determination of the disconnected contributions to the
leading-order hadronic vacuum polarization (HVP) to the muon anomalous
magnetic moment in the so-called short and intermediate time-distance windows.
We employ gauge ensembles produced by the Extended Twisted Mass Collaboration
(ETMC) with $N_{f}=2+1+1$ flavours of Wilson twisted-mass clover-improved
quarks with masses approximately tuned to their physical value. We take the
continuum limit employing three lattice spacings at about 0.08, 0.07 and 0.06
fm.

##  1 Introduction

In Ref. [1] we have presented our full calculation of the short and
intermediate time-distance hadronic vacuum polarization (HVP) contributions to
the muon magnetic moment using twisted-mass fermions on ETMC ensembles. In
this work, we present an extract that focuses on disconnected contributions,
summarizing the results of the manuscript and providing some additional
details.

In our calculation, we have adopted the time momentum representation [2] and
evaluate the HVP contribution to the muon anomalous magnetic moment
$a_{\mu}^{\rm HVP}$ as

$a_{\mu}^{\rm{HVP}}=2\alpha_{em}^{2}\int_{0}^{\infty}~{}dt\,t^{2}\,K(m_{\mu}t)\,V(t)~{},~{}$
(1)

where $t$ is the Euclidean time and the kernel function $K(m_{\mu}t)$ is
defined as111The leptonic kernel $K(z)$ is proportional to $z^{2}$ at small
values of $z$ and it goes to $1$ for $z\to\infty$.

$K(z)=2\int_{0}^{1}dy(1-y)\left[1-j_{0}^{2}\left(\frac{z}{2}\frac{y}{\sqrt{1-y}}\right)\right]~{},\qquad
j_{0}(y)=\frac{\sin{(y)}}{y}~{}.~{}$ (2)

The Euclidean vector correlator $V(t)$ is defined as

$V(t)\equiv-\frac{1}{3}\sum_{i=1,2,3}\int d^{3}{x}~{}\langle
J_{i}(\vec{x}_{f},t_{f})J_{i}(\vec{x}_{i},t_{i})\rangle$ (3)

with $J_{\mu}(x)$ being the electromagnetic current operator

$J_{\mu}(x)\equiv\sum_{f=u,d,s,c,...}q_{f}~{}\overline{\psi}_{f}(x)\gamma_{\mu}\psi_{f}(x)~{}$
(4)

and $q_{f}$ the electric charge for the quark flavour $f$ (in units of the
absolute value of the electron charge). Clearly, the vector correlator $V(t)$
give rise to both connected and disconnected contributions, as depicted in
Fig. 1. The latter are the focus of this proceeding.

Figure 1: Connected (left) and disconnected (right) contributions to the
vector correlator $V(t)$.

###  1.1 The RBC/UKQCD windows in the time-momentum representation

Following the analysis of the RBC/UKQCD Collaboration [3], we separate the
whole HVP contribution in three terms,

$a_{\mu}^{\rm HVP}\equiv a_{\mu}^{\rm SD}+a_{\mu}^{\rm W}+a_{\mu}^{\rm
LD}~{},~{}$ (5)

that can be obtained from Eq. (1) with integration kernel $K(m_{\mu}t)$
multiplied by suitably smoothed Heaviside step-functions, namely

$a_{\mu}^{w}=2\alpha_{em}^{2}\int_{0}^{\infty}~{}dt\,t^{2}\,K(m_{\mu}t)\,\Theta^{w}(t)\,V(t)~{}\qquad
w=\\{\rm SD,W,LD\\}~{},~{}$ (6)

where the time-modulating function $\Theta^{w}(t)$ is given by

$\displaystyle\Theta^{\rm SD}(t)$ $\displaystyle\equiv$ $\displaystyle
1-\frac{1}{1+e^{-2(t-t_{0})/\Delta}}~{},~{}$ (7) $\displaystyle\Theta^{\rm
W}(t)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{1+e^{-2(t-t_{0})/\Delta}}-\frac{1}{1+e^{-2(t-t_{1})/\Delta}}~{},~{}$
(8) $\displaystyle\Theta^{\rm LD}(t)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{1+e^{-2(t-t_{1})/\Delta}}~{}$ (9)

with the parameters $t_{0},t_{1},\Delta$ chosen [3] to be equal to

$t_{0}=0.4~{}\rm{fm}~{},~{}\qquad t_{1}=1~{}{\rm
fm}~{},~{}\qquad\Delta=0.15~{}{\rm fm}~{}.~{}$ (10)

##  2 Lattice setup

In this work we compute disconnected contributions on three ensembles with
$N_{f}=2+1+1$ flavours of Wilson twisted-mass clover-improved quarks with
masses approximately tuned to their physical value. The parameters of the
ensembles are give in Table 1. For the full calculation in Ref. [1], a fourth
ensemble, cB211.072.96, at $\beta=1.778$ and lattice volume
$V/a^{4}=96^{3}\cdot 192$ was used for estimating finite size effects (FSEs).
However, due to the high cost of the calculation, we did not compute
disconnected contributions using the larger volume cB211.072.96 ensemble,
since FSEs are expected to be within statistical errors.

ensemble |  $\beta$ |  $V/a^{4}$ |  $a$ (fm) |  $a\mu_{\ell}$ |  $M_{\pi}$ (MeV) |  $L$ (fm) | $M_{\pi}L$
---|---|---|---|---|---|---|---
cB211.072.64 | $1.778$ | $64^{3}\cdot 128$ | $0.07961~{}(13)$ | $0.00072$ | $140.2~{}(0.2)$ | $5.09$ | $3.62$
cC211.060.80 | $1.836$ | $80^{3}\cdot 160$ | $0.06821~{}(12)$ | $0.00060$ | $136.7~{}(0.2)$ | $5.46$ | $3.78$
cD211.054.96 | $1.900$ | $96^{3}\cdot 192$ | $0.05692~{}(10)$ | $0.00054$ | $140.8~{}(0.2)$ | $5.46$ | $3.90$
Table 1: Parameters of the ETMC ensembles used in this work. We give the
light-quark bare mass, $a\mu_{\ell}=a\mu_{u}=a\mu_{d}$, the pion mass
$M_{\pi}$, of the lattice size $L$ and the product $M_{\pi}L$.

##  3 Noise reduction techniques and statistics

The disconnected contributions are computed for the light-, strange- and
charm-quark masses. Various noise-reduction techniques are employed to improve
the signal-to-noise ratio of disconnected loops. These are the one-end-trick
[4], exact deflation of low-modes [5] and hierarchical probing [6]. The one-
end-trick is used for all loops; hierarchical probing with distance 8 is used
for all loops, except the charm-quark loops for the cB211.072.64 ensemble,
where distance 4 is used; and deflation of the low-modes is used for the light
quark loops for the cB211.072.64 and cC211.060.80 ensembles. The latter is not
employed for the cD211.054.96 ensemble because of the prohibitively large
memory requirements. Indeed, the number of low-modes to be deflated should be
increased with the volume, making the costs of this technique to scale with
volume-squared. For this reason deflation is not used on the larger volume
and, instead, multiple stochastic sources are utilized. In Table 2 we
summarize the statistics used for the three ensembles and three quark
flavours.

| cB211.072.64 | cC211.060.80 | cD211.054.96
---|---|---|---
Flavour | $N_{\rm defl}$ | $N_{\rm r}$ | $N_{\rm Had}$ | $N_{\rm vect}$ | $N_{\rm defl}$ | $N_{\rm r}$ | $N_{\rm Had}$ | $N_{\rm vect}$ | $N_{\rm defl}$ | $N_{\rm r}$ | $N_{\rm Had}$ | $N_{\rm vect}$
Light | 200 | 1 | 512 | 6144 | 450 | 1 | 512 | 6144 | 0 | 8 | 512 | 49152
Strange | 0 | 2 | 512 | 12288 | 0 | 4 | 512 | 24576 | 0 | 4 | 512 | 24576
Charm | 0 | 12 | 32 | 4608 | 0 | 1 | 512 | 6144 | 0 | 1 | 512 | 6144
$N_{\rm confs}$ |  $\times$750 configurations |  $\times$400 configurations |  $\times$500 configurations
Table 2: Noise reduction techniques and statistics used for the disconnected
quark loops. For each ensemble, the columns are in order: i) the number of
deflated eigenvectors $N_{\rm defl}$, ii) the number of stochastic sources
$N_{\rm r}$, iii) the number of Hadamard vectors $N_{\rm Had}$, and iv) the
total number of computed vectors $N_{\rm vect}=12\times N_{\rm r}\times N_{\rm
Had}$. Figure 2: Statistical errors of the integrand in the short (SD),
intermediate (W) and long (LD) distance windows for the light (left panel) and
strange (right panel) contributions to the disconnected loops on the
cB211.072.64 ensemble. In orange are the values used in the final calculation,
i.e. those reported in Table 2, while in blue are the same quantities computed
with an alternative setup. Namely, we used 2 stochastic sources (instead of 1)
but no deflation for the light loops and used 12 stochastic sources (instead
of 2) and 32 Hadamard vectors (instead of 512) for the strange loops. In the
text we report about the gain in cost-to-error.

For certain flavour of quark loops we have also available a calculation made
with a different setup. E.g. on the cB211.072.64, for the light disconnected
contributions, we have also computed loops without deflation and two
stochastic sources, or, for the strange disconnected contributions, we have
employed 12 stochastic sources and 32 Hadamard vectors. These different
combinations allow us to estimate the improvements due to the noise reduction
techniques adopted. We analyse this in Fig. 2 where we compare the error on
disconnected contributions for the various windows using the aforementioned
statistics for the light- and strange-quark loops. Looking at the light-quark
loops, the cost for one stochastic source with deflation is about 35% cheaper
than computing two stochastic sources without deflation, and its error is also
35% smaller. Summing these up, deflation reduces the cost-to-error ratio by a
factor of 3.7$\times$ for this specific quantity and ensemble. Looking at the
strange-quark loops, 2 sources with 512 Hadamard vectors require $2.7\times$
more computational resources than 12 sources with 32 Hadamard vector and
reduce the errors squared by a factor of $3\times$, resulting in a 11%
reduction of the cost-to-error ratio.

##  4 The strange- and charm-quark loops

The strange- and charm-quark loops are computed at a quark mass obtained by
tuning the $\Omega$ and $\Lambda_{c}$ baryons, respectively, to their physical
value. The values of the bare masses for the strange, $a\mu_{s}$, and for the
charm, $a\mu_{c}$, quarks are listed in Table 3. In Fig. 3, we show the
continuum limit of the renormalized strange and charm quark masses in the
$\overline{MS}$(2 GeV) and $\overline{MS}$(3 GeV) scheme [7], respectively. We
compare them against the results computed in the continuum limit in Ref. [7].
We note that the values of $a\mu_{s}$ do not show sizable cut-off effects,
while $a\mu_{c}$ does.

Ensemble | $a\mu_{s}$ | $a\mu_{c}$
---|---|---
cB211.072.64 | 0.01860 | 0.249
cC211.060.80 | 0.01615 | 0.206
cD211.054.96 | 0.01360 | 0.166
Table 3: Values of the bare quark masses $a\mu_{s}$ and $a\mu_{c}$ used for
the calculation of strange and charm disconnected contributions.

Figure 3: Renormalized strange (left) and charm (right) quark mass, given
respectively in the $\overline{MS}$(2 GeV) and $\overline{MS}$(3 GeV) scheme
[7, 8], tuning the $\Omega$ and $\Lambda_{c}$ baryon masses, versus the
squared lattice spacing. The red stars are the results of the continuum limit
extrapolation carried out in Ref. [7].

##  5 Disconnected contributions

The quark disconnected contributions to the vector correlator $V(t)$ (see Eq.
(3)) is the sum of the six relevant quark disconnected correlators weighted by
the appropriate charge factors, namely

$\displaystyle V_{\rm disc.}(x)$ $\displaystyle=$
$\displaystyle\sum_{i=1,2,3}\Bigg{(}+\frac{1}{9}\left\langle
J_{i}^{\ell\ell}(x)[J_{i}^{\ell\ell}]^{\dagger}(0)\right\rangle+\frac{1}{9}\left\langle
J_{i}^{ss}(x)[J_{i}^{ss}]^{\dagger}(0)\right\rangle+\frac{4}{9}\left\langle
J_{i}^{cc}(x)[J_{i}^{cc}]^{\dagger}(0)\right\rangle$ $\displaystyle-$
$\displaystyle\frac{1}{9}\left\langle
J_{i}^{\ell\ell}(x)[J_{i}^{ss}]^{\dagger}(0)+{\rm
hc}\right\rangle+\frac{2}{9}\left\langle
J_{i}^{\ell\ell}(x)[J_{i}^{cc}]^{\dagger}(0)+{\rm
hc}\right\rangle-\frac{2}{9}\left\langle
J_{i}^{ss}(x)[J_{i}^{cc}]^{\dagger}(0)+{\rm hc}\right\rangle\Bigg{)}\,,$

where the renormalized vector currents $J_{i}^{ff}$, which are written in
terms of Osterwalder-Seiler valence lattice quarks [9, 10], read as

$\displaystyle
J_{\mu}^{\ell\ell}(x)=Z_{V}\bar{\ell}(x)\gamma_{\mu}\ell(x)\;,\quad
J_{\mu}^{ss}(x)=Z_{V}\bar{s}(x)\gamma_{\mu}s(x)\;,\quad
J_{\mu}^{cc}(x)=Z_{V}\bar{c}(x)\gamma_{\mu}c(x)\;.\quad$ (12)

For details on the calculation of the renormalization constant $Z_{V}$ we
refer to Ref. [1]. From the vector correlator $V_{disc.}(t)$ the values of
$a_{\mu}^{\rm SD}(disc.)$ and $a_{\mu}^{\rm W}(disc.)$ are straightforwardly
evaluated according to Eq. (6). The results for the diagonal and off-diagonal
disconnected contributions are summarized in Table 4 for $a_{\mu}^{\rm SD}$
and in Table 5 for $a_{\mu}^{\rm W}$.

Ensemble | $\ell\ell$ | $ss$ | $cc$ | $\ell s$ | $\ell c$ | $sc$
---|---|---|---|---|---|---
cB211.072.64 | $-3.37~{}(13)$ | $-2.090~{}(59)$ | $-1.18~{}(14)$ | $+5.29~{}(15)$ | $-1.52~{}(24)$ | $+1.67~{}(13)$
cC211.060.80 | $-3.36~{}(16)$ | $-2.090~{}(73)$ | $-0.78~{}(11)$ | $+5.53~{}(17)$ | $-1.48~{}(20)$ | $+1.37~{}(15)$
cD211.054.96 | $-3.54~{}(16)$ | $-2.084~{}(75)$ | $-0.71~{}(14)$ | $+5.60~{}(18)$ | $-1.51~{}(21)$ | $+1.27~{}(18)$
Table 4: Summary of the various flavour contributions to $a_{\mu}^{\rm SD}$(disc.) in units of $10^{-12}$ for the cB211.072.64, cC211.060.80 and cD211.054.96 ensembles. The symbols $\ell\ell$, $ss$ and $cc$ denote respectively the flavour-diagonal light, strange and charm contributions, while $\ell s,$ $\ell c$ and $sc$ denote the off-diagonal light-strange, light-charm and strange-charm contributions, respectively.  Ensemble | $\ell\ell$ | $ss$ | $cc$ | $\ell s$ | $\ell c$ | $sc$
---|---|---|---|---|---|---
cB211.072.64 | $-1.087~{}(49)$ | $-0.149~{}(22)$ | $-0.030~{}(53)$ | $+0.635~{}(58)$ | $+0.00~{}(8)$ | $-0.02~{}(6)$
cC211.060.80 | $-1.300~{}(69)$ | $-0.159~{}(27)$ | $-0.033~{}(49)$ | $+0.726~{}(81)$ | $-0.03~{}(7)$ | $+0.04~{}(7)$
cD211.054.96 | $-1.201~{}(73)$ | $-0.149~{}(29)$ | $+0.018~{}(54)$ | $+0.627~{}(81)$ | $+0.02~{}(8)$ | $-0.02~{}(7)$
Table 5: The same as in Table 4, but for the various flavour contributions to $a_{\mu}^{\rm W}$(disc.) in units of $10^{-10}$. Ensemble |  $a_{\mu}^{\rm SD}$(disc.) |  $a_{\mu}^{\rm W}$(disc.)
---|---|---
cB211.072.64 | $-1.20~{}(23)\cdot 10^{-12}$ | $-0.651~{}(93)\cdot 10^{-10}$
cC211.060.80 | $-0.80~{}(18)\cdot 10^{-12}$ | $-0.762~{}(75)\cdot 10^{-10}$
cD211.054.96 | $-0.96~{}(20)\cdot 10^{-12}$ | $-0.701~{}(80)\cdot 10^{-10}$
Continuum | $-0.6~{}(5)\cdot 10^{-12}$ | $-0.78~{}(21)\cdot 10^{-10}$
Table 6: Final results for the disconnected contributions to the short and
intermediate distance windows

