Fair Value Measurements - Additional Information (Detail) - USD ($)	Sep. 30, 2024	Sep. 30, 2023
Debt Instrument Fair Value Carrying Value [Line Items]
Transfers among the Level 1, Level 2 and Level 3 categories	$ 0	$ 0

Amount of transfers of assets measured on a recurring basis among Level 1, Level 2 and Level 3.

No definition available.

Name:	sage_FairValueAssetsTransfersAmongLevel1Level2AndLevel3Amount
Namespace Prefix:	sage_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant