Does PostgreSQL support global indexes on partitioned tables?

PostgreSQL does not support Global Indexes, i.e indexes that spans all partitions in a partitioned table. The documentation states:

Although all partitions must have the same columns as their partitioned parent, partitions may have their own indexes, constraints and default values, distinct from those of other partitions.

and

Create an index on the key column(s), as well as any other indexes you might want, on the partitioned table. ... This automatically creates a matching index on each partition, and any partitions you create or attach later will also have such an index. An index or unique constraint declared on a partitioned table is “virtual” in the same way that the partitioned table is: the actual data is in child indexes on the individual partition tables. A proposal to add such feature seem to have been suggested back in 2019, but currently no implementation exists.

I also found this comparison between Oracle and PostgreSQL that elaborates on the topic a bit more.

How to rebuild an index partition

To limit a rebuild to a specific index partition only, use the following syntax:

ALTER INDEX SH.SALES_IDX4 
REBUILD PARTITION SYS_P51490 
TABLESPACE DATA04 ONLINE;

Notice the use of the ONLINE keyward, which will allow DML against the table (and thus update the index). To generate rebuild-statements, one index at a time, I have used the script below. It takes as arguments

1. The index owner 
2. The index name 
3. The new tablespace where you want to place your partitions 
4. The old tablespace from which you want to move out

set termout off
select 'alter session set nls_language=''american'';' from dual;
set termout on
accept index_owner prompt 'Index owner: '
accept index_name prompt 'Index name: '
accept new_tabspc prompt 'New tablespace: '
accept old_tabspc prompt 'Old tablespace: '

set lines 300
set heading off
set feedback off
set verify off
set echo off
set pages 0
set trimspool on
spool exec_move_part_&&index_owner..&&index_name..sql
select 'alter session set nls_language=''american'';' from dual;

select 'alter session force parallel ddl;' from dual;
select 'set timing on' from dual;
select 'set lines 200' from dual;
select 'set trimspool on' from dual;
select 'spool exec_rebuild_part_&&index_owner..&&index_name..log' from dual;
select 'exec DBMS_APPLICATION_INFO.SET_MODULE(module_name=>''Move'', action_name=>''Rebuild_part_&&index_name'');' from dual;
select 'exec DBMS_APPLICATION_INFO.SET_CLIENT_INFO (client_info=>''Rebuild_part_t_&&index_name'');' from dual;
select 'exec DBMS_SESSION.SET_IDENTIFIER (client_id=>''Rebuild_part_&&index_name''); ' from dual;

select 'prompt moving the index partitions from &&old_tabspc to &&new_tabspc,' from dual;
select 'prompt and setting default attributes for the table ' from dual;
SELECT 'alter index ' || idx.table_owner || '.' ||  idx.index_name || ' rebuild partition ' || idxpart.partition_name || ' TABLESPACE &&new_tabspc ONLINE;'
FROM    dba_indexes idx, dba_ind_partitions idxpart
WHERE   idx.table_owner = '&&index_owner'
and     idxpart.TABLESPACE_NAME = '&&old_tabspc'
AND     idx.index_name = idxpart.index_name
AND     idx.index_name = '&&index_name'
AND     idxpart.subpartition_count=0
ORDER BY idx.table_owner, idx.index_name;

select distinct 'alter index ' ||   IDXPART.INDEX_OWNER || '.' || IDXPART.INDEX_NAME || ' modify default attributes tablespace &&new_tabspc;'
FROM   DBA_IND_PARTITIONS IDXPART JOIN DBA_SEGMENTS S
ON     (IDXPART.PARTITION_NAME = S.PARTITION_NAME )
WHERE  IDXPART.INDEX_NAME='&&index_name'
AND    IDXPART.TABLESPACE_NAME='&&old_tabspc';

select 'exit' from dual;
exit

Run it as a user with access to the data dictionary: 

sqlplus / as sysdba @gen_move_idx_part.sql 

It will generate a new script called exec_move_part_<username>.<index_name>.sql which will perform the actual rebuild: 

sqlplus / as sysdba @exec_move_part_<username>.<index_name>.sql

I also added som calls to dbms_application_info so that the session can be easily identfied in v$session. Remember, sometimes it's desirable to set the PARALLEL degree and the NOLOGGING options during rebuild, to complete it as fast as possible. See this post for more info.