In Table 6 we give the sum of all contributions and the result of the
continuum limit that we take as final value. In Fig. 4, we show the continuum
limit extrapolation for the disconnected contributions to $a_{\mu}^{\rm SD}$
and $a_{\mu}^{\rm W}$. Qualitatively, for $a_{\mu}^{\rm W}$ the light-light
contribute +150% of the total disconnected contribution, the strange-light
-80% and the strange-strange +30%. All other combinations are consistent with
zero within the errors. We do not observe sizable cutoff effects at this level
of precision. The disconnected contribution to $a_{\mu}^{\rm SD}$ is very
small, being approximately forty times smaller as compared to our error on the
light-connected contribution to $a_{\mu}^{\rm SD}$. Given the available data,
we perform only two continuum extrapolation, using either a constant fit
Ansatz or a linear one in $a^{2}$. The latter fit Ansatz is adopted in view of
the expected automatic O(a) improvement [10] of the relevant two-point
correlation functions of the renormalized vector currents in Eq. (12). The
extrapolated values using these two fitting procedures are in agreement with
each other. The one obtained from the linear fit in $a^{2}$ has a larger
statistical uncertainty and conservatively we take it as our final estimate.

Figure 4: Top panel: The quark-loop disconnected contribution to the short time-distance window, $a_{\mu}^{\rm SD}$, versus the squared lattice spacing $a^{2}$ in physical units. Bottom panel: the same as in the top panel, but for the intermediate window $a_{\mu}^{\rm W}$. The blue band corresponds to the extrapolation performed using a linear fit Ansatz in $a^{2}$. Ref. | $a_{\mu}^{\rm W}(disc.)$
---|---
this work | $~{}-0.78~{}(0.21)\cdot 10^{-10}$
BMW [11] | $~{}-0.85~{}(0.06)\cdot 10^{-10}$
CLS/Mainz [12] | $~{}-0.81~{}(0.09)\cdot 10^{-10}$
average | $~{}-0.83~{}(0.05)\cdot 10^{-10}$
Table 7: Disconnected contribution to the intermediate time-distance window
$a_{\mu}^{\rm W}$ obtained in this work and in Refs. [11, 12]. The last row
lists the average of all lattice results made following the PDG approach.

##  6 Conclusions

In Table 7 we compare our results with those obtained by BMW [11] and
CLS/Mainz [12] on the disconnected contribution to the intermediate window.
The largest systematic effect on our error comes from the continuum
extrapolation. We observe a remarkable agreement among all available lattice
results. In the future we plan to extended this calculation with a further
lattice spacing in order to improve our continuum extrapolation.

## Acknowledgments

We thank all members of ETMC for the most enjoyable collaboration. We thank
the developers of the QUDA [13, 14, 15] library for their continued support,
without which the calculations for this project would not have been possible.
S.B. and J.F. are supported by the H2020 project PRACE 6-IP (grant agreement
No. 82376) and the EuroCC project (grant agreement No. 951740). We acknowledge
support by the European Joint Doctorate program STIMULATE grant agreement No.
765048. P.D. acknowledges support from the European Unions Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 813942 (EuroPLEx) and also support from INFN under the research
project INFN-QCDLAT. K.H. is co-funded by the European regional development
fund and the Republic of Cyprus through the Research and Innovation Foundation
under contract number POST-DOC/0718/0100, under contract number CULTURE-AWARD-
YR/0220/0012 and by the EuroCC project (grant agreement No. 951740). R.F.
acknowledges partial support from the University of Tor Vergata program
“Beyond Borders/ Strong Interactions: from Lattice QCD to Strings, Branes and
Holography". F.S., G.G. and S.S. are supported by the Italian Ministry of
University and Research (MIUR) under grant PRIN20172LNEEZ. F.S. and G.G. are
supported by INFN under GRANT73/CALAT. This work is supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) and the NSFC through
the funds provided to the Sino-German Collaborative Research Center CRC 110
“Symmetries and the Emergence of Structure in QCD” (DFG Project-ID 196253076 -
TRR 110, NSFC Grant No. 12070131001). The authors gratefully acknowledge the
Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding the
project pr74yo by providing computing time on the GCS Supercomputer SuperMUC
at Leibniz Supercomputing Centre (www.lrz.de), as well as computing time
projects on the GCS supercomputers JUWELS Cluster and JUWELS Booster [16] at
the Jülich Supercomputing Centre (JSC) and time granted by the John von
Neumann Institute for Computing (NIC) on the supercomputers JURECA and JURECA
Booster [17], also at JSC. Part of the results were created within the EA
program of JUWELS Booster also with the help of the JUWELS Booster Project
Team (JSC, Atos, ParTec, NVIDIA). We further acknowledge computing time
granted on Piz Daint at Centro Svizzero di Calcolo Scientifico (CSCS) via the
project with id s702. The authors acknowledge the Texas Advanced Computing
Center (TACC) at The University of Texas at Austin for providing HPC resources
that have contributed to the research results. The authors gratefully
acknowledge PRACE for awarding access to HAWK at HLRS within the project with
Id Acid 4886.

## References

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# memorAIs: an Optical Character Recognition and Rule-Based Medication Intake
Reminder-Generating Solution

Eden Shaveet  Dept. Computer Science
Columbia University
New York, USA
<EMAIL_ADDRESS>Utkarsh Singh  Dept. Computer Science
Stony Brook University
Stony Brook, USA
<EMAIL_ADDRESS>Nicholas Assaderaghi  Dept. Computer Science
Columbia University
New York, USA
<EMAIL_ADDRESS>Maximo Librandi  Dept. Computer Science
Columbia University
New York, USA
<EMAIL_ADDRESS>
###### Abstract

Memory-based medication non-adherence is an unsolved problem that is
responsible for considerable disease burden in the United States. Digital
medication intake reminder solutions with minimal onboarding requirements that
are usable at the point of medication acquisition may help to alleviate this
problem by offering a low barrier way to help people remember to take their
medications. In this paper, we propose memorAIs, a digital medication intake
reminder solution that mitigates onboarding friction by leveraging optical
character recognition strategies for text extraction from medication bottles
and rule based expressions for text processing to create configured medication
reminders as local device calendar invitations. We describe our ideation and
development process, as well as limitations of the current implementation.
memorAIs was the winner of the Patient Safety award at the 2023 Columbia
University DivHacks Hackathon, presented by the Patient Safety Technology
Challenge, sponsored by the Pittsburgh Regional Health Initiative.

