update a set a.A3=b.B3 from A表 a join B表 b on a.A1=b.B1
-- 是用的a.A1=b.B1做为条件，看楼主是不是要这个条件。

------解决方案--------------------

SQL code
update a set a.A3=b.B3 from A表 a join (select B2，sum(B3) from b group by b2) b on a.A2=b.B2

------解决方案--------------------
SQL code

create table a(a1 int,a2 varchar(10),a3 int)
insert into a 
select 1,'a',0 union all
select 2,'b',0

create table b(b1 int,b2 varchar(10),b3 int)
insert into b 
select 1,'a',10 union all
select 2,'b',2 union all
select 3,'b',1

update a set a.A3=(select sum(b.b3) from b where b.b2=a.a2)

--drop table a
--drop table b
------------------------
1    a    10
2    b    3