##  I Introduction

###  I-A Medication Adherence

Medication adherence is the extent to which a patient follows a stipulatory
medication treatment plan [1]. Over half of U.S. adults do not take
medications as prescribed, which is responsible for an estimated 33%–69% of
hospital admissions and 125,000 deaths annually [1]. When medications are
appropriately prescribed and made accessible to patients in a timely fashion,
the value proposition of outpatient medication adherence is seen across
several disease progression use cases [2], [3]. For instance, adherence to
disease-modifying medication regimens and therapies for the treatment and
control of Multiple Sclerosis in early stages of disease onset is associated
with decreased disease severity over time [4].

As more medications are taken outside of healthcare settings than in U.S.
hospitals and clinics combined, better home medication management and intake
practices may be critical to improving medication adherence overall [5]. In
2003, the World Health Organization (WHO) corroborated this claim by stating
that increased effectiveness of medication adherence interventions may have “a
far greater impact on the health of the population than any improvement in
specific medical treatments” [6].

###  I-B Barriers to Adherence

There are several well-documented barriers to medication adherence, including
high costs of medications and routine care [7]; lack of access to prescribers
and pharmacies [8]; medical mistrust of clinical guidance or medications [9];
and memory-based adherence problems [10]. Memory-based non-adherence is a
prevalent reason for home medication non-adherence, specifically for patients
with cognitive impairments [11], and has led to the development of several
reminder-based solutions, including user-facing digital solutions [12].
However, of the several digital reminder-based solutions on the market, none
have demonstrated significant improvements in adherence among their users
[13][14][15]. While there may be several reasons for this lack of efficacy,
one proposed reason is under-utilization.

Known barriers to digital health solution utilization at large include low
digital literacy among users [16], meaning that users with limited experience
with other digital tools may find digital health tools equally difficult to
use, and onboarding friction [17], meaning that the effort exerted to
configure the solution outweighs the perceived benefits of its use. For
instance, patients with complex medication regimens, who would be required to
manually input information for each medication, along with their corresponding
routine details, before being able to meaningfully use a digital medication
reminder application, may perceive the setup of the application as cumbersome,
potentially discouraging its use.

In an effort to address the challenge of onboarding friction, we developed a
solution that offers the benefit of reminder-based solutions while mitigating
onboarding friction attributed to user-facing configuration tasks. In this
paper, we present a prototype implementation of an automated medication
reminder application, memorAIs, that extracts text from prescription
medication containers using optical character recognition models and
identifies medication intake time, duration, and frequency information using
text classification-based strategies. The application returns this information
in the form of an ics (iCalendar) file directly to a user’s device,
eliminating the need for manual configuration.

memorAIs was conceptualized, designed, developed, and implemented in less than
36 hours at the 2023 DivHacks Hackathon at Columbia University. It was awarded
the first-place Patient Safety Award from the Patient Safety Technology
Challenge, sponsored by the Pittsburgh Regional Health Initiative.

###  I-C Solution Overview

Our application prototype, memorAIs, is a platform that allows users to scan
their on-the-bottle prescription intake directions using their device’s camera
to generate and download a ics (iCalendar) file pre-configured with intake
frequency, duration, and time information straight to their device in use. ics
files are calendar files stored in a universal calendar format used by all
major email and calendar programs, including Microsoft Outlook, Google
Calendar, and Apple Calendar [18].

To use the application, a user can access our front-end interface and has the
option to either upload a photo of their prescription label, including the
intake directions from the pharmacy, or capture one using their device’s
camera. Our system processes the photo using the following methods:

  1. 1.

Optical character recognition (OCR) models, which extract text from .png and
.jpg files and return relevant substrings given text probability and character
coordinates.

  2. 2.

Regular-expression-based text interpretation that associates keywords with the
specific frequency, duration, and time details required to create a calendar
reminder.

The extracted details are then used to generate and export an ics file
directly to the user’s current device. All they need to do is download the
file, and their medication reminders will be seamlessly integrated into a
calendar of their choice.

##  II Methodology

###  II-A Solution Design

We began by considering patient-facing medication-acquisition and intake
process flows and the point at which a digital reminder may be useful. For
example, some patients may acquire their medications in-person from a
pharmacy. Others may use a mail delivery service. Others may rely on a third-
party service for hand-delivery. To minimize opportunities for memory-based
non-adherence, we determined that our solution needed to be usable at the
point of medication acquisition in any of these circumstances. In order for
the solution to be usable at the point of medication acquisition, the
solution’s requirements of a user would need to be minimal, quick, and easy.

Recognizing that prescription medication bottles typically provide intake
instructions and that individuals using digital intake reminder solutions
often possess devices such as smartphones equipped with cameras, we opted to
create a solution integrating these elements. The optimal process flow we
envisioned for the solution depended on patients capturing an image of the
medication intake instructions and developing an underlying program that would
extract, process, and return the information contained therein as a digital
reminder.

###  II-B Front-End Design & Development

We scripted the front-end interface in HTML, CSS, & JavaScript with background
styling using a gray to black linear gradient with white features (Figure 1).
The page layout featured left and right sections, where the left contained
user instructions and the right contained an image upload form. A semi-
transparent spinner was programmed to show during background tasks.

The front-end JavaScript functionality captured and handled user interactions
and communicated with the remote server. It also sent the image file to the
server and handled the server response. Finally, it dynamically generated a
download link when the server provided the expected ics file.

Figure 1: memorAIs User Interface.

###  II-C Back-End Development & Rule-Based Text Processing

We developed the back-end using Python and employed a Flask server for the
memorAIs application. The back-end architecture consisted of a well-organized
set of scripts, configuration files, and modules, tailored to facilitate
different facets of our application, such as supporting the Flask server,
implementing web services, managing the application programming interface
(API), and handling data.

The rule-based text processing was handled through a single script that
extracted text from images of medication instructions from medication
containers and generated medication reminder events in ics format. This
process was broken up into four parts: 1.) text extraction, 2.) text
preprocessing, 3.) text interpretation, and 4.) information formatting and
return.

###  II-D Text Extraction

Text extraction from images uploaded on the platform were handled via optical
character recognition (OCR). OCR is a computer vision task that converts text
on images or printed documents into machine-readable text [19]. There are
several ways to implement OCR, including rule-based OCR, template matching,
feature-based OCR, and machine learning OCR to name a few. While developing
memorAIs, we tested two OCR implementation packages from the Python library:
1.) pytesseract, a Python wrapper for Google’s Tesseract-OCR Engine which uses
a blend of traditional computer vision techniques and modern deep learning
methods for character recognition [20], and 2.) PaddleOCR, an OCR
implementation which leverages PaddlePaddle’s deep learning framework [21]
(Figure 2).

We chose to use the PaddleOCR package for memorAIs, citing higher accuracy and
the package’s ability to return String bounding box coordinates for each
detected text element. We determined that this feature would be helpful in our
later implementation of regular expressions that would consider character
coordinates in placement probabilities.

Figure 2: PaddleOCR Bounding Box Example (from PaddleOCR GitHub)

###  II-E Text Preprocessing

Text was preprocessed in two ways: 1.) bounding box coordinate comparison, and
2.) standardized character representation. Leveraging the bounding box
coordinate information provided by the PaddleOCR package, we wrote a function
to compare the coordinates of all detected bounding boxes to determine the
order in which they appear on the bottle. Next, we converted all String
representations of numbers to numeric form (e.g., ”two” as 2) for uniform
representation.

###  II-F Text Interpretation

In recognition of the fact that U.S. prescription pill bottle instructions
tend to contain similar text elements concerning medication intake time,
duration, and frequency, we opted for a regular-expression-based approach to
text interpretation. To inform medication reminder components, we developed 20
regular expressions as frequency indicators and nine regular expressions to
determine the duration of intake.

Example frequency regular expressions included:

  * •

"regex": r"every other day", "frequency": 0.5, "frequency_unit": "days"

  * •

"regex": r"twice per day", "frequency": 2, "frequency_unit": "days"

  * •

"regex": r"every night", "frequency": 1, "frequency_unit": "days",
"time_of_days": ["evening"]

Example frequency duration expressions included:

  * •

"regex": r"for ([0-9]+) days", "duration_unit": "days"

  * •

"regex": r"after ([0-9]+) months", "duration_unit": "months"

The general rule-based classification algorithm was as follows:

Input: Preprocessed Text (label_text)
Output: Medication Schedule Parameters

Parameters:

  * •

frequency: How often the medication should be taken

  * •

frequency unit: Unit of frequency (e.g., hours, days, weeks)

  * •

duration: Duration for which the medication should be taken

  * •

duration unit: Unit of duration (e.g., days, weeks, months)

  * •

time of day: Times of day for medication intake

Rule-based Classification:

  * •

For each rule in frequency_indicator, duration_indicator:

    * –

Apply the regular expression to label_text

    * –

If a match is found:

      * *

Update the corresponding parameter based on the rule

###  II-G Source Code & Materials

Front-End GitHub Repository: Click here
Back-End GitHub Repository: Click here
Web Application: Click here

Note: In order to be functional, the current memorAIs implementation relies on
making a call to a server that is currently turned off. If the product gains
traction, we will search for a long term server solution so that anyone,
anywhere can use it.

###  II-H Information Formatting & Return

Once the text had been interpreted based on our indicator rules, the
information was stored as values in a JSON file. From there, an empty Calendar
object was created and values extracted from the JSON file were used to map
values to Calendar object format using the icalendar Python library. The
populated calendar object, configured with the appropriate reminder
information (see examples in Figure 3) was returned and sent to the front-end,
which then generated the ics download link to the user’s device.

Figure 3: Example ics File Content

##  III Discussion

###  III-A Limitations

There is no shortage of limitations in short-term hackathon projects; however,
one of the most prominent included our strategy for text processing. Our text
processing methodology was conducted via hardcoded regular expressions as
opposed to more generalizable approaches, such as natural language processing
methods. Because of this, our approach lacked generalizability and was only
useful up to a certain point of instruction phrasing complexity. Additionally,
using a language model would have been useful for more meaningfully
integrating other indicators, aside from frequency and duration, including
medication name, and intake routine details.

Another substantial limitation was that our approach lacked a way to remove or
overwrite old ics files when they became outdated. One of the express appeals
of a solution like memorAIs is the ability to reduce burden when medications
are updated. In a more developed implementation, we would have included a way
to remove outdated files.

###  III-B Conclusion

Memory-based medication non-adherence is an unsolved problem that is
responsible for considerable disease burden in the United States. Digital
medication intake reminder solutions with minimal onboarding requirements that
are usable at the point of medication acquisition may help to alleviate this
problem by offering a low barrier way to help people remember to intake their
medications. In this paper, we proposed memorAIs, a digital medication intake
reminder solution that mitigated onboarding friction by leveraging OCR for
text extraction from medication bottles and rule based expressions for text
processing to create configured medication reminders as local device calendar
invitations. While there were several limitations to our implementation,
including the lack of generalizability of our text interpretation beyond
hardcoded regular expressions and an inability to replace outdated ics files,
memorAIs was awarded the first-place Patient Safety award at the 2023 DivHacks
Hackathon by the Patient Safety Technology Challenge, sponsored by the
Pittsburgh Regional Health Initiative. In a formal implementation of memorAIs
as a viable product, we would improve our text processing and interpretation
methodology, integrate a way to remove outdated ics files from the user’s
device, and improve our UI for a more seamless and enjoyable user experience.

## Acknowledgment

We would like to thank the 2023 DivHacks team at Columbia University for
organizing the hackathon at which memorAIs was conceived and the Pittsburgh
Regional Health Initiative for awarding us with the Patient Safety award.

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# Rigid integral representations of quivers revisited

William Crawley-Boevey  Fakultät für Mathematik, Universität Bielefeld, 33501
Bielefeld, Germany<EMAIL_ADDRESS>
###### Abstract.

In earlier work, the author classified rigid representations of a quiver by
means of finitely generated free modules over a principal ideal ring. We show
that the classification of exceptional pointwise free lattices can be extended
to arbitrary commutative rings and that the classification of rigid lattices
can be extended to reduced commutative rings.

###### Key words and phrases:

Quiver representations, Rigid representations, Lattices over orders

###### 2020 Mathematics Subject Classification:

Primary 16G20; Secondary 16G30,16H20,13C10.

Partially supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – SFB-TRR 358/1 2023 – 491392403

##  1\. Introduction

In [3] we studied rigid representations of a quiver over a principal ideal
domain. Here we generalize the results to more general commutative base rings.

Let $R$ be a commutative ring (always non-zero). Given an $R$-module $X$ and a
homomorphism $R\to S$ we write $X^{S}$ for the induced $S$-module
$S\otimes_{R}X$. Recall that a finitely generated projective $R$-module $P$
has _constant rank $n$_ if $P_{\mathfrak{p}}$ is a free
$R_{\mathfrak{p}}$-module of rank $n$ for each prime ideal $\mathfrak{p}$ of
$R$, or equivalently if $\dim_{K}P^{K}=n$ for all homomorphisms $R\to K$ with
$K$ a field, which we may take to be algebraically closed.

Let $Q=(Q_{0},Q_{1},h,t)$ be a finite quiver and let $RQ$ be the path algebra
of $Q$ over $R$, with a trivial path $e_{i}$ for each vertex $i\in Q_{0}$. By
an $RQ$-_lattice_ we mean an $RQ$-module which is finitely generated
projective as an $R$-module. We say that an $RQ$-module $X$ is _pointwise
free_ if $e_{i}X$ is a free $R$-module for all $i$. We say that an
$RQ$-lattice $X$ has _rank vector $\alpha\in\mathbb{N}^{Q_{0}}$_ if $e_{i}X$
has constant rank $\alpha_{i}$ for each $i$, and that $X$ has _pointwise
constant rank_ if it has rank vector $\alpha$ for some $\alpha$. If $R\to S$
is a homomorphism, then $X^{S}$ is naturally an $SQ$-module.

We say that an $RQ$-module $X$ is _rigid_ if
$\operatorname{Ext}^{1}_{RQ}(X,X)=0$, and _exceptional_ if also the natural
map $R\to\operatorname{End}_{RQ}(X)$ is an isomorphism. If $K$ is an
algebraically closed field, then by definition the possible dimension vectors
of exceptional $KQ$-modules are the _real Schur roots_ for $Q$. We call the
possible dimension vectors of rigid $KQ$-modules _rigid dimension vectors_. By
results of [1] or [2] these do not depend on the field $K$. If $X$ is a rigid
(respectively exceptional) $RQ$-lattice of rank vector $\alpha$, then $X^{K}$
is a rigid (respectively exceptional) $KQ$-module of dimension vector $\alpha$
for any homomorphism $R\to K$ with $K$ a field (see Lemmas 3.1,3.2), and so
$\alpha$ must be a rigid dimension vector (respectively real Schur root).

######  Theorem A.

For any commutative ring $R$ and real Schur root $\alpha$ for $Q$, there is a
unique rigid pointwise free $RQ$-lattice with rank vector $\alpha$ and it is
exceptional.

The existence follows easily from [3], and the uniqueness follows from the
next result. As mentioned, the case when $R$ is a principal ideal domain is in
[3]. The case when $R$ is a truncated polynomial ring
$K[\epsilon]/(\epsilon^{n})$ is a special case of [5, Theorem 1.2]. Recall
that a commutative ring $R$ is reduced if it has no nilpotent elements.

######  Theorem B.

If $R$ is reduced, then any rigid $RQ$-lattice $X$ of pointwise constant rank
has a decomposition

$X\cong(X_{1}\otimes_{R}P_{1})\oplus\dots\oplus(X_{r}\otimes_{R}P_{r})$

where the $X_{i}$ are pairwise non-isomorphic exceptional pointwise free
$RQ$-lattices satisfying $\operatorname{Ext}_{RQ}^{1}(X_{i},X_{j})=0$ for all
$i,j$ and the $P_{i}$ are finitely generated projective $R$-modules of
constant rank. Moreover this decomposition is unique up to isomorphism and
reordering. In particular any exceptional $RQ$-lattice is of the form
$Y\otimes_{R}P$ with $Y$ an exceptional pointwise free $RQ$-module and $P$ a
projective $R$-module of constant rank 1.

Note that in general one doesn’t have uniqueness for rigid pointwise free
lattices of a given rank vector. For example let $Q$ be the quiver $1\to 2$
and $R$ a reduced ring with a stably free projective module $P$ which is not
free, say $P\oplus R^{n}\cong R^{m}$. Then
$(RQe_{2}\otimes_{R}P)\oplus(RQe_{1}\otimes_{R}R^{n})$ is pointwise free, but
not isomorphic to
$(RQe_{2}\otimes_{R}R^{m-n})\oplus(RQe_{1}\otimes_{R}R^{n})$.

##  2\. Commutative rings

Concerning our definitions and results, note that every finitely generated
projective $R$-module has constant rank if and only if $\operatorname{Spec}R$
is connected, or equivalently $R$ has no idempotents other than 0 and 1, see
for example Exercise 20.12 in [4].

######  Lemma 2.1.

Let $R$ be a commutative ring.

  * (i)

If $X$ is a finitely generated $R$-module, then $X=0$ if and only if $X^{K}=0$
for all homomorphisms $R\to K$ with $K$ an algebraically closed field.

  * (ii)

If $\theta:X\to Y$ is a homomorphism of $R$-modules and $Y$ is finitely
generated, then $\theta$ is onto if and only if $\theta^{K}$ is onto for all
$R\to K$ with $K$ an algebraically closed field.

  * (iii)

If $R$ is reduced and $X$ is a finitely generated $R$-module, then $X$ is
projective of constant rank $n$ if and only if $\dim_{K}X^{K}=n$ for all
homomorphisms $R\to K$ with $K$ an algebraically closed field.

  * (iv)

If $R$ is reduced and $\theta:X\to Y$ is a homomorphism with $X,Y$ finitely
generated projective of constant rank, then $\theta$ is a split monomorphism
if and only if $\theta^{K}$ is a monomorphism for all $R\to K$ with $K$ an
algebraically closed field.

###### Proof.

(i) Since $X$ is finitely generated, if non-zero it has a map onto some
$R/\mathfrak{m}$, and taking $K$ to be the algebraic closure of this field
gives a surjective map $X^{K}\to R/\mathfrak{m}\otimes_{R}K\cong K$.

(ii) Clear.

(iii) See Exercise 20.13 in [4].

(iv) If $\theta^{K}$ is a monomorphism for all $R\to K$, then
$\operatorname{Coker}\theta^{K}$ has constant dimension, so
$\operatorname{Coker}\theta$ is projective of constant rank. Then
$\operatorname{Im}\theta$ is a summand of $Y$, so projective. But then
$0\to\operatorname{Ker}\theta\to X\to\operatorname{Im}\theta\to 0$ splits, so
is exact after inducing to $K$. Thus $\operatorname{Ker}\theta^{K}=0$. Also
$\operatorname{Ker}\theta$ is summand of $X$, so finitely generated. Thus
$\operatorname{Ker}\theta$ is zero. ∎

######  Lemma 2.2.

Any commutative ring $R$ can be written as a quotient $R\cong\tilde{R}/I$ with
$\tilde{R}$ an integral domain and $I$ is an ideal contained in the Jacobson
radical of $\tilde{R}$.

###### Proof.

Let $A=\mathbb{Z}[x_{r}:r\in R]$ be the polynomial ring over $\mathbb{Z}$ with
indeterminates indexed by the elements of $R$. There is a surjective
homomorphism $\theta:A\to R$ sending each $x_{r}$ to $r$ and the set

$S=\\{a\in A:\text{$\theta(a)$ is invertible}\\}$

is a multiplicative subset. We set $\tilde{R}$ to be the localization $A_{S}$,
an integral domain. The homomorphism $\theta$ extends to a surjective
homomorphism $\tilde{\theta}:\tilde{R}\to R$.

Let $x\in\operatorname{Ker}\tilde{\theta}$. If $y\in\tilde{R}$, then
$xy\in\operatorname{Ker}\tilde{\theta}$. Write $xy=as^{-1}$ with $a\in A$ and
$s\in S$. Then $\theta(a)=0$, so $s+a\in S$. Thus $1+xy=(s+a)s^{-1}$ is
invertible in $\tilde{R}$. Since this is true for all $y$, it follows that $x$
is in the Jacobson radical of $\tilde{R}$. ∎

##  3\. Lattices

Let $R$ be a commutative ring and $Q$ a quiver. Lemma 1 of [3] generalizes
immediately to this setting.

######  Lemma 3.1.

If $X$ is an $RQ$-lattice, then $\operatorname{proj.dim}X\leq 1$ and

$\operatorname{Ext}^{1}_{RQ}(X,Y)^{S}\cong\operatorname{Ext}^{1}_{SQ}(X^{S},Y^{S})$

for any homomorphism $R\to S$ and any $RQ$-module $Y$. In particular if $X$ is
a rigid $RQ$-lattice, then $X^{S}$ is a rigid $SQ$-lattrice for all
homomorphisms $R\to S$. Conversely, if $X$ is an $RQ$-lattice and $X^{K}$
rigid for all homomorphisms $R\to K$ with $K$ an algebraically closed field,
then $X$ is rigid.

###### Proof.

Letting $S$ be the $R$-subalgebra of $RQ$ with basis the trivial paths $e_{i}$
and $B$ be the free $R$-submodule of $RQ$ with basis the arrows, the path
algebra $RQ$ is isomorphic to the tensor algebra of $B$ over $S$, so there is
a standard resolution

$0\to RQ\otimes_{S}B\otimes_{S}RQ\to RQ\otimes_{S}RQ\to RQ\to 0,$

or equivalently

$0\to\bigoplus_{a\in Q_{1}}RQe_{h(a)}\otimes_{R}e_{t(a)}RQ\to\bigoplus_{i\in
Q_{0}}RQe_{i}\otimes_{R}e_{i}R\to RQ\to 0.$

Tensoring with the $RQ$-module $X$ gives

$0=\operatorname{Tor}_{1}^{RQ}(RQ,X)\to\bigoplus_{a\in
Q_{1}}RQe_{h(a)}\otimes_{R}e_{t(a)}X\to\bigoplus_{i\in
Q_{0}}RQe_{i}\otimes_{R}e_{i}X\to X\to 0.$

and this is a resolution of $X$ by finitely generated projective $RQ$-modules.
We denote it by $0\to P_{1}\to P_{0}\to X\to 0$. Since $X$ is projective over
$R$, the induced sequence

$0\to P_{1}^{S}\to P_{0}^{S}\to X^{S}\to 0$

is exact, so a resolution of $X^{S}$ by projective $SQ$-modules. Now if $P$ is
a finitely generated projective $RQ$-module, then

$\operatorname{Hom}_{RQ}(P,Y)^{S}\cong\operatorname{Hom}_{SQ}(P^{S},Y^{S}).$

Thus we get a commutative diagram with exact rows

$\begin{CD}\operatorname{Hom}_{RQ}(P_{0},Y)^{S}@>{}>{}>\operatorname{Hom}_{RQ}(P_{1},Y)^{S}@>{}>{}>\operatorname{Ext}^{1}_{RQ}(X,Y)^{S}@>{}>{}>0\\\
@V{}V{}V@V{}V{}V\\\
\operatorname{Hom}_{SQ}(P_{0}^{S},Y^{S})@>{}>{}>\operatorname{Hom}_{SQ}(P_{1}^{S},Y^{S})@>{}>{}>\operatorname{Ext}^{1}_{SQ}(X^{S},Y^{S})@>{}>{}>0\end{CD}$

in which the vertical maps are isomorphisms, and hence an isomorphism

$\operatorname{Ext}^{1}_{RQ}(X,Y)^{S}\cong\operatorname{Ext}^{1}_{SQ}(X^{S},Y^{S}).$

Now $\operatorname{Ext}^{1}_{RQ}(X,X)$ is a finitely generated $R$-module, so
if $\operatorname{Ext}^{1}_{RQ}(X,X)^{K}=0$ for all homomorphisms from $R$ to
an algebraically closed field $K$ then $\operatorname{Ext}^{1}_{RQ}(X,X)=0$,
that is, $X$ is rigid. ∎

Over an algebraically closed field $K$, the isomorphism classes of
representations $X$ of $Q$ of dimension vector $\alpha$ correspond to orbits
$O_{X}$ of a group $\operatorname{GL}(\alpha)$ acting on an affine space
$\operatorname{Rep}(Q,\alpha)$. Moreover $X$ is rigid if and only if $O_{X}$
is open orbit. Since the affine space it irreducible, it follows that there is
at most one rigid module of dimension $\alpha$, up to isomorphism.

The general dimension of $\operatorname{Hom}(X,Y)$ and
$\operatorname{Ext}^{1}(X,Y)$ for $X$ of dimension $\alpha$ and $Y$ of
dimension $\beta$ is denoted $\hom(\alpha,\beta)$ and
$\operatorname{ext}(\alpha,\beta)$. In particular this is these are the
dimensions for $X$ and $Y$ rigid. Based on work of Schofield [11], it is
proved in [2] that $\hom(\alpha,\beta)$ and $\operatorname{ext}(\alpha,\beta)$
do not depend on the algebraically closed field $K$.

######  Lemma 3.2.

If $R$ is reduced, and if $X,Y$ are rigid $RQ$-lattices of pointwise constant
rank $\alpha,\beta$, then

$\operatorname{Hom}_{RQ}(X,Y)^{S}\cong\operatorname{Hom}_{SQ}(X^{S},Y^{S})$

for all homomorphisms $R\to S$. Moreover $\operatorname{Ext}^{1}_{RQ}(X,Y)$
and $\operatorname{Hom}_{RQ}(X,Y)$ are finitely generated projective
$R$-modules of constant rank $\operatorname{ext}(\alpha,\beta)$ and
$\hom(\alpha,\beta)$.

###### Proof.

For any $R\to K$ with $K$ an algebraically closed field,
$\operatorname{Ext}^{1}_{RQ}(X,Y)^{K}$ has constant dimension
$\operatorname{ext}(\alpha,\beta)$, so $\operatorname{Ext}^{1}_{RQ}(X,Y)$ is
projective over $R$ of this rank by Lemma 2.1(iii). Now the projective
resolution $0\to P_{1}\to P_{0}\to X\to 0$ of $X$ gives an exact sequence

$0\to\operatorname{Hom}_{RQ}(X,Y)\to\operatorname{Hom}_{RQ}(P_{0},Y)\to\operatorname{Hom}_{RQ}(P_{1},Y)\to\operatorname{Ext}^{1}_{RQ}(X,Y)\to
0.$

The last three are projective $R$-modules of constant rank, so this sequence
splits. Thus the first module is also projective over $R$. Moreover the
sequence remains exact under induction using a homomorphism $R\to S$. It
follows that

$\operatorname{Hom}_{RQ}(X,Y)^{S}\cong\operatorname{Hom}_{SQ}(X^{S},Y^{S}).$

Considering the case when $S$ is a field, we see that
$\operatorname{Hom}_{RQ}(X,Y)$ has constant rank $\hom(\alpha,\beta)$. ∎

The following result is well known.

######  Lemma 3.3.

The full subquiver of $Q$ on the support of any rigid dimension vector
$\alpha$ has no oriented cycles.

###### Proof.

If not, one can easily construct representations of $Q$ of dimension $\alpha$
over an algebraically closed field $K$ on which the trace of the oriented
cycle is arbitrary, but the variety of representations has a dense orbit, so
the trace must be constant. (This argument is already used in [3, Theorem 1]
for real Schur roots.) ∎

##  4\. Mutations

######  Lemma 4.1.

If $R$ is a commutative ring and $\alpha$ is real Schur root for $Q$, then
there exists an exceptional pointwise free $RQ$-lattice of dimension vector
$\alpha$.

###### Proof.

By [3, Theorem 2], there is a (unique) exceptional $\mathbb{Z}Q$-lattice
$X_{0}$ of rank vector $\alpha$. Now we obtain an $RQ$-lattice
$X^{R}=X\otimes_{\mathbb{Z}}R$, and it is exceptional by Lemmas 3.1 and 3.2. ∎

The cited theorem [3, Theorem 2] depends on a braid group action for
exceptional sequences of lattices, defined using mutations of exceptional
pairs. The result about mutations in turn depends on the construction of
mutations of pairs of representations of $KQ$ where $K$ is a field, for which
[3] cites [10] and [1] cites [6]. We take this opportunity to give another
reference.

A sequence of exceptional $RQ$-lattices $(X_{1},\dots,X_{r})$ is called an
_exceptional sequence_ provided that
$\operatorname{Hom}_{RQ}(X_{i},X_{j})=\operatorname{Ext}^{1}_{RQ}(X_{i},X_{j})=0$
for all $i>j$. The result about mutations is as follows.

######  Theorem 4.2.

Suppose $R$ is reduced. If $(X,Y)$ is an exceptional pair of $RQ$-lattices,
then there are exceptional pairs $(L_{X}Y,X)$ and $(Y,R_{Y}X)$ given as
follows. If $\operatorname{Hom}_{RQ}(X,Y)=0$ then $L_{X}Y$ and $R_{Y}X$ are
given by the universal exact sequences

$0\to Y\to L_{X}Y\to X\otimes_{R}\operatorname{Ext}^{1}_{RQ}(X,Y)\to 0,$ $0\to
Y\otimes_{R}D\operatorname{Ext}^{1}_{RQ}(X,Y)\to R_{Y}X\to X\to 0.$

where $D=\operatorname{Hom}_{R}(-,R)$. If $\operatorname{Hom}_{RQ}(X,Y)\neq
0$, then the universal map

$f:X\otimes_{R}\operatorname{Hom}_{RQ}(X,Y)\to Y$

is an epimorphism or a monomorphism, and $L_{X}Y$ is its kernel or cokernel,
and the universal map

$g:X\to\operatorname{Hom}_{R}(\operatorname{Hom}_{RQ}(X,Y),Y)\cong
Y\otimes_{R}D\operatorname{Hom}_{RQ}(X,Y)$

is an epimorphism or a monomorphism and $R_{Y}X$ is its kernel or cokernel.

###### Proof.

Since $R$ is reduced and the lattices are rigid, the Hom and Ext spaces are
projective over $R$ of constant rank, independent of $R$. Moreover using Lemma
2.1 we can, as in [3, Lemma 4], reduce to the case when $R=K$ is an
algebraically closed field. For this case, see [8, §4]. ∎

##  5\. Classification of rigids

We prove the following result. Once Theorem A is proved, it gives Theorem B.
The proof of the theorem is the same as Theorem 2(ii) of [3].

######  Theorem 5.1.

Let $R$ be reduced. For each real Schur root $\alpha$, we choose an
exceptional pointwise free $RQ$-lattice $X(\alpha)$ of rank vector $\alpha$.
Any rigid $RQ$-lattice $X$ of pointwise constant rank has a decomposition

$X\cong(X(\alpha^{1})\otimes_{R}P_{1})\oplus\dots\oplus(X(\alpha^{r})\otimes_{R}P_{r})$

where the $\alpha^{1},\dots,\alpha^{r}$ are distinct real Schur roots,
$\operatorname{ext}(\alpha^{i},\alpha^{j})=0$ for all $i,j$ and the $P_{i}$
are finitely generated projective $R$-modules of constant rank. Moreover this
decomposition is unique up to isomorphism and reordering. In particular any
exceptional $RQ$-lattice of rank vector $\alpha$ is of the form
$X(\alpha)\otimes_{R}P$ with $X$ exceptional pointwise free and $P$ a
projective $R$-module of constant rank 1.

###### Proof.

By Lemma 3.3, we may assume that $Q$ has no oriented cycles. We fix
temporarily a homomorphism $R\to K$ with $K$ an algebraically closed field.
Then $X^{K}$ is rigid, so decomposes as direct sum

$X^{K}\cong M_{1}^{m_{1}}\oplus\dots\oplus M_{r}^{m_{r}}.$

with the $M_{i}$ pairwise non-isomorphic exceptional $KQ$-modules,
$\operatorname{Ext}^{1}_{KQ}(M_{i},M_{j})=0$ for all $i,j$ and all $m_{i}>0$.
By [7, Corollary 4.2], we can order the $M_{i}$ so that $(M_{1},\dots,M_{r})$
is an exceptional sequence, so $\operatorname{Hom}(M_{i},M_{j})=0$ for $i>j$.
Let $M_{i}$ have dimension vector $\alpha^{i}$. We have
$\operatorname{Hom}(X(\alpha^{i}),X(\alpha^{j}))=0$ for $i>j$ since
$\dim\operatorname{Hom}(M_{i},M_{j})=\hom(\alpha^{i},\alpha^{j})$.

Now $P_{r}=\operatorname{Hom}_{RQ}(X(\alpha^{r}),X)$ is a finitely generated
projective $R$-module of constant rank. Consider the evaluation map
$\theta:X(\alpha^{r})\otimes_{R}P_{r}\to X$ and let $C$ be its cokernel. For
an arbitrary homomorphism $R\to K$ with $K$ an algebraically closed field (no
longer the one fixed above), using Lemma 3.2, we can identify $\theta^{K}$
with the evaluation map

$X(\alpha^{r})^{K}\otimes_{K}\operatorname{Hom}_{KQ}(X(\alpha^{r})^{K},X^{K})\to
X$

but we know that

$X^{K}\cong(X(\alpha^{1})^{K})^{m_{1}}\oplus\dots\oplus(X(\alpha^{r})^{K})^{m_{r}}$

since both sides are rigid $KQ$-modules of the same dimension vector. Thus
$\theta^{K}$ is the inclusion of $(X(\alpha^{r})^{K})^{m_{r}}$ as a direct
summand of $X^{K}$. Thus by Lemma 2.1, $\theta$ is a split monomorphism of
$R$-modules, so $C$ is projective over $R$. Also

$C^{K}\cong\operatorname{Coker}(\theta^{K})\cong(X(\alpha^{1})^{K})^{m_{1}}\oplus\dots\oplus(X(\alpha^{r-1})^{K})^{m_{r-1}}$

so $\operatorname{Ext}^{1}_{KQ}(C^{K},C^{K})=0$ and
$\operatorname{Ext}^{1}_{KQ}(C^{K},X(\alpha^{r})^{K})=0$. Thus $C$ is rigid
and $\operatorname{Ext}^{1}(C,X_{r})=0$. Thus $X\cong
X(\alpha^{r})\otimes_{R}P_{r}\oplus C$. Now the result follows by induction
(on $r$ or on the total rank of $X$).

For uniqueness, note that if we have another decomposition

$X=(X(\beta^{1})\otimes P_{1}^{\prime})\oplus\dots\oplus(X(\beta^{s})\otimes
P_{s}^{\prime})$

with the $\beta^{i}$ being distinct real Schur roots and the $P_{i}^{\prime}$
projective $R$-modules of constant rank, then using a fixed homomorphism $R\to
K$ and the uniqueness of rigid $KQ$-modules of a given dimension vector, we
see that $s=r$ and that the $\beta^{i}$ are a permutation of the $\alpha^{i}$.
Thus we reduce to the case that $\beta^{i}=\alpha^{i}$. But then

$P_{r}^{\prime}\cong\operatorname{Hom}_{RQ}(X(\alpha^{r}),X)\cong P_{r}$

and then the corresponding cokernels of the evaluation maps are isomorphic, so
by induction $P_{i}^{\prime}\cong P_{i}$ for all $i$.

The special case of exceptional lattices follows. ∎

##  6\. Uniqueness for exceptional pointwise frees

We now prove Theorem A. The existence part was Lemma 4.1. Suppose $X$ is a
pointwise free lattice for $RQ$ of rank vector $\alpha$, a real Schur root. By
Lemma 2.2 we can write $R\cong\tilde{R}/I$ with $\tilde{R}$ reduced and $I$
contained in the Jacobson radical of $\tilde{R}$.

We define a pointwise free $\tilde{R}Q$-lattice $\tilde{X}$ by giving it as a
representation of $Q$ by free $\tilde{R}$-modules as follows. For each vertex
$i$ we take $e_{i}\tilde{X}$ to be a free $\tilde{R}$-module of rank
$\alpha_{i}$, and we fix an isomorphism

$(e_{i}\tilde{X})/I(e_{i}\tilde{X})\cong(e_{i}\tilde{X})^{R}\cong e_{i}X.$

For each arrow $a\in Q_{1}$, multiplication by $a$ induces an $R$-linear map
$e_{t(a)}X\to e_{h(a)}X$, and we lift this to an $\tilde{R}$-linear map
$e_{t(a)}\tilde{X}\to e_{h(a)}\tilde{X}$. This defines $\tilde{X}$ as a
representation of $Q$ and clearly we have $\tilde{X}^{R}\cong X$.

Let $E=\operatorname{Ext}^{1}_{\tilde{R}Q}(\tilde{X},\tilde{X})$, a finitely
generated $\tilde{R}$-module. Now
$E^{R}\cong\operatorname{Ext}^{1}_{RQ}(X,X)=0$ by Lemma 3.1, so $E=IE$, so by
Nakayama’s Lemma $E=0$. Thus $\tilde{X}$ is rigid. (This idea of lifting $X$
to a rigid representation $\tilde{X}$ is taken from [5].)

By Theorem B we have $\tilde{X}\cong X(\alpha)\otimes_{\tilde{R}}\tilde{P}$,
where $X(\alpha)$ is a chosen exceptional pointwise free $\tilde{R}Q$-lattice
of rank vector $\alpha$ and $\tilde{P}$ is a finitely generated projective
$\tilde{R}$-module of constant rank 1.

Tensoring with $R$, we get $X\cong Y\otimes_{R}P$ where $Y=X(\alpha)^{R}$ is
an exceptional pointwise free $RQ$-lattice and $P=\tilde{P}^{R}$ is a finitely
generated projective $R$-module of constant rank 1.

Now since $X$ and $Y$ are pointwise free of rank vector $\alpha$, we have

$R^{\alpha_{i}}\cong e_{i}X\cong e_{i}Y\otimes_{R}P\cong
R^{\alpha_{i}}\otimes_{R}P\cong P^{\alpha_{i}}$

for all $i$. Since $\alpha$ is a real root for $Q$, it is indivisible, that
is, its components are coprime. Thus we can find $a_{i},b_{i}\in\mathbb{N}$
such that

$1+\sum_{i}a_{i}\alpha_{i}=\sum_{i}b_{i}\alpha_{i}.$

Then

$P\oplus R^{\sum_{i}a_{i}\alpha_{i}}\cong
P\oplus\bigoplus_{i}(R^{\alpha_{i}})^{a_{i}}\cong
P\oplus\bigoplus_{i}(P^{\alpha_{i}})^{a_{i}}\cong\bigoplus_{i}(P^{\alpha_{i}})^{b_{i}}\cong\bigoplus_{i}(R^{\alpha_{i}})^{b_{i}}\cong
R^{\sum_{i}b_{i}\alpha_{i}}.$

Thus $P$ is stably free. Now any stably free projective module of constant
rank 1 for a commutative ring is free, see for example [9, Theorem 4.11], so
$P\cong R$. Thus $X\cong Y$. We have already observed that $Y$ is exceptional,
hence so is $X$.